bollu.github.io

Table of contents:

Geometric proof of Cauchy Schwarz inequality

geometric-proof-cauchy-schwarz

Dataflow analysis using Grobner basis

This was a quick experiment in using Grobner basis to model situations. We can represent our dataflow analysis constraints in terms of polynomial rewrites over $F_2$.

Given the program:

p = { 0: ["=", "x", 'y'], 
      1: ['br', 2, 100], 
      2: ['=', 'z', 'x'], 
      3: ['br', 2],
      100: ['ret', 'z'] }

whose semantics I hope are fairly straightforward — the dictionary represents instruction locations. Instructions proceed sequentially. branch moves control flow around. Note that br can branch to multiple locations, since we are not control-flow sensitive.

The idea is that since in a dataflow analysis, we need information at each variable at each program point, we can create a ring of polynomials over $F_2$ for each variable at each program point. So in this case, we wold need:

R = F_2[x0, y0, z0, x1, y1, z1, x2, y2, z2, x3, y3, z3, x100, y100, z100]

We then add elements into the ideal that represents our constraints. For example, to perform dataflow analysis, we need to add constraints about how if a variable z is alive, all variables that are used to compute z at 100 are alive. This sets up equations that may have cycles (in the case of loops).

These are usually resolved using the Kildall algorithm.

However, we can also ask SAGE to kindly solve the Grobner basis. I hypothesize that the “easy” dataflow problems out to be toric ideals which admit much faster solutions.

Fenwick trees and orbits

I learnt of a nice, formal way to prove the correctness of Fenwick trees in terms of orbits that I wish to reproduce here.

One can use a Fenwick tree to perform cumulative sums $Sum(n) \equiv \sum_i^n A[i]$, and updates $Upd(i, v) \equiv A[i] += v$. Naively, cumulative sums can take $O(n)$ time and updates take $O(1)$ time.

A Fenwick tree can perform both in $\log(n)$. In general, we can perform any monoid-based catenation and update in $\log(n)$.

organization

We allow indexes $[1, 2, \dots n]$. The node with factorization $i \equiv 2^k \times l$, $2 \not \vert l$ (that is, $k$ is the highest power of $2$ in $i$) is responsible for the interval $[i-2^k+1, i] = (i-2^k, i]$.

I’m going to state all the manipulations in terms of prime factorizations, since I find it far more intuitive than bit-fiddling. In general, I want to find a new framework to discover and analyze bit-fiddling heavy algorithms.

Some examples of the range of responsibility of an index are:

fenwick-structure

query

To perform a cumulative sum, we need to read from the correct overlap regions that cover the full array. For example, to read from $15$, we would want to read:

So we need to read the indices:

At each location, we strip off the value $2^r$. We can discover this value with bit-fiddling: We claim that $a \& (-a) = 2^r$.

Let $a = x 1 0^r$. Now, $-a = \lnot a + 1 = x01^r + 1 = \overline{x}10^r$. Hence, $a \& (-a) = a \& (\lnot a + 1) = (x 10^r) \& (\overline{\x}10^r) = 0^{|\alpha|}10^r = 2^r$

So the full implementation of query is:

#define LSB(x) x&(-x)
int a[N];
int q(int i) {
    int s = 0;
    while (i > 0) { s += a[i]; i -= LSB(i); }
    return s;
}

update

To perform an update at $i$, we need to update all locations which on querying overlap with $i$. For example, to update the location $9$, we would want to update:

So we need to update the indices:

We use the same bit-fiddling technique as above to strip off the value $2^r$

#define LSB(x) x&(-x)
int tree[N];
int u(int i, int v) {
    while (i < N) { tree[i] += v; i += LSB(i); }
}

correctness

We wish to analyze the operations $Query(q) \equiv \sum_{i=1}^q a[i]$, and $Update(i, val) \equiv a[i]~\texttt{+=}~val$. To do this, we are allowed to maintain an auxiliary array $d$ which we will manipuate. We will stipulate the conditions of operations on $d$ such that they will reflect the values of $Query$ and $Update$, albeit much faster.

We will analyze the algorithm in terms of orbits. We have two operators, one for update called $U$, and one for query called $Q$. Given an index $i$, repeatedly applying the query operator gives us the indeces we need to read and accumulate from the underlying array $a$ to get the total sum $a[0..i]$:

Given an index $u$, repeatedly applying the update operator $U$ gives us all the indeces we need to add the change to update:

For query and update to work, we need the condition that:

That is, if and only if the query index $q$ includes the update location $u$, will the orbits intersect.

The intuition is that we want updates at an index $u$ to only affect queries that occur at indeces $q \geq u$. Hence, we axiomatise that for an update to be legal, it must the orbits of queries that are at indeces greater than it.

We will show that our operators:

do satisfy the conditions above.

For a quick numerical check, we can use the code blow to ensure that the orbits are indeed disjoint:

# calculate orbits of query and update in fenwick tree

def lsb(i): return i & (-i)
def U(i): return i + lsb(i)
def Q(i): return i - lsb(i)
def orbit(f, i):
    s = set()
    while i not in s and i > 0 and i < 64:
        s.add(i); i = f(i)
    return s

if __name__ == "__main__":
    for q in range(1, 16):
        for u in range(1, 16):
            qo = orbit(Q, q); uo = orbit(U, u)
            c = qo.intersection(uo)
            print("q:%4s | u:%4s | qo: %20s | uo: %20s | qu: %4s" % 
                  (q, u, qo, uo, c))

        print("--")
Case 1: $q = u$

We note that $Q$ always decreases the value of $q$, and $u$ always increases it. Hence, if $q = u$, they meet at this point, and $\forall i, j \geq 1, \quad Q^i (q) \neq U^j(u)$. Hence, they meet exactly once as required.

Case 2: $q < u$

As noted above, $q$ always decreases and $u$ always increases, hence in this case they will never meet as required.

Case 3: $q > u$

Let the entire array have size $2^N$.
Let $q = \texttt{e1$f_q$}, u = \texttt{e0$f_u$}$, where $\texttt{e},f_q, f_u$ may be empty strings.

Notice that $Q$ will always strip away rightmost ones in $f_q$, leading to $q = \texttt{e10…0}$ at some point.

Similarly, $U$ will keep on adding new rightmost ones, causing the state to be $u = \texttt{e01…10…0} \xrightarrow{U} \texttt{e100…}$.

Hence, at some point $q = u$.

References

Dirichlet inversion

Incunabulum for the 21st century: Making the J interpreter compile in 2020

This is me trying to understand the fabled interpreter of the J language working, so I could absorb Arthur Whitney’s style of writing C: it’s cryptic, short, and fits in a page. I learnt of this from the J language page, which comes with the quote:

One summer weekend in 1989, Arthur Whitney visited Ken Iverson at Kiln Farm and produced—on one page and in one afternoon—an interpreter fragment on the AT&T 3B1 computer. I studied this interpreter for about a week for its organization and programming style; and on Sunday, August 27, 1989, at about four o’clock in the afternoon, wrote the first line of code that became the implementation described in this document. Arthur’s one-page interpreter fragment is as follows:

The original source

typedef char C;typedef long I;
typedef struct a{I t,r,d[3],p[2];}*A;
#define P printf
#define R return
#define V1(f) A f(w)A w;
#define V2(f) A f(a,w)A a,w;
#define DO(n,x) {I i=0,_n=(n);for(;i<_n;++i){x;}}
I *ma(n){R(I*)malloc(n*4);}mv(d,s,n)I *d,*s;{DO(n,d[i]=s[i]);}
tr(r,d)I *d;{I z=1;DO(r,z=z*d[i]);R z;}
A ga(t,r,d)I *d;{A z=(A)ma(5+tr(r,d));z->t=t,z->r=r,mv(z->d,d,r);
 R z;}
V1(iota){I n=*w->p;A z=ga(0,1,&n);DO(n,z->p[i]=i);R z;}
V2(plus){I r=w->r,*d=w->d,n=tr(r,d);A z=ga(0,r,d);
 DO(n,z->p[i]=a->p[i]+w->p[i]);R z;}
V2(from){I r=w->r-1,*d=w->d+1,n=tr(r,d);
 A z=ga(w->t,r,d);mv(z->p,w->p+(n**a->p),n);R z;}
V1(box){A z=ga(1,0,0);*z->p=(I)w;R z;}
V2(cat){I an=tr(a->r,a->d),wn=tr(w->r,w->d),n=an+wn;
 A z=ga(w->t,1,&n);mv(z->p,a->p,an);mv(z->p+an,w->p,wn);R z;}
V2(find){}
V2(rsh){I r=a->r?*a->d:1,n=tr(r,a->p),wn=tr(w->r,w->d);
 A z=ga(w->t,r,a->p);mv(z->p,w->p,wn=n>wn?wn:n);
 if(n-=wn)mv(z->p+wn,z->p,n);R z;}
V1(sha){A z=ga(0,1,&w->r);mv(z->p,w->d,w->r);R z;}
V1(id){R w;}V1(size){A z=ga(0,0,0);*z->p=w->r?*w->d:1;R z;}
pi(i){P("%d ",i);}nl(){P("\n");}
pr(w)A w;{I r=w->r,*d=w->d,n=tr(r,d);DO(r,pi(d[i]));nl();
 if(w->t)DO(n,P("< ");pr(w->p[i]))else DO(n,pi(w->p[i]));nl();}

C vt[]="+{~<#,";
A(*vd[])()={0,plus,from,find,0,rsh,cat},
 (*vm[])()={0,id,size,iota,box,sha,0};
I st[26]; qp(a){R  a>='a'&&a<='z';}qv(a){R a<'a';}
A ex(e)I *e;{I a=*e;
 if(qp(a)){if(e[1]=='=')R st[a-'a']=ex(e+2);a= st[ a-'a'];}
 R qv(a)?(*vm[a])(ex(e+1)):e[1]?(*vd[e[1]])(a,ex(e+2)):(A)a;}
noun(c){A z;if(c<'0'||c>'9')R 0;z=ga(0,0,0);*z->p=c-'0';R z;}
verb(c){I i=0;for(;vt[i];)if(vt[i++]==c)R i;R 0;}
I *wd(s)C *s;{I a,n=strlen(s),*e=ma(n+1);C c;
 DO(n,e[i]=(a=noun(c=s[i]))?a:(a=verb(c))?a:c);e[n]=0;R e;}

main(){C s[99];while(gets(s))pr(ex(wd(s)));}

It’s a lot to take in — it’s quite breathtaking really, the way it all hangs together in one page.

The attempt to get it run

Unfortunately, this does not work if we try to get it to run in 2020. I decided to read the code and understand what would happen. I managed to read enough to understand that the code a=~3 ought to create an array with values [0 1 2]. On attempting to run this however, we get:

$ gcc -O0 -g -std=c89 -fsanitize=address -fsanitize=undefined incunabulum.c -o bin/incunabulum && ./bin/incunabulum
...
(many many GCC warnings elided)
...
a=~3
=================================================================
==23726==ERROR: AddressSanitizer: heap-buffer-overflow on address 
  0x60300000eff0 at pc 0x000000402be3 bp 0x7ffe6dde6b70 sp 0x7ffe6dde6b60
WRITE of size 8 at 0x60300000eff0 thread T0
    #0 0x402be2 in wd /home/bollu/work/w/incunabulum.c:40
    #1 0x402d28 in main /home/bollu/work/w/incunabulum.c:42
    #2 0x7f7ae901082f in __libc_start_main (/lib/x86_64-linux-gnu/libc.so.6+0x2082f)
    #3 0x400ca8 in _start (/home/bollu/w/bin/incunabulum+0x400ca8)
...
SUMMARY: AddressSanitizer: heap-buffer-overflow /home/bollu/work/w/incunabulum.c:40 wd
...
==23726==ABORTING

oops! The code uses a lot of punning between int and int*. These assumptions break now that we’re in 64-bit. The patch to get this to work is:

the patch

diff --git a/incunabulum.c b/incunabulum.c
index 2cae744..778e35a 100644
--- a/incunabulum.c
+++ b/incunabulum.c
@@ -1,11 +1,11 @@
-typedef char C;typedef long I;
+typedef char C;typedef long long I;
 typedef struct a{I t,r,d[3],p[2];}*A;
 #define P printf
 #define R return
 #define V1(f) A f(w)A w;
 #define V2(f) A f(a,w)A a,w;
 #define DO(n,x) {I i=0,_n=(n);for(;i<_n;++i){x;}}
-I *ma(n){R(I*)malloc(n*4);}mv(d,s,n)I *d,*s;{DO(n,d[i]=s[i]);}
+I *ma(n){R(I*)malloc(n*8);}mv(d,s,n)I *d,*s;{DO(n,d[i]=s[i]);}
 tr(r,d)I *d;{I z=1;DO(r,z=z*d[i]);R z;}
 A ga(t,r,d)I *d;{A z=(A)ma(5+tr(r,d));z->t=t,z->r=r,mv(z->d,d,r);
  R z;}
@@ -34,9 +34,10 @@ I st[26]; qp(a){R  a>='a'&&a<='z';}qv(a){R a<'a';}
 A ex(e)I *e;{I a=*e;
  if(qp(a)){if(e[1]=='=')R st[a-'a']=ex(e+2);a= st[ a-'a'];}
  R qv(a)?(*vm[a])(ex(e+1)):e[1]?(*vd[e[1]])(a,ex(e+2)):(A)a;}
-noun(c){A z;if(c<'0'||c>'9')R 0;z=ga(0,0,0);*z->p=c-'0';R z;}
-verb(c){I i=0;for(;vt[i];)if(vt[i++]==c)R i;R 0;}
+I noun(c){A z;if(c<'0'||c>'9')R 0;z=ga(0,0,0);*z->p=c-'0';R z;}
+I verb(c){I i=0;for(;vt[i];)if(vt[i++]==c)R i;R 0;}
 I *wd(s)C *s;{I a,n=strlen(s),*e=ma(n+1);C c;
  DO(n,e[i]=(a=noun(c=s[i]))?a:(a=verb(c))?a:c);e[n]=0;R e;}
 
 main(){C s[99];while(gets(s))pr(ex(wd(s)));}
+

It runs!

After applying the patch, we manage to get the interpreter to run!

./bin/incunabulum
a=~3
3 
0 1 2 

The lock screen

I liked it so much that I took a screenshot and made it my lock screen.

screenshot-j-incunabulum

Thoughts

I’m really fascinated by the code. I loved I could simply stare the screenshot to absorb the code. There was no scrolling involved. The variables are well-named (to the extent I understand the code), and it’s clearly extremely well thought out. If there’s someone who understands some of the thorny aspects of the code:

I’d be really glad to know the details. Please leave an issue or a pull request against the repo.

I’m going write a dissection of the code once I fully understand it, since I couldn’t find explanaing the code on the internet.

Until then, enjoy the monolith of code!

An example of a sequence whose successive terms get closer together but isn’t Cauchy (does not converge)

The problem

Provide an example of a sequence $a_n: \mathbb N \rightarrow \mathbb R$ such that $\lim_{n \rightarrow \infty} \vert a_{n+1} - a_n \vert \rightarrow 0$, but $\lim_{n, m \rightarrow \infty, m > n} |a_n - a_m| \neq 0$. That is, proide a series where the distances between successive terms converges to zero, but where distances between terms that are “farther apart than 1” does not converge to 0. That is, the sequence is not Cauchy.

Regular solution: Harmonic numbers

The usual solution is to take the harmonic numbers, $H_n \equiv \sum_{i=1}^n 1/i$. Then, we show that:

Memorable solution: logarithm

We can much more simply choose $a_n = \log(n)$. This yields the simple calculation:

while on the other hand,

I find this far cleaner conceptually, since it’s “obvious” to everyone that $a_n = \log(n)$ diverges, while the corresponding fact for $H_n$ is hardly convincing. We also get straightforward equalities everywhere, instead of inequalities.

I still feel that I don’t grok what precisely fails here, in that, my intuition still feels that the local condition ought to imply the Cauchy condition: if $a_n$ tells $a_{n+1}$ to not be too far, and $a_{n+1}$ tells $a_{n+2}$, surely this must be transitive?

I have taught my instincts to not trust my instincts on analysis, which is a shitty solution :) I hope to internalize this someday.

Krylov subspace method

This is a class of methods used to solve $Ax = b$, where $A$ is sparse. Krylov subspace methods are a class of methods which use the idea of a Krylov subspace. There is conjugate gradient (CG), GMRES (Generalized minimal residual method).

Clearly, $K_m \subseteq K_{m+1}$, and there is a maximum $K_N$ that we can span (the full vector space). We are interested in the smallest index $M$ such that $K_M = K_{M+1}$.

We notice that $K_M$ is invariant under the action of $A$.

Now, let’s consider:

We learnt that $Ax = b$ has a solution in $K_m(A, b)$. Using this, we can build solvers that exploit the Krylov subspace. We will describe GMRES and CG.

Generalized minimal residual — GMRES

We wish to solve $Ax = b$ where $A$ is sparse and $b$ is normalized. The $n$th Krylov subspace is $K_n(A, b) \equiv span~{b, Ab, A^2b, \dots, A^nb }$.

We approximate the actual solution with a vector $x_n \in K_n(A, b)$. We define the residual as $r_n \equiv A x_n - b$.

