Functional Reachability

Transcription

1 Functional Reachability Luke Ong Nikos Tzevelekos Oxford University Computing Laboratory 24th Symposium on Logic in Computer Science Los Angeles, August Ong & Tzevelekos Functional Reachability 1

2 The Problem Reachability in functional computation. Consider a term M of a higher-order functional programming language. Now consider a point p inside M. Is there a program context C such that the computation of C[M] reaches p? The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 2

3 The Problem Reachability in functional computation. Consider a term M of a higher-order functional programming language. Now consider a point p inside M. Is there a program context C such that the computation of C[M] reaches p? The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 2

4 The Problem Reachability in functional computation. Consider a term M of a higher-order functional programming language. Now consider a point p inside M. Is there a program context C such that the computation of C[M] reaches p? Surprisingly, (Contextual) Reachability per se had not been studied in HO functional languages. The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 2

5 Relevant work Control Flow Analysis: Approximate at compile time the flow of control to happen at run time. In a HO-setting, the crucial element is that of closures. Reynolds ( 70), Jones ( 80), Shivers ( 90),... Malacaria & Hankin (late 90 s). CFA > Reach: more general. Reach > CFA: open vs closed world approach. Useless code detection, etc. The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 3

6 The examined language: PCF Types: Terms: A,B ::= o A B M,N ::= x x.m MN t f if M N 1 N 2 Y A Contexts: C ::=... Ong & Tzevelekos Functional Reachability 4

7 The examined language: PCF Types: Terms: A,B ::= o A B M,N ::= x x.m MN t f if M N 1 N 2 Y A Contexts: C ::=... Reductions: (x.m)n M{N/x} if t xy.x YM M(YM) if f xy.y M N = E[M] E[N] Ev. Contexts: E ::= [ ] E M if E Ong & Tzevelekos Functional Reachability 4

8 The examined language: PCF Types: Terms: A,B ::= o A B M,N ::= x x.m MN t f if M N 1 N 2 Y A Contexts: C ::=... Reductions: Call-by-name -calculus + if + Y Write (A 1,...,A n,o) for A 1 A n o. Divergence definable, e.g. := Y o (x.x). Finitary restrictions (i.e. no Y): fpcf, fpcf. Ong & Tzevelekos Functional Reachability 4

9 Reachability Given a PCF-term M and a coloured subterm L of M, Is there a program context C such that C[M] E[L ] with L coloured? The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 5

10 Reachability Given a PCF-term M and a coloured subterm L of M, Is there a program context C such that C[M] E[L ] with L coloured? Equivalently: Given a closed PCF-term M : (A 1,...,A n,o) and a coloured subterm L of M, Are there closed PCF-terms N 1,...,N n such that M N E[L ] The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on with L coloured? Ong & Tzevelekos Functional Reachability 5

11 PCF-with-error: PCF Take base type o = {t, f, } with an error constant: -Reachability: E[ ] Given a closed PCF -term M : (A 1,...,A n,o) that has exactly one occurrence of, are there closed PCF-terms N 1,...,N n such that M N? The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 6

12 PCF-with-error: PCF Take base type o = {t, f, } with an error constant: -Reachability: E[ ] Given a closed PCF -term M : (A 1,...,A n,o) that has exactly one occurrence of, are there closed PCF-terms N 1,...,N n such that M N? Lemma: Reachability = -Reachability. The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 6

13 Reach template For v {t, f, } and L 1, L 2 PCF : v-reach [L 1, L 2 ]: Given a closed L 1 -term M : (A 1,...,A n,o), are there closed L 2 -terms N 1,...,N n such that M N v? e.g. -Reachability = -Reach [PCF 1, PCF]. Ong & Tzevelekos Functional Reachability 7

14 Reach template For v {t, f, } and L 1, L 2 PCF : v-reach [L 1, L 2 ]: Given a closed L 1 -term M : (A 1,...,A n,o), are there closed L 2 -terms N 1,...,N n such that M N v? Three classes of problems: Reachability -Reachability -Reach [PCF 1, PCF] -Reach [PCF 1, fpcf] -Reach [fpcf 1, fpcf] -Reach [fpcf, fpcf]... -Reach [fpcf 1, fpcf] -Reach [fpcf, fpcf]... Ong & Tzevelekos Functional Reachability 7

