Abstract Parsing for Two-staged Language. Soonho Kong May 1, 2009

Transcription

1 Astract arsing for Two-staged Language May 1, 009

2 Contents Motivation Language - Syntax & Semantics LR arsing Concrete arsing Astraction Instantiation / 5

3 Motivation rogram generates another program. How can we make sure that the generated program is syntactically correct? 3 / 5

4 Language: Syntax We design a small & simple language to focus on the property we want to verify e Exp :: x let xe 1 e or e 1 e re xe 1 e e 3 `f f F rag :: x let or re f 1.f,e 4 / 5

5 Language: Example code let x `a let y ` or x y let x `a let y ` `x.y.,y re x `a ` x > `a ` > `x y > `a ` re x `a `a.,x re x `a `,x.,x > `a `a a `a a a `a a a a... > `a `a a `a a a a `a a a a a a a a... 5 / 5

6 Language: Collecting Semantics σ Env : ar Code 0 : Exp Env Code 1 : F rag Env Code 0 x Σ {σ(x) σ Σ} 0 (let xe 1 e )Σ 0 e {σ[x c] σ Σ} c 0 e 1 Σ 0 (or e 1 e )Σ 0 e 1 Σ 0 e Σ 0 (re xe 1 e e 3 )Σ 0 e 3 {σ[x c] c fix λc. 0 e 1 {σ} 0 e {σ[x c ] c C}} σ Σ 0 (`f)σ 1 fσ 1 (f 1.f )Σ {xy x 1 f 1 Σ,y 1 f Σ} 1 (,e)σ 0 eσ 1 or Σ { or } 1 let Σ { let } 1 re Σ { re } 1 xσ { x } 6 / 5

7 Language: Astract Collecting Semantics α Env d Env λσ.λx.{σ(x) σ Σ} σ Ênv : ar Code 0 : Exp Ênv Code 1 : F rag Ênv Code 0 x σ σ(x) 0 (or e 1 e ) σ 0 e 1 σ 0 e σ 0 (let xe 1 e ) σ 0 e ( σ[x 0 e 1 σ]) 0 (re xe 1 e e 3 ) σ 0 e 3 ( σ[x fix λk. 0 e 1 σ 0 e ( σ[x k])]) 0 (`f) σ 1 f σ 1 (f 1.f ) σ {xy x 1 f 1 σ y 1 f σ} 1 (,e) σ 0 e σ 1 or σ { or } 1 let σ { let } 1 re σ { re } 1 x σ { x } 7 / 5

8 LR arsing arsing determines whether the string conforms to the grammar or not Grammar String arser O/X 8 / 5

9 LR arsing Scan Left-to-right, derive Rightmost derivation tree. Uses parsing stack and parse_action function Token parse_action parse action : T oken 9 / 5

10 LR arsing Algorithm 1 parse action algorithm 1: procedure parse action(p, a) : t the state on top of stack p 3: if ACT ION[t, a] shift s then 4: push s onto the stack p 5: return p 6: else if ACT ION[t, a] reduce A β then 7: pop β symol off the stack p 8: t the state on top of stack p 9: push GOT O[t, A] onto the stack p 10: return parse action(p, a) 11: end if 1: end procedure Shift Reduce t s s GOTO[t,A] 10 / 5

11 LR arsing arsing is composition of parse_action. Code parse Token Token Token parse_action parse_action parse_action parse : Code 11 / 5

12 Astract arsing: Idea Instead of executing the program and parsing the result, 0 eσ {c 1,c,..., c n } parse(c i )O/X Execute the program on astract semantics. 0 e Σ {O, X} 1 / 5

13 Concrete arsing: alue What should e the one corresponding with Code in parsing domain? It is easy to think that it is (parse c p init ): Code c s 0 p init parse 13 / 5

14 Concrete arsing: alue rolem: Code Concatenation : Easy c1 + c c1 c arse Stack Concatenation : Hard +? 14 / 5

15 Concrete arsing: alue Solution Let it e the function (parse c) : Code c parse 15 / 5

16 Concrete arsing: alue Then the concatenation is handled as function composition c1 c + 16 / 5

17 Astraction Concrete arsing α Code λc.{λp.parse(p, c) c C} Semantics { σ Env : ar 0 : Exp Env 1 : F rag Env 0 xσ σ (x) 0 (or e 1 e )σ 0 e 1 σ 0 e σ 0 (let xe 1 e )σ 0 e (σ [x 0 e 1 σ ]) 0 (re xe 1 e e 3 )σ 0 e 3 (σ [x fix λk. 0 e 1 σ 0 e (σ [x k])]) 0 (`f)σ 1 fσ 1 (f 1.f )σ {p p 1 p 1 1 f 1 σ p 1 f σ } 1 (,e)σ 0 eσ 1 a σ {λp.parse action a p} 17 / 5

18 Astraction First, we astract to y, Semantics is α ( ) λf λ.{f(p) f F p } { ( ) σ Env : ar ( ) 0 : Exp Env ( ) 1 : F rag Env ( ) 0 xσ σ (x) 0 (or e 1 e )σ 0 e 1 σ 0 e σ 0 (let xe 1 e )σ 0 e (σ [x 0 e 1 σ ]) 0 (re xe 1 e e 3 )σ 0 e 3 (σ [x fix λk. 0 e 1 σ 0 e (σ [x k])]) 0 (`f)σ 1 fσ 1 (f 1.f )σ 1 f σ 1 f 1 σ 1 (,e)σ 0 eσ 1 a σ λ.{parse action a p p } 18 / 5