Conjugate gradient descent

Good reference to the Rete pattern matching algorithm

The Rete pattern matching algorithm is an algorithm that allows matching a huge number of rules with a huge database of “facts”.

MLIR (“multi-language intermediate representation”) is a new technology that hopes to centralize much of the research and development of various compilers into a single authoritative source. The major claim-to-fame is that it allows one to mix various “dialects” (much as Racket does). So, to a first order approximation, MLIR is “JSON for compiler intermediate representations”.

What MLIR gets right is tooling. They take the experience that the LLVM project has mined for the last decade and a half, and bake many of the good stuff that came with LLVM right in. For example, MLIR comes in-built with a pretty printer, a notion of types, a notion of “attributes”, SSA, enforced provenance tracking of code (so one can always know what the original source code was that led to some assembly). Sound engineering might see MLIR succeed where many others failed.

I was reminded of Rete since the MLIR folks are trying to solve the pattern matching problem in general for their Generic DAG Rewriter.

They currently just use a worklist based algorithm. I’m trying to understand if Rete can be used instead. Rete is famous for being hard to understand, so I began a quest to look for good sources to implement it. I found a great PhD thesis written by Robert B. Doorenboos, which quips:

Since the Rete match algorithm provides the starting point for much of the work in this thesis, this chapter describes Rete. Unfortunately, most of the descriptions of Rete in the literature are not particularly lucid,1 which is perhaps why Rete has acquired \a reputation for extreme differentialculty.”(Perlin, 1990b) To remedy this situation, this chapter describes Rete in a tutorial style, rather than just briey reviewing it and referring the reader to the literature for a full description. We will rst give an overview of Rete, and then discuss the principle data structures and procedures commonly used to implement it. High-level pseudocode will be given for many of the structures and procedures, so that this chapter may serve as a guide to readers who want to implement Rete (or some variant) in their own systems.

I now have a reference to an accessible description of this stuff. I might implement Rete to understand it, so that it’s part of my toolkit if I ever need it.

Leapfrog Integration

We have a system we wish to simulate using hamilton’s equations:

We want to simulate a system using these differential equations. We will begin with some initial position and momentum $(q_0, p_0)$, evaluate $\frac{\partial q}{\partial t} \rvert_{(q_0, p_0)}$, $\frac{\partial p}{\partial t} \rvert_{(q_0, p_0)}$, and use these to find $(q_{next}, p_{next})$. An integrator is a general algorithm that produces the next position and momentum using current information:

The design of $I$ is crucial: different choices of $I$ will have different trade-offs for accuracy and performance. Another interesting property we might want is for $I$ to be a symplectic integrator — that is, it preserves the total energy of the system. For example, here’s a plot of the orbits of planets using two integrators, one that’s symplectic (leapfrog) and one that isn’t (Euler)

leapfrog-integration

Notice that since leapfrog attempts to keep energy conserved, the orbits stay as orbits! On the other hand, the euler integrator quickly spirals out, since we lose energy during the integration. Note that this is not an issue of numerical precision: The euler integrator is ineherently such that over long timescales, it will lose energy. On the other hand, the leapfrog integrator will always remain stable, even with very large timesteps and low precision.

I present the equations of the leapfrog integrator, a proof sketch that it is symplectic, and the code listing that was used to generate the above plot. Often, code makes most ideas very clear!

The integrator

Code listing

Incantations
# Run HMC with a particular choice of potential
import numpy as np
from matplotlib.animation import FuncAnimation
import matplotlib.pyplot as plt
import numpy.linalg
# dq/dt = dH/dp|_{p0, q0}
# dp/dt = -dH/dq|_{p0, q0}
def leapfrog(dhdp, dhdq, q0, p0, dt):
    p0 += -dhdq(q0, p0) * 0.5 * dt 

    # full step position
    # q += dt * p 
    q0 += dhdp(q0, p0) * dt
    
    # half step position
    p0 += -dhdq(q0, p0) * 0.5 * dt 
    return (q0, p0)

For reference, we also implement an euler integrator, that uses the derivative to compute the position and momentum of the next timestep independently.

def euler(dhdp, dhdq, q, p, dt):
    pnew = p + -dhdq(q, p) * dt
    qnew = q + dhdp(q, p) * dt
    return (qnew, pnew)

Finally, we implement planet(integrator, n, dt) which simulates gravitational potential and usual kinetic energy, using the integrator given by integrator for n steps, with each timestep taking dt.

def planet(integrator, n, dt):
    STRENGTH = 0.5

    # minimise potential V(q): q, K(p, q) p^2
    q = np.array([0.0, 1.0])
    p = np.array([-1.0, 0.0])
    
    # H = STRENGTH * |q| (potential) + p^2/2 (kinetic)
    def H(qcur, pcur): return STRENGTH * np.linalg.norm(q) + np.dot(p, p) / 2
    def dhdp(qcur, pcur): return p
    def dhdq(qcur, pcur): return STRENGTH * 2 * q / np.linalg.norm(q)
    
    qs = []
    for i in range(n):
        print("q: %10s | p: %10s | H: %6.4f" % (q, p, H(q, p)))
        (q, p) = integrator(dhdp, dhdq, q, p, dt)
        qs.append(q.copy())
    return np.asarray(qs)

We plot the simulations using matplotlib and save them.

NITERS = 15
TIMESTEP = 1

print("planet simulation with leapfrog")
planet_leapfrog = planet(leapfrog, NITERS, TIMESTEP)

plt.rcParams.update({'font.size': 12, 'font.family':'monospace'})
fig, ax = plt.subplots()
print(planet_leapfrog)
ax.plot(planet_leapfrog[:, 0], planet_leapfrog[:, 1], label='leapfrog',
        linewidth=3, color='#00ACC1')
print("planet simulation with euler")
planet_euler = planet(euler, NITERS, TIMESTEP)
ax.plot(planet_euler[:, 0], planet_euler[:, 1], label='euler',
        linewidth=3, color='#D81B60')

legend = plt.legend(frameon=False)
ax.set_title("leapfrog v/s euler: NITERS=%s dt=%s" % (NITERS, TIMESTEP))
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.spines['bottom'].set_visible(False)
ax.spines['left'].set_visible(False)
plt.savefig("leapfrog-vs-euler.png")
plt.show()

Comparison of forward and reverse mode AD

Quite a lot of ink has been spilt on this topic. My favourite reference is the one by Rufflewind.

However, none of these examples have a good stock of examples for the diference. So here, I catalogue the explicit computations between computing forward mode AD and reverse mode AD.

In general, in forward mode AD, we fix how much the inputs wiggle with respect to a parameter $t$. We figure out how much the output wiggles with respect to $t$. If $output = f(input_1, input_2, \dots input_n)$, then $\frac{\partial output}{\partial t} = \sum_i \frac{\partial f}{\partial input_i} \frac{\partial input_i}{\partial dt}$.

In reverse mode AD, we fix how much the parameter $t$ wiggles with respect to the output. We figure out how much the parameter $t$ wiggles with respect to the inputs. If $output_i = f_i(input, \dots)$, then $\frac{\partial t}{\partial input} = \sum_i \frac{\partial t}{\partial output_i} \frac{\partial f_i}{input}$. This is a much messier expression, since we need to accumulate the data over all outputs.

Essentially, deriving output from input is easy, since how to compute an output from an input is documented in one place. deriving input from output is annoying, since many outputs can depent on a single output.

The upshot is that if we have few “root outputs” (like a loss function), we need to run AD once with respect to this, and we will get the wiggles of all inputs at the same time with respect to this output, since we compute the wiggles output to input.

The first example of z = max(x, y) captures the essential difference between the two approached succinctly. Study this, and everything else will make sense.

Maximum: z = max(x, y)

We can compute $\frac{\partial z}{\partial x}$ by setting $t = x$. That is, $\frac{\partial x}{\partial t} = 1, \frac{\partial y}{\partial t} = 0$. Similarly, can compute $\frac{\partial z}{\partial y}$ by setting $t = y$. That is, $\frac{\partial x}{\partial t} = 1, \frac{\partial y}{\partial t} = 0$. If we want both gradients $\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}$, we will have to rerun the above equations twice with the two initializations.

In our equations, we are saying that we know how sensitive the inputs $x, y$ are to a given parameter $t$. We are deriving how sensitive the output $z$ is to the parameter $t$ as a composition of $x, y$. If $x > y$, then we know that $z$ is as sensitive to $t$ as $x$ is.

We can compute $\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}$ in one shot by setting $t = z$. That is, $\frac{\partial z}{\partial t} = 1$.

In our equations, we are saying that we know how sensitive the parameter $t$ is to a given output $z$. We are trying to see how sensitive $t$ is to the inputs $x, y$. If $x$ is active (ie, $x > y$), then $t$ is indeed sensitive to $x$ and $\frac{\partial t}{\partial x} = 1$. Otherwise, it is not sensitive, and $\frac{\partial t}{\partial x} = 0$.

sin: z = sin(x)

We can compute $\frac{\partial z}{\partial x}$ by setting $t = x$. That is, setting $\frac{\partial x}{\partial t} = 1$.

We can compute $\frac{\partial z}{\partial x}$ by setting $t = z$. That is, setting $\frac{\partial z}{\partial t} = 1$.

addition: z = x + y:

multiplication: z = xy

subtraction: z = x - y:

An invitation to homology and cohomology, Part 1 — Homology

There are many introductions to homology on the internet, but none of them really met my criteria for being simple, picture filled, and getting the basic ideas across. I feel that Hatcher might come closest to what I want (and where I originally learnt the material), but their description of homology is surrounded by the context of Algebraic Topology, while really, simplicial homology is accessible to anyone who has seen some linear algebra and group theory. This is my attempt to get the ideas across.

Let’s first try to understand what we’re trying to do here. We want to detect holes in a space, broadly construed. We focus in simplicial complexes, which are collections of triangles and triangle-like objects in higher (and lower) dimensions. We define what holes are for these simplicial complexes, and we then try to find algebraic objects that allow us to “detect” these holes.

Simplices

Simplicial complexes

A simplicial complex $K$ is a collection of simplices where:

Examples of simplicial complexes:

points

unfilled-tri

complex-unfilled-butterfly

complex-half-filled-butterfly

Non-examples of simplicial complexes are:

non-simplex-1

non-simplex-2

Holes in a space: Homology of a triangle

Let’s consider the simplest possible case of computing the homlogy, and we do so, we will expand on what homology is, and what we’re trying to do.

homology-triangle-edges

Look at the triangle above. We have the red, green, and blue vertices, which I’ll notate $r, g, b$. We also have the edges that are orange ($o$), cyan ($c$), and magenta ($m$).

What we are interested in doing is to be able to detect the “hole” in the triangle in between the edges o-m-c. That is, we want some algorithm which when offered the representation of the triangle, can somehow detect the hole. Note that the hole doesn’t really depend on the length of the edges. We can “bend and stretch” the triangle, and the hole will still exist. The only way to destroy the hole is to either cut the triangle, or fill in the triangle.

We first need to agree on an abstract representation of the triangle, which ideally does not change if we were to stretch out the edges, before we can discuss how we can detect the existence of the hole.

Representation of the triangle: boundary operators

We first describe the shape of the triangle in terms of two sets, $E$ and $V$ representing the edges and the vertices, and a function $\partial_{EV}$, called as the boundary operator, which tells us how edges are glued to vertices.

We first have a ground set of edges $E \equiv \{o, m, c\}$ and a set of vertices $V \equiv \{r, g, b \}$.

What we now need is to know how the edges are connected to the vertices, since that’s what really makes a triangle. We would like to say something like “the boundary of the edge $o$ has points $r, g$”. In fact, we have slightly more information than that: the orientation. So what we really ought to be saying is “the edge $o$ points from $g$ to $r$”.

To do this, we create a map from $o$ to $r - g$, where we think of $o$ as a “vector”, pointing from $g$ to $r$. But hang on, what is $r - g$? we don’t have a mathematical structure on $V$ that lets us add and subtract vertices. So, we create a new set $\mathcal V$, which represents linear combinations of vertices in $V$. Similarly, anticipating some future development, we also create a new set $\mathcal E$ of linear combinations of edges $E$.

Formal definition of the boundary operator

We define $\mathcal E \equiv \mathbb Z \times \mathbb Z \times \mathbb Z$ that represents linear combinations of edges. For example, $(1, 2, 3) \in \mathcal E$ represents $o + 2m + 3c$ — that is, take 1 copy of the orange edge, 2 copies of the magenta edge, and 3 copies of the cyan edge.

We define $\mathcal V \equiv \mathbb Z \times \mathbb Z \times \mathbb Z$ which represents linear combinations of vertices. For example, $(1, -1, 2) \in V$ represents $r - g + 2b$ — that is, take a copy of the red vertex, subtract the green vertex, and add two copies of the blue vertex.

The boundary operator $\partial_{EV}: \mathcal{E} \rightarrow \mathcal V$ is depicted in the picture. This operator sends edges to their boundary, and is therefore called the boundary operator. The boundary of an edge describes the edge in terms of vertices, just like we would describe a direction vector (to borrow physics parlance) by subtracting points.

The action of the operator on a linear combination of edges is:

Now, notice that to traverse the cycle, we should traverse the orange edge, then the magenta edge, then the cyan edge, in that direction. That is, the cycle can be thought of as $o + m + c$. However, how do we detect this cycle? The key idea is that if we look at the image of the cycle $o + m + c$ under the boundary operator $\partial_{EV}$, we will get $0$! For us to have completed a cycle, we must have both entered and exited each vertex, so the total sum must be $0$.

Formally:

Formal definition of cycles

This is very nice, since we have converted the topological invariant of a hole in the space into an algebraic invariant of “linear combination of edges that map to 0”. That is, we want to consider all thoose loops that belong to the kernel of $\partial_{EV}$. (Terminology: the kernel of a linear transformation is the set of all things in the domain which map to zero)

So, we define (tentatively) the first homology group:

If we try to compute this, we will have to have:

So, we know that we have a $\mathbb Z$ worth of cycles in our triangle, which makes sense: We can go clockwise (positive numbers) and counter-clockwise (negative numbers) around the triangle, and we can go as many times as we wish, so we have $\mathbb Z$ as the number of cycles.

that is, it’s the linear combination of edges that map to zero through the boundary map. Note that this also includes combinations such as two loops around the triangle, such as $o + m + c + o + m + c$.

(No) Holes in a space: Homology of a filled triangle

homology-triangle-faces

In this case, notice that the triangle is filled with a face $f$. Therefore, the “hole” that we had previously is now filled up, and does not count anymore. So, we now need to amend our previous definition of $H_1$ to kill the hole we had detected.

The idea is that the hole we had previously is now the boundary of the new face $f$. Since it is the boundary of a “filled in” region, it does not count anymore, since we can “shrink the hole” across the face to make it a non-loop. Hence, we need to quotient our $H_1$ with the boundary of the face.

Formally, what we do is we create another group $\mathcal F \equiv \mathbb Z$, which counts copies of our face $f$, and we define another boundary operator, such that the boundary of the face $f$ is $o + m + c$.

Now, we should notice that the image of $\partial_{FE}$ is a loop $(o + m + c)$, which lies ie the kernel of $\partial_{EV}$. This is a general feature of homology, so it bears repeating:

Now, since the image of $\partial_{FE}$ lies entirely in the kernel of $\partial_{EV}$, we can construct $H_1$ as:

A complicated space: Homology of a butterfly

An invitation to homology and cohomology, Part 2 — Cohomology

cohomology-triangle-vertices

Once again, we have our humble triangle with vertices $V = \{r, g, b\}$, edges $E = \{o, m, c \}$, faces $F = \{ f \}$ with boundary maps $\partial_{EV}$, $\partial_{FE}$:

We define a function $h_v: V \rightarrow \mathbb R$ on the vertices as:

We now learn how to extend this function to the higher dimensional objects, the edges and the faces of the triangle.

To extend this function to the edges, we define a new function:

Expanded out on the example, we evaluate $h_v$ as:

More conceptually, we have created an operator called $d$ (the coboundary operator) which takes functions defined on vertices to functions defined on edges. This uses the boundary map on the edges to “lift” a function on the vertices to a function on the edges. It does so by assigning the “potential difference” of the vertices to the edges.

We can repeat the construction we performed above, to construct another operator $d : (E \rightarrow \mathbb R) \rightarrow (F \rightarrow \mathbb R)$, defined in exactly the same way as we did before. For example, we can evaluate:

What we have is a chain:

Where we notice that $d^2 = d \circ d = 0$, since the function $h_f$ that we have gotten evaluates to zero on the face $f$. We can prove this will happen in general, for any choice of $h_v$. (it’s a good exercise in definition chasing).

Introducing some terminology, A differential form $f$ is said to be a closed differential form iff $df = 0$.

In our case, $h_e$ is closed, since $d h_e = h_f = 0$. On the other hand $h_v$ is not closed, since $d h_v = h_e \neq 0$.

The intuition for why this is called “closed” is that its coboundary vanishes.

