Programming by Sketching

Armando Solar-Lezama, Liviu Tancau, Gilad Arnold, Rastislav Bodik, Sanjit Seshia UC Berkeley, Rodric Rabbah MIT, Kemal Ebcioglu, Vijay Saraswat, Vivek Sarkar IBM

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int[] mergeSort (int[] input, int n) {return merge( mergeSort (input[0::n/2]),

mergeSort (input[n/2+1::n]) , n);}int[] merge (int[] a, int b[], int n) {

int j=0, k=0;for (int i = 0; i < n; i++)

if ( a[j] < b[k] ) { result[i] = a[j++];

} else { result[i] = b[k++];

}}return result;

}

Merge sort

looks simple to code, but there is a bug

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Merge sortint[] mergeSort (int[] input, int n) {

return merge( mergeSort (input[0::n/2]), mergeSort (input[n/2+1::n]) , n);

}int[] merge (int[] a, int b[], int n) {

int j, k;for (int i = 0; i < n; i++)

if ( j<n && ( !(k<n) || a[j] < b[k]) ) { result[i] = a[j++];

} else { result[i] = b[k++];

}}return result;

}

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The sketching experience

sketch implementation (completed sketch)

spec

specification

+

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The spec: bubble sort

int[] sort (int[] input, int n) { for (int i=0; i<n; ++i)

for (int j=i+1; j<n; ++j) if (input[j] < input[i])

swap(input, j, i);}

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int[] mergeSort (int[] input, int n) {return merge( mergeSort (input[0::n/2]),

mergeSort (input[n/2+1::n]) , n);}int[] merge (int[] a, int b[], int n) {

int j, k;for (int i = 0; i < n; i++)

if ( expression( ||, &&, <, !, [] ) ) { result[i] = a[j++];

} else { result[i] = b[k++];

}}return result;

}

Merge sort: sketched

hole

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Merge sort: synthesizedint[] mergeSort (int[] input, int n) {

return merge( mergeSort (input[0::n/2]), mergeSort (input[n/2::n]) );

}int[] merge (int[] a, int b[], int n) {

int j, k;for (int i = 0; i < n; i++)

if ( j<n && ( !(k<n) || a[j] < b[k]) ) { result[i] = a[j++];

} else { result[i] = b[k++];

}}return result;

}

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Sketching: spec vs. sketch

• Specification– executable: easy to debug, serves as a

prototype– a reference implementation: simple and

sequential– written by domain experts: crypto, bio, MPEG

committee

• Sketched implementation– program with holes: filled in by synthesizer– programmer sketches strategy: machine

provides details– written by performance experts: vector

wizard; cache guru

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How sketching fits into autotuning• Autotuning: two methods for obtaining code

variants1. optimizing compiler: transform a “spec” in various ways2. custom generator: for a specific algorithm

• We seek to simplify the second approach

• Scenario 1: library of variants stores resolved sketches

– as if written by hand

• Scenario 2: library has unresolved, flexible sketches– sketch works for a variety of specifications:

e.g., a class of stencils

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SKETCH

• A language with support for sketching-based synthesis– like C without pointers – two simple synthesis constructs

• restricted to finite programs:– input size known at compile time, terminates on all inputs

• most high-performance kernels are finite:– matrix multiply: yes – binary search tree: no

• we’re already working on relaxing the fineteness restriction– later in this talk

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Ex1: Isolate rightmost 0-bit. 1010 0111 0000 1000bit[W] isolate0 (bit[W] x) { // W: word size

bit[W] ret = 0;for (int i = 0; i < W; i++)

if (!x[i]) { ret[i] = 1; break; } return ret;

}

bit[W] isolate0Fast (bit[W] x) implements isolate0 { return ~x & (x+1);

}

bit[W] isolate0Sketched (bit[W] x) implements isolate0 {return ~(x + ??) & (x + ??);

}

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Programmer’s view of sketches

• the ?? operator replaced with a suitable constant

• as directed by the implements clause.

• the ?? operator introduces non-determinism

• the implements clause constrains it.

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Beyond synthesis of literals

• Synthesizing values of ?? already very useful– parallelization machinery: bitmasks, tables

in crypto codes– array indices: A[i+??,j+??]