15 An undecidability result Lemma: -Reach [fpcf 1, fpcf] is undecidable. Proof: By reduction of solvability of systems of fpcf -equations (proved undecidable by [Loader 01]). Reachability -Reachability -Reach [PCF 1, PCF] -Reach [PCF 1, fpcf] -Reach [fpcf 1, fpcf] -Reach [fpcf, fpcf]... -Reach [fpcf 1, fpcf] -Reach [fpcf, fpcf]... Ong & Tzevelekos Functional Reachability 8

16 Our approach We focus on v-reach [fpcf, fpcf]. For fpcf -term P : o, Computations of P Traversals over its computation tree, (P) Runs of an Alternating Tree Automaton (ATA) on (P) Ong & Tzevelekos Functional Reachability 9

17 Our approach We focus on v-reach [fpcf, fpcf]. For fpcf -term P : o, Computations of P Traversals over its computation tree, (P) Runs of an Alternating Tree Automaton (ATA) on (P) P v iff an ATA accepts (P) on initial state with value v. Ong & Tzevelekos Functional Reachability 9

18 Computation trees Starting from a fpcf -term M, take its η-long form, add application symbols (@), view the result as a tree, (M). Ong & Tzevelekos Functional Reachability 10

19 Computation trees Starting from a fpcf -term M, take its η-long form, add application symbols (@), view the result as a tree, (M). (Φz. Φ(y. ify z)t)(ϕx.ϕx) t Φz ϕx ϕ y t if t x y z Ong & Tzevelekos Functional Reachability 10

20 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Semantics. Ong & Tzevelekos Functional Reachability 11

21 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

22 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

23 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

24 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

25 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

26 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

27 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

28 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

29 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

30 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

31 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

32 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

33 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

34 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

35 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

36 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

37 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 11

38 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Φz ϕx Φ ϕ t y if t x y z -complete traversal Ong & Tzevelekos Functional Reachability 11

39 Traversals A traversal [Blum, Ong] over a computation tree, follows the flow of control within it, seen from the perspective of Game Semantics. A traversal is v-complete if every question (red visit) has been answered (green visit), and the root question has been answered with v. Theorem: For any P : o and value v, P v iff there is a complete v-traversal over (P). Ong & Tzevelekos Functional Reachability 11

40 Alternating Tree Automata An ATA is a quadruple A = Q, Σ,q 0, where: Q is a finite set of states, Σ is a finite ranked alphabet, q 0 Q is the initial state, is a finite transition relation: q s (Q 1,...,Q k ). s Σ q Q Q 1,..., Q k Q Ong & Tzevelekos Functional Reachability 12

41 Alternating Tree Automata An ATA is a quadruple A = Q, Σ,q 0, where: Q is a finite set of states, Σ is a finite ranked alphabet, q 0 Q is the initial state, is a finite transition relation: q s (Q 1,...,Q k ). s Σ q Q Q 1,..., Q k Q Ong & Tzevelekos Functional Reachability 12 s s 1 s 2... s k

42 Alternating Tree Automata An ATA is a quadruple A = Q, Σ,q 0, where: Q is a finite set of states, Σ is a finite ranked alphabet, q 0 Q is the initial state, is a finite transition relation: q s (Q 1,...,Q k ). s Σ q Q Q 1,..., Q k Q Ong & Tzevelekos Functional Reachability 12 s A(q) s 1 s 2... s k

43 Alternating Tree Automata An ATA is a quadruple A = Q, Σ,q 0, where: Q is a finite set of states, Σ is a finite ranked alphabet, q 0 Q is the initial state, is a finite transition relation: q s (Q 1,...,Q k ). s Σ q Q Q 1,..., Q k Q Ong & Tzevelekos Functional Reachability 12 s A(Q 1 ) A(Q 2 ) A(Q k ) s 1 s 2... s k

44 Traversal-simulating ATA s How can we simulate a complete traversal by an ATA? ϕx Φ ϕ y t if t x y z Ong & Tzevelekos Functional Reachability 13