Exploring the structure of functions defined on the edges

Here, we try to understand what functions defined on the edges can look like, and their relationship with the $d$ operator. We discover that there are some functions $g_e: E \rightarrow \mathbb R$ which can be realised as the differential of another function $g_v: V \rightarrow \mathbb R$. The differential forms such as $g_e$ which can be generated a $g_v$ through the $d$ operator are called as exact differential forms. That is, $g_e = d g_v$ exactly, such that there is no “remainder term” on applying the $d$ operator.

cohomology-triangle-edges

We take an example of a differential form that is not exact, which has been defined on the edges of the triangle above. Let’s call it $h_e$.

It is defined on the edges as:

We can calcuate $h_f = d h_e$ the same way we had before:

Since $d h_e \neq 0$, this form is not exact.

Let’s also try to generate $h_e$ from a potential. We arbitrarily fix the potential of $b$ to $0$. That is, we fix $h_v(b) = 0$, and we then try to see what values we are forced to values of $h_v$ across the rest of the triangle.

Hence, there can exist no such $h_v$ such that $h_e \equiv d h_v$. The interesting thing is, when we started out by assigning $h_v(b) = 0$, we could make local choices of potentials that seemed like they would fit together, but they failed to fit globally throughout the triangle. This failure of locally consistent choices to be globally consistent is the essence of cohomology.

Cohomology of half-filled butterfly

cohomology-half-filled-butterfly

Here, we have vertices $V \equiv \{ r, g, b, b, p \}$, edges $E \equiv \{rb, gr, bg, m, o, c \}$ and faces $F \equiv \{ f \}$.

Here, we see a differential form $h_e$ that is defined on the edges, and also obeys the equation $dh_e = 0$ (Hence is closed). However, it does not have an associated potential energy to derive it from. That is, there cannot exist a certain $h_v$ such that $d h_v = h_e$.

So, while every exact form is closed, not every closed form is exact.

Hence, this $g$ that we have found is a non-trivial element of $Kernel(d_{FE}) / Image(d_{EV})$, since $dh_e = 0$, hence $h_e \in Kernel(d_{FE})$, while there does not exist a $h_v$ such that $d h_v = h_e$, hence it is not quotiented by the image of $d_{EV}$.

So the failure of the space to be fully filled in (ie, the space has a hole), is measured by the existence of a function $h_e$ that is closed but not exact!

This reveals a deep connection between homology and cohomology, which is made explicit by the Universal Coefficient Theorem

Stuff I learnt in 2019

I write these retrospective blog posts every year since 2017. I tend to post a collection of papers, books, and ideas I’ve stumbled across that year. Unfortunately, this year, the paper list will be sparser, since I lost some data along the way to the year, and hence I don’t have links to everything I read. So this is going to be a sparser list, consisting of things that I found memorable.

I also re-organised my website, letting the link die, since keeping it up was taking far too many cycles (In particular, CertBot was far too annoying to maintain, and the feedback of hugo was also very annoying). I now have a single file, the README.mdof the bollu/bollu.github.io repo, to which I add notes on things I find interesting. I’ve bound the i alias (for idea) on all my shells everywhere, to open the README.md file, wait for me to write to it, run a git commit so I can type out a commit, and then push. This has been massive for what I manage to write down: I feel like I’ve managed to write down a lot of one-liners / small facts that I’ve picked up which I would not otherwise. I’m attempting to similarly pare down other friction-inducing parts of my workflow. Suggestions here would be very welcome!

If there’s a theme of the year (insofar as my scattered reading has a theme…), it’s “lattices and geometry”. Geometry in terms of differential geometry, topology, and topoi. Lattices in the sense of a bunch of abstract interpretation and semantics.

Course work: optimisation theory, quantum computation, statistics

My course work was less interesting to me this time, due to the fact that I had chosen to study some wild stuff earlier on, and now have to take reasonable stuff to graduate. However, there were courses that filled in a lot of gaps in my self-taught knowledge for me, and the ones I listed were the top ones in that regard.

I wound up reading Boyd on optimisation theory, Nielsen and Chuang for quantum computation, where I also solved a bunch of exercises in Q# which was very fun and rewarding. I’m beginning to feel that learning quantum computation is the right route to grokking things like entanglement and superposition, unlike the physics which is first of all much harder due to infinite dimensionality, and less accessible since we can’t program it.

Formal research work: Compilers, Formal verification, Programming languages

My research work is on the above topics, so I try to stay abreast of what’s going on in the field. What I’ve read over the past year on these topics is:

Internship at Tweag.io over the summer: Hacking on Asterius (Haskell -> WebAssembly compiler)

I really enjoyed my time at Tweag! It was fun, and Shao Cheng was a great mentor. I must admit that I was somewhat distracted, by all the new and shiny things I was learning thanks to all the cool people there :) In particular, I wound up bugging Arnaud Spiwack, Simeon Carstens, and Matthias Meschede quite a bit, about type theory, MCMC sampling, and signal processing of storm clouds.

I wound up reading a decent chunk of GHC source code, and while I can’t link to specifics here, I understood a lot of the RTS much better than I did before. It was an enlightening experience, to say the least, and being paid to hack on a GHC backend was a really fun way to spend the summer.

It also led me to fun discoveries, such as how does one debug debug info?

I also really loved Paris as a city. My AirBnb host was a charming artist who suggest spots for me around the city, which I really appreciated. Getting around was disorienting for the first week or so, due to the fact that I could not (and still do not) really understand how to decide in which direction to walk inside the subways to find a particular line going in a particular direction.

The city has some great spots for quiet work, though! In particular, the Louvre Anticafe was a really nice place to hang out and grab coffee. The model is great: you pay for hours spent at the Anticafe, with coffee and snacks free. They also had a discount for students which I gratefully used. I bumped into interesting artists, programmers, and students who were open for conversation there. I highly recommend hanging out there.

Probabilistic programming & giving a talk at FunctionalConf

This was the first talk I’d ever given, and it was on probabilistic programming in haskell. In particular, I explained the monad-bayes approach of doing this, and why this was profitable. The slides are available here.

It was a fun experience giving a talk, and I’d like to do more of it, since I got a lot out of attempting to explain the ideas to people. I wish I had more time, and had a clearer idea of who the audience was. I got quite a bit of help from Michael Snoyman to whip the talk into shape, which I greatly appreciated.

The major ideas of probabilistic programming as I described it are from Adam Scibior’s thesis:

Along the way, I and others at tweag read the other major papers in the space, including:

Presburger Arithmetic

Since we use a bunch of presburger arithmetic for polyhedral compilation which is a large research interest of mine, I’ve been trying to build a “complete” understanding of this space. So this time, I wanted to learn how to build good solvers:

Open questions for which I want answers

I want better references to being able to regenerate the inequalities description from a given automata which accepts the presburger set automata. This will allow one to smoothly switch between the geometric description and the algebraic description. There are some operations that only work well on the geometry (such as optimisation), and others that only work well on the algebraic description (such as state-space minimisation). I have not found any good results for this, only scattered fragments of partial results. If nothing else, I would like some kind of intuition for why this is hard.

Having tried my stab at it, the general impression that I have is that the space of automata is much larger than the things that can be encoded as presburger sets. Indeed, it was shown that automata accept numbers which are ultimately periodic.

But yes, it’s known that automata accept a language that’s broader than just first order logic + “arithmetic with +”, which means it’s hard to dis-entangle the presburger gits from the non-presburger bits of the automata.

Prolog

I wanted to get a better understading of how prolog works under the hood, so I began re-implementing the WAM: warren abstract machine. It’s really weird, this is the only stable reference I can find to implementing high-performance prolog interpreters. I don’t really understand how to chase the paper-trail in this space, I’d greatly appreciate references. My implementation is at bollu/warren-cpp. Unfortunately, I had to give up due to a really hard-to-debug bug.

It’s crazy to debug this abstract machine, since the internal representation gets super convoluted and hard to track, due to the kind of optimised encoding it uses on the heap.

If anyone has a better/cleaner design for implementing good prologs, I’d love to know.

Another fun paper I found in this space thanks to Edward Kmett was the Rete matching algorithm, which allows one to declare many many pattern matches, which are then “fused” together into an optimal matcher that tries to reuse work across failed matchers.

General Relativity

This was on my “list of things I want to understand before I die”, so I wound up taking up an Independent Study in university, which basically means that I study something on my own, and visit a professor once every couple weeks, and am graded at the end of the term. For GR, I wound up referencing a wide variety of sources, as well as a bunch of pure math diffgeo books. I’ve read everything referenced to various levels. I feel I did take away the core ideas of differential and Riemannian geometry. I’m much less sure I’ve grokked general relativity, but I can at least read the equations and I know all the terms, so that’s something.

Discrete differential geometry

I can’t recall how I ran across this: I think it was because I was trying to get a better understanding of Cohomology, which led me to Google for “computational differential geometry”, that finally led me to Discrete differential geometry.

It’s a really nice collection of theories that show us how to discretize differential geometry in low dimensions, leading to rich intuitions and a myriad of applications for computer graphics.

Synthetic differential geometry

It was Arnaud Spiwack who pointed me to this. It’s a nice axiomatic system of differential geometry, where we can use physicist style proofs of “taking stuff upto order dx”, and having everything work upto mathematical rigor.

The TL;DR is that we want to add a new number called dx into the reals, such that dx^2 = 0. But if such a number did exist, then clearly dx = 0. However, the punchline is that to prove that dx^2 = 0 => dx = 0 requires the use of contradiction!

So, if we banish the law of excluded middle (and therefore no longer use proof by contradiction), we are able to postulate the existence of a new element dx, which obeys dx^2 = 0. Using this, we can build up the whole theory of differential geometry in a pleasing way, without having to go through the horror that is real analysis. (I am being hyperbolic, but really, real analytic proofs are not pleasant).

I began formalizing this in Coq and got a formalism going: bollu/diffgeo.

Once I was done with that, I realised I don’t know how to exhibit models of the damn thing! So, reading up on that made me realise that I need around 8 chapters worth of a grad level textbook (the aptly named Models of Smooth Infinitesimal Analysis).

I was disheartened, so I asked on MathOverflow (also my first ever question there), where I learnt about tangent categories and differential lambda calculus. Unfortunately, I don’t have the bandwidth to read another 150-page tome, so this has languished.

Optimisation on Manifolds

I began reading Absil: Optimisation on matrix manifolds which describes how to perform optimisation / gradient descent on arbitrary Riemannian manifolds, as well as closed forms for well-known manifolds. The exposition in this book is really good, since it picks a concrete manifold and churns out all the basic properties of it manually. The only problem I had with the books was that there were quite a few gaps (?) in the proofs – perhaps I missed a bunch.

This led me to learn Lie theory to some degree, since that was the natural setting for many of the proofs. I finally saw why anyone gives a shit about the tangent space at the identity: because it’s easier to compute! For a flavour of this, consider this question on math.se by me that asks about computing tangent spaces of $O(n)$.

AIRCS workshop

I attended the AI risk for computer scientists workshop hosted by MIRI (Machine intelligence research institute) in December. Here, a bunch of people were housed at a bed & breakfast for a week, and we discussed AI risk, why it’s potentially the most important thing to work on, and anything our hearts desired, really. I came away with new branches of math I wanted to read, a better appreciation of the AI risk community and a sense of what their “risk timelines” were, and some explanations about sheaves and number theory that I was sorely lacking. All in all, it was a great time, and I’d love to go back.

P-adic numbers

While I was on a particularly rough flight back from the USA to India when coming back from the AIRCS workshop, I began to read the textbook Introduction to p-adic numbers by Fernando Gouvea, which fascinated me, so I then wrote up the cool parts introduced in the first two chapters as a blog post. I wish to learn more about the p-adics and p-adic analysis, since they seem to be deep objects in number theory.

In particular, a question that I thought might have a somewhat trivial answer (why do the p-adics use base p in defining norm) turned out to have answers that were quite deep, which was something unexpected and joyful!

Topology of functional programs

Both of these describe a general method to transfer all topological ideas as statements about computability in a way that’s far more natural (at least for me, as a computer scientist). The notion of what a continuous function “should be” (keeping inverse images of open sets open) arises naturally from this computational viewpoint, and many proofs of topology amount to finding functional programs that fit a certain type. It’s a great read, and I feel gave me a much better sense of what topology is trying to do.

Philosophy

I’ve wanted to understand philosophy as a whole for a while now, at least enough to get a general sense of what happened in each century. The last year, I meandered through some philosophy of science, which led me to some really wild ideas (such as that of Paul Feyerabend’s ‘science as an anarchic enterprise’ which I really enjoyed).

I also seem to get a lot more out of audio and video than text in general, so I’ve been looking for podcasts and video lectures. I’ve been following:

I also attempted to read a bunch of philosophers, but the only ones I could make a good dent on were the ones listed below. I struggled in writing this section, since it’s much harder to sanity check my understanding of philosophy, versus mathematics, since there seems to be a range of interpretations of the same philosophical work, and the general imprecise nature of language doesn’t help here at all. So please take all the descriptions below with some salt to taste.

Information theory

I’ve been on a quest to understand information theory far better than I currently do. In general, I feel like this might be a much better way to internalize probability theory, since it feels like it states probabilistic objects in terms of “couting” / “optimisation of encodings”, which is a perspective I find far more natural.

Towards this aim, I wound up reading:

Building intuition for Sheaves, Topoi, Logic

I don’t understand the trifecta of sheaves, topoi, geometry, and logic, and I’m trying to attack this knot from multiple sides at once.

All of these provide geometric viewpoints of what sheaves are, in low-dimensional examples of graphs which are easy to visualize. I’m also trudging through the tome:

which appears to follow the “correct path” of the algebraic geometers, but this requires a lot of bandwidth.

This is a hardcore algebraic geometry textbook, and is arguably great for studying sheaves because of it. Sheaves are Chapter 2, and allows one to see them be developed in their “true setting” as it were. In that Grothendeick first invented sheaves for algebraic geometry, so it’s good to see them in the home they were born in. Once again, this is a book I lack bandwidth for except to breezily read it as I go to bed. I did get something out from doing this. I’m considering taking this book up as an independent study, say the first four chapters. I’ll need someone who knows algebraic geometry to supervise me, though, which is hard to find in an institute geared purely for computer science. (If anyone on the internet is kind enough to volunteer some of their time to answer questions, I’ll be very glad! Please email me at rot13(fvqqh.qehvq@tznvy.pbz))

The attic

This section contains random assortments that I don’t recall how I stumbled across, but too cool to not include on the list. I usually read these in bits and pieces, or as bedtime reading right before I go to bed to skim. I find that skimming such things gives me access to knowing about tools I would not have known otherwise. I like knowing the existence of things, even if I don’t recall the exact thing, since knowing that something like X exists has saved me from having to reinvent X from scratch.

Conclusions

I always feel a little wrong posting this at the end of every year, since I feel that among the things I cover under “read”, I’ve internalized some things far better than others: For example, I feel I understannd Riemannian geometry far better than I do General Relativity. I try to put up the caveats at the beginning of each section, but I’d really like a way to communicate my confidence without reducing readability.

The final thing that I wish for is some kind of reading group? It’s hard to maintain a group when my interests shift as rapidly as they do, which was one of the reason I really loved the AIRCS workshop: They were people who were working on formal methods, compilers, type theory, number theory, embedded systems, temporal logic… It was very cool to be in a group of people who had answers and intuitions to questions that had bugged me for some time now. I wonder if attending courses at a larger research university feels the same way. My uni is good, but we have quite small population, which almost by construction means reduced diversity.

I also wish that I could openly add more references to repos I’ve been working on for a while now, but I can’t due to the nature of academia and publishing. This one bums me out, since there’s a long story of a huge number of commits and trial-by-fire that I think I’ll be too exhausted to write about once the thing is done.

Sometimes, I also wish that I could spend the time I spend reading disparate topics on focused reading on one topic. Unfortunately, I feel like I’m not wired this way, and the joy I get from sampling many things at the same time and making connections is somehow much deeper than the joy I get by deeply reading one topic (in exclusion of all else). I don’t know what this says about my chances as a grad student in the future :).

A motivation for p-adic analysis

I’ve seen the definitions of p-adic numbers scattered around on the internet, but this analogy as motivated by the book p-adic numbers by Fernando Gouvea really made me understand why one would study the p-adics, and why the definitions are natural. So I’m going to recapitulate the material, with the aim of having somoene who reads this post be left with a sense of why it’s profitable to study the p-adics, and what sorts of analogies are fruitful when thinking about them.

We wish to draw an analogy between the ring $\mathbb C[X]$, where $(X - \alpha)$ are the prime ideals, and $\mathbb Z$ where $(p)$ are the prime ideals. We wish to take all operations one can perform with polynomials, such as generating functions ($1/(X - \alpha) = 1 + X + X^2 + \dots$ ), taylor expansions (expanding aronund $(X - \alpha)$), and see what their analogous objects will look like in $\mathbb Z$ relative to a prime $p$.

Perspective: Taylor series as writing in base $p$:

Now, for example, given a prime $p$, we can write any positive integer $m$ in base $p$, as $(m = \sum_{i=0}^n a_i p^i)$ where $(0 \leq a_i \leq p - 1)$.

For example, consider $m = 72, p = 3$. The expansion of 72 is $72 = 0\times 1 + 0 \times 3 + 2 \times 3^2 + 2 \times 3^3$. This shows us that 72 is divisible by $3^2$.

