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Intro. Specification Correctness&. . . Programs Final Logic + Control: An example or SAT solver of Howe & King as a logic program (File ./LIPIcs/29.pdf ) W lodzimierz Drabent Institute of Computer Science, Polish Academy of

  1. Intro. Specification Correctness&. . . Programs Final Logic + Control: An example or SAT solver of Howe & King as a logic program (File ./LIPIcs/29.pdf ) W� lodzimierz Drabent Institute of Computer Science, Polish Academy of Sciences Link¨ oping University (Sweden) http://www.ipipan.waw.pl/~drabent ICLP’12, 6th September 2012 Version compiled on September 10, 2012 1 / 25

  2. Intro. Specification Correctness&. . . Programs Final This file contains extra material, not intended to be shown within a short presentation. In particular, such are all the slides with their titles in parentheses. 2 / 25

  3. Intro. Specification Correctness&. . . Programs Final logic Is there in actual Logic Programming ? “logic” To which extent LP is declarative/logical ? 3 / 25

  4. Intro. Specification Correctness&. . . Programs Final Representation How to reason about logic programs? We present a construction of a practical Prolog program (SAT solver of Howe&King). Most of the reasoning done at the declarative level (formally) abstracting from any operational semantics. Plan ◮ Specification ◮ Logic programs 1, 2, 3 ◮ Proving correctness & ◮ Adding control completeness ◮ Conclusions 4 / 25

  5. Intro. Specification Correctness&. . . Programs Final Representation How to reason about logic programs? We present a construction of a practical Prolog program (SAT solver of Howe&King). Most of the reasoning done at the declarative level (formally) abstracting from any operational semantics. Plan ◮ Specification ◮ Logic programs 1, 2, 3 ◮ Proving correctness & ◮ Adding control completeness ◮ Conclusions 4 / 25

  6. Intro. Specification Correctness&. . . Programs Final Representation How to reason about logic programs? We present a construction of a practical Prolog program (SAT solver of Howe&King). Most of the reasoning done at the declarative level (formally) abstracting from any operational semantics. Plan ◮ Specification ◮ Logic programs 1, 2, 3 ◮ Proving correctness & ◮ Adding control completeness ◮ Conclusions 4 / 25

  7. Intro. Specification Correctness&. . . Programs Final Representation Preliminaries Definite programs. To describe relations to be defined by program predicates: Specification – a Herbrand interpretation S . Specified atom – a p ( t 1 , . . . , t n ) ∈ S . 5 / 25

  8. Intro. Specification Correctness&. . . Programs Final Representation Representation of propositional formulae for a SAT solver [Howe&King] Literals x ¬ x as pairs true-X false-X CNF formulae ( . . . ∧ ( . . . ∨ Literal ij ∨ . . . ) ∧ . . . ) as lists of lists [ . . . , [ . . . , Pair ij , . . . ] , . . . ] CNF formula [ f 1 , . . . , f n ] is satisfiable iff it has an instance [ f 1 θ, . . . , f n θ ] where ∀ i f i θ ∈ L 0 1 = { [ t 1 - u 1 , . . . , u - u, . . . , t n - u n ] ∈ H } . CNF formula f is satisfiable iff some fθ is in L 0 2 = { [ f 1 θ, . . . , f n θ ] | as above } . A program defining L 0 2 is a SAT solver. 6 / 25

  9. Intro. Specification Correctness&. . . Programs Final Towards Approximate specifications (Spec. 1) Specifying a SAT solver So apparently a SAT solver should compute L 0 2 . 7 / 25

  10. Intro. Specification Correctness&. . . Programs Final Towards Approximate specifications (Spec. 1) Specifying a SAT solver So apparently a SAT solver should compute L 0 2 . Computing exact L 0 2 unnecessary. E.g. nobody uses append/3 defining the list appending relation exactly! ❘ Common in LP: relations to be computed known approximately. 7 / 25

  11. Intro. Specification Correctness&. . . Programs Final Towards Approximate specifications (Spec. 1) Specifying a SAT solver So may a SAT solver should compute L 0 2 . Computing exact L 0 2 unnecessary. E.g. nobody uses append/3 defining the list appending relation exactly! ❘ Common in LP: relations to be computed known approximately. 7 / 25

  12. Intro. Specification Correctness&. . . Programs Final Towards Approximate specifications (Spec. 1) Specifying a SAT solver So may a SAT solver should compute L 0 2 . Also it may compute a certain L 2 ⊇ L 0 2 . L 2 = { s ∈ H | if s is a list of lists of pairs then s ∈ L 0 2 } . ❘ Common in LP: relations to be computed known approximately. 7 / 25

  13. Intro. Specification Correctness&. . . Programs Final Towards Approximate specifications (Spec. 1) Specifying a SAT solver So may a SAT solver should compute L 0 2 . Also it may compute a certain L 2 ⊇ L 0 2 . L 2 = { s ∈ H | if s is a list of lists of pairs then s ∈ L 0 2 } . Any set L 0 2 ⊆ L ′ 2 ⊆ L 2 will do: a CNF formula f is satisfiable iff some fθ is in L ′ 2 . ❘ Common in LP: relations to be computed known approximately. 7 / 25

