Games and Full Abstraction for Java programs

Transcription

1 Games and Full Abstraction for Java programs Andrzej Murawski University of Warwick Nikos Tzevelekos Queen Mary Uni. of London PLSeminar 2013

2 What this talk is about Semantics of programs: Operational (abstract machines) Denotational (abstract domains) and the notion of Full Abstraction Full abstraction: storyline, sequentiality problems, solution with Game Semantics A fully abstract game semantics for Java programs (Interface Middleweight Java)

3 Operational Semantics Interpret programs in abstract machines: (s, x = 1; P) (s[x 1], P) concrete step-wise description 'intuitive' semantics

4 Operational Equivalence Closed programs: P P' if (s 0, P) * (s, V) (s 0, P') * (s', V) Open programs: P P' if (s 0, C[P]) * (s, V) (s 0, C[P']) * (s', V)

5 Denotational Semantics Translate programs into a domain of 'functions': program: P L denotation: P M abstract mathematical description 'high-level' semantics

6 Power of denotations Abstract away from implementation details Compositional translation: divide-and-conquer modular approach modelling of components in isolation Useful for understanding & analysing programs

7 Full Abstraction Desired properties of the translation: Correctness: P P' P = P' Soundness: P P' P = P' Full abstraction: P P' P = P'

8 Full abstraction pictorially program: P L. M denotation: P

9 The quest for full abstraction 1977 [Milner, Plotkin]: Formulation of the problem (λ-calculus, PCF) Functions cannot capture sequentiality 's: Function stability [Berry, Bucciarelli, Erhard] Sequential algorithms [Berry, Currien] 1993 [AJM, HO/N *]: Game semantics (PCF) 'Functions' with operational content (games) * Abramsky, Jagadeesan, Malacaria; Hyland, Ong; Nickau

10 From PCF to realistic languages Full Abstraction for PCF (early 90's) Two groups in the UK, one in Germany Roots in Mathematical Logic First stage ( ) Models for various programming features Program analysis resources? Nominal game semantics (2004-) Fragments of ML, now Java (IMJ)

11 Interface Middleweight Java (IMJ) Object calculus based on MJ [Bierman, Parkinson, Pitts] Objects, inheritance, subtyping, casting, interfaces

12 Game Semantics Computation is modelled as a 2-player game between: Opponent (the environment), aka O Proponent (the program), aka P Qualitative games ( Game Theory) Computations = plays of a specified game Programs = strategies for P Families (i.e. categories) of games

13 Examples x : Var x.val + 1 : int Var : { val : int } Fun : { val : int int }

14 Examples x : Var x.val + 1 : int Var : { val : int } Fun : { val : int int } O : (x.val = 5) x

15 Examples x : Var x.val + 1 : int Var : { val : int } Fun : { val : int int } O : (x.val = 5) x P : (x.val = 5) 6

16 Examples x : Var x.val + 1 : int Var : { val : int } Fun : { val : int int } O : x (x.val = 5) P : 6 (x.val = 5) O : x (x.val = 8)

17 Examples x : Var x.val + 1 : int Var : { val : int } Fun : { val : int int } O : (x.val = 5) x O : x (x.val = 8) P : 6 (x.val = 5) P : (x.val = 8) 9

18 Examples x : Var x.val + 1 : int Var : { val : int } Fun : { val : int int } O : (x.val = 5) x O : x (x.val = 8) O : x (x.val = 3)... P : 6 (x.val = 5) P : 9 (x.val = 8) P : 4 (x.val = 3)...

19 Examples x : Var x.val + 1 : int Var : { val : int } Fun : { val : int int } O : (x.val = 5) x O : x (x.val = 8) O : x (x.val = 3)... P : 6 (x.val = 5) P : 9 (x.val = 8) P : 4 (x.val = 3)... x : Var x.val + 1 : int = { x (x.val = i) (i +1) (x.val = i) }

20 Examples x : Var, f : Fun Var : { val : int } f. val (x.val) + 1 : int Fun : { val : int int }

21 Examples x : Var, f : Fun Var : { val : int } f. val (x.val) + 1 : int Fun : { val : int int } O : (x.val = 5) (x,f) P : (x.val = 5) call f.val(5) O : (x.val = 42) ret f.val(8) P : (x.val = 42) 9

22 Examples x : Var, f : Fun Var : { val : int } f. val (x.val) + 1 : int Fun : { val : int int } O : (x.val = 5) (x,f) P : (x.val = 5) call f.val(5) O : (x.val = 42) ret f.val(8) P : (x.val = 42) 9 O : (x.val = 7) (x,f) P : (x.val = 7) call f.val(7) O : (x.val = 7) ret f.val(13) P : (x.val = 7) 14

23 Examples x : Var, f : Fun Var : { val : int } f. val (x.val) + 1 : int Fun : { val : int int } O : (x.val = 5) (x,f) P : (x.val = 5) call f.val(5) O : (x.val = 42) ret f.val(8) P : (x.val = 42) 9 O : (x.val = 7) (x,f) P : (x.val = 7) call f.val(7) O : (x.val = 7) ret f.val(13) P : (x.val = 7) 14 x : Var, f : Fun f. val (x.val) + 1 : int = = { (x,f) (x.val = i ) call f.val(i) (x.val = i ) ret f.val(j) (x.val = i' ) ( j +1) (x.val = i' ) }

24 Games in detail x 1 :I 1,, x n :I n M : I M : I 1,, I n I

25 Games in detail free variables program x 1 :I 1,, x n :I n M : I output interface input interfaces M : I 1,, I n I

26 Games in detail free variables program x 1 :I 1,, x n :I n M : I output interface input interfaces M : I 1,, I n I strategy arenas

27 Arenas, moves M : I 1,, I n I strategy arenas Arenas : sets of names with assigned interfaces Moves : value moves are tuples of names, integers, etc.) method moves are calls and returns (of methods, using values) 42, (5,*,x,f), call f.val(42), ret p.set(5,4),

28 Plays, strategies M : I 1,, I n I strategy arenas Plays : sequences of moves-with-store call f.val(5) (x.val=5), Strategies : sets of plays A series of sanity conditions is used, e.g. - moves have polarities (O/P), which alternate - P calls methods of O, and viceversa - dually for returns - calls and returns adhere to interfaces of names - strategies are closed wrt to O-subtyping -

29 Composition Compound programs translated compositionally Strategy composition: play one strategy against the other M M 2 1 I 1 M 1 I 2 M 2 I 3

30 Full abstraction for IMJ Theorem. The game translation is correct & sound Lemma. Every finitary strategy is IMJ-definable Theorem. The game model is fully abstract

31 Further on Program analysis for IMJ Algorithmic representations Automata over infinite alphabets! Model checking Further effects Exceptions (cf. FoSSaCS'14) Concurrency (multi-threading, cf. Laird'06)

32 Further on Program analysis for IMJ Algorithmic representations Automata over infinite alphabets! Model checking thanks! Further effects Exceptions (cf. FoSSaCS'14) Concurrency (multi-threading, cf. Laird'06)