This perspective to take is that this us the information local to prime $p$, about what order the number $m$ is divisible by $p$, just as the taylor expansion tells us around $(X - \alpha)$ of a polynomial $P(X)$ tells us to what order $P(X)$ vanishes at a point $\alpha$.

Perspective: rational numbers and rational functions as infinite series:

Now, we investigate the behaviour of expressions such as

We know that the above formula is correct formally from the theory of generating functions. Hence, we take inspiration to define values for rational numbers.

Let’s take $p \equiv 3$, and we know that $4 = 1 + 3 = 1 + p$.

We now calculate $1/4$ as:

However, we don’t really know how to interpret $(-1 \cdot p)$, since we assumed the coefficients are always non-negative. What we can do is to rewrite $p^2 = 3p$, and then use this to make the coefficient positive. Performing this transformation for every negative coefficient, we arrive at:

We can verify that this is indeed correct, by multiplying with $4 = (1 + p)$ and checking that the result is $1$:

What winds up happening is that all the numbers after $1$ end up being cleared due to the carrying of $(3p^i \mapsto p^{i+1})$.

This little calculation indicates that we can also define take the $p$-adic expansion of rational numbers.

Perspective: -1 as a p-adic number

We next want to find a p-adic expansion of -1, since we can then expand out theory to work out “in general”. The core idea is to “borrow” $p$, so that we can write -1 as $(p - 1) - p$, and then we fix $-p$, just like we fixed $-1$. This eventually leads us to an infinite series expansion for $-1$. Written down formally, the calculation proceeds as:

This now gives us access to negative numbers, since we can formally multiply the series of two numbers, to write $-a = -1 \cdot a$.

Notice that this definition of $-1$ also curiously matches the 2s complement definition, where we have $-1 = 11\dots 1$. In this case, the expansion is infinite, while in the 2s complement case, it is finite. I would be very interested to explore this connection more fully.

What have we achieved so far?

We’ve now managed to completely reinterpret all the numbers we care about in the rationals as power series in base $p$. This is pretty neat. We’re next going to try to complete this, just as we complete the rationals to get the reals. We’re going to show that we get a different number system on completion, called $\mathbb Q_p$.

To perform this, we first look at how the $p$-adic numbers help us solve congruences mod p, and how this gives rise to completions to equations such as $x^2 - 2 = 0$, which in the reals give us $x = \sqrt 2$, and in $\mathbb Q_p$ give us a different answer!

Solving $X^2 \equiv 25 \mod 3^n$

Let’s start by solving an equation we already know how to solve: $X^2 \equiv 25 \mod 3^n$.

We already know the solutions to $X^2 \equiv 25 \mod 3^n$ in $\mathbb Z$ are $X \equiv \pm 5 \mod 3^n$.

Explicitly, the solutions are:

This was somewhat predictable. We move to a slightly more interesting case.

Solving $X = -5 \mod 3^n$

The solution sets are:

This gives us the infinite 3-adic expansion:

Note that we can’t really predict the digits in the 3-adic sequence of -5, but we can keep expanding and finding more digits.

Also see that the solutions are “coherent”. In that, if we look at the solution mod 9, which is $4$, and then consider it mod 3, we get $1$. So, we can say that given a sequence of integers $0 \leq \alpha_n \leq p^n - 1$, $\alpha_n$ is p-adically coherent sequence iff:

Viewpoint: Solution sets of $X^2 = 25 \mod 3^n$

Since our solution sets are coherent, we can view the solutions as a tree, with the expansions of $X = 5, X = -5 \mod 3$ and then continuing onwards from there. That is, the sequences are

Solving $X^2 \equiv 2 \mod 7^n$

We now construct a solution to the equation $X^2 = 1$ in the 7-adic system, thereby showing that $\mathbb Q_p$ is indeed strictly larger than $\mathbb Q$, since this equation does not have rational roots.

For $n=1$, we have the solutions as $X \equiv 3 \mod 7$, $X \equiv 4 \equiv -3 \mod 7$.

To find solutions for $n = 2$, we recall that we need our solutions to be consistent with those for $n = 1$. So, we solve for:

Solving the first of these:

This gives the solution $X \equiv 10 \mod 49$. The other branch ($X = 4 + 7k$) gives us $X \equiv 39 \equiv -10 \mod 49$.

We can continue this process indefinitely (exercise), giving us the sequences:

We can show that the sequences of solutions we get satisfy the equation $X^2 = 2 \mod 7$. This is so by construction. Hence, $\mathbb Q_7$ contains a solution that $\mathbb Q$ does not, and is therefore strictly bigger, since we can already represent every rational in $\mathbb Q$ in $\mathbb Q_7$.

Use case: Solving $X = 1 + 3X$ as a recurrence

Let’s use the tools we have built so far to solve the equation $X = 1 + 3X$. Instead of solving it using algebra, we look at it as a recurrence $X_{n+1} = 1 + 3X_n$. This gives us the terms:

In $\mathbb R$, this is a divergent sequence. However, we know that the solution so $1 + X + X^2 + \dots = 1/(1-X)$, at least as a generating function. Plugging this in, we get that the answer should be:

which is indeed the correct answer.

Now this required some really shady stuff in $\mathbb R$. However, with a change of viewpoint, we can explain what’s going on. We can look at the above series as being a series in $\mathbb Q_3$. Now, this series does really converge, and by the same argument as above, it converges to $-1/2$.

The nice thing about this is that a dubious computation becomes a legal one by changing one’s perspective on where the above series lives.

Viewpoint: ‘Evaluation’ for p-adics

The last thing that we need to import from the theory of polynomials is the ability to evaluate them: Given a rational function $F(X) = P(X)/Q(X)$, where $P(X), Q(X)$ are polynomials, we can evaluate it at some arbitrary point $x_0$, as long as $x_0$ is not a zero of the polynomial $Q(X)$.

We would like a similar function, such that for a fixed prime $p$, we obtain a ring homomorphism from $\mathbb Q \rightarrow \mathbb F_p^x$, which we will denote as $p(x_0)$, where we are imagining that we are “evaluating” the prime $p$ against the rational $x_0$.

We define the value of $x_0 = a/b$ at the prime $p$ to be equal to $ab^{-1} \mod p$, where $b b^{-1} \equiv 1 \mod p$. That is, we compute the usual $ab^{-1}$ to evaluate $a/b$, except we do this $(\mod p)$, to stay with the analogy.

Note that if $b \equiv 0 \mod p$, then we cannot evaluate the rational $a/b$, and we say that $a/b$ has a pole at $p$. The order of the pole is the number of times $p$ occurs in the prime factorization of $b$.

I’m not sure how profitable this viewpoint is, so I asked on math.se, and I’ll update this post when I recieve a good answer.

Perspective: Forcing the formal sum to converge by imposing a new norm:

So far, we have dealt with infinite series in base $p$, which have terms $p^i, i \geq 0$. Clearly, these sums are divergent as per the usual topology on $\mathbb Q$. However, we would enjoy assigning analytic meaning to these series. Hence, we wish to consider a new notion of the absolute value of a number, which makes it such that $p^i$ with large $i$ are considered small.

We define the absolute value for a field $K$ as a function $|\cdot |: K \rightarrow \mathbb R$. It obeys the axioms:

  1. $\lvert x \rvert = 0 \iff x = 0$
  2. $\lvert xy \rvert = \lvert x \rvert \lvert y \rvert$ for all $x, y \in K$
  3. $\lvert x + y \rvert \leq \lvert x \rvert + \lvert y \rvert$, for all $x, y \in K$.

We want the triangle inequality so it’s metric-like, and the norm to be multiplicative so it measures the size of elements.

The usual absolute value $\lvert x \rvert \equiv \{ x : x \geq 0; -x : ~ \text{otherwise} \}$ satisfies these axioms.

Now, we create a new absolute value that measures primeness. We first introduce a gadget known as a valuation, which measures the $p$-ness of a number. We use this to create a norm that makes number smaller as their $p$-ness increases. This will allow infinite series in $p^i$ to converge.

p-adic valuation: Definition

First, we introduce a valuation $v_p: \mathbb Z - \{0\} \rightarrow \mathbb R$, where $v_p(n)$ is the power of the prime $p^i$ in the prime factorization of $n$. More formally, $v_p(n)$ is the unique number such that:

The valuation gets larger as we have larger powers of $p$ in the prime factorization of a number. However, we want the norm to get smaller. Also, we need the norm to be multiplicative, while $v_p(nm) = v_p(n) + v_p(m)$, which is additive.

To fix both of these, we create a norm by exponentiating $v_p$. This converts the additive property into a multiplicative property. We exponentiate with a negative sign so that higher values of $v_p$ lead to smaller values of the norm.

p-adic abosolute value: Definition

Now, we define the p-adic absolute value of a number $n$ as $|n|_p \equiv p^{-v_p(n)}$.

So $|n|_p$ is indeed a norm, which measures $p$-ness, and is smaller as $i$ gets larger in the power $p^i$ of the factorization of $n$, causing our infinite series to converge.

There is a question of why we chose a base $p$ for $|n|_p = p^{v_p(n)}$. It would appear that any choice of $|n|_p = c^{v_p(n)}, c > 1$ would be legal. I asked this on math.se, and the answer is that this choosing a base $p$ gives us the nice formula

That is, the product of all $p$ norms and the usual norm (denoted by $\lvert x \rvert_\infty $ ) give us the number 1. The reason is that the $ \lvert x\rvert_p $ give us multiples $p^{-v_p(x)}$, while the usual norm $\lvert x \rvert_\infty$ contains a multiple $p^{v_p(x)}$, thereby cancelling each other out.

Conclusion

What we’ve done in this whirlwind tour is to try and draw analogies between the ring of polynomials $\mathbb C[X]$ and the ring $\mathbb Z$, by trying to draw analogies between their prime ideals: $(X - \alpha)$ and $(p)$. So, we imported the notions of generating functions, polynomial evaluation, and completions (of $\mathbb Q$) to gain a picture of what $\mathbb Q_p$ is like.

We also tried out the theory we’ve built against some toy problems, that shows us that this point of view maybe profitable. If you found this interesting, I highly recommend the book p-adic numbers by Fernando Gouvea.

Line of investigation to build physical intuition for semidirect products

To quote wikipedia:

In crystallography, the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group is symmorphic

The if and only if is interesting: The geometry ofthe crystal lattice truly appears to capture the structure of the semidirect product. It’s a discrete object as well, which makes it way easier to visualize. I’m going to hunt down the definitions involved so I can finally feel like I truly understand semidirect products from the “action” perspective.

Topology is really about computation — part 2

Here, we’re going to describe whatever I’ve picked up of sheaves in the past couple of weeks. I’m trying to understand the relationship between sheaves, topoi, geometry, and logic. I currently see how topoi allows us to model logic, and how sheaves allow us to model geometry, but I see nothing about the relationship! I’m hoping that writing this down will allow me to gain some perspective on this.

What is a sheaf?

Let’s consider two sets $P, A$, $P \subseteq A$. Now, given a function $f: A \rightarrow X$, we can restrict this function to $ A_P: P \rightarrow X $. So, we get to invert the direction:

.

We should now try to discover some sort of structure to this “reversal” business. Perhaps we will discover a contravariant functor! (Spoiler: we will).

Topology is really about computation — part 1

Most people believe that topology is about some notion of “nearness” or “closeness”, which has been abstracted out from our usual notion of continuity that we have from a metric space. Here, I make the claim that topology is really about computation, and more specifically, decidability. These are not new ideas. I learnt of this from a monograph The topology of computation, but this does not seem very well known, so I decided to write about it.

The idea is this: We have turing machines which can compute things. We then also have a set $S$. Now, a topology $\tau \subset 2^S$ precisely encodes which of the subsets of $S$ can be separated from the rest of the space by a turing machine. Thus, a discrete space is a very nice space, where every point can be separated from every other point. An indescrete space is one where no point can be separated.

Something like the reals is somewhere in between, where we can separate stuff inside an open interval from stuff clearly outside, but there’s some funny behaviour that goes on at the boundary due to things like (0.999... = 1), which we’ll see in detail in a moment.

Semidecidability

A subset $Q\subseteq S$ is semidecidable, if there exists a turing machine $\hat Q: Q \rightarrow { \bot, \top }$, such that:

Where $\top$ signifies stopping at a state and returning TRUE, and $\bot$ signifies never halting at all!. So, the subset $Q$ is semidedicable, in that, we will halt and say TRUE if the element belongs in the set. But if an element does not belong in the set, we are supposed to never terminate.

Deep dive: semidecidability of the interval $(1, 2)$

Let’s start with an example. We consider the interval $I = (1, 2)$, as a subset of $\mathbb{R}$.Let the turing machine recieve the real number as a function $f: \mathbb N \rightarrow {0, 1, \dots 9}$, such that given a real number ${(a_0 \cdot a_1 \cdot a_2 \dots)}$, this is encoded as a function ${f_a(i) = a_i}$.

We now build a turing machine $\hat I$ which when given the input the function $f_a$, semi-decides whether ${a \in I}$.

Let’s consider the numbers in $I$:

So, we can write a turing machine (ie, some code) that tries to decide whether a real number $a$’s encoding $f_a$ belongs to the interval $I = (1, 2)$ as follows:

def decide_number_in_open_1_2(f):
  # if the number is (1.abcd)
  if f(0) == 1:
    # (1.99...99x) | look for the x.
    # If the number is 1.999..., do not terminate.
    # if the number is any other number of the form 1.99..x, terminate
    i = 1
    while f(i) != 9: i += 1
    return
  # if the number is not 1.abcd, do not terminate
  while True: pass

Hence, we say that the interval $I = (1, 2)$ is semi-decidable, since we have a function $\hat I \equiv \texttt{decide-number-in-open-1-2}$ such that $\hat I (f_a) \text{ terminates } \iff a \in I$. We don’t make any claim about what happens if $a \notin I$. This is the essence of semidecidability: We can precisely state when elements in the set belong to the set, but not when they don’t.

Semi decidability in general

To put this on more solid ground, we define a topology on a set $S$ by considering programs which recieve as input elements of $S$, suitably encoded. For example, the way in which we encoded real numbers as functions from the index to the digit. Similarly, we encode other mathematical objects in some suitable way.

Now, we define:

These are just two viewpoints on the same object. In one, we define the set based on the program. In the other, we define the program based on the set.

Semi decidability of the empty set and the universe set.

The empty set is semi-decidable, due to the existence of the program:

def semidecide_empty(x):
  while True: continue

The universe set is semi-decidable, due to the existence of the program:

def semidecide_univ(x): return

Semi decidability of the union of sets

infinite unions of sets are semi decidable, since we can “diagonalize” on the steps of all programs. That way, if any program halts, we will reach the state where it halts in our diagonalized enumeration.

Let A00, A01... A0n be the initial states of the machines. We are trying to semidecide whether any of them halt. We lay out the steps of the machines in an imaginary grid:

A00 A01 A02 ... A0n
A10 A11 A12 ... A1n
A20 A21 A22 ... A2n
Am0 Am1 Am2 ... Amn

For example, machine A0 has states:

A00 -> A10 -> .. -> Am0

We can walk through the combined state-space of the machines as:

A00
A01 A10
A02 A11 A20
A03 A12 A21 A30
...

Where on the k‘th line, we collect all states $A_{ij}$ such that $(i + j = k)$.

Now, if any of the machines have a state that is HALT, we will reach the state as we enumerate the diagonals, and the machine that explores the combined state space can also return HALT.

Semi decidability of the intersection of sets

infinite intersections of sets are not semi decidable, since by running these programs in parallel, we cannot know if an infinite number of programs halt in finite time. We can tell if one of them halts, but of if all of them halt.

For example, consider the sequence of machines produced by machine_creator:

# creates a machine that stops after n steps
def machine_creator(n):
    # f terminates after n steps
    def f(x):
      for _ in range(n):
        continue
    return

    return f

We wish to check if the intersection of all machine_creator(n) halt, for all $n \geq 0, n \in \mathbb N$. Clearly, the answer is an infinite number of steps, even though every single machine created by machine_creator halts in a finite number of steps.

PSLQ algorithm: finding integer relations between reals

An algorithm to find integer relations between real numbers. It was apparently named “algorithms of the century” by Computing in science and engineering.

Geometric characterization of normal subgroups

$Stab(Orb(x)) = Stab(x) \iff Stab(x) \text{ is normal}$

$\forall x’ \in Orb(x), Stab(x’) = Stab(x) \iff Stab(x) \text{ is normal}$

Forward: if the stabilizer is normal, then all elements in the orbit have the same stabilizer

Let a group $G$ act on a set $X$ with action $(~\dot~) : G \times X \rightarrow X$.
Let $H \subseteq G$ be the stabilizer of a point $x \in X$. Now, let $K = kHk^{-1}$, a conjugacy class of $H$. Clearly, the element $(k \cdot x)$ in the orbit of $x$ is stabilized by $K$.

If the group $H$ is normal, then $K = H$. So every element in the orbit of $x$ is stabilized by $H$.

Interaction of stablizer and the orbit:

$Stab(g \cdot x) = g Stab(x) g^{-1}$

$g^{-1} Stab(g \cdot x) g = Stab(x)$

Hence, both containments are proved.

