Programs Synthesis from Polymorphic Refinement Types Nadia - - PowerPoint PPT Presentation

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Programs Synthesis from Polymorphic Refinement Types Nadia - - PowerPoint PPT Presentation

Sep 19, 2022 231 likes •653 views

Programs Synthesis from Polymorphic Refinement Types Nadia Polikarpova Ivan Kuraj Armando Solar-Lezama Program synthesis Make a list with n copies of x declarative specification Synthesizer ? 2 50 replicate n x = if if n 0

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SLIDE 1

Programs Synthesis from Polymorphic Refinement Types

Nadia Polikarpova Ivan Kuraj Armando Solar-Lezama

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SLIDE 2

Program synthesis

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“Make a list with n copies of x” replicate n x = if if n ≤ 0 th then Nil els lse Cons x (replicate (dec n) x)

declarative specification executable program

250 ⊨ ?

Synthesizer

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SLIDE 3

Modular verification for synthesis

3

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SLIDE 4

Specifications for synthesis

4

?

replicate n x = if if n ≤ 0 th then Nil els lse Cons x (replicate (dec n) x) Synthesizer

  • 1. supports automatic, modular

verification

  • 2. abstract and concise
  • 3. sufficiently expressive

refinement types

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SLIDE 5

Demo: replicate

  • - Specification:

replicate :: n: Nat → x: α → {ν: List α | len ν = n} replicate = ??

  • - Components:

zero :: {ν: Int | ν = 0} inc :: x: Int → {ν: Int | ν = x + 1} dec :: x: Int → {ν: Int | ν = x - 1} leq :: x: Int → y: Int → {Bool | ν = (x ≤ y) } neq :: x: Int → y: Int → {Bool | ν = (x ≠ y) }

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SLIDE 6

Synthesis from refinement types

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Γ ⊢ ?? :: T

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SLIDE 7

Synthesis from refinement types

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Γ ⊢ ?? :: T

x1 :: T1; ... φ1; ...

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SLIDE 8

Synthesis from refinement types

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Γ ⊢ ?? :: T

  • I. top-down enumerative search

x1 :: T1; ... φ1; ...

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SLIDE 9

Synthesis from refinement types

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Γ ⊢ ?? :: T

?? :: U ?? :: V

  • I. top-down enumerative search

x1 :: T1; ... φ1; ... :: T

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SLIDE 10

Synthesis from refinement types

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Γ ⊢ ?? :: T

?? :: U ?? :: V

  • I. top-down enumerative search

x1 :: T1; ... φ1; ... :: T

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SLIDE 11

Synthesis from refinement types

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Γ ⊢ ?? :: T

?? :: _ ?? :: _

  • I. top-down enumerative search

x1 :: T1; ... φ1; ... :: T’

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SLIDE 12

Synthesis from refinement types

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Γ ⊢ ?? :: T

?? :: _ ?? :: _

  • I. top-down enumerative search

x1 :: T1; ... φ1; ... :: T’

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SLIDE 13

Synthesis from refinement types

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Γ ⊢ ?? :: T

?? :: _

  • I. top-down enumerative search
  • II. round-trip type checking

x1 :: T1; ... φ1; ... ?? :: U

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SLIDE 14

Synthesis from refinement types

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Γ ⊢ ?? :: T

?? :: _

  • I. top-down enumerative search
  • II. round-trip type checking

x1 :: T1; ... φ1; ... ?? :: U

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SLIDE 15

Synthesis from refinement types

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Γ ⊢ ?? :: T

?? :: _ ?? :: _

  • I. top-down enumerative search
  • II. round-trip type checking

x1 :: T1; ... φ1; ... ?? :: U ?? :: V

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SLIDE 16

Synthesis from refinement types

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Γ ⊢ ?? :: T

?? :: _ ?? :: _

  • I. top-down enumerative search
  • II. round-trip type checking

x1 :: T1; ... φ1; ... :: T’ ?? :: U ?? :: V

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SLIDE 17

Synthesis from refinement types

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Γ ⊢ ?? :: T

  • I. top-down enumerative search
  • II. round-trip type checking

x1 :: T1; ... φ1; ... if f then else

?? :: Bool

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SLIDE 18

Synthesis from refinement types

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Γ ⊢ ?? :: T

  • I. top-down enumerative search
  • II. round-trip type checking

x1 :: T1; ... φ1; ... if f then else

P⊢?? :: T

  • III. condition abduction
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SLIDE 19

