Synthesis with the Sketch System D AY 1 Armando Solar-Lezama

The Challenge Computer should help make programming easier Problem  Programming requires insight and experience  Computers are not that smart Interaction between programmers and tools is key

The sketching approach Let the programmer control the implementation strategy Focus the synthesizer on the low-level details Key design principle:  Exploit familiar programming concepts

Framing the synthesis problem Automate challenging programming steps Leverage programmer knowledge VisualBasic Python Fortran? MundaneScience Fiction ( )

Fundamental question How to combine human insight and machine power Synergy between human and machine is key

Lessons from verification Embrace partial specifications  Because there will never be a spec for PowerPoint Speak to programmers in their own language  Test harnesses good, high order logic bad Don’t be afraid of exhaustive exploration  But do it intelligently  deductive reasoning is a double-edged sword You don’t have to do everything statically  For most problems it is too hard

Example You want to partition N elements over P procs  How many elements should a processor get? Obvious answer is N/P Obvious answer is wrong! N = 18 P = 5

void partition(int p, int P, int N, ref int ibeg, ref int iend){ if(p< {$ p, P, N, N/P, N%P : *, + $} ){ iend = {$ p, P, N, N/P, N%P : *, + $}; ibeg = {$ p, P, N, N/P, N%P : *, + $}; }else{ iend = {$ p, P, N, N/P, N%P : *, + $}; ibeg = {$ p, P, N, N/P, N%P : *, + $}; } Synthesizing a partition function What do we know?  The interface to the function we want  Not all processors will get the same # of elements  The kind of expressions we expect p P N N/P N%P * +

harness void testPartition(int p, int N, int P){ if(p>=P || P < 1){ return; } int ibeg, iend; partition(p, P, N, ibeg, iend); assert iend - ibeg < (N/P) + 2; if(p+1 < P){ int ibeg2, iend2; partition(p+1, P, N, ibeg2, iend2); assert iend == ibeg2; } if(p==0){ assert ibeg == 0; } if(p==P-1){ assert iend == N; } } Synthesizing a partition function How does the system know what a partition is? Partitions should be balanced Adjacent partitions should match First and last partition should go all the way to the ends

DEMO

Solution void partition(int p, int P, int N, ref int ibeg, ref int iend){ if(p < (N % P)){ iend = ((N / P) + 1) * (1 + p); ibeg = p + ((N / P) * p); }else{ iend = ((N / P) * p) + ((N / P) + (N % P)); ibeg = (N % P) + ((N / P) * p); }

THE SKETCH LANGUAGE

Sketch language basics Sketches are programs with holes  write what you know  use holes for the rest

Specifications Idea: Use unit tests as specification  Programmers know how to write those Two mechanisms  assertions  function equivalence blockedMatMul(Mat a, Mat b) implements matMul assert x > y;

Holes Holes are placeholders for the synthesizer  synthesizer replaces hole with concrete code fragment  fragment must come from a set defined by the user Defining sets of code fragments is the key to Sketching effectively

Language Design Strategy Extend base language with one construct Constant hole: ?? Synthesizer replaces ?? with a constant High-level constructs defined in terms of ?? int bar (int x) { int t = x * ??; assert t == x + x; return t; } int bar (int x) { int t = x * 2; assert t == x + x; return t; }

Integer Holes  Sets of Expressions Expressions with ?? == sets of expressions  linear expressions x*?? + y*??  polynomials x*x*?? + x*?? + ??  sets of variables ?? ? x : y

Integer Holes  Sets of Expressions Example: Least Significant Zero Bit   Trick:  Adding 1 to a string of ones turns the next zero to a 1  i.e = int W = 32; bit[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; return ret; } } !(x + ??) & (x + ??) !(x + 1) & (x + 0) !(x + 0) & (x + 1) !(x + 1) & (x + 0xFFFF) !(x + 0xFFFF) & (x + 1) 

Integer Holes  Sets of Expressions Example: Least Significant Zero Bit   int W = 32; bit[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; return ret; } } bit[W] isolateSk (bit[W] x) implements isolate0 { return !(x + ??) & (x + ??) ; }

Integer Holes  Sets of Expressions Expressions with ?? == sets of expressions  linear expressions x*?? + y*??  polynomials x*x*?? + x*?? + ??  sets of variables ?? ? x : y Semantically powerful but syntactically clunky  Regular Expressions are a more convenient

Regular Expression Generators {| RegExp |} RegExp supports choice ‘|’ and optional ‘?’  can be used arbitrarily within an expression - to select operands {| (x | y | z) + 1 |} - to select operators {| x (+ | -) y |} - to select fields {| n(.prev |.next)? |} - to select arguments {| foo( x | y, z) |} Set must respect the type system  all expressions in the set must type-check  all must be of the same type

Sets of statements Statements with holes = sets of statements Higher level constructs for Statements too  repeat bit[W] tmp=0; repeat(3){ {| x | tmp |} = {| (!)?((x | tmp) (& | +) (x | tmp | ??)) |}; } return tmp;

repeat Avoid copying and pasting  repeat(n){ s}  s;s;…s;  each of the n copies may resolve to a distinct stmt  n can be a hole too. n bit[W] tmp=0; repeat(??){ {| x | tmp |} = {| (!)?((x | tmp) (& | +) (x | tmp | ??)) |}; } return tmp;

GENERATORS

User defined generators Mechanism to define sets of code fragments They look like functions  But with a few caveats

Key features of generators Different dynamic invocations  different code Recursive generators = grammar of expressions generator bit[W] gen(bit[W] x, int bnd){ assert bnd > 0; if(??) return x; if(??) return ??; if(??) return ~gen(x, bnd-1); if(??){ return {| gen(x, bnd-1) (+ | & | ^) gen(x, bnd-1) |}; } bit[W] isolate0sk (bit[W] x) implements isolate0 { return gen(x, 3); }

CONSTRAINT BASED SYNTHESIS PRIMER

Synthesis problem at the high level Game theoretic view of synthesis For every move of the environment Synthesized program makes a counter move

The challenge of synthesis

Quantifying over programs

31 Insight Sketches are not arbitrary constraint systems  They express the high level structure of a program A small number of inputs can be enough  focus on corner cases This is an inductive synthesis problem ! where E = {x 1, x 2, …, x k }

Insert your favorite checker here CEGIS SynthesizeCheck

Insert your favorite checker here CEGIS SynthesizeCheck