Backward: if all elements in the orbit have the same orbit, then the stabilizer is normal.

From the above equation $Stab(g \cdot x) = g Stab(x) g^{-1}$. If the entire orbit has the same stabilizer, $Stab (g \cdot x) = Stab(x)$. Hence, we get $Stab(x) = g Stab(x) g^{-1}$, proving that it’s normal.

Handy characterization of adding an element into an ideal, proof that maximal ideal is prime

The characterization

Let $I$ be an ideal. The ideal generated by adding $(a \in R)$ to $I$ is defined as $A \equiv (I \cup { a})$. We prove that $A = I + aR$.

Quotient based proof that maximal ideal is prime

An ideal $P$ is prime iff the quotient ring $R/P$ is an integral domain. An ideal $M$ is maximal $R/M$ is a field. Every field is an integral domain, hence:

$M \text{ is maximal } \implies R/M \text{ is a field } \implies R/M \text {is an integral domain} \implies M \text{ is prime}$.

I was dissatisfied with this proof, since it is not ideal theoretic: It argues about the behaviour of the quotients. I then found this proof that argues purly using ideals:

Ideal theoretic proof that maximal ideal is prime

Sketch

Let $I$ be a maximal ideal. Let $a, b \in R$ such that $ab \in I$. We need to prove that $a \in I \lor b \in I$. If $a \in I$, the problem is done. So, let $a \notin I$. Build ideal $A = (I \cup {a})$. $I \subsetneq A$. Since $I$ is maximal, $A = R$. Hence, there are solutions for $1_R \in A \implies 1_r \in I + aR \implies \exists i \in I, r \in R, 1_R = i + ar$. Now, $b = b \cdot 1_R = b(i + ar) = bi + (ba)r \in I + IR = I$. ($ba \in I$ by assumption). Hence, $b \in I$.

Details

let $i$ be a maximal ideal. let $a, b \in r$ such that $ab \in i$. we need to prove that $a \in i \lor b \in i$.

if $a \in i$, then the problem is done. so, let $a \notin i$. consider the ideal $A$ generated by adding $a$ into $I$. $A \equiv (I \cup {a})$.

We have shown that $A = I + aR$. Hence, $I + a0 = I \subset A$. Also, $0 + ac \dot 1 = a \in A$, $a \neq I$ \implies $A \neq I$. Therefore, $I \subsetneq A$. Since $I$ is maximal, this means that $A = R$

Therefore, $I + aR = R$. Hence, there exists some $i \in I, r \in R$ such that $i + ar = 1_R$.

Now, $b = b \cdot 1_R = b \cdot (i + ar) = bi + (ba) r \in I + IR = I$ Hence, $b \in I$.

Radical ideals, nilpotents, and reduced rings

Radical Ideals

A radical ideal of a ring $R$ is an ideal such that $\forall r \in R, r^n \in I \implies r \in I$. That is, if any power of $r$ is in $I$, then the element $r$ also gets “sucked into” $I$.

Nilpotent elements

A nilpotent element of a ring $R$ is any element $r$ such that there exists some power $n$ such that $r^n = 0$.

Note that every ideal of the ring contains $0$. Hence, if an ideal $I$ of a ring is known to be a radical ideal, then for any nilpotent $r$, since $\exists n, r^n = 0 \in I$, since $I$ is radical, $r \in I$.

That is, a radical ideal with always contain all nilpotents! It will contain other elements as well, but it will contain nilpotents for sure.

Radicalization of an ideal

Given a ideal $I$, it’s radical idea $\sqrt I \equiv { r \in R, r^n \in I }$. That is, we add all the elements $I$ needs to have for it to become a radical.

Notice that the radicalization of the zero ideal $I$ will precisely contain all nilpotents. that is, $\sqrt{(0)} \equiv { r \in R, r^n = 0}$.

Reduced rings

A ring $R$ is a reduced ring if the only nilpotent in the ring is $0$.

creating reduced rings (removing nilpotents) by quotienting radical ideals

Tto remove nilpotents of the ring $R$, we can create $R’ \equiv R / \sqrt{(0}$. Since $\sqrt{(0)}$ is the ideal which contains all nilpotents, the quotient ring $R’$ will contain no nilpotents other than $0$.

Similarly, quotienting by any larger radical ideal $I$ will remove all nilpotents (and then some), leaving a reduced ring.

A ring modulo a radical ideal is reduced

Integral domains

a Ring $R$ is an integral domain if $ab = 0 \implies a = 0 \lor b = 0$. That is, the ring $R$ has no zero divisors.

Prime ideals

An ideal $I$ of a ring $R$ is a prime ideal if $\forall xy \in R, xy \in I \implies x \in I \lor y \in I$. This generalizes the notion of a prime number diving a composite: $p | xy \implies p | x \lor p | y$.

creating integral domains by quotenting prime ideals

Recall that every ideal contains a $0$. Now, if an ideal $I$ is prime, and if $ab = 0 \in I$, then either $a \in I$ or $b \in I$ (by the definition of prime).

We create $R’ = R / I$. We denote $\overline{r} \in R’$ as the image of $r \in R$ in the quotient ring $R’$.

The intuition is that quotienting by a $I$, since if $ab = 0 \implies a \in I \lor b \in I$, we are “forcing” that in the quotient ring $R’$, if $\overline{a} \overline{b} = 0$, then either $\overline{a} = 0$ or $\overline{b} = 0$, since $(a \in I \implies \overline a = 0)$, and $b \in I \implies \overline b = 0)$.

A ring modulo a prime ideal is an integral domain.

I learnt of this explanation from this excellent blog post by Stefano Ottolenghi.

My disenchantment with abstract interpretation

When I first ran across the theory of abstract interpretation, it seemed magical: Define two functions, check that they’re monotone maps, and boom, we have on our hands an analysis.

However, the problem appears to be that in reality, it’s not as simple. Here is the list of issues I’ve run across when trying to use abstract interpretation for a “real world” use-case:

First of all, all interesting lattices are infinte height, requiring some choice of widening. Defining a good widening is a black art. Secondly, while there is a lot of theory on combining abstract domains (reduced products and the like), it seems hard to deploy the theory in the real world.

I read a fair bit into the theory of abstract acceleration, where the idea is that instead of widening indiscriminately, if our theory is powerful enough to compute an exact closed form, we choose to do so. However, the problem is that this regime does not “fit well” into abstract interpretation: We have the abstract interpreter on the one hand, and then the acceleration regime on the other, which is a separate algorithm. So the full analysis looks something like:

def analyze(program):
  analysis = {}
  for loop in inner to outer:
     loop_data = abstract_interpret(loop)
     analaysis.append(accelerate(loop))
  return analysis

That is, what used to be a nice theory of just “do things in any order and it will converge”, now becomes a new algorithm, that uses abstract interpretation as a subroutine. This was not the hope I had! I wanted to get away from having to do proofs by analyzing an algorithm, this was the entire promise: Define a lattice well enough and the proof is guaranteed. Rather, what I had imagined was:

def analyze(program):
  return abstract_interpret_using_acceleration_domain(program)

Now this acceleration_domain maybe frightfully complicated, but I’m willing to pay that price, as long as it’s an honest-to-god abstract interpretation. This was a huge bummer for me to find out that this is not the case.

Computing equivalent gate sets using grobner bases

Here’s a fun little problem, whose only solution I know involves a fair bit of math and computer algebra:

We are given the grammar for a language L:

E = T +_mod8 E | T -_mod8 E | T
T = V | V ^ V | V ^ V ^ V
V = 'a1' | 'a2' | ...

where +_mod8 is addition modulo 8, -_mod8 is subtraction modulo 8, and ^ is XOR.

This language is equipped with the obvious evaluation rules, corresponding to those of arithmetic. We are guaranteed that during evaluation, the variables a_i will only have values 0 and 1. Since we have addition, we can perform multiplication by a constant by repeated addition. So we can perform 3*a as a+a+a.

We are then given the input expression (a0 ^ a1 ^ a2 ^ a3). We wish to find an equivalent expression in terms of the above language L.

We think of E as some set of logic gates we are allowed to use, and we are trying to express the operation (a0 ^ a1 ^ a2 ^ a3) in terms of these gates.

The first idea that I thought was that of employing a grobner basis, since they essentially embody rewrite rules modulo polynomial equalities, which is precisely our setting here.

In this blog post, I’m going to describe what a grobner basis is and why it’s natural to reach for them to solve this problem, the code, and the eventual solution.

As a spolier, the solution is:

a^b^c^d =
-a - b + c + 3*d - 3*axorb - axorc
+ axord - bxorc + bxord + 3*cxord 
- 3*axorbxorc - axorbxord 
+ axorcxord + bxorcxord

Clearly, this contains only additions/subtractions and multiplications by a constant.

If there’s some principled way to derive this (beyond throwing symbolic algebra machinery), I’d really love to know — Please raise an issue with the explanation!

What the hell is Grobner Basis?

The nutshell is that a grobner basis is a way to construct rewrite rules which also understand arithmetic (I learnt this viewpoint from the book “Term rewriting and all that”. Fantastic book in general). Expanding on the nutshell, assume we have a term rewriting system:

A -> -1*B -- (1)
C -> B^2  -- (2)

over an alphabet {A, B, C}.

Now, given the string C + AB, we wish to find out if it can be rewritten to 0 or not. Let’s try to substitute and see what happens:

C + AB -2-> B^2 + AB -1-> B^2 + (-1*B)B

At this point, we’re stuck! we don’t have rewrite rules to allow us to rewrite (-1*B)B into -B^2. Indeed, creating such a list would be infinitely long. But if we are willing to accept that we somehow have the rewrite rules that correspond to polynomial arithmetic, where we view A, B, C as variables, then we can rewrite the above string to 0:

B^2 + (-1*B)B -> B^2 - B^2 -> 0

A Grobner basis is the algorithmic / mathematical machine that allows us to perform this kind of substitution.

In this example, this might appear stupid: what is so special? We simply substituted variables and arrived at 0 by using arithmetic. What’s so complicated about that? To understand why this is not always so easy, let’s consider a pathological, specially constructed example

A complicated example that shatters dreams

Here’s the pathological example:

A -> 1     -- (1)
AB -> -B^2 -- (2)

And we consider the string S = AB + B^2. If we blindly apply the first rule, we arrive at:

S = AB + B^2 -1-> 1B + B^2 = B + B^2 (STUCK)

However, if we apply (2) and then (1):

S = AB + B^2 -2-> -B^2 + B^2 -> 0

This tells us that we can’t just apply the rewrite rules willy-nilly. It’s sensitive to the order of the rewrites! That is, the rewrite system is not confluent.

The grobner basis is a function from rewrite systems to rewrite systems. When given a rewrite system R, it produces a new rewrite system R' that is confluent. So, we can apply the rewrite rules of R' in any order, and we guaranteed that we will only get a 0 from R' if and only if we could have gotten a 0 from R for all strings.

We can then go on to phrase this whole rewriting setup in the language of ideals from ring theory, and that is the language in which it is most often described. I’ve gone into more depth on that perspective here: “What is a grobner basis? polynomial factorization as rewrite systems”.

Now that we have a handle on what a grobner basis is, let’s go on to solve the original problem:

An explanation through a slightly simpler problem

I’ll first demonstrate the idea of how to solve the original problem by solving a slightly simpler problem:

Rewrite a^b^c in terms of a^b, b^c, c^a and the same +_mod8 instruction set as the original problem. The only difference this time is that we do not have T -> V ^ V ^ V.

The idea is to construct the polynomial ring over Z/8Z (integers modulo 8) with variables as a, b, c, axorb, bxorc, axorc. Now, we know that a^b = a + b - 2ab. So, we setup rewrite rules such that a + b - 2ab -> axorb, b + c - 2bc -> bxorb, c + a - 2ca -> cxora.

We construct the polynomial f(a, b, c) = a^b^c, which has been written in terms of addition and multiplication, defined as f_orig(a, b, c) = 4*a*b*c - 2*a*b - 2*a*c - 2*b*c + a + b + c. We then rewrite f_orig with respect to our rewrite rules. Hopefully, the rewrite rules should give us a clean expression in terms of one variable and two-variable xors. There is the danger that we may have some term such as a * bxorc, and we do get such a term (2*b*axorc) in this case, but it does not appear in the original problem.

# Create ring with variables a, b, c, axorb, bxorc, axorc
R = IntegerModRing(8)['a, b, c, axorb, bxorc, axorc']
(a, b, c, axorb, bxorc, axorc) = R.gens()


# xor of 2 numbers as a polynomial
def xor2(x, y): return x + y - 2*x*y

# xor of 3 numbers as a polynomial
def xor3(x, y, z): return xor2(x, xor2(y, z))

# define the ideal which contains relations:
# xor2(a, b) -> axorb, xor2(b, c) -> bxorc, xor2(a, c) -> axorc
# we also add the relation (a^2 - a = 0 => a = 0 or a = 1)
# since we know that our variables are only {0, 1}
I = ideal((axorb - xor2(a, b), bxorc - xor2(b, c), axorc - xor2(a, c), a*a-a, b*b-b, c*c-c))

# the polynomial representing a^b^c we wish to reduce
f_orig = xor3(a, b, c)

# we take the groebner basis of the ring to reduce the polynomial f.
IG = I.groebner_basis()

# we reduce a^b^c with respect to the groebner basis.
f_reduced = f_orig.reduce(IG)

print("value of a^b^c:\n\t%s\n\treduced: %s" % (f_orig, f_reduced))

# Code to evaluate the function `f` on all inputs to check correctness
def evalxor2(f):
    for (i, j, k) in [(i, j, k) for i in [0, 1] for j in [0, 1] for k in [0, 1]]:
      ref = i^^j^^k
      eval = f.substitute(a=i, b=j, c=k, axorb=i^^j, bxorc=j^^k, axorc=i^^k)
      print("%s^%s^%s: ref(%s) =?= f(%s): %s" % 
        (i, j, k, ref, eval, ref == eval))

# check original formulation is correct
print("evaulating original f for sanity check:")
evalxor2(f_orig)

# Check reduced formulation is correct
print("evaulating reduced f:")
evalxor2(f_reduced)

Running the code gives us the reduced polynomial -2*b*axorc + b + axorc which unfortunately contains a term that is b * axorc. So, this approach does not work, and I was informed by my friend that she is unaware of a solution to this problem (writing a^b^c in terms of smaller xors and sums).

The full code output is:

value of a^b^c:
	4*a*b*c - 2*a*b - 2*a*c - 2*b*c + a + b + c
	reduced: -2*b*axorc + b + axorc
evaulating original f for sanity check:
0^0^0: ref(0) =?= f(0): True
0^0^1: ref(1) =?= f(1): True
0^1^0: ref(1) =?= f(1): True
0^1^1: ref(0) =?= f(0): True
1^0^0: ref(1) =?= f(1): True
1^0^1: ref(0) =?= f(0): True
1^1^0: ref(0) =?= f(0): True
1^1^1: ref(1) =?= f(1): True
evaulating reduced f:
0^0^0: ref(0) =?= f(0): True
0^0^1: ref(1) =?= f(1): True
0^1^0: ref(1) =?= f(1): True
0^1^1: ref(0) =?= f(0): True
1^0^0: ref(1) =?= f(1): True
1^0^1: ref(0) =?= f(0): True
1^1^0: ref(0) =?= f(0): True
1^1^1: ref(1) =?= f(1): True

That is, both the original polynomial and the reduced polynomial match the expected results. But the reduced polynomial is not in our language L, since it has a term that is a product of b with axorc.

Tacking the original problem.

We try the exact same approach to the original problem of expressing a ^ b ^ c ^ d. We find that the reduced polynomial is

-a - b + c + 3*d - 3*axorb - axorc
+ axord - bxorc + bxord + 3*cxord 
- 3*axorbxorc - axorbxord 
+ axorcxord + bxorcxord

which happily has no products between terms! It also passes our sanity check, so we’ve now found the answer.