Synthesis from refinement types

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Γ ⊢ ?? :: T

  • I. top-down enumerative search
  • II. round-trip type checking

x1 :: T1; ... φ1; ... if f then else

?? :: Bool P⊢?? :: T

  • III. condition abduction

??::{Bool|ν=P}

¬P⊢??::T

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Round-trip type checking

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⊢ ?? :: {List Neg | len ν ≥ 5}

Γ

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SLIDE 21

Round-trip type checking

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Nil ; 0 ; 5 ; -5 zeros replicate ; Cons ⊢ ?? :: {List Neg | len ν ≥ 5}

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SLIDE 22

Round-trip type checking

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Nil ; 0 ; 5 ; -5 zeros replicate ; Cons ⊢ ?? :: {List Neg | len ν ≥ 5}

Nil :: {List Neg|len ν = 0}

Nil :: {List a | len ν = 0}

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SLIDE 23

Round-trip type checking

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Nil ; 0 ; 5 ; -5 zeros replicate ; Cons ⊢ ?? :: {List Neg | len ν ≥ 5}

?? :: _ → {List Neg | len ν ≥ 5}

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SLIDE 24

Round-trip type checking

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Nil ; 0 ; 5 ; -5 zeros replicate ; Cons ⊢ ?? :: {List Neg | len ν ≥ 5} zeros :: n:Nat → {List Zero | len ν = n}

zeros :: n:Nat → {List Zero | len ν = n} ?? :: _ → {List Neg | len ν ≥ 5}

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SLIDE 25

Nil ; 0 ; 5 ; -5 zeros replicate Cons

Round-trip type checking

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⊢ ?? :: {List Neg | len ν ≥ 5} ?? :: _ → _ → {List Neg|len ν ≥ 5}

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SLIDE 26

Nil ; 0 ; 5 ; -5 zeros replicate Cons

Round-trip type checking

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⊢ ?? :: {List Neg | len ν ≥ 5} replicate :: n: Nat → x: Neg → {List Neg | len ν = n} ?? :: Nat ?? :: Neg

replicate :: n: Nat → x: a → {List a | len ν = n}

?? :: _ → _ → {List Neg|len ν ≥ 5}

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SLIDE 27

Nil ; 0 ; 5 ; -5 zeros replicate Cons

Round-trip type checking

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⊢ ?? :: {List Neg | len ν ≥ 5} replicate :: n: Nat → x: Neg → {List Neg | len ν = n} ?? :: Nat 0 :: { ν = 0 }

replicate :: n: Nat → x: a → {List a | len ν = n}

?? :: _ → _ → {List Neg|len ν ≥ 5}

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SLIDE 28

Nil ; 0 ; 5 ; -5 zeros replicate Cons

Round-trip type checking

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⊢ ?? :: {List Neg | len ν ≥ 5} replicate :: n: Nat → x: Neg → {List Neg | len ν = n} ?? :: Nat ?? :: Neg 0 :: { ν = 0 } :: {List Neg | len ν = 0}

replicate :: n: Nat → x: a → {List a | len ν = n}

?? :: _ → _ → {List Neg|len ν ≥ 5}

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SLIDE 29

Nil ; 0 ; 5 ; -5 zeros replicate Cons

Round-trip type checking

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⊢ ?? :: {List Neg | len ν ≥ 5} replicate :: n: Nat → x: Neg → {List Neg | len ν = n} ?? :: Nat ?? :: Neg 0 :: { ν = 0 } 5 :: { ν = 5 } :: {List Neg | len ν = 0} :: {List Neg | len ν = 5}

