Synthesis with the Sketch System

DAY 2

Armando Solar-Lezama

The challenge of synthesis

•For functions, the environment controls the inputs i.e. whatever we synthesize must work for all inputs

•Modeled with a doubly quantified constraint

What does it mean to quantify over programs?

Quantifying over programs

•Synthesis as curve fitting we want a function that satisfies some properties

•It’s hard to do curve fitting with arbitrary curves Instead, people use parameterized families of curves Quantify over parameters instead of over functions

•A sketch is just a way of describing these families

4

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 !

∃𝑐 ∀ 𝑖𝑛∈𝐸𝑄(𝑖𝑛 ,𝑐 )where E = {x1, x2, …, xk}

Insert your favorite checker here

CEGIS

{𝒊𝒏𝒊}

∃𝒄 𝒔 .𝒕 .𝑪𝒐𝒓𝒓𝒆𝒄𝒕 (𝑷 𝒄 , 𝒊𝒏𝒊) ∃ 𝒊𝒏𝒔 .𝒕 .¬𝑪𝒐𝒓𝒓𝒆𝒄𝒕 (𝑷𝒄 , 𝒊𝒏𝒊)

Synthesize Check𝒄

𝒊𝒏

•Constraints for each follow from semantics

Loops handled through simple unrolling

Insert your favorite checker here

CEGIS

{𝒊𝒏𝒊}

∃𝒄 𝒔 .𝒕 .𝑪𝒐𝒓𝒓𝒆𝒄𝒕 (𝑷 𝒄 , 𝒊𝒏𝒊)

Synthesize Check𝒄

𝒊𝒏

CEGIS in Detail

𝑸 (𝒄 ,𝒊𝒏)+

+

+

b

+

+

++

a c d

A

+

+

+ ++

A A A

{𝒊𝒏𝒊}

SynthesizeCheck𝒄

𝒊𝒏

CEGIS in Detail

𝑸𝟎(𝒄 , 𝒊𝒏)𝑸𝟎 (𝒄 , 𝒊𝒏 )⇒𝑸𝟏(𝒄 , 𝒊𝒏)𝑸𝟏(𝒄 , 𝒊𝒏)𝑸𝟏 (𝒄 , 𝒊𝒏)⇒𝑸𝟐(𝒄 , 𝒊𝒏)𝑸𝟐(𝒄 , 𝒊𝒏)

+

+

+

b

+

+

++

a c d

A

+

+

+ ++

A A A

{𝒊𝒏𝒊}

SynthesizeCheck𝒄

𝒊𝒏

Bits

2

3

45

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 expressionsgenerator 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);}

bit[W] reverseSketch(bit[W] in) implements reverse {

bit[W] t = in; int s = 1; repeat(??){ bit[W] tmp1 = (t << s); bit[W] tmp2 = (t >> s); t = tmp1 {|} tmp2; s = s*??; } return t;}

Gens + Closures = Extensible Language•Redefining Repeat

Gens + Closures = Extensible Language•Redefining Repeat

generator void rep(int n, fun f){ if(n>0){ f(); rep(n-1, f); } }

bit[W] reverseSketch(bit[W] in) implements reverse {

bit[W] t = in; int s = 1; generator void tmp(){ bit[W] tmp1 = (t << s); bit[W] tmp2 = (t >> s); t = tmp1 {|} tmp2; s = s*??; } rep(??, tmp); return t;}

Karatsuba Multiplication

•Recursive grade-school multiplication

𝑥=𝑥2∗𝑝𝑛2+𝑥1 𝑦=𝑦2∗𝑝

𝑛2 +𝑦1

𝑥∗ 𝑦=𝑎+𝑏∗𝑝𝑛2+𝑐 ∗𝑝

𝑛2+𝑑∗𝑝𝑛

Karatsuba Multiplication

•Smarter Karatsuba Multiplication

𝑥=𝑥1∗𝑝𝑛2+𝑥2 𝑦=𝑦1∗𝑝

𝑛2+𝑦2

𝑥∗ 𝑦=…