The full output is:

value of a^b^c^d:
	4*a*b*c + 4*a*b*d + 4*a*c*d + 4*b*c*d 
      - 2*a*b - 2*a*c - 2*b*c - 2*a*d 
      - 2*b*d - 2*c*d + a + b + c + d
reduced: -a - b + c + 3*d - 3*axorb 
      - axorc + axord - bxorc + bxord 
      + 3*cxord - 3*axorbxorc 
      - axorbxord + axorcxord + bxorcxord
evaluating original a^b^c^d
0^0^0^0: ref(0) =?= f(0): True
0^0^0^1: ref(1) =?= f(1): True
0^0^1^0: ref(1) =?= f(1): True
0^0^1^1: ref(0) =?= f(0): True
0^1^0^0: ref(1) =?= f(1): True
0^1^0^1: ref(0) =?= f(0): True
0^1^1^0: ref(0) =?= f(0): True
0^1^1^1: ref(1) =?= f(1): True
1^0^0^0: ref(1) =?= f(1): True
1^0^0^1: ref(0) =?= f(0): True
1^0^1^0: ref(0) =?= f(0): True
1^0^1^1: ref(1) =?= f(1): True
1^1^0^0: ref(0) =?= f(0): True
1^1^0^1: ref(1) =?= f(1): True
1^1^1^0: ref(1) =?= f(1): True
1^1^1^1: ref(0) =?= f(0): True
evaluating reduced a^b^c^d
0^0^0^0: ref(0) =?= f(0): True
0^0^0^1: ref(1) =?= f(1): True
0^0^1^0: ref(1) =?= f(1): True
0^0^1^1: ref(0) =?= f(0): True
0^1^0^0: ref(1) =?= f(1): True
0^1^0^1: ref(0) =?= f(0): True
0^1^1^0: ref(0) =?= f(0): True
0^1^1^1: ref(1) =?= f(1): True
1^0^0^0: ref(1) =?= f(1): True
1^0^0^1: ref(0) =?= f(0): True
1^0^1^0: ref(0) =?= f(0): True
1^0^1^1: ref(1) =?= f(1): True
1^1^0^0: ref(0) =?= f(0): True
1^1^0^1: ref(1) =?= f(1): True
1^1^1^0: ref(1) =?= f(1): True
1^1^1^1: ref(0) =?= f(0): True
code for a^b^c^d reduction:
def xor3(x, y, z): return xor2(x, xor2(y, z))

R = IntegerModRing(8)['a, b, c, d, axorb, axorc, axord, bxorc, bxord, cxord, axorbxorc, axorbxord, axorcxord, bxorcxord']

(a, b, c, d, axorb, axorc, axord, bxorc, bxord, cxord, axorbxorc, axorbxord, axorcxord, bxorcxord) = R.gens()
I = ideal((axorb - xor2(a, b),
           axorc - xor2(a, c),
           axord - xor2(a, d),
           bxorc - xor2(b, c),
           bxord - xor2(b, d),
           cxord - xor2(c, d),
           axorbxorc - xor3(a, b, c),
           axorbxord - xor3(a, b, d),
           axorcxord - xor3(a, c, d),
           bxorcxord - xor3(b, c, d),
           a*a-a,
           b*b-b,
           c*c-c,
           d*d-d
           ))
IG = I.groebner_basis()
f_orig = (xor2(a, xor2(b, xor2(c, d))))
f_reduced = f_orig.reduce(IG)
print("value of a^b^c^d:\n\t%s\n\treduced: %s" % (f_orig, f_reduced))

def evalxor3(f):
    for (i, j, k, l) in [(i, j, k, l) for i in [0, 1] for j in [0, 1] for k in [0, 1] for l in [0, 1]]:
      ref = i^^j^^k^^l
      eval = f.substitute(a=i, b=j, c=k, d=l,
                          axorb=i^^j, axorc=i^^k,
                          axord=i^^l, bxorc=j^^k,
                          bxord=j^^l, cxord=k^^l,
                          axorbxorc=i^^j^^k, axorbxord=i^^j^^l,
                          axorcxord=i^^k^^l, bxorcxord=j^^k^^l)
      print("%s^%s^%s^%s: ref(%s) =?= f(%s): %s" % 
        (i, j, k, l, ref, eval, ref == eval))

print("evaluating original a^b^c^d")
evalxor3(f_orig)


print("evaluating reduced a^b^c^d")
evalxor3(f_reduced)
Closing thoughts

This was a really fun exercise: Around a hundred lines of code illuminates the use of machinery such as grobner basis for solving real-world problems! I really enjoyed hacking this up and getting nerd sniped.

The janus programming language — Time reversible computation

I found out it’s called Janus, since Janus is the god of doorways in greek mythology. Hence, he is also the god of duality and transitions — he literally looks both into the future and into the past.

He is usually depicted as having two faces, since he looks to the future and to the past.

An apt name for the language!

A = B — A book about proofs of combinatorial closed forms

The book explains algorithms on solving closed forms for combinatorial recurrences, by means of Zeilberger’s algorithm.

The book is written by Zeilberger himself, and supposedy also teaches one Maple. I’d like to learn the algorithm, since it might be useful eventually for Groebner basis / loop analysis shenanigans I like to play as part of my work on compilers.

Generating k bitsets of a given length n:

The problem is to generate all bitvectors of length n that have k bits set. For example, generate all bitvectors of length 5 that have 3 bits set.

I know that an algorithm exists in Hacker’s delight, but I’ve been too sick to crack open a book, so I decided to discover the algorithm myself. The one I came up with relies on looking at the numbers moving at a certain velocity, and them colliding with each other. For example, let us try to generate all 5C3 combinations of bits.

We start wih:

#1           count of position
a b c d e    positions
1 1 1 0 0    bitset 
< - - - -    velocity

Where the < represents that the 1 at position a is moving leftwards. Our arena is circular, so the leftmost 1 can wrap around to the right. This leads to the next state

#2
a b c d e
0 1 1 0 1
- - - - <

We continue moving left peacefully.

#3
a b c d e
0 1 1 1 0
- - - < -

whoops, we have now collided with a block of 1s. Not to worry, we simply transfer our velocity by way of collision, from the 1 at d to the 1 at b.

I denote the transfer as follows:

#3
a b c d e
0 1 1 1 0  original state
- - - < -
- < < < -  transfer of velocity
- < - - -  final state after transfer of velocity

The 1 at b proceeds along its merry way with the given velocity

#4
a b c d e
1 0 1 1 0
< - - - -

Once again, it wraps around, and suffers a collision

#5
a b c d e
0 0 1 1 1
- - - - - < (collision, transfer)
- - < < < transfer of velocity
- - < - - final state after transfer of velocity

This continues:

0 1 0 1 1  #6
- < - - -
1 0 0 1 1  #7
< - - - - (collision, transfer velocity)
< - - < <
- - - < -
1 0 1 0 1 #8
- - < - -
1 1 0 0 1 #9
- < - - - (colision, transfer velocity
< < - - <
- - - - <
1 1 0 1 0 #10
- - - < - 
1 1 1 0 0 #11: wrap around to initial state

I don’t have a proof of correctness, but I have an intuition that this should generate all states. Does anyone have a proof?

EDIT: this algorithm does not work, since it will keep clusters of $k-1$ bits next to each other, when a bit hits a cluster of $k - 1$ bits. For completeness, I’m going to draft out the usual algorithm in full:

Usual Algorithm

Let’s consider the same example of 5C3:

   a b c d e
1| 0 0 1 1 1 (LSB)

We start with all bits at their lowest position. Now, we try to go to the next smallest number, which still has 3 bits toggled. Clearly, we need the bit at position b to be 1, since that’s the next number. Then, we can keep the lower 2 bits d, e set to 1, so that it’s still as small a number as possible.

   a b c d e
2| 0 1 0 1 1 (LSB)

Once again, we now move the digit at d to the digit at c, while keeping the final digit at e to make sure it’s still the smallest possible.

   a b c d e
3| 0 1 1 0 1 (LSB)

Now, we can move the 1 at e to d, since that will lead to the smallest increase:

   a b c d e
4| 0 1 1 1 0 (LSB)

At this point, we are forced to move to location a, since we have exhausted all smaller locations. so we move the 1 at b to a, and then reset all the other bits to be as close to the LSB as possible:

   a b c d e
5| 1 0 0 1 1 (LSB)

Continuing this process gives us the rest of the sequence:

    a b c d e
5 | 1 0 0 1 1
6 | 1 0 1 0 1
7 | 1 0 1 1 0
8 | 1 1 0 0 1 (note the reset of d!)
9 | 1 1 0 1 0
10| 1 1 1 0 0

Bondi k-calculus

An alternative formalism to derive special relativity geometrically, resting purely on hypotehses about the way light travels.

However, I’ve not been able to prove the correctness of the assumptions made, by using coordinate geometry. I suspect this is because I will need to use hyperbolic geometry for the “side lengths” to work out.

Indeed, I found another source, called as The k-calculus fiddle which attempts to discredit k-calculus. The author of the above blog writes at the end:

In asking Ray D’Inverno’s permission to use his book as the example of k-calculus, he was kind enough to point out that the arguments I have given are invalid. Chapter 2 of his book should be read through to the end and then reread in the light of the fact that the geometry of space and time is Minkowskian. Euclidean geometry should not be used in interpreting the diagrams because their geometry is Minkowskian.

which seems to imply that we need to use hyperbolic geometry for this.

Topology as an object telling us what zero-locus is closed:

Vivado toolchain craziness

I found this file as I was cleaning up some old code, for a project to implement a fast K/V store on an FPGA, so I thought I should put this up for anyone else who stumbles on the same frustrations / errors. I’m not touching this particular toolchain again with a 10-foot pole till the tools stabilize by a lot.

Vivado HLS issues
[BD 41-241] Message from IP propagation TCL of /blk_mem_gen_7: set_property
error: Validation failed for parameter 'Write Width A(Write_Width_A)' for BD
Cell 'blk_mem_gen_7'. Value '8' is out of the range (32,1024) Customization
errors found on 'blk_mem_gen_7'. Restoring to previous valid configuration.
struct S {
    bool b;
    int16 x;
    int16 y;
}

This gets generated as 3 ports for memory, of widths 1, 16, 16. Ideally, I wanted one port, of width 16+16+1=33, for each struct value. However, what was generated were three ports of widths 1, 16, and 16 which I cannot connect to BRAM.

struct Foo {...};
void f (Foo conflict) {
    #pragma HLS interface bram port=conflict
}

void g (Foo conflict) {
    #pragma HLS interface bram port=conflict
}
// HLS side
struct Vec2  { int x; int y};
void f(Vec2 points[NUM_POINTS]) {
	#pragma HLS DATA_PACK variable=points
    #pragma HLS INTERFACE bram port=points

    points[0] = {2, 3};
}

// Host side
Vec2 *points = (Vec2 *)(0xPOINTER_LOCATION_FROM_VIVADO);

int main() {
    // points[0] will *not* be {2, 3}!
}

SDAccel bugs

link to tutorial we were following

At some point, I gave up on the entire enterprise.

What the hell is a Grobner basis? Ideals as rewrite systems

A motivating example

The question a Grobner basis allows us to answer is this: can the polynomial $p(x, y) = xy^2 + y$ be factorized in terms of $a(x, y) = xy + 1, b(x, y) = y^2 - 1$, such that $p(x, y) = f(x, y) a(x, y) + g(x, y) b(x, y)$ for some arbitrary polynomials $f(x, y), g(x, y) \in R[x, y]$.

One might imagine, “well, I’ll divide and see what happens!” Now, there are two routes to go down:

So, clearly, the order in which we perform of factorization / division starts to matter! Ideally, we want an algorithm which is not sensitive to the order in which we choose to apply these changes. $x^2 + 1$.

The rewrite rule perspective

An alternative viewpoint of asking “can this be factorized”, is to ask “can we look at the factorization as a rewrite rule”? For this perspective, notice that “factorizing” in terms of $xy + 1$ is the same as being able to set $xy = -1$, and then have the polynomial collapse to zero. (For the more algebraic minded, this relates to the fact that $R[x] / p(x) \sim R(\text{roots of p})$). The intuition behind this is that when we “divide by $xy + 1$”, really what we are doing is we are setting $xy + 1 = 0$, and then seeing what remains. But $xy + 1 = 0 \iff xy = -1$. Thus, we can look at the original question as:

How can we apply the rewrite rules $xy \rightarrow -1$, $y^2 \rightarrow 1$, along with the regular rewrite rules of polynomial arithmetic to the polynomial $p(x, y) = xy^2 + y$, such that we end with the value $0$?

Our two derivations above correspond to the application of the rules:

That is, our rewrite rules are not confluent

The grobner basis is a mathematical object, which is a a confluent set of rewrite rules for the above problem. That is, it’s a set of polynomials which manage to find the rewrite $p(x, y) \xrightarrow{\star} 0$, regardless of the order in which we apply them. It’s also correct, in that it only rewrites to $0$ if the original system had some way to rewrite to $0$.

The buchberger’s algorithm

We need to identify critical pairs, which in this setting are called as S-polynomials.

Let $f_i = H(f_i) + R(f_i)$ and $f_j = H(f_j) + R(f_j)$. Let $m = lcm(H(f_i), H(f_j))$, and let $m_i, m_j$ be monomials such that $m_i \cdot H(f_i) = m = m_j \cdot H(f_j)$. The S-polynomial induced by $f_i, f_j$ is defined as $S(f_i, f_j) = m_i f_i - m_i f_j$.

References

Lie bracket versus torsion

torsion-vs-parallel-transport

This picture finally made the difference between these two things clear. The lie bracket moves along the flow, while the torsion moves along parallel transport.

This is why the sides of the parallelogram that measure torsion form, well, a parallelogram: we set them up using parallel transport.

On the other hand, the lie bracket measures the actual failure of the parallelogram from being formed.

Blog post: Weekend paper replication of STOKE, the stochastic superoptimizer

Click the title to go to the post. We replicate the STOKE paper in haskell, to implement a superoptimiser based on MCMC methods.

Collapsing BlockId, Label, Unique:

We have this hiearchy of BlockId, Label, and Unique that can be collapsed.

Spatial partitioning data structures in molecular dynamics

appear to be version of spatial hierarchical data structures for fast interaction computation. Apparently, multipole expansions are not useful in this case since multipole expansions are useful to take into account long range effects, but not short range effects.

Vector: Arthur Whitney and text editors

Representing CPS in LLVM using the @coro.* intrinsics

This is part of a larger thread — Adding CPS call support to LLVM where there is a large discussion on the correct design of how to teach LLVM about CPS.

Gor Nishanov proided the above example of encoding CPS using the llvm coro instructions.

Bug in the LLVM code generator: Lowering of MO_Add2 and MO_AddWordC

Both of these are lowered the same way, but they should be different.

In particular, GHC.Prim explains:

Honestly, this is confusing, but I guess there’s some story to having two separate primops for this?

Discrete random distributions with conditioning in 20 lines of haskell

newtype D a = D { unD :: [(a, Double)] } deriving(Eq, Show, Ord)

instance Functor D where
    -- fmap :: (a -> b) -> D a -> D b
    fmap f (D xs) = D $ fmap (\(a, d) -> (f a, d)) xs

instance Monad D where
    return x = D $ [(x, 1.0)]
    -- f :: a -> (D b)
    (D as) >>= f = D $ do -- list monad
                      (a, p) <- as
                      (b, p2) <- unD (f a)
                      return $ (b, p * p2)

-- [(a, 0.5), (b, 0.5)]
-- [(a, 0.3), (a, 0.2), (b, 0.1), (b, 0.4)]
--
instance Applicative D where
    pure = return
    ff <*> fa = do
        f <- ff
        a <- fa
        return $ f  a

condition :: Bool -> D ()
condition True = D [((), 1.0)]
condition False = D [((), 0.0)]


dice :: D Int
dice = let p = 1.0 / 6 in D $ [(x, p) | x <- [1..6]]


dice_hard :: D Int
dice_hard = do
    x <- dice
    condition $ x > 3
    return $ x


main :: IO ()
main = do
    print dice
    print dice_hard

This gives the output:

D {unD = [(1,0.16666666666666666),
          (2,0.16666666666666666),
          (3,0.16666666666666666),
          (4,0.16666666666666666),
          (5,0.16666666666666666),
          (6,0.16666666666666666)]}
          
D {unD = [(1,0.0),
          (2,0.0),
          (3,0.0),
          (4,0.16666666666666666),
          (5,0.16666666666666666),
          (6,0.16666666666666666)]}

Notice that D a ~= WriterT (Product Float) []!

Everything you know about word2vec is wrong

The classic explanation of word2vec, in skip-gram, with negative sampling, in the paper and countless blog posts on the internet is as follows:

while(1) {
   1. vf = vector of focus word
   2. vc = vector of context word
   3. train such that (vc . vf = 1)
   4. for(0 <= i < negative samples):
           vneg = vector of word *not* in context
           train such that (vf . vneg = 0)
}

Indeed, if I google “word2vec skipgram”, the results I get are:

The original word2vec C implementation does not do what’s explained above, and is drastically different. Most serious users of word embeddings, who use embeddings generated from word2vec do one of the following things:

  1. They invoke the original C implementation directly.
  2. They invoke the gensim implementation, which is transliterated from the C source to the extent that the variables names are the same.

Indeed, the gensim implementation is the only one that I know of which is faithful to the C implementation.

The C implementation

The C implementation in fact maintains two vectors for each word, one where it appears as a focus word, and one where it appears as a context word. (Is this sounding familiar? Indeed, it appears that GloVe actually took this idea from word2vec, which has never mentioned this fact!)