replicate :: n: Nat → x: a → {List a | len ν = n}

?? :: _ → _ → {List Neg|len ν ≥ 5}

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SLIDE 30

0 :: { ν = 0 } 5 :: { ν = 5 }

Nil ; 0 ; 5 ; -5 zeros replicate Cons

Round-trip type checking

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⊢ ?? :: {List Neg | len ν ≥ 5} replicate :: n: Nat → x: Neg → {List Neg | len ν = n} ?? :: Nat ?? :: Neg 0 :: { ν = 0 } 5 :: { ν = 5 } :: {List Neg | len ν = 0} :: {List Neg | len ν = 5}

  • 5 :: { ν = -5 }

replicate :: n: Nat → x: a → {List a | len ν = n}

?? :: _ → _ → {List Neg|len ν ≥ 5}

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SLIDE 31

Condition abduction

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Nil ; 0 ; -5 ; n :: Nat (≤) ; (≠) ⊢ ?? :: {List Neg | len ν = n} P

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SLIDE 32

Condition abduction

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Nil ; 0 ; -5 ; n :: Nat (≤) ; (≠) ⊢ ?? :: {List Neg | len ν = n} Nil :: {List Neg |len ν = 0} P n ≤ 0

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SLIDE 33

Condition abduction

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Nil ; 0 ; -5 ; n :: Nat (≤) ; (≠) ⊢ ?? :: {List Neg | len ν = n} Nil :: {List Neg |len ν = 0} P n ≤ 0 if if n ≤ 0 th then Nil els lse Γ;¬(n ≤ 0) ⊢ ?? :: {List Neg | len ν = n}

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SLIDE 34

Liquid abduction

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n ≥ 0 ∧ len ν = 0 ∧ P ⇒ len ν = n Nil :: {List a | len ν = 0} n :: Nat

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SLIDE 35

Liquid abduction

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n ≥ 0 ∧ len ν = 0 ∧ P ⇒ len ν = n ★ ≤ ★ ★ ≠ ★ ∧ ¬(len ν = n)

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SLIDE 36

Liquid abduction

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n ≥ 0 ∧ len ν = 0 ∧ P ⇒ len ν = n ★ ≤ ★ ★ ≠ ★ n ≤ 0 n ≤ -5

  • 5 ≤ n

n ≠ 0 n ≠ -5

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SLIDE 37

Liquid abduction

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n ≥ 0 ∧ len ν = 0 ∧ P ⇒ len ν = n ★ ≤ ★ ★ ≠ ★ n ≤ 0 n ≤ -5

  • 5 ≤ n

n ≠ 0 n ≠ -5 ∧ ¬(len ν = n) UNSAT core [

]

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SLIDE 38

Liquid abduction

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n ≥ 0 ∧ len ν = 0 ∧ P ⇒ len ν = n ★ ≤ ★ ★ ≠ ★ n ≤ 0 n ≤ -5

  • 5 ≤ n

n ≠ 0 n ≠ -5 ∧ ¬(len ν = n) UNSAT core [

]

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SLIDE 39

Evaluation

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take, drop, delete, zip with, reverse, de- duplicate, fold, length/append with fold, ...

Lists

insertion s., selection s., merge s., quick s.

Sorting

member, insert, delete

Binary Search Trees

RBT & AVL insertion, AVL deletion

Balanced trees

AST desugaring, address book

Custom datatypes

20 s

6 s

64 benchmarks

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> > 120 s No roundtrip type checking

37 27

Naive liquid abduction

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SLIDE 40

Synthesis of recursive programs

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easy to verify hard to verify strong guarantees weak guarantees

pre-/post- conditions input-output examples

[Escher: CAV’13] [Myth: PLDI’15] [λ2: PLDI’15] [Leon: OOPSLA’13]

refinement types

[Myth+, POPL’16]

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SLIDE 41

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http://tiny.cc/synquid