• We can synthesize more than constants– semi-permutations: functions that select and

shuffle bits– polynomials: over one or more variables– actually, arbitrary expressions, programs

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Synthesizing polynomials

int spec (int x) {return 2*x*x*x*x + 3*x*x*x + 7*x*x + 10;

}

int p (int x) implements spec {return (x+1)*(x+2)*poly(3,x);

}

int poly(int n, int x) {if (n==0) return ??; else return x * poly(n-1, x) + ??;

}

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Karatsuba’s multiplication

x = x1*b + x0 y = y1*b + y0 b=2k

x*y = b2*x1*y1 + b*(x1*y0 + x0*y1) + x0*y0

x*y = poly(??,b) * x1*y1 + + poly(??,b) * poly(1,x1,x0,y1,y0)*poly(1,x1, x0, y1, y0) + poly(??,b) * x0*y0

x*y = (b2 +b) * x1*y1

+ b * (x1 - x0)*(y1 - y0) + (b+1) * x0*y0

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Sketch of Karatsubabit[N*2] k<int N>(bit[N] x, bit[N] y) implements mult {

if (N<=1) return x*y;

bit[N/2] x1 = x[0:N/2-1]; bit[N/2+1] x2 = x[N/2:N-1]; bit[N/2] y1 = y[0:N/2-1]; bit[N/2+1] y2 = y[N/2:N-1];

bit[2*N] t11 = x1 * y1; bit[2*N] t12 = poly(1, x1, x2, y1, y2) * poly(1, x1, x2, y1, y2);bit[2*N] t22 = x2 * y2;

return multPolySparse<2*N>(2, N/2, t11) // log b = N/2 + multPolySparse<2*N>(2, N/2, t12) + multPolySparse<2*N>(2, N/2, t22);

}bit[2*N] poly<int N>(int n, bit[N] x0, x1, x2, x3) {

if (n<=0) return ??;else return (??*x0 + ??*x1 + ??*x2 + ??*x3) * poly<N>(n-1, x0, x1, x2, x3);

}bit[2*N] multPolySparse<int N>(int n, int x, bit[N] y) {

if (n<=0) return 0;else return y << x*?? + multPolySparse<N>(n-1, x, y);

}

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Semantic view of sketches

• a sketch represents a set of functions:– the ?? operator modeled as reading from an

oracle

int f (int y) { int f (int y, bit[][K] oracle) {x = ??; x = oracle[0];loop (x) { loop (x) {

y = y + ??; y = y + oracle[1];} }return y; return y;

} }

Synthesizer must find oracle satisfying f implements g

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Synthesis algorithm: overview

1. translation: represent spec and sketch as circuits

2. synthesis: find suitable oracle3. code generation: specialize sketch wrt

oracle

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Ex : Population count. 0010 0110 3int pop (bit[W] x){ int count = 0; for (int i = 0; i < W; i++)

{ if (x[i]) count++; } return count;}

x count 0 0 0 0 one 0 0 0 1

+

mux

count

+

count

mux

+

count

+

count

mux

mux

F(x) =

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Synthesis as generalized SAT• The sketch synthesis problem is an instance of 2QBF:

o . x . P(x) = S(x,o)

• Counter-example driven solver:

I = {}x = random()do

I = I U {x}c = synthesizeForSomeInputs(I)if c = nil then exit(“buggy sketch'')x = verifyForAllInputs(c) // x: counter-example

while x != nilreturn c

S(x1, c)=P(x1) … S(xk, c)=P(xk)

I ={ x1, x2, …, xk }

S(x, c) P(x)

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Case study

• Implemented AES– the modern block-cipher standard – 14 rounds: each has table lookup, permutation, GF-

multiply– a good implementation collapses each round into

table lookups

• Our results– we synthesized 32Kbit oracle!– synthesis time: about 1 hour– counterexample-driven synthesizer iterated 655

times– performance of synthesized code within 10% of

hand-tuned

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Finite programs

• In theory, SKETCH is complete for all finite programs:– specification can specify any finite program– sketch can describe any implementation over given

instructions– synthesizer can resolve any sketch

• In practice, SKETCH scales for small finite programs– small finite programs: block ciphers, small kernels– large finite: big-integer multiplication, matrix

multiplication

• Solution: – synthesize for a small input size– prove (or examine) that result of synthesis works for

bigger inputs

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Lossless abstraction

• Problem– does result of synthesis for a small matrix work

for all matrices?

• Approach– spec, sketch have unbounded-input/output– abstract them into finite functions, with the same

abstraction– synthesize– obtained oracle works for original sketch

• Stencil kernels– concrete: matrix A[N] matrix B[N]– abstract: A[e(i)], i, N B[i]

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Example: divide and conquer parallelization• Parallel algorithm:

– Data rearrangement + parallel computation

• spec: – sequential version of the program

• sketch: – parallel computation

• automatically synthesized: – Rearranging the data (dividing the data

structure)