The setup is incredibly well done in the C code:

https://github.com/tmikolov/word2vec/blob/20c129af10659f7c50e86e3be406df663beff438/word2vec.c#L369
  for (a = 0; a < vocab_size; a++) for (b = 0; b < layer1_size; b++) {
    next_random = next_random * (unsigned long long)25214903917 + 11;
    syn0[a * layer1_size + b] = 
       (((next_random & 0xFFFF) / (real)65536) - 0.5) / layer1_size;
  }

https://github.com/tmikolov/word2vec/blob/20c129af10659f7c50e86e3be406df663beff438/word2vec.c#L365
for (a = 0; a < vocab_size; a++) for (b = 0; b < layer1_size; b++)
  syn1neg[a * layer1_size + b] = 0;
if (negative > 0) for (d = 0; d < negative + 1; d++) {
  // if we are performing negative sampling, in the 1st iteration,
  // pick a word from the context and set the dot product target to 1
  if (d == 0) {
    target = word;
    label = 1;
  } else {
    // for all other iterations, pick a word randomly and set the dot
    //product target to 0
    next_random = next_random * (unsigned long long)25214903917 + 11;
    target = table[(next_random >> 16) % table_size];
    if (target == 0) target = next_random % (vocab_size - 1) + 1;
    if (target == word) continue;
    label = 0;
  }
  l2 = target * layer1_size;
  f = 0;

  // find dot product of original vector with negative sample vector
  // store in f
  for (c = 0; c < layer1_size; c++) f += syn0[c + l1] * syn1neg[c + l2];

  // set g = sigmoid(f) (roughly, the actual formula is slightly more complex)
  if (f > MAX_EXP) g = (label - 1) * alpha;
  else if (f < -MAX_EXP) g = (label - 0) * alpha;
  else g = (label - expTable[(int)((f + MAX_EXP) * (EXP_TABLE_SIZE / MAX_EXP / 2))]) * alpha;

  // 1. update the vector syn1neg,
  // 2. DO NOT UPDATE syn0
  // 3. STORE THE syn0 gradient in a temporary buffer neu1e
  for (c = 0; c < layer1_size; c++) neu1e[c] += g * syn1neg[c + l2];
  for (c = 0; c < layer1_size; c++) syn1neg[c + l2] += g * syn0[c + l1];
}
// Finally, after all samples, update syn1 from neu1e
https://github.com/tmikolov/word2vec/blob/20c129af10659f7c50e86e3be406df663beff438/word2vec.c#L541
// Learn weights input -> hidden
for (c = 0; c < layer1_size; c++) syn0[c + l1] += neu1e[c];

Why random and zero initialization?

Once again, since none of this actually explained in the original papers or on the web, I can only hypothesize.

My hypothesis is that since the negative samples come from all over the text and are not really weighed by frequency, you can wind up picking any word, and more often than not, a word whose vector has not been trained much at all. If this vector actually had a value, then it could move the actually important focus word randomly.

The solution is to set all negative samples to zero, so that only vectors that have occurred somewhat frequently will affect the representation of another vector.

It’s quite ingenious, really, and until this, I’d never really thought of how important initialization strategies really are.

Why I’m writing this

I spent two months of my life trying to reproduce word2vec, following the paper exactly, reading countless articles, and simply not succeeding. I was unable to reach the same scores that word2vec did, and it was not for lack of trying.

I could not have imagined that the paper would have literally fabricated an algorithm that doesn’t work, while the implementation does something completely different.

Eventually, I decided to read the sources, and spent three whole days convinced I was reading the code wrong since literally everything on the internet told me otherwise.

I don’t understand why the original paper and the internet contain zero explanations of the actual mechanism behind word2vec, so I decided to put it up myself.

This also explains GloVe’s radical choice of having a separate vector for the negative context — they were just doing what word2vec does, but they told people about it :).

Is this academic dishonesty? I don’t know the answer, and that’s a heavy question. But I’m frankly incredibly pissed, and this is probably the last time I take a machine learning paper’s explanation of the algorithm seriously again — from next time, I read the source first.

Hamiltonian monte carlo, leapfrog integrators, and sympletic geometry

This is a section that I’ll update as I learn more about the space, since I’m studying differential geometry over the summer, I hope to know enough about “sympletic manifolds”. I’ll make this an append-only log to add to the section as I understand more.

31st May
Simulating orbits with large timesteps

euler-vs-leapfrog

Clearly, the leapfrog integrator preserves energy and continues to move in an orbit, while the euler integrator goes batshit and causes orbits to spiral outwards. Full code is available below. More of the code is spent coaxing matplotlib to look nice, than doing the actual computation.

import numpy as np
import matplotlib.pyplot as plt
import numpy.linalg

# dq/dt = dH/dp | dp/dt = -dH/dq (a = -del V)
def leapfroge(dhdp, dhdq, q, p, dt):
    p += -dhdq(q, p) * 0.5 * dt # halfstep momentum
    q += dhdp(q, p) * dt # fullstep position
    p += -dhdq(q, p) * 0.5 * dt # halfstep momentum
    return (q, p)

def euler(dhdp, dhdq, q, p, dt):
    pnew = p + -dhdq(q, p) * dt
    qnew = q + dhdp(q, p) * dt

def planet(integrator, n, dt):
    STRENGTH = 0.5

    q = np.array([0.0, 1.0]); p = np.array([-1.0, 0.0])
    
    # H = STRENGTH * |q| (potential) + p^2/2 (kinetic)
    def H(qcur, pcur): return STRENGTH * np.linalg.norm(q) + np.dot(p, p) / 2
    def dhdp(qcur, pcur): return p
    def dhdq(qcur, pcur): return STRENGTH * 2 * q / np.linalg.norm(q)
    
    qs = []
    for i in range(n):
        (q, p) = integrator(dhdp, dhdq, q, p, dt)
        qs.append(q.copy())
    return np.asarray(qs)

NITERS = 15
TIMESTEP = 1

plt.rcParams.update({'font.size': 22, 'font.family':'monospace'})
fig, ax = plt.subplots()

planet_leapfrog = planet(leapfroge, NITERS, TIMESTEP)
ax.plot(planet_leapfrog[:, 0], planet_leapfrog[:, 1], label='leapfrog',
        linewidth=3, color='#00ACC1')
planet_euler = planet(euler, NITERS, TIMESTEP)
ax.plot(planet_euler[:, 0], planet_euler[:, 1], label='euler',
        linewidth=3, color='#D81B60')

legend = plt.legend(frameon=False)
ax.set_title("leapfrog v/s euler: NITERS=%s dt=%s" % (NITERS, TIMESTEP))
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.spines['bottom'].set_visible(False)
ax.spines['left'].set_visible(False)
plt.show()
plt.savefig("leapfrog-vs-euler.png")

Small Haskell MCMC implementation

We create a simple monad called PL which allows for a single operation: sampling from a uniform distribution. We then exploit this to implement MCMC using metropolis hastings, which is used to sample from arbitrary distributions. Bonus is a small library to render sparklines in the CLI.

For next time:

Source code

{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE DeriveFunctor #-}
import System.Random
import Data.List(sort, nub)
import Data.Proxy
import Control.Monad (replicateM)
import qualified Data.Map as M


-- | Loop a monadic computation.
mLoop :: Monad m =>
      (a -> m a) -- ^ loop
      -> Int -- ^ number of times to run
      -> a -- initial value
      -> m a -- final value
mLoop _ 0 a = return a
mLoop f n a = f a >>= mLoop f (n - 1)


-- | Utility library for drawing sparklines

-- | List of characters that represent sparklines
sparkchars :: String
sparkchars = "_▁▂▃▄▅▆▇█"

-- Convert an int to a sparkline character
num2spark :: RealFrac a => a -- ^ Max value
  -> a -- ^ Current value
  -> Char
num2spark maxv curv =
   sparkchars !!
     (floor $ (curv / maxv) * (fromIntegral (length sparkchars - 1)))

series2spark :: RealFrac a => [a] -> String
series2spark vs =
  let maxv = if null vs then 0 else maximum vs
  in map (num2spark maxv) vs

seriesPrintSpark :: RealFrac a => [a] -> IO ()
seriesPrintSpark = putStrLn . series2spark

-- Probabilities
-- ============
type F = Float
-- | probability density
newtype P = P { unP :: Float } deriving(Num)

-- | prob. distributions over space a
newtype D a = D { runD :: a -> P }

uniform :: Int -> D a
uniform n =
  D $ \_ -> P $ 1.0 / (fromIntegral $ n)

(>$<) :: Contravariant f => (b -> a) -> f a  -> f b
(>$<) = cofmap

instance Contravariant D where
  cofmap f (D d) = D (d . f)

-- | Normal distribution with given mean
normalD :: Float ->  D Float
normalD mu = D $ \f -> P $ exp (- ((f-mu)^2))

-- | Distribution that takes on value x^p for 1 <= x <= 2.  Is normalized
polyD :: Float -> D Float
polyD p = D $ \f -> P $ if 1 <= f && f <= 2 then (f ** p) * (p + 1) / (2 ** (p+1) - 1) else 0

class Contravariant f where
  cofmap :: (b -> a) -> f a -> f b

data PL next where
    Ret :: next -> PL next -- ^ return  a value
    Sample01 :: (Float -> PL next) -> PL next -- ^ sample uniformly from a [0, 1) distribution

instance Monad PL where
  return = Ret
  (Ret a) >>= f = f a
  (Sample01 float2plnext) >>= next2next' =
      Sample01 $ \f -> float2plnext f >>= next2next'

instance Applicative PL where
    pure = return
    ff <*> fx = do
        f <- ff
        x <- fx
        return $ f x

instance Functor PL where
    fmap f plx = do
         x <- plx
         return $ f x

-- | operation to sample from [0, 1)
sample01 :: PL Float
sample01 = Sample01 Ret


-- | Run one step of MH on a distribution to obtain a (correlated) sample
mhStep :: (a -> Float) -- ^ function to score sample with, proportional to distribution
  -> (a -> PL a) -- ^ Proposal program
  -> a -- current sample
  -> PL a
mhStep f q a = do
 	a' <- q a
 	let alpha = f a' / f a -- acceptance ratio
 	u <- sample01
 	return $ if u <= alpha then a' else a

-- Typeclass that can provide me with data to run MCMC on it
class MCMC a where
    arbitrary :: a
    uniform2val :: Float -> a

instance MCMC Float where
	arbitrary = 0
	-- map [0, 1) -> (-infty, infty)
	uniform2val v = tan (-pi/2 + pi * v)


{-
-- | Any enumerable object has a way to get me the starting point for MCMC
instance (Bounded a, Enum a) => MCMC a where
     arbitrary = toEnum 0
     uniform2val v = let
        maxf = fromIntegral . fromEnum $ maxBound
        minf = fromIntegral . fromEnum $ minBound
        in toEnum $ floor $ minf + v * (maxf - minf)
-}


-- | Run MH to sample from a distribution
mh :: (a -> Float) -- ^ function to score sample with
 -> (a -> PL a) -- ^ proposal program
 -> a -- ^ current sample
 -> PL a
mh f q a = mLoop (mhStep f q) 100  $ a

-- | Construct a program to sample from an arbitrary distribution using MCMC
mhD :: MCMC a => D a -> PL a
mhD (D d) =
    let
      scorer = (unP . d)
      proposal _ = do
        f <- sample01
        return $ uniform2val f
    in mh scorer proposal arbitrary


-- | Run the probabilistic value to get a sample
sample :: RandomGen g => g -> PL a -> (a, g)
sample g (Ret a) = (a, g)
sample g (Sample01 f2plnext) = let (f, g') = random g in sample g' (f2plnext f)


-- | Sample n values from the distribution
samples :: RandomGen g => Int -> g -> PL a -> ([a], g)
samples 0 g _ = ([], g)
samples n g pl = let (a, g') = sample g pl
                     (as, g'') = samples (n - 1) g' pl
                 in (a:as, g'')

-- | count fraction of times value occurs in list
occurFrac :: (Eq a) => [a] -> a -> Float
occurFrac as a =
    let noccur = length (filter (==a) as)
        n = length as
    in (fromIntegral noccur) / (fromIntegral n)

-- | Produce a distribution from a PL by using the sampler to sample N times
distribution :: (Eq a, Num a, RandomGen g) => Int -> g -> PL a -> (D a, g)
distribution n g pl =
    let (as, g') = samples n g pl in (D (\a -> P (occurFrac as a)), g')


-- | biased coin
coin :: Float -> PL Int -- 1 with prob. p1, 0 with prob. (1 - p1)
coin p1 = do
    Sample01 (\f -> Ret $ if f < p1 then 1 else 0)


-- | Create a histogram from values.
histogram :: Int -- ^ number of buckets
          -> [Float] -- values
          -> [Int]
histogram nbuckets as =
    let
        minv :: Float
        minv = minimum as
        maxv :: Float
        maxv = maximum as
        -- value per bucket
        perbucket :: Float
        perbucket = (maxv - minv) / (fromIntegral nbuckets)
        bucket :: Float -> Int
        bucket v = floor (v / perbucket)
        bucketed :: M.Map Int Int
        bucketed = foldl (\m v -> M.insertWith (+) (bucket v) 1 m) mempty as
     in map snd . M.toList $ bucketed


printSamples :: (Real a, Eq a, Ord a, Show a) => String -> [a] -> IO ()
printSamples s as =  do
    putStrLn $ "***" <> s
    putStrLn $ "   samples: " <> series2spark (map toRational as)

printHistogram :: [Float] -> IO ()
printHistogram samples = putStrLn $ series2spark (map fromIntegral . histogram 10 $  samples)


-- | Given a coin bias, take samples and print bias
printCoin :: Float -> IO ()
printCoin bias = do
    let g = mkStdGen 1
    let (tosses, _) = samples 100 g (coin bias)
    printSamples ("bias: " <> show bias) tosses



-- | Create normal distribution as sum of uniform distributions.
normal :: PL Float
normal =  fromIntegral . sum <$> (replicateM 5 (coin 0.5))


main :: IO ()
main = do
    printCoin 0.01
    printCoin 0.99
    printCoin 0.5
    printCoin 0.7

    putStrLn $ "normal distribution using central limit theorem: "
    let g = mkStdGen 1
    let (nsamples, _) = samples 1000 g normal
    -- printSamples "normal: " nsamples
    printHistogram nsamples


    putStrLn $ "normal distribution using MCMC: "
    let (mcmcsamples, _) = samples 1000 g (mhD $  normalD 0.5)
    printHistogram mcmcsamples

    putStrLn $ "sampling from x^4 with finite support"
    let (mcmcsamples, _) = samples 1000 g (mhD $  polyD 4)
    printHistogram mcmcsamples

Output

***bias: 1.0e-2
   samples: ________________________________________█_█_________________________________________________________
***bias: 0.99
   samples: ████████████████████████████████████████████████████████████████████████████████████████████████████
***bias: 0.5
   samples: __█____█__███_███_█__█_█___█_█_██___████████__█_████_████_████____██_█_██_____█__██__██_██____█__█__
***bias: 0.7
   samples: __█__█_█__███_█████__███_█_█_█_██_█_████████__███████████_████_█_███_████_██__█_███__██_███_█_█__█_█
normal distribution using central limit theorem: 
_▄▇█▄_
normal distribution using MCMC: 
__▁▄█▅▂▁___
sampling from x^4 with finite support
▁▁▃▃▃▄▅▆▇█_

The smallest implementation of reverse mode AD (autograd) ever:

{-# LANGUAGE GeneralizedNewtypeDeriving #-}
import qualified Data.Map.Strict as M

-- | This file can be copy-pasted and will run!

-- | Symbols
type Sym = String
-- | Environments
type E a = M.Map Sym a
-- | Newtype to represent deriative values
type F = Float
newtype Der = Der { under :: F } deriving(Show, Num)

infixl 7 !#
-- | We are indexing the map at a "hash" (Sym)
(!#) :: E a -> Sym -> a
(!#) = (M.!)

-- | A node in the computation graph
data Node = 
  Node { name :: Sym -- ^ Name of the node
       , ins :: [Node] -- ^ inputs to the node
       , out :: E F -> F -- ^ output of the node
       , der :: (E F, E (Sym -> Der)) 
                  -> Sym -> Der -- ^ derivative wrt to a name
       }

-- | @ looks like a "circle", which is a node. So we are indexing the map
-- at a node.
(!@) :: E a -> Node -> a 
(!@) e node = e M.! (name node)

-- | Given the current environments of values and derivatives, compute
-- | The new value and derivative for a node.
run_ :: (E F, E (Sym -> Der)) -> Node -> (E F, E (Sym -> Der))
run_ ein (Node name ins out der) = 
  let (e', ed') = foldl run_ ein ins -- run all the inputs
      v = out e' -- compute the output
      dv = der (e', ed') -- and the derivative
  in (M.insert name v e', M.insert name dv ed')  -- and insert them

-- | Run the program given a node 
run :: E F -> Node -> (E F, E (Sym -> Der))
run e n = run_ (e, mempty) n

-- | Let's build nodes
nconst :: Sym -> F -> Node
nconst n f = Node n [] (\_ -> f) (\_ _ -> 0)

-- | Variable
nvar :: Sym -> Node 
nvar n = Node n [] (!# n) (\_ n' -> if n == n' then 1 else 0)
  
-- | binary operation
nbinop :: (F -> F -> F)  -- ^ output computation from inputs
 -> (F -> Der -> F -> Der -> Der) -- ^ derivative computation from outputs
 -> Sym -- ^ Name
 -> (Node, Node) -- ^ input nodes
 -> Node
nbinop f df n (in1, in2) = 
  Node { name = n
       , ins = [in1, in2]
       , out = \e -> f (e !# name in1) (e !# name in2)
       , der = \(e, ed) n' -> 
                 let (name1, name2) = (name in1, name in2)
                     (v1, v2) = (e !# name1, e !# name2)
                     (dv1, dv2) = (ed !# name1 $ n', ed !# name2 $ n')
                     in df v1 dv1 v2 dv2
       }

nadd :: Sym -> (Node, Node) -> Node
nadd = nbinop (+) (\v dv v' dv' -> dv + dv')

nmul :: Sym -> (Node, Node) -> Node
nmul = nbinop (*) (\v (Der dv) v' (Der dv') -> Der $ (v*dv') + (v'*dv))

main :: IO ()
main = do
  let x = nvar "x" :: Node
  let y = nvar "y"
  let xsq = nmul "xsq" (x, x)
  let ten = nconst "10" 10
  let xsq_plus_10 = nadd "xsq_plus_10" (xsq, ten)
  let xsq_plus_10_plus_y = nadd "xsq_plus_10_plus_y"  (xsq_plus_10, y)
  let (e, de) = run (M.fromList $ [("x", 2.0), ("y", 3.0)]) xsq_plus_10_plus_y
  putStrLn $ show e
  putStrLn $ show $ de !@ xsq_plus_10_plus_y $ "x"
  putStrLn $ show $ de !@ xsq_plus_10_plus_y $ "y"

Yeah, in ~80 lines of code, you can basically build an autograd engine. Isn’t haskell so rad?

