A Logic Your Typechecker Can Count On: Unordered Tree Types in Practice

Transcription

1 A Logic Your Typechecker Can Count On: Unordered Tree Types in Practice Nate Foster (Penn) Benjamin C. Pierce (Penn) Alan Schmitt (INRIA Rhône-Alpes) µx. {} (hd[t ]+tl[x ]) PLAN-X 07 2 φ(x 0,.., x 4 ), 3 hd[t ], hd[ T ], 4 tl[x ], tl[ X ], 5 {hd, tl}[true]

2 µx. {} (hd[t ]+tl[x ]) φ(x 0,.., x 4 ), hd[t ], hd[ T ], tl[x ], tl[ X ], {hd, tl}[true]

3 Types in A O Sync B Harmony A B T A generic synchronization framework Architecture takes two replicas + original updated replicas. Data model is deterministic trees: unordered, edge-labeled trees.

4 Types in A O Sync B A B T Harmony: Typed Synchronization [DBPL 05] Behavior of synchronizer guided by type. If inputs well-typed, so are outputs. Required operations: membership of trees in type [also sets of names].

5 Types in A O Sync B A B T Harmony: Lenses [POPL 05] Pre-/post-process replicas using bi-directional programs. Facilitates heterogeneous synchronization. Types in conditionals, run-time asserts, static checkers. Required operations: membership, inclusion, equivalence, emptiness, [projection, injection, etc.].

6 Deterministic Tree Types Syntax T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ]

7 Deterministic Tree Types Syntax Semantics T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Singleton denoting the unique tree with no children: {}

8 Deterministic Tree Types Syntax Semantics T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Atoms: trees with single child n and subtree in T : If T, then n n[t ] t t

9 Deterministic Tree Types Syntax Semantics T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Commutative concatenation operator: If t T and t T, then t t T +T

10 Deterministic Tree Types Syntax Semantics T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Boolean operations and recursion: X 1 =. T 1 X n = T n

11 Deterministic Tree Types Syntax Semantics T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] If m {n 1,.., n k } and t T, then t m!\{n 1,.., n k }[T ]

12 Deterministic Tree Types Syntax Semantics T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] If m 1,.., m k {n 1,.., n k } and t 1.. t k T, then m m k *\{n 1,.., n k }[T ] t 1 t k

13 Deterministic Tree Types Syntax T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Example: hd[true]+tl[true] hd tl

14 Deterministic Tree Types Syntax T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Example: {} (hd[true]+tl[true]) hd tl or

15 Deterministic Tree Types Syntax T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Example: X = {} (hd[true]+tl[x ]) hd tl or hd tl hd tl

16 Deterministic Tree Types Syntax T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Example:![True]+![True]

17 Deterministic Tree Types Syntax T ::= {} n[t ] T +T T T ~T X!\{n 1,.., n k }[T ] *\{n 1,.., n k }[T ] Example: ~(![True]+![True]) or or or... Can eliminate negations, and use direct algorithms, but types get large...

18 Sheaves Formulas Formulas S = φ(x 0,.., x k ), [r 0 [S 0 ],.., r k [S k ]] where φ is a Presburger formula and r i a set of names. [Dal Zilio, Lugiez, Meyssonnier, POPL 04]

19 Sheaves Formulas Formulas S = φ(x 0,.., x k ), [r 0 [S 0 ],.., r k [S k ]] where φ is a Presburger formula and r i a set of names. φ(x 0, x 1 ), [b[true], {a, c}[true]] 0 0 a b c

20 Sheaves Formulas Formulas S = φ(x 0,.., x k ), [r 0 [S 0 ],.., r k [S k ]] where φ is a Presburger formula and r i a set of names. φ(x 0, x 1 ), [b[true], {a, c}[true]] 0 1 a b c

21 Sheaves Formulas Formulas S = φ(x 0,.., x k ), [r 0 [S 0 ],.., r k [S k ]] where φ is a Presburger formula and r i a set of names. φ(x 0, x 1 ), [b[true], {a, c}[true]] 1 1 a b c

22 Sheaves Formulas Formulas S = φ(x 0,.., x k ), [r 0 [S 0 ],.., r k [S k ]] where φ is a Presburger formula and r i a set of names. φ(x 0, x 1 ), [b[true], {a, c}[true]] 1 2 a b c

23 Sheaves Formulas Formulas S = φ(x 0,.., x k ), [r 0 [S 0 ],.., r k [S k ]] where φ is a Presburger formula and r i a set of names. φ(x 0, x 1 ), [b[true], {a, c}[true]] 1 2? = φ(1, 2)

24 Sheaves Formulas Formulas S = φ(x 0,.., x k ), [r 0 [S 0 ],.., r k [S k ]] where φ is a Presburger formula and r i a set of names. φ(x [ 0, x 1, x 2 ), ] b[true], {a, c}[true], {a, b, c}[true] For coherence: r i [S i ] must partition set of atoms. Note: does not ensure determinism.

