Semantics of an Intermediate Language for Program Transformation

Transcription

1 Semantics of an Intermediate Language for Program Transformation Sigurd Schneider Master Thesis Proposal Talk Advisors: Prof. Dr. Sebastian Hack, Prof. Dr. Gert Smolka Saarland University Graduate School of Computer Science November 11, 2011 Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 1 / 26

2 Overview 1 Introduction Static Single Assignment Form Control Flow Graph 2 Intermediate Language (IL) Syntax Semantics 3 Equivalence 4 Transformation 5 Conclusion Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 2 / 26

3 Introduction Motivation Static Single Assignment Form (SSA Form) standard (libfirm, llvm, hotspot, gcc) eases specification and implementation of program transformations Does SSA form ease the correctness proofs of program transformations, too? Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 3 / 26

4 Introduction Static Single Assignment Form Static Single Assignment Form Alpern, Cytron, Ferrante, Rosen, Wegman, Zadeck [6, 1, 2] developed in the context of data flow analysis SSA condition every name has exactly one assignment in the program join points of control flow are mediated by phi nodes (e.g. loops) Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 4 / 26

5 Example Introduction Example Program 1 // n i n p u t 2 e n t r y : i = 1 ; 3 l o o p : i f n = 0 then 4 r e t u r n i 5 e l s e 6 i = n i ; 7 n = n 1 ; 8 goto l o o p Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 5 / 26

6 Example Introduction Example Program 1 // n i n p u t 2 e n t r y : i = 1 ; 3 l o o p : i f n = 0 then 4 r e t u r n i 5 e l s e 6 i = n i ; 7 n = n 1 ; 8 goto l o o p Program with variables renamed (broken) 1 // n0 i n p u t 2 e n t r y : i 0 = 1 ; 3 l o o p : i f n0 = 0 then 4 r e t u r n i 0 5 e l s e 6 i 1 = n0 i 0 ; 7 n1 = n0 1 ; 8 body : goto l o o p Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 6 / 26

7 Example Introduction Example Program 1 // n i n p u t 2 e n t r y : i = 1 ; 3 l o o p : i f n = 0 then 4 r e t u r n i 5 e l s e 6 i = n i ; 7 n = n 1 ; 8 goto l o o p Program in SSA form 1 // n0 i n p u t 2 e n t r y : i 0 = 1 ; 3 l o o p : ( n2, i 2 ) = 4 p h i ( e n t r y : ( n0, i 0 ), 5 body : ( n1, i 1 ) ) ; 6 i f n2 = 0 then 7 r e t u r n i 2 8 e l s e 9 i 1 = n2 i 2 ; 10 n1 = n2 1 ; 11 body : goto l o o p Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 7 / 26

8 Control Flow Graph Introduction Control Flow Graph entry : i0=1 loop: return i2 n2 == 0 n2,i2 = phi(entry:(n0,i0 ),body:(n1,i1 )); n2!= 0 body: i1 = n2 i2; n1 = n2 1; Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 8 / 26

9 Introduction Control Flow Graph revisited Control Flow Graph entry n0 i0=1 (n0,i0) loop (n2, i2) return i2 n2 == 0 n2!= 0 (n1,i1) i1 = n2 i2; n1 = n2 1; Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 9 / 26

10 Intermediate Language (IL) Syntax Syntax We assume expressions e of integer type with tuples. Program equations f 0 x 0 = s 0 f 1 x 1 = s 1 shortly written as. Statement language s, t ::= x = e; s assignment if e s t conditional goto l x goto return x return f x = s Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 10 / 26

11 Example Intermediate Language (IL) Example Program in SSA form 1 // n0 i n p u t 2 e n t r y : i 0 = 1 ; 3 l o o p : ( n2, i 2 ) = 4 p h i ( e n t r y : ( n0, i 0 ), 5 body : ( n1, i 1 ) ) ; 6 i f n2 = 0 then 7 r e t u r n i 2 8 e l s e 9 i 1 = n2 i 2 ; 10 n1 = n2 1 ; 11 body : goto l o o p Program equations in IL 1 // n0 i n p u t 2 e n t r y n0 = 3 i 0 = 1 ; 4 goto l o o p ( n0, i 0 ) 5 6 l o o p ( n2, i 2 ) = 7 i f n2 = 0 then 8 r e t u r n i 2 9 e l s e 10 i 1 = n2 i 2 ; 11 n1 = n2 1 ; 12 goto l o o p ( n1, i 1 ) Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 11 / 26

12 Semantics for the IL 1 Intermediate Language (IL) Semantics The semantics is defined on state tuples Π σ, s where Π σ s program equations f x = s environment current statement Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 12 / 26

13 Intermediate Language (IL) Semantics Semantics for the IL 2 Assign Π σ[x := [[e]] σ], s v Π σ, x = e; s v Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 13 / 26

14 Semantics for the IL 2 Intermediate Language (IL) Semantics Assign Π σ[x := [[e]] σ], s v Π σ, x = e; s v Cond [[e]] σ = b Π σ, s b v Π σ, if e s 0 s 1 v Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 13 / 26

15 Semantics for the IL 2 Intermediate Language (IL) Semantics Assign Π σ[x := [[e]] σ], s v Π σ, x = e; s v Cond [[e]] σ = b Π σ, s b v Π σ, if e s 0 s 1 v Return σ x = v Π σ, return x v Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 13 / 26

16 Semantics for the IL 2 Intermediate Language (IL) Semantics Assign Π σ[x := [[e]] σ], s v Π σ, x = e; s v Cond [[e]] σ = b Π σ, s b v Π σ, if e s 0 s 1 v Return σ x = v Π σ, return x v Goto f x = s [x i := σ y], s i v f x = s σ, goto f i y v Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 13 / 26