Timings of passes in GHC, and low hanging fruit in the backend:

A comment from this test case tells us why the function debugBelch2 exists:

ghc/testsuite/tests/rts/T7160.hs
-- Don't use debugBelch() directly, because we cannot call varargs functions
-- using the FFI (doing so produces a segfault on 64-bit Linux, for example).
-- See Debug.Trace.traceIO, which also uses debugBelch2.
foreign import ccall "&debugBelch2" fun :: FunPtr (Ptr () -> Ptr () -> IO ())

The implementation is:

ghc/libraries/base/cbits/PrelIOUtils.c

void debugBelch2(const char*s, char *t)
{
    debugBelch(s,t);
}
ghc/rts/RtsMessages.c

RtsMsgFunction *debugMsgFn  = rtsDebugMsgFn;
...

void
debugBelch(const char*s, ...)
{
  va_list ap;
  va_start(ap,s);
  (*debugMsgFn)(s,ap);
  va_end(ap);
}

Debugging debug info in GHC

I wanted to use debug info to help build a better debugging experience within tweag/asterius. So, I was reading through the sources of cmm/Debug.hs.
I’d never considered how to debug debug-info, and I found the information tucked inside a cute note in GHC (Note [Debugging DWARF unwinding info]):

This makes GDB produce a trace of its internal workings. Having gone this far, it’s just a tiny step to run GDB in GDB. Make sure you install debugging symbols for gdb if you obtain it through a package manager.

GHC LLVM code generator: Switch to unreachable

The switch to out of range code generator switches to the first label. It should be more profitable to switch to a unreachable block. That way, LLVM can take advantage of UB.

Concurrency in Haskell

Great link to the GHC wiki that describes the concurrency primitives “bottom up”: https://gitlab.haskell.org/ghc/ghc/wikis/lightweight-concurrency

Handy list of differential geometry definitions

There are way too many objects in diffgeo, all of them subtly connected. Here I catalogue all of the ones I have run across:

Manifold

A manifold $M$ of dimension $n$ is a topological space. So, there is a topological structure $T$ on $M$. There is also an Atlas, which is a family of _Chart_s that satisfy some properties.

Chart

A chart is a pair $(O \in T , cm: O -> \mathbb R^n$. The $O$ is an open set of the manifold, and $cm$ (“chart for “m”) is a continuous mapping from $O$ to $\mathbb R^n$ under the subspace topology for $U$ and the standard topology for $\mathbb R^n$.

Atlas

An Atlas is a collection of _Chart_s such that the charts cover the manifold, and the charts are pairwise compatible. That is, $A = { (U_i, \phi_i) }$, such that $\cup{i} U_i = M$, and $\phi_j \circ phi_i^{-1}$ is smooth.

Differentiable map

$f: M \to N$ be a mapping from an $m$ dimensional manifold to an $n$ dimensional manifold. Let $frep = cn \circ f \circ cm^{-1}: \mathbb R^m -> \mathbb R^n$ where $cm: M \to \mathbb R^m$ is a chart for $M$, $cn: N \to \mathbb R^n$ is a chart for $N$. $frep$ is $f$ represented in local coordinates. If $frep$ is smooth for all choices of $cm, cn$, then $f$ is a differentiable map from $M$ to $N$.

Curve:

Let $I$ be an open interval of $\mathbb R$ which includes the point 0. A Curve is a differentiable map $C: (a, b) \to M$ where $a < 0 < b$.

Function: (I hate this term, I prefer something like Valuation):

A differentiable mapping from $M$ to $R$.

Directional derivative of a function f(m): M -> R with respect to a curve c(t): I -> M, denoted as c[f].

Let g(t) = (f . c)(t) :: I -c-> M -f-> R = I -> R. This this is the value dg/dt(t0) = (d (f . c) / dt) (0).

Tangent vector at a point p:

On a m dimensional manifold M, a tangent vector at a point p is an equivalence class of curves that have c(0) = p, such that c1(t) ~ c2(t) iff :

  1. For a (all) charts (O, ch) such that c1(0) ∈ O, d/dt (ch . c1: R -> R^m) = d/dt (ch . c2: R -> R^m).

That is, they have equal derivatives.

Tangent space(TpM):

The set of all tangent vectors at a point p forms a vector space TpM. We prove this by creating a bijection from every curve to a vector R^n.

Let (U, ch: U -> R) be a chart around the point p, where p ∈ U ⊆ M. Now, the bijection is defined as:

forward: (I -> M) -> R^n
forward(c) = d/dt (c . ch)

reverse: R^n -> (I -> M)
reverse(v)(t) = ch^-1 (tv)
Cotangent space(TpM*): dual space of the tangent space / Space of all linear functions from TpM to R.
Pushforward push(f): TpM -> TpN

Given a curve c: I -> M, the pushforward is the curve f . c : I -> N. This extends to the equivalence classes and provides us a way to move curves in M to curves in N, and thus gives us a mapping from the tangent spaces.

This satisfies the identity:

push(f)(v)[g] === v[g . f]
Pullback pull(f): TpN* -> TpM*

Given a linear functional wn : TpN -> R, the pullback is defined as ` wn . push(f) : TpM -> R`.

This satisfies the identity:

(pull wn)(v) === wn (push v)
(pull (wn : TpN->R): TpM->R) (v : TpM) : R  = (wn: TpN->R) (push (v: TpM): TpN) : R
Vector field as derivation

TODO

Lie derivation
Lie derivation as lie bracket

Lazy programs have space leaks, Strict programs have time leaks

Stumbled across this idea while reading some posts on a private discourse.

This is an interesting perspective that I’ve never seen articulated before, and somehow helps make space leaks feel more… palatable? Before, I had no analogue to a space leak in the strict world, so I saw them as a pathology. But with this new perspective, I can see that the strict world’s version of a space leak is a time leak.

Presburger arithmetic can represent the Collatz Conjecture

An observation I had: the function

f(x) = x/2      if (x % 2 == 0)
f(x) = 3x + 1   otherwise

is a Presburger function, so by building better approximations to the transitive closure of a presburger function, one could get better answers to the Collatz conjecture. Unfortunately, ISL (the integer set library) of today is not great against the formidable foe.

The code:

#include <isl/set.h>
#include <isl/version.h>
#include <isl/map.h>
#include <isl/aff.h>
#include <isl/local_space.h>
#include <isl/constraint.h>
#include <isl/space.h>

int main() {
    isl_ctx *ctx = isl_ctx_alloc();
    const char *s = "{ [x] -> [x / 2] : x % 2 = 0; [x] -> [3 * x + 1] : x % 2 = 1}";

    isl_map *m = isl_map_read_from_str(ctx, s);

    isl_map_dump(m);

    isl_bool b;
    isl_map *p = isl_map_transitive_closure(m, &b);
    printf("exact: %d\n", b);
    printf("map:\n");
    isl_map_dump(p);

}

Produces the somewhat disappointing, and yet expected output:

$ clang bug.c -lisl -Lisl-0.20/.libs -o bug -I/usr/local/include/
$ ./bug
{ [x] -> [o0] : 2o0 = x or (exists (e0 = floor((1 + x)/2): o0 = 1 + 3x and 2e0 = 1 + x)) }
exact: 0
map:
{ [x] -> [o0] }

I find it odd that it is unable to prove anything about the image, even that it is non-negative, for example. This is an interesting direction in which to improve the functions isl_map_power and isl_map_transitive_closure though.

Using compactness to argue about covers

I’ve always seen compactness be used by starting with a possibly infinite coverm and then filtering it into a finite subcover. This finite subcover is then used for finiteness properties (like summing, min, max, etc.).

I recently ran across a use of compactness when one starts with the set of all possible subcovers, and then argues about why a cover cannot be built from these subcovers if the set is compact. I found it to be a very cool use of compactness, which I’ll record below:

Theorem:

If a family of compact, countably infinite sets S_a have all finite intersections non-empty, then the intersection of the family S_a is non-empty.

Proof:

Let S = intersection of S_a. We know that S must be compact since all the S_a are compact, and the intersection of a countably infinite number of compact sets is compact.

Now, let S be empty. Therefore, this means there must be a point p ∈ P such that p !∈ S_i for some arbitrary i.

Cool use of theorem:

We can see that the cantor set is non-empty, since it contains a family of closed and bounded sets S1, S2, S3, ... such that S1 ⊇ S2 ⊇ S3 ... where each S_i is one step of the cantor-ification. We can now see that the cantor set is non-empty, since:

  1. Each finite intersection is non-empty, and will be equal to the set that has the highest index in the finite intersection.

  2. Each of the sets Si are compact since they are closed and bounded subsets of R

  3. Invoke theorem.

Japanese Financial Counting system

Japanese contains a separate kanji set called daiji, to prevent people from adding strokes to stuff previously written.

#  |Common |Formal
1  |一     |壱
2  |二     |弐
3  |三     |参

Stephen wolfram’s live stream

I’ve taken to watching the live stream when I have some downtime and want some interesting content.

The discussions of Wolfram with his group are great, and they bring up really interesting ideas (like that of cleave being very irregular).

Cleave as a word has some of the most irregular inflections

McCune’s single axiom for group theory

Single Axioms for Groups and Abelian Groups with Various Operations provides a single axiom for groups. This can be useful for some ideas I have for training groups, where we can use this axiom as the loss function!

Word2Vec C code implements gradient descent really weirdly

I’ll be posting snippets of the original source code, along with a link to the Github sources. We are interested in exploring the skip-gram implementation of Word2Vec, with negative sampling, without hierarchical softmax. I assume basic familiarity with word embeddings and the skip-gram model.

Construction of the sigmoid lookup table

// https://github.com/tmikolov/word2vec/blob/master/word2vec.c#L708

expTable = (real *)malloc((EXP_TABLE_SIZE + 1) * sizeof(real));
for (i = 0; i < EXP_TABLE_SIZE; i++) {
  expTable[i] = exp((i / (real)EXP_TABLE_SIZE * 2 - 1) *
                    MAX_EXP);  // Precompute the exp() table
  expTable[i] =
      expTable[i] / (expTable[i] + 1);  // Precompute f(x) = x / (x + 1)
}

Here, the code constructs a lookup table which maps [0...EXP_TABLE_SIZE-1] to [sigmoid(-MAX_EXP)...sigmoid(MAX_EXP)]. The index i first gets mapped to (i / EXP_TABLE_SIZE) * 2 - 1, which sends 0 to -1 and EXP_TABLE_SIZE to 1. This is then rescaled by MAX_EXP.

Layer initialization

// https://github.com/imsky/word2vec/blob/master/word2vec.c#L341
a = posix_memalign((void **)&syn0, 128,
               (long long)vocab_size * layer1_size * sizeof(real));
...

// https://github.com/imsky/word2vec/blob/master/word2vec.c#L355
for (a = 0; a < vocab_size; a++)
        for (b = 0; b < layer1_size; b++) {
            next_random = next_random * (unsigned long long)25214903917 + 11;
            syn0[a * layer1_size + b] =
                (((next_random & 0xFFFF) / (real)65536) - 0.5) / layer1_size;
        }
// https://github.com/imsky/word2vec/blob/master/word2vec.c#L350
a = posix_memalign((void **)&syn1neg, 128,
                   (long long)vocab_size * layer1_size * sizeof(real));
...
for (a = 0; a < vocab_size; a++)
    for (b = 0; b < layer1_size; b++) syn1neg[a * layer1_size + b] = 0;
// https://github.com/imsky/word2vec/blob/master/word2vec.c#L370
real *neu1e = (real *)calloc(layer1_size, sizeof(real));

Backpropogation

Throughout word2vec, no 2D arrays are used. Indexing of the form arr[word][ix] is manually written as arr[word * layer1_size + ix]. So, I will call word * layer1_size as the “base address”, and ix as the “offset of the array index expression henceforth.

Here, l1 is the base address of the word at the center of window (the focus word). l2 is the base address of either the word that is negative sampled from the corpus, or the word that is a positive sample from within the context window.

label tells us whether the sample is a positive or a negative sample. label = 1 for positive samples, and label = 0 for negative samples.

// zero initialize neu1e
// https://github.com/imsky/word2vec/blob/master/word2vec.c#L419
for (c = 0; c < layer1_size; c++) neu1e[c] = 0;
...
// loop through each negative sample
// https://github.com/imsky/word2vec/blob/master/word2vec.c#L508
if (negative > 0)  for (d = 0; d < negative + 1; d++) {
  ...
  // https://github.com/imsky/word2vec/blob/master/word2vec.c#L521
  // take the dot product: f=  syn0[focus] . syn1neg[context]
  for (c = 0; c < layer1_size; c++) f += syn0[c + l1] * syn1neg[c + l2];
  
  // compute: g = (label - sigmoid(2f - 1)) * alpha
  // g is computed using lookups into a lookup table and clamping for
  // efficiency.
  if (f > MAX_EXP) g = (label - 1) * alpha;
  else if (f < -MAX_EXP) g = (label - 0) * alpha;
  else
  g = (label - expTable[(int)((f + MAX_EXP) *
                              (EXP_TABLE_SIZE /
                               MAX_EXP / 2))]) * alpha;
  // Now that we have computed the gradient:
  // `g = (label - output) * learningrate`,
  // we need to perform backprop. This is where the code gets weird.

  for (c = 0; c < layer1_size; c++) neu1e[c] += g * syn1neg[c + l2];
  for (c = 0; c < layer1_size; c++) syn1neg[c + l2] += g * syn0[c + l1];
  } // end loop through negative samples
// Learn weights input -> hidden
for (c = 0; c < layer1_size; c++) syn0[c + l1] += neu1e[c];

The paper does not mentioned these implementation details, and neither does any blog post that I’ve read. I don’t understand what’s going on, and I plan on updating this section when I understand this better.

Arthur Whitney: dense code

How does one work with arrays in a linear language?

Given an array of qubits xs: Qubit[], I want to switch to little endian. Due to no-cloning, I can’t copy them! I suppose I can use recursion to build up a new “list”. But this is not the efficient array version we know and love and want.

The code that I want to work but does not:

function switchEndian(xs: Qubit[]): Unit {
    for(i in 0..Length(xs) - 1) {
        Qubit q = xs[i]; // boom, this does not work!
        xs[i] = xs[Length(xs) - 1 - i]
        xs[Length(xs) - 1 - i] = q;
    }
}

On the other hand, what does work is to setup a quantum circuit that performs this flipping, since it’s a permutation matrix at the end of the day. But this is very slow, since it needs to simulate the “quantumness” of the solution, since it takes 2^n basis vectors for n qubits.

However, the usual recursion based solution works:

function switchEndian(xs: Qubit[]): Qubit[] {
    if(Length(xs) == 1) {
        return xs;
    } else {
        switchEndian(xs[1..(Length(xs) - 1)] + xs[0]
    }
}

This is of course, suboptimal.

I find it interesting that in the linear types world, often the “pure” solution is forced since mutation very often involves temporaries / copying!

(I’m solving assignments in qsharp for my course in college)

Linear optimisation is the same as linear feasibility checking

Core building block of effectively using the ellipsoid algorithm.

Quantum computation without complex numbers

I recently learnt that the Toeffili and Hadamard gates are universal for quantum computation. The description of these gates involve no complex numbers. So, we can write any quantum circuit in a “complex number free” form. The caveat is that we may very well have input qubits that require complex numbers.

Even so, a large number (all?) of the basic algorithms shown in Nielsen and Chaung can be encoded in an entirely complex-number free fashion.

I don’t really understand the ramifications of this, since I had the intuition that the power of quantum computation comes from the ability to express complex phases along with superposition (tensoring). However, I now have to remove the power from going from R to C in many cases. This is definitely something to ponder.

Linguistic fun fact: Comparative Illusion

I steal from wikipedia:

Comparative Illusion, which is a grammatical illusion where certain sentences seem grammatically correct when you read them, but upon further reflection actually make no sense.

For example: “More people have been to Berlin than I have.”

Long-form posts:

Reading

Haskell

Simplexhc (STG -> LLVM compiler) progress

GSoC (2015)