Beyond Verification. Software Synthesis

Size: px

Start display at page:

Download "Beyond Verification. Software Synthesis"

Tobias Walsh
6 years ago
Views:

1 Beynd Verificatin Sftware Synthesis 1

2 What d we mean by synthesis We want t get cde frm high-level specs - Pythn and VB are pretty high level, why is that nt synthesis? Supprt cmpsitinal and incremental specs - Pythn and VB dn t have this prperty If I dn t like the way the pythn cmpiler/runtime is implementing my prgram, I am ut f luck. - Lgical specificatins d I can always add additinal prperties that my system can satisfy - Specs are nt nly functinal Structural specificatins play a big rle in synthesis Hw is my algrithm ging t lk like. 2

3 The fundamental challenge The fundamental challenge f synthesis is dealing with an uncperative envirnment - Fr reactive systems, peple mdel this as a game Fr every mve f the adversary (ever actin f the envirnment), the synthesized prgram must make a cunter-mve that keeps the system wrking crrectly. The game can be mdeled with an autmata 3

4 The fundamental challenge The fundamental challenge f synthesis is dealing with an uncperative envirnment - If we are synthesizing functins, the envirnment prvides the inputs i.e. whatever we synthesize must wrk crrectly fr all inputs - This is mdeled with a dubly quantified cnstraint E.g. if the spec is given as pre and pst cnditins, we have P σ σ pre (σ WP(P, pst )) - What des it mean t quantify ver the space f prgrams? 4

5 Quantifying ver prgrams Synthesis in the functinal setting can be seen as curve fitting - i.e. we want t find a curve that satisfies sme prperties It s very hard t d curve fitting when yu have t cnsider arbitrary curves - Instead, peple use parameterized families f curves - This means yu quantify ver parameters instead f ver functins This is the first fundamental idea in sftware synthesis - Peple call these Sketches, scafflds, templates, - They are all the same thing 5

6 The Sketch Language Define parameterized prgrams explicitly - Think f the parameterized prgrams as prgrams with hles Example: Hell Wrld f Sketching spec: int f (int x) { return x + x; } sketch: int bar (int x) implements f { return x *??; } Integer Hle 6

7 Integer Hles Sets f Expressins Expressins with?? == sets f expressins - linear expressins x*?? + y*?? - plynmials x*x*?? + x*?? +?? - sets f variables??? x : y 7

8 Integer Hles Sets f Expressins Example: Least Significant Zer Bit int W = 32; bit[w] islate0 (bit[w] x) { // W: wrd size bit[w] ret = 0; fr (int i = 0; i < W; i++) if (!x[i]) { ret[i] = 1; return ret; } } Trick: - Adding 1 t a string f nes turns the next zer t a 1 - i.e = !(x +??) & (x +??)!(x + 1) & (x + 0)!(x + 0) & (x + 1)!(x + 1) & (x + 0xFFFF)!(x + 0xFFFF) & (x + 1) 8

9 Integer Hles Sets f Expressins Example: Least Significant Zer Bit int W = 32; bit[w] islate0 (bit[w] x) { // W: wrd size bit[w] ret = 0; fr (int i = 0; i < W; i++) if (!x[i]) { ret[i] = 1; return ret; } } bit[w] islatesk (bit[w] x) implements islate0 { } return!(x +??) & (x +??) ; 9

10 Integer Hles Sets f Expressins Expressins with?? == sets f expressins - linear expressins x*?? + y*?? - plynmials x*x*?? + x*?? +?? - sets f variables??? x : y Semantically pwerful but syntactically clunky - Regular Expressins are a mre cnvenient way f defining sets 10

11 Regular Expressin Generatrs { RegExp } RegExp supprts chice and ptinal? - can be used arbitrarily within an expressin t select perands { (x y z) + 1 } t select peratrs { x (+ -) y } t select fields { n(.prev.next)? } t select arguments { f( x y, z) } Set must respect the type system - all expressins in the set must type-check - all must be f the same type 11

12 Least Significant One revisited Hw did I knw the slutin wuld take the frm!(x +??) & (x +??). What if all yu knw is that the slutin invlves x, +, & and!. bit[w] tmp=0; { x tmp } = { (!)?((x tmp) (& +) (x tmp??)) }; { x tmp } = { (!)?((x tmp) (& +) (x tmp??)) }; return tmp; This is nw a set f statements (and a really big ne t) 12

13 Sets f statements Statements with hles = sets f statements Higher level cnstructs fr Statements t - repeat bit[w] tmp=0; repeat(3){ { x tmp } = { (!)?((x tmp) (& +) (x tmp??)) }; } return tmp; 13

14 repeat Avid cpying and pasting - repeat(n){ s} s;s; s; - each f the n cpies may reslve t a distinct stmt - n can be a hle t. n bit[w] tmp=0; repeat(??){ { x tmp } = { (!)?((x tmp) (& +) (x tmp??)) }; } return tmp; Keep in mind: - the synthesizer wn t try t minimize n 14

15 Slving fr a parameterized prgram At a high level, tw fundamental appraches: - Search and Test - Derive in ne sht Usually by means f abstractin. 15

16 The CEGIS apprach Synthesis reduces t cnstraint satisfactin φ. x. A E Q(x, φ) Cnstraints are t hard fr standard techniques - Universal quantificatin ver inputs - T many inputs - T many cnstraints - T many hles 16

17 Insight Sketches are nt arbitrary cnstraint systems - They express the high level structure f a prgram A small number f inputs can be enugh - fcus n crner cases φ. x in E. Q(x, φ) A E where E = {x 1, x 2,, x k } This is an inductive synthesis prblem! 17

18 CEGIS Synthesis algrithm Synthesize c Check Insert yur favrite checker here c s. t. Crrect(P c, in i ) in s. t. Crrect(P c, in i ) {in i } in 18

19 CEGIS 19

20 A sketch as a cnstraint system int lin(int x){ if(x >?? 1 ) return?? 2 *x +?? 3 ; else return?? 4 *x; }?? 2 *x +?? 3 x >?? 1 x+1 >?? 1?? 2 *(x+1) +?? 3 vid main(int x){ int t1 = lin(x); int t2 = lin(x+1);?? 4 *x mux mux?? 4 *(x+1) } if(x<4) assert t1 >= x*x; if(x>=3) assert t2-t1 == 1; x>=4 r >= x*x - = r 1 x<3 and 20

21 Ex : Ppulatin cunt x cunt ne cunt mux mux + cunt + cunt mux F(x) = cunt mux + 21

22 x & >> & + mux x & >> & + mux x & >> + & S(x,) = x mux 22

23 MIT OpenCurseWare Fundamentals f Prgram Analysis Fall 2015 Fr infrmatin abut citing these materials r ur Terms f Use, visit:

Introduction to Sketching

Introduction to Sketching IAP 2008 Armando Solar-Lezama What is sketching? A program synthesis system - generates small fragments of code - checks their validity against a specification A programming aid