25 Examples as Sheaves Formulas X = ({} hd[true]+tl[x ]) X = (x 0 =x 1 =x 2 =x 3 =0) [ (x 0 =x 1 =1 x 2 =x 3 =0), ] hd[true], tl[x ], tl[ X ], {hd, tl}[true]

26 Examples as Sheaves Formulas X = ({} hd[true]+tl[x ]) X = (x 0 =x 1 =x 2 =x 3 =0) [ (x 0 =x 1 =1 x 2 =x 3 =0), ] hd[true], tl[x ], tl[ X ], {hd, tl}[true] ~(![True]+![True]) x [ 0 2, ] {}[True]

27 Challenges and Strategies Blowup in naive compilation from types to formulas. Syntactic optimizations avoid blowup in common cases. Backtracking in top-down, non-deterministic traversal. Incremental algorithm avoids useless paths. Presburger arithmetic requires double-exponential time. Compile Presburger formulas to MONA representation. Hash-consing allocation + aggressive memoization.

28 Challenges and Strategies Blowup in naive compilation from types to formulas. Syntactic optimizations avoid blowup in common cases. Backtracking in top-down, non-deterministic traversal. Incremental algorithm avoids useless paths. Presburger arithmetic requires double-exponential time. Compile Presburger formulas to MONA representation. Hash-consing allocation + aggressive memoization. Contributions Strategies and algorithms; Implementation in Harmony; Experimental results.

29 Incremental Algorithm φ(x 0,.., x k ), [r 0 [S 0 ],..r k [S k ]] n 1 n 2.. n k 1 n k..

30 Incremental Algorithm φ(x 0,.., x k ), [r 0 [S 0 ],..r k [S k ]] (φ) n 1 n 2.. n k 1 n k..

31 Incremental Algorithm φ(x 0,.., x k ), [r 0 [S 0 ],..r k [S k ]] (φ ψ dom ) n 1 n 2.. n k 1 n k..

32 Incremental Algorithm φ(x 0,.., x k ), [r 0 [S 0 ],..r k [S k ]] (φ ψ dom ψ 1 ) n 1 n 2.. n k 1 n k..

33 Incremental Algorithm φ(x 0,.., x k ), [r 0 [S 0 ],..r k [S k ]] (φ ψ dom ψ 1 ψ 2 ) n 1 n 2.. n k 1 n k..

34 Incremental Algorithm φ(x 0,.., x k ), [r 0 [S 0 ],..r k [S k ]] (φ ψ dom ψ 1.. ψ k 1 ) n 1 n 2.. n k 1 n k..

35 Incremental Algorithm φ(x 0,.., x k ), [r 0 [S 0 ],..r k [S k ]] (φ ψ dom ψ 1.. ψ k ) n 1 n 2.. n k 1 n k..

36 Hash-Consing and Memoization Thousands of formulas and trees, but many repeats. Suggests hash-consed allocation: Sheaves formulas; Presburger formulas; Trees. Memoization of intermediate results: MONA representations of Presburger formulas; Satisfiability of Presburger formulas; Membership results; Partially-evaluated member functions.

37 Experiments Programs: Structured text parser; Address book validator; icalendar lens. Experimental setup: structures populated with snippets of Joyce s Ulysses; 1.4GHz Intel Pentium III, 2GB RAM, SuSE Linux OS kernel ; execution times collected from POSIX functions.

38 Experiments: Address Book Validator 150 base base-memo Time(seconds) Input Size (# lines) States Formulas Sat Trees % % %

39 Experiments: Address Book Validator 150 base base-memo incr-all-off incr Time(seconds) Input Size (# lines) States Formulas Sat Trees % % %

40 Experiments: Address Book Validator 150 base base-memo incr-all-off incr-phi-off incr-member-off incr Time(seconds) Input Size (# lines) States Formulas Sat Trees % % %

41 Experiments: Structured Text Parser base base-memo incr-all-off incr-phi-off incr-member-off incr Time(seconds) Input Size (# lines) States Formulas Sat Trees % % %

42 Experiments: icalendar Lens Time(seconds) base base-memo incr-all-off incr-phi-off incr-member-off incr Input Size (# lines) States Formulas Sat Trees % % %

43 Related Work Types and Automata: TQL [Cardelli and Ghelli, ESOP 01] A Logic You Can Count On [Dal Zilio, Lugiez, Meyssonnier, POPL 04] Counting In Trees For Free [Seidl, Schwentick, Muscholl, Habermehl, ICALP 04] Survey and Foundations: [Boneva and Talbot, RTA 05, LICS 05] Implementations: Static Checkers for Tree Structrures and Heaps [Hague 04] Boolean Operations and Inclusion Test for Attribute Element Constraints [Hosoya and Murata, ICALP 03]

44 Conclusions and Future Work Summary Strategies and algorithms; Implemented in Harmony; Reasonable performance. Tune algorithm, hash-consing, memoization parameters. Determinize sheaves formulas. Implement Presburger arithmetic directly, optimized for adding constraints incrementally; also restricted fragments. Extend to new structures and types: multitrees, ordered trees, also horizontal recursion, adjoint operators, etc.

45 Acknowledgements Haruo Hosoya, Christian Kirkegaard, Stéphane Lescuyer, Thang Nguyen, Val Tannen, Penn PLClub and DB Group. harmony/