Synthesis with the Sketch System
DAY 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
Example
•You

want to partition N elements over P procs
How many elements should a processor get?
N = 18
P=5
•Obvious
answer is N/P
•Obvious
answer is wrong!
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
void partition(int
if(p< {$ p, P,
iend = {$
ibeg = {$
}else{
iend = {$
ibeg = {$
}
}
p,
N,
p,
p,
int P, int N, ref int ibeg, ref int iend){
N/P, N%P : *, + $} ){
P, N, N/P, N%P : *, + $};
P, N, N/P, P
N%P : *, + N%P
$};
N
p, P, N,* N/P, +
N%P : *, + $};
p
N/P
p, P, N, N/P, N%P : *, + $};
Synthesizing a partition function
•How
does the system know what a partition is?
harness void testPartition(int p, int N, int P){
if(p>=P || P < 1){ return; }
Partitions should be
int ibeg, iend;
balanced
partition(p, P, N, ibeg, iend);
assert iend - ibeg < (N/P) + 2;
Adjacent partitions
if(p+1 < P){
should match
int ibeg2, iend2;
partition(p+1, P, N, ibeg2, iend2);
assert iend == ibeg2;
}
if(p==0){ assert ibeg == 0; }
First and last
if(p==P-1){ assert iend == N; }
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:

Programmers know how to write those
•Two

Use unit tests as specification
mechanisms
assertions
assert x > y;

function equivalence
blockedMatMul(Mat a, Mat b) implements matMul
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: ??
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;
}
Synthesizer replaces ?? with a constant
High-level constructs defined in terms of ??
Integer Holes  Sets of Expressions
•Expressions
with ?? == sets of expressions
linear expressions
 polynomials
 sets of variables

x*?? + y*??
x*x*?? + x*?? + ??
?? ? x : y
Integer Holes  Sets of Expressions
•Example: Least Significant
 0010 0101  0000 0010
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;
}
}
•Trick:
 Adding 1 to a string of ones turns the next zero to a 1
 i.e. 000111 + 1 = 001000
!(x + ??) & (x + ??)

!(x + 1) & (x + 0)
!(x + 1) & (x + 0xFFFF)
!(x + 0) & (x + 1)
!(x + 0xFFFF) & (x + 1)
Integer Holes  Sets of Expressions
•Example:

Least Significant Zero Bit
0010 0101  0000 0010
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



linear expressions
polynomials
sets of variables
•Semantically

with ?? == sets of expressions
x*?? + y*??
x*x*?? + x*?? + ??
?? ? x : y
powerful but syntactically clunky
Regular Expressions are a more convenient
Regular Expression Generators
•{|
RegExp |}
•RegExp

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


supports choice ‘|’ and optional ‘?’
must respect the type system
all expressions in the set must type-check
all must be of the same type
Sets of statements
•Statements
•Higher

with holes = sets of statements
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;
n
each of the n copies may resolve to a distinct stmt
 n can be a hole too.

bit[W] tmp=0;
repeat(??){
{| x | tmp |} = {| (!)?((x | tmp) (& | +) (x | tmp | ??)) |};
}
return tmp;
GENERATORS
User defined generators
•Mechanism
•They

to define sets of code fragments
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)
return gen(x, 3);
}
implements isolate0 {
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
•For

functions, the environment controls the inputs
i.e. whatever we synthesize must work for all inputs
•Modeled

with a doubly quantified constraint
∃ 𝑃 ∀ 𝑖𝑛 𝑖𝑛, 𝑃 ⊨ 𝑆𝑝𝑒𝑐
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
∃ 𝑐 ∀ 𝑖𝑛
𝑖𝑛, 𝑆𝑘(𝑐) ⊨ 𝑆𝑝𝑒𝑐
∃𝑐 ∀𝑖𝑛 𝑃(𝑖𝑛, 𝑐)
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
∃𝑐 ∀𝑖𝑛 ∈ 𝐸 𝑄(𝑖𝑛, 𝑐)
where E = {x1, x2, …, xk}
This is an inductive synthesis problem !
31
CEGIS
Synthesize
𝒄
Insert your favorite
∃𝒊𝒏 𝒔. 𝒕.
¬𝑪𝒐𝒓𝒓𝒆𝒄𝒕(𝑷𝒄 , 𝒊𝒏𝒊 )
checker here
∃𝒄 𝒔. 𝒕. 𝑪𝒐𝒓𝒓𝒆𝒄𝒕(𝑷𝒄 , 𝒊𝒏𝒊 )
{𝒊𝒏𝒊 }
Check
𝒊𝒏
CEGIS
Synthesize
𝒄
Insert your favorite
checker here
∃𝒄 𝒔. 𝒕. 𝑪𝒐𝒓𝒓𝒆𝒄𝒕(𝑷𝒄 , 𝒊𝒏𝒊 )
{𝒊𝒏𝒊 }
•Constraints

Check
𝒊𝒏
for each 𝒊𝒏𝒊 follow from semantics
Loops handled through simple unrolling