Tabled higher-order logic programming Brigitte Pientka Department of Computer Science Carnegie Mellon University Pittsburgh, PA, 15213-3891, USA Thesis Committee: Frank Pfenning (Chair) Robert Harper Dana Scott David Warren, University of New York at Stony Brook Tabled higher-order logic programming – p.1/47
� � � Outline Logical frameworks and certified code Tabled higher-order logic programming - Basic idea and challenges - Experiments and Evaluation - Improving efficiency Conclusion and future work Tabled higher-order logic programming – p.2/47
� � Deductive systems and logical frameworks Deductive systems are plentiful computer science. Axioms and inference rules Examples: operational semantics, type system, logic, etc. Tabled higher-order logic programming – p.3/47
� � � � � Deductive systems and logical frameworks Deductive systems are plentiful computer science. Axioms and inference rules Examples: operational semantics, type system, logic, etc. Logical framework: meta-language for deductive systems High-level specifications (e.g. type system) Execution via logic programming interpretation (e.g. type checker) Meta-reasoning via theorem prover combining induction and logic programming search (e.g. type preservation) Tabled higher-order logic programming – p.3/47
✆ ✟ ✆ ✟ ✆ ✁ ✠ ✠✡ ✁ ✁ ✝ ✠ ✁ ✠ ✁ ✡ ☛ � ✞ Declarative description of subtyping types :: = zero pos nat bit �✄✂ �✄☎ Example: and nat Tabled higher-order logic programming – p.4/47
✞ ✟ ✆ ✆ ✆ ✁ ✠ ✠✡ ✁ ✁ ✝ ✠ ✁ ✠ ✁ ✡ ☛ � ✟ Declarative description of subtyping types :: = zero pos nat bit �✄✂ �✄☎ Example: and nat pn zn nb zero nat pos nat nat bit Tabled higher-order logic programming – p.4/47
✠ ✆ ☛ ✡ � ✠ ✟ ✂ ✠✡ ✠ ✟ ✞ ✝ � ✆ ✆ � ✁ � � ✁ ☎ ✁ � ✁ ✂ ✁ � ☎ � � Declarative description of subtyping types :: = zero pos nat bit �✄✂ �✄☎ Example: and nat pn zn nb zero nat pos nat nat bit tr refl Tabled higher-order logic programming – p.4/47
✆ ☎ � ✆ � � � ✆ ✂ ✄ � � � ✂ ✝ ✁ ☎ � ✂ ✄ � ✁ ☎ ✄ ✄ ✁ � ☎ ✟ ✁ � � ✁ � ✂ � ✂ ✁ ✆ � ✁ ☎ � ✂ � ☎ ✂ Typing rules for Mini-ML expressions ::= 0 1 fun app tp-sub ☎ �✄☎ tp-fun fun Tabled higher-order logic programming – p.5/47
Implementation of subtyping zn: sub zero nat. pn: sub pos nat. nb: sub nat bit. refl: sub T T. tr: sub T S <- sub T R <- sub R S. Tabled higher-order logic programming – p.6/47
Implementation of subtyping zn: sub zero nat. ?- sub zero bit. pn: sub pos nat. nb: sub nat bit. refl: sub T T. tr: sub T S <- sub T R <- sub R S. Tabled higher-order logic programming – p.6/47
Implementation of subtyping zn: sub zero nat. ?- sub zero bit. pn: sub pos nat. nb: sub nat bit. yes refl: sub T T. tr: sub T S Proof: (tr nb zn) <- sub T R <- sub R S. Tabled higher-order logic programming – p.6/47
� Implementation of typing rules tp sub: of E T <- of E T’ <- sub T’ T. tp fun: of (fun x.E x) (T1 => T2) <-( x:exp.of x T1 -> of (E x) T2). “forall x:exp , assume of x T1 and show of (E x) T2 ” Tabled higher-order logic programming – p.7/47
� � � � Higher-order logic programming Higher-order data-types: -abstraction – – dependent types Dynamic program clauses Explicit proof objects Tabled higher-order logic programming – p.8/47
� � � � � Higher-order logic programming Higher-order data-types: -abstraction – – dependent types Dynamic program clauses Explicit proof objects Different approaches: Prolog, Isabelle, Twelf Tabled higher-order logic programming – p.8/47
� � � � Application: certified code Code Producer Code Consumer Program Safety policy Safety policy Certificate Generate Certificate Check Certificate proof Foundational proof-carrying code : [Appel, Felty 00] Temporal-logic proof carrying code [Bernard,Lee02] Foundational typed assembly language : [Crary 03] Proof-carrying authentication: [Felten, Appel 99] Tabled higher-order logic programming – p.9/47
� � � Application: certified code Code Producer Code Consumer Program Safety policy Safety policy Certificate Generate Certificate Check Certificate proof Large-scale applications Typical code size: 70,000 – 100,000 lines includes data-type definitions and proofs Higher-order logic program: 5,000 lines Over 600 – 700 clauses Tabled higher-order logic programming – p.9/47