45 Traversal-simulating ATA s How can we simulate a complete traversal by an ATA? By guessing the number of visits of each node. By guessing the profile of each variable per visit. ϕx By verifying these guesses. Φ ϕ y t if t x y z Ong & Tzevelekos Functional Reachability 13

46 Variable profiles Introduced by [Ong 06]. VP(A 1,...,A n,o) := Var Val P( n i=1 VP(A i)) Notation: (x,v), (x,v π 1,...,π n ) Ong & Tzevelekos Functional Reachability 14

47 Variable Φz ϕx Φ ϕ t y if t x y z Ong & Tzevelekos Functional Reachability 14

48 Variable Φz Φϕx Φ ϕ t ϕ y x y if t x y z Ong & Tzevelekos Functional Reachability 14

49 Variable Φz Φ ϕx Φ ϕ t ϕ y x y t t t if t t x t t y z Ong & Tzevelekos Functional Reachability 14

50 Variable Φz Φϕx Φ ϕ t ϕ y x y if t t x t y z Ong & Tzevelekos Functional Reachability 14

51 Variable profiles Ong & Tzevelekos Functional Reachability Φz Φϕx Φ ϕ t ϕ y x y if t (x, t) x y z (y, t)

52 Variable profiles Ong & Tzevelekos Functional Reachability Φz Φϕx Φ ϕ t (ϕ, (y, t)) ϕ y x y if t (x, t) x y z (y, t)

53 Variable profiles Ong & Tzevelekos Functional Reachability Φz Φϕx (Φ, (ϕ, (y, t)), (x, t)) Φ ϕ t (ϕ, (y, t)) ϕ y x y if t (x, t) x y z (y, t)

54 ATA correspondence Given a finite fpcf -alphabet Σ, the states of the traveral-simulating ATA A Σ are: Q := Val P(VP Σ ) P(VP Σ ) Ong & Tzevelekos Functional Reachability 15

55 ATA correspondence Given a finite fpcf -alphabet Σ, the states of the traveral-simulating ATA A Σ are: Q := Val P(VP Σ ) P(VP Σ ) M N v iff A Σ accepts (M N) on initial state with value v. Any tree accepted by ÃΣ is a closed fpcf-term. Ong & Tzevelekos Functional Reachability 15

56 Results Theorem: M v-reach [fpcf Σ, fpcf Σ ] iff there is an initial state q 0 with value v such that: A Σ (q o ) accepts (M), i, the language accepted by ÃΣ(q 0 A i ) is non-empty. The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 16

57 Results Theorem: M v-reach [fpcf Σ, fpcf Σ ] iff there is an initial state q 0 with value v such that: A Σ (q o ) accepts (M), i, the language accepted by ÃΣ(q 0 A i ) is non-empty. Corollary: -Reach [fpcf, fpcf(n)] is decidable. Corollary: -Reach [fpcf, fpcf] is decidable up to order 3. The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 16

58 Alternating Dependency Tree Automata For the general case we can use Alternating Dependency Tree Automata [Stirling 09]. Corollary: Emptiness problem is undecidable for ADTA s. The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 17

59 Conclusion and on A new kind of Reachability problems. Some undecidability results. Some technology from game semantics. Characterisation by ATA s and ADTA s. Some (relativised) decidability results. The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 18

60 Conclusion and on A new kind of Reachability problems. Some undecidability results. Some technology from game semantics. Characterisation by ATA s and ADTA s. Some (relativised) decidability results. Revisit (semantic) CFA? Reachability through intersection types? Conjecture: -Reach [fpcf, fpcf]? The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on Ong & Tzevelekos Functional Reachability 18

61 Conclusion and on A new kind of Reachability problems. Some undecidability results. Some technology from game semantics. Characterisation by ATA s and ADTA s. Some (relativised) decidability results. Revisit (semantic) CFA? Reachability through intersection types? Conjecture: -Reach [fpcf, fpcf]? The Problem Relevant work The examined language: PCF Reachability PCF-with-error: PCF Reach template An undecidability result Our approach Computation trees Traversals Alternating Tree Automata Traversal-simulating ATA s Variable profiles ATA correspondence Results Alternating Dependency Tree Automata Conclusion and on THANKS! Ong & Tzevelekos Functional Reachability 18