17 Equivalence Contextual Equivalence (Morris [5]) Two programs are contextually equivalent if, whenever they are each inserted into a hole in a larger program of integer type, the resulting programs either both converge or both diverge. [3] C ::= [] x = e; C if 1 e C s if 2 e s C C v, Π, C[s] v, C[s ] v Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 14 / 26

18 Equivalence Statement Equivalence Π s s σ v, Π σ, s v σ, s v Congruence Π s s Π C[s] C[s ] Completeness C v, Π, C[s] v, C[s ] v Π s s Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 15 / 26

19 Program Equivalence Equivalence Π f g Π goto f x goto g x Extend Π s s g fresh Π, g x = t s s Extract σ, Π g σ, goto g x v Π σ, goto g x v where Π g denotes the set equation names possibly reachable from g. Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 16 / 26

20 Equivalence Example f x = y = x; return y is equivalent to f x = return x Extend Π s s g fresh Π, g x = t s s Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 17 / 26

21 Equivalence Example f x = y = x; return y is equivalent to f x = return x Extend Π s s g fresh Π, g x = t s s Π := f x = y = x; return y g x = return x Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 17 / 26

22 Equivalence Example f x = y = x; return y is equivalent to f x = return x Extend Π s s g fresh Π, g x = t s s Π := f x = y = x; return y g x = return x f x = s s i s j f x = s goto f i x goto f j x Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 17 / 26

23 Equivalence Example f x = y = x; return y is equivalent to f x = return x Extend Π s s g fresh Π, g x = t s s Π := f x = y = x; return y g x = return x f x = s s i s j f x = s goto f i x goto f j x y = x; return y return x Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 17 / 26

24 Equivalence Example f x = y = x; return y is equivalent to f x = return x Extend Π s s g fresh Π, g x = t s s Π := f x = y = x; return y g x = return x f x = s s i s j f x = s goto f i x goto f j x y = x; return y return x y = x; return y return x Assumption Π y = x; return y return x Extend Π goto f x goto g x Lemma Π f g Def. Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 17 / 26

25 Transformation SSA Construction How to translate from an imperative language to IL? Appel/Kelsey embedding [4] Any SSA program can be syntactically translated into an equivalent continuation passing style program Via Appel/Kelsey embedding, if in SSA form How can we investigate the SSA construction in our setting? Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 18 / 26

26 Imperative Semantics Transformation Imperative Semantics Goto f x = s σ[x i := σ y], s i v f x = s σ, goto f i y v Subset Π σ, s v Π σ, s v Counter-example for Superset property: f x = return y [y := 1], goto f y 1 f x = return y [y := 1], goto f y 1 Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 19 / 26

27 Transformation Imperative Semantics Imperative and Functional Semantics Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 20 / 26

28 Transformation Imperative Semantics Imperative and Functional Semantics s v h t v Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 20 / 26

29 Transformation Imperative Semantics Imperative and Functional Semantics s v h t v essentially h is an SSA construction algorithm Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 20 / 26

30 Conclusion Conclusion IMP s v h t v o t v M h : SSA construction algorithm o : optimizing program transformation Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 21 / 26

31 Conclusion Thank you! Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 22 / 26

32 References [1] Bowen Alpern, Mark N. Wegman, and F. Kenneth Zadeck. Detecting Equality of Variables in Programs. In: POPL. 1988, pp [2] Ron Cytron et al. An Efficient Method of Computing Static Single Assignment Form. In: POPL. 1989, pp [3] Andrew D. Gordon. A Tutorial on Co-induction and Functional Programming. In: Glasgow Functional Programming Workshop. Springer, 1994, pp [4] Richard A. Kelsey. A correspondence between continuation passing style and static single assignment form. In: SIGPLAN Not. 30 (3 Mar. 1995), pp issn: [5] J.H. Morris. Lambda-calculus models of programming languages. PhD Thesis. MIT, [6] Barry K. Rosen, Mark N. Wegman, and F. Kenneth Zadeck. Global Value Numbers and Redundant Computations. In: POPL. 1988, pp Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 23 / 26

33 Example Appendix Program in SSA form Program 1 // n i n p u t 2 e n t r y : i = 1 ; 3 l o o p : i f n = 0 then 4 r e t u r n i 5 e l s e 6 i = n i ; 7 n = n 1 ; 8 goto l o o p 1 // n i n p u t 2 e n t r y = 3 i = 1 ; 4 goto l o o p ( ) 5 6 l o o p = 7 i f n = 0 then 8 r e t u r n i 9 e l s e 10 i = n i ; 11 n = n 1 ; 12 goto l o o p ( ) Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 24 / 26

34 Appendix Conjecture Conjecture η, f x = s σ, t Goto η, f x = s η[x n := σ y], s n v η, f x = s σ, goto f n y v Goto f x = s σ[x i := σ y], s i v f x = s σ, goto f i y v Conjecture For all σ, s, Π in SSA form, σ, Π σ, s v σ, Π σ, s v Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 25 / 26

35 Appendix Conjecture Example Goal: f x = y = x; return y is equivalent to f x = return x. Extract σ, Π g σ, goto g x v Π σ, goto g x v Π := f x = y = x; return y g x = return x By Extract and Def., we get for all σ, g x = return x σ, goto g x v Π σ, goto g x v Π σ, goto f x v f x = y = x; return y σ, goto f x v Extract Π f g Extract Sigurd Schneider (UdS) Semantics of an IL for PT 11/11/11 26 / 26