  14. Intro. Specification Correctness&. . . Programs Final Towards Approximate specifications (Spec. 1) Approximate specifications S 0 � �� � required incorrect � �� � S , where S 0 ⊆ S . Approximate specification – ( S 0 , S ) ↑ ↑ for completeness for correctness S 0 ⊆ M P ⊆ S . S 0 – what has to be computed. Intention: S – what may be computed. 8 / 25

  15. Intro. Specification Correctness&. . . Programs Final Towards Approximate specifications (Spec. 1) Approximate specifications S 0 � �� � required incorrect � �� � S Approximate specification for SAT solver: ( S 0 1 , S 1 ) , 2 : L 0 states that predicate sat cnf defines a set L ′ 2 ⊆ L ′ 2 ⊆ L 2 . [Details � the paper] 8 / 25

  16. Intro. Specification Correctness&. . . Programs Final Towards Approximate specifications (Spec. 1) (Details – 1st specification for SAT solver) ( S 0 Specification: 1 , S 1 ) with the specified atoms S 0 1 : S 1 : where t ∈ L 0 sat cnf ( t ) , where t ∈ L 2 , sat cnf ( t ) , 2 , sat cl ( s ) , s ∈ L 0 1 , sat cl ( s ) , s ∈ L 1 , x = x, x = x, x ∈ H x ∈ H L 0 1 = { [ t 1 - u 1 , . . . , u - u, . . . , t n - u n ] ∈ H } , L 0 2 = { [ s 1 , . . . , s n ] | s 1 , . . . , s n ∈ L 0 1 } , L 1 = { t ∈ H | if t is a list of pairs then t ∈ L 0 1 } , L 2 = { s ∈ H | if s is a list of lists of pairs then s ∈ L 0 2 } . 9 / 25

  17. Intro. Specification Correctness&. . . Programs Final Correctness & completeness of programs Correctness (imperative programming) ւ ց Correctness Completeness (logic programming) M P ⊆ S S ⊆ M P Completeness: Everything required by the spec. is computed. Correctness: Everything computed is compatible with the spec. P semi-complete w.r.t. S = P complete for terminating queries (under some selection rule). [Details � the paper] 10 / 25

  18. Intro. Specification Correctness&. . . Programs Final Correctness & completeness of programs Correctness (imperative programming) ւ ց Correctness Completeness (logic programming) M P ⊆ S S ⊆ M P Completeness: Everything required by the spec. is computed. Correctness: Everything computed is compatible with the spec. P semi-complete w.r.t. S = P complete for terminating queries (under some selection rule). [Details � the paper] 10 / 25

  19. Intro. Specification Correctness&. . . Programs Final Correctness & completeness, sufficient conditions Th . (Clark 1979): P correct w.r.t. S when for each ( H ← B ) ∈ ground ( P ) , B ⊆ S ⇒ H ∈ S . (Out of correct atoms, the clauses produce only correct atoms.) Th .: P semi-complete w.r.t. S when for each H ∈ S , exists ( H ← B ) ∈ ground ( P ) where B ⊆ S . (Each required atom can be produced out of required atoms.) Semi-complete + terminating ⇒ complete. 11 / 25

  20. Intro. Specification Correctness&. . . Programs Final Correctness & completeness, sufficient conditions Th . (Clark 1979): P correct w.r.t. S when for each ( H ← B ) ∈ ground ( P ) , B ⊆ S ⇒ H ∈ S . (Out of correct atoms, the clauses produce only correct atoms.) Th .: P semi-complete w.r.t. S when for each H ∈ S , exists ( H ← B ) ∈ ground ( P ) where B ⊆ S . (Each required atom can be produced out of required atoms.) Semi-complete + terminating ⇒ complete. 11 / 25

  21. Intro. Specification Correctness&. . . Programs Final Program 1 (2 (2a) 3) 23 Control (Control details) SAT solver 1 P 1 : sat cnf ([ ]) . sat cnf ([ Clause | Clauses ]) ← sat cl ( Clause ) , sat cnf ( Clauses ) . sat cl ([ Pol - V ar | Pairs ]) ← Pol = V ar. sat cl ([ H | Pairs ]) ← sat cl ( Pairs ) . Can be constructed guided by the sufficient conditions above, and specification ( S 0 1 , S 1 ) . Correct w.r.t. S 1 , complete w.r.t. S 0 1 . [Details � the paper] Inefficient backtracking search. 12 / 25

  22. Intro. Specification Correctness&. . . Programs Final Program 1 (2 (2a) 3) 23 Control (Control details) SAT solver 1 P 1 : sat cnf ([ ]) . sat cnf ([ Clause | Clauses ]) ← sat cl ( Clause ) , sat cnf ( Clauses ) . sat cl ([ Pol - V ar | Pairs ]) ← Pol = V ar. sat cl ([ H | Pairs ]) ← sat cl ( Pairs ) . Can be constructed guided by the sufficient conditions above, and specification ( S 0 1 , S 1 ) . Correct w.r.t. S 1 , complete w.r.t. S 0 1 . [Details � the paper] Inefficient backtracking search. 12 / 25

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