19 Some astract domainsarsing are naturally derived from. Theorem 1. Astract Framework 4. Framework Theorem 1. Astract arsing Framework Let : : May 1, 009 ROAS Let : 1. Astract arsing Framework Theorem Env : ar Let : Astraction,, ) e CO. 1. ( ",. and e", galois via α and γ. (,, ) e CO. connected 4. 1.Framework Galois connections etween derived domains and derived astract domains are 1. Some (,, ", ) e CO. astract domains are naturally derived from. Some astract domains are naturally derived from. We can astract to and γframework Theorem 1. from Astract. derived α arsing Let : Some astractdomains are naturally derived from. : : if satisfies the following conditions. α (F (γ )) λf.λ.α : ar ar )) γ λfenv.λ.γ (: F (α Env Env : ar 1.1. (,, ", ) e is CO. CO αenv Env λσ.λx.α (σ(x)) Some astract domains are naturally derived from. and e galois connected via α. and (... and galois connected via α and γ. and galois connected via α and γ σ σ.γ. and eeare connected via γgalois λ.λx.γ (x)) Env Env derived : domains and domains Galois connections etween derived derived astract doma Galois connections etween and derived ast Galois connections etween derived domains and derived as is a sound astraction of parse derived of action. Fo action : T oken e a sound astraction parse Env : ar. from α and γ. derived from α and γ derived from α. and γ.α a T oken. action(p, a) (γ p }) ({parse α is, λf.λ.α (F )) parse action : T oken. That. and e galois connected via α and γ. α (F (γ(γ λf.λ.α α (F λf.λ.α parse action (α ( ), a) Galois domains and derived astract domains.λ.γ λ F (F (α )) a Tconnections oken. etween. γderived.λ.γ (F(α γ λ F defined. from α and γof 4. derived astract counterpart, as elow. e a γ λ F.λ.γ ( F (α parse α ({parse action p p }) # action a α ( ) αenva Env λσ.λx.α (σ(x)) α λσ.λx.α (σ(x)) Env )) Env α (F (γ λf.λ.α αenv λσ.λx.α (σ(x) Env γenv λ σ.λx.γ ( σ (x)) Env σ Env γ λ σ.λx.γ ( σ (x)).λ.γ (F(α Env )) γenv λ F γ λ σ.λx.γ ( σ (x) Env Env : gm Env T Env oken eλσ.λx.α a sound astraction of parse acti 3. parse action : αenv (σ(x)) Tell 1 May, 009 Show & ROAS Weekly 19 / 5

20 We define 0 and 1. elow σ Env Astraction 0 : Exp Env ( ) 1 : F rag Env ( ) 0 xσ σ (x) 0 (or e 1 e )σ 0 e 1 σ 0 e σ 0 (let xe 1 e )σ 0 e (σ [x 0 e 1 σ ]) 0 (re xe 1 e e 3 )σ 0 e 3 (σ [x fix λk. 0 e 1 σ 0 e (σ [x k])]) 0 `fσ 1 fσ 1 (f 1 f )σ 1 f σ 1 f 1 σ 1 (,e)σ 0 eσ 1 a σ λ.parse action a Then is a sound approximation of. 0 elow e Exp. σ Env. α ( ) ( )( 0 e σ ) 0 e α Env Env (σ ) / 5

21 Instantiation of : Astract arsing Stack with k-cutting IEA : limit the length of parsing stack with k s1 s s3 s4 3-cutting s1 s _ : { p p } α id γ λ. ( ) d (if d then {d} else ) 1 / 5

22 Instantiation of : Astract arsing Stack with k-cutting Algorithm parse action algorithm 1: procedure parse action( p, a) : if p then 3: return p 4: end if 5: t the state on top of stack p 6: if ACT ION[t, a] shift s then 7: push s onto the stack p 8: return p 9: else if ACT ION[t, a] reduce A β then 10: pop β symol off the stack p 11: t the state on top of stack p 1: if t then 13: return p 14: end if 15: push GOT O[t, A] onto the stack p 16: return parse action(a, p) 17: end if 18: end procedure s1 s s3 pop _ / 5

23 Instantiation of : Astract arsing Stack with k-cutting We only perform k-cutting at loop head y defining widening operator A. A B : A B {cut k ( p) p A B} Algorithm 3 cut k algorithm 1: procedure cut k ( p s 1... s n ) : if n k then 3: return p 4: else 5: return s 1... s k : end if 7: end procedure s1 s s1 s s3 s4 3-cutting 3-cutting s1 s s1 s _ 3 / 5

24 Reference Kyung-Goo oh, Hyunha Kim and avid Schmidt. Astract parsing: static analysis of dynamically generated string output using LR-parsing technology. SAS 009: The 16th International Static Analysis Symposium, LA, August 7-9, 009. (to appear) 4 / 5

25 Thank you 5 / 5