� � � Some limitations in practice Straightforward specifications are not executable. Redundancy severely hampers performance. Meta-reasoning capabilities limited in practice. Overcome some of these limitations using tabelling and other optimizations! Tabled higher-order logic programming – p.10/47
� � This thesis Tabled higher-order logic programming allows us to efficiently execute logical systems (interpreter using tabled search) automate the reasoning with and about them. (meta-theorem prover using tabled search) This is a significant step towards applying logical frameworks in practice. Tabled higher-order logic programming – p.11/47
� � � � � � Contributions Tabled higher-order logic programming Characterization based on uniform proofs (ICLP’02) Implementation of a tabled interpreter Case studies (parsing, refinement types, rewriting)(LFM’02) Efficient data-structures and algorithms Foundation for meta-variables (LFM’03) Optimizing higher-order unification (CADE’03) Higher-order term indexing (ICLP’03) Meta-reasoning based on tabled search Tabled higher-order logic programming – p.12/47
� � � Outline Logical frameworks and certified code Tabled higher-order logic programming - Basic idea and challenges - Experiments and Evaluation - Improving efficiency Conclusion and future work Tabled higher-order logic programming – p.13/47
� � � Outline Logical frameworks and certified code Tabled higher-order logic programming - Basic idea and challenges - Experiments and Evaluation - Improving efficiency Conclusion and future work Tabled higher-order logic programming – p.13/47
The idea “...it is very common for the proofs to have repeated sub-proofs that should be hoisted out and proved only once ...” [Necula,Lee97] Tabled higher-order logic programming – p.14/47
The idea “...it is very common for the proofs to have repeated sub-proofs that should be hoisted out and proved only once ...” [Necula,Lee97] Redundant computation Tabled higher-order logic programming – p.14/47
The idea “...it is very common for the proofs to have repeated sub-proofs that should be hoisted out and proved only once ...” [Necula,Lee97] Redundant computation Tabled higher-order logic programming – p.14/47
The idea “...it is very common for the proofs to have repeated sub-proofs that should be hoisted out and proved only once ...” [Necula,Lee97] Redundant computation Infinite computation Tabled higher-order logic programming – p.14/47
� Recall...subtyping tp sub: of E T <- of E T’ <- sub T’ T. tp fun: of (fun x.E x) (T1 => T2) <-( x:exp.of x T1 -> of (E x) T2). “forall x:exp , assume of x T1 and show of (E x) T2 ” Tabled higher-order logic programming – p.15/47
� ✁ � Proof tree λ of (fun x. x) T tp_sub tp_fun λ of (fun x. x) T u: of x T 2 of x T 3 1 sub T 1 T tp_sub u T 2 = S u: of x T 2 of x ( T 4 x u) T 3 = S sub ( T 4 x u) T 3 T = S S Tabled higher-order logic programming – p.16/47
� � ✁ Proof tree of (fun λ x. x) T tp_sub tp_fun λ x. x) of (fun T u: of x T 2 of x T 3 1 sub T 1 T tp_sub u T 2 = S u: of x T 2 of x ( T 4 x u) T 3 = S sub ( T 4 x u) T 3 T = S S Loop detection Tabled higher-order logic programming – p.16/47
✁ � � Proof tree of (fun λ x. x) T tp_sub tp_fun λ x. x) of (fun T u: of x T 2 of x T 3 1 sub T 1 T tp_sub u T 2 = S u: of x T 2 of x ( T 4 x u) T 3 = S sub ( T 4 x u) T 3 T = S S Loop detection How can we detect loops? Tabled higher-order logic programming – p.16/47
� ☎ � Loops modulo strengthening Dependencies among terms u:of x T of x (T x u) Tabled higher-order logic programming – p.17/47
� ☎ � ☎ � Loops modulo strengthening Dependencies among terms u:of x T of x (T x u) strengthen u:of x T of x T Tabled higher-order logic programming – p.17/47
☎ ☎ � ☎ � � � � � Loops modulo strengthening Dependencies among terms u:of x T of x (T x u) strengthen u:of x T of x T Dependencies among propositions u:of x T sub (T x u) T Tabled higher-order logic programming – p.17/47
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