Writing Declarative Specifications for Clauses Martin Gebser 1 , 2 , - PowerPoint PPT Presentation

Writing Declarative Specifications for Clauses Martin Gebser 1 , 2 , Tomi Janhunen 1 , Roland Kaminski 2 , Torsten Schaub 2 , 3 , Shahab Tasharrofi 1 1 ) Aalto University, Finland 2 ) University of Potsdam, Germany 3 ) INRIA Rennes, France GTTV’15, Lexington, Kentucky, September 27, 2015

Background: Boolean Satisfiability Satisfiability (SAT) solvers provide an efficient implementation of classical propositional logic. ◮ SAT solvers expect their input in the conjunctive normal form (CNF), i.e., a conjunction of clauses l 1 ∨ . . . ∨ l k . ◮ Clauses can be viewed as “machine code” for expressing constraints and representing knowledge. GTTV’15, September 27, 2015 2/48

Background: Boolean Satisfiability Satisfiability (SAT) solvers provide an efficient implementation of classical propositional logic. ◮ SAT solvers expect their input in the conjunctive normal form (CNF), i.e., a conjunction of clauses l 1 ∨ . . . ∨ l k . ◮ Clauses can be viewed as “machine code” for expressing constraints and representing knowledge. ◮ Typically clauses are either — generated using a procedural program or — obtained when more complex formulas are translated. GTTV’15, September 27, 2015 3/48

Background: Boolean Satisfiability Satisfiability (SAT) solvers provide an efficient implementation of classical propositional logic. ◮ SAT solvers expect their input in the conjunctive normal form (CNF), i.e., a conjunction of clauses l 1 ∨ . . . ∨ l k . ◮ Clauses can be viewed as “machine code” for expressing constraints and representing knowledge. ◮ Typically clauses are either — generated using a procedural program or — obtained when more complex formulas are translated. ◮ First-order formulas are prone to combinatorial effects: ¬ edge ( X , Y ) ∨ ¬ edge ( Y , Z ) ∨ ¬ edge ( Z , X ) ∨ ( X = Y ) ∨ ( X = Z ) ∨ ( Y = Z ) GTTV’15, September 27, 2015 4/48

Analogue: Assembly Languages smodels: testl %eax, %eax pushq %rbp je .L2 movq %rsp, %rbp movl $0, %eax subq $32, %rsp jmp .L3 movq %rdi, -24(%rbp) .L2: movq %rsi, -32(%rbp) movq -24(%rbp), %rax movl $0, -4(%rbp) movq %rax, %rdi movq -32(%rbp), %rdx movl $0, %eax movq -24(%rbp), %rax call complete movq %rdx, %rsi testl %eax, %eax movq %rax, %rdi je .L4 call propagate movl $-1, %eax movq %rax, -24(%rbp) jmp .L3 movq -24(%rbp), %rax ... movq %rax, %rdi .L3: movl $0, %eax leave call conflict ret GTTV’15, September 27, 2015 5/48

How to Generate Machine Code? 1. Assembly language GTTV’15, September 27, 2015 6/48

How to Generate Machine Code? 1. Assembly language 2. Assembly language + macros [ tigcc.ticalc.org ] .long 0 .macro sum from=0, to=5 .long 1 .long \from .long 2 .if \to-\from �− → .long 3 sum "(\from+1)",\to .long 4 .endif .long 5 .endm GTTV’15, September 27, 2015 7/48

How to Generate Machine Code? 1. Assembly language 2. Assembly language + macros [ tigcc.ticalc.org ] .long 0 .macro sum from=0, to=5 .long 1 .long \from .long 2 .if \to-\from �− → .long 3 sum "(\from+1)",\to .long 4 .endif .long 5 .endm 3. High level language (C, C++, scala, ...) + compilation GTTV’15, September 27, 2015 8/48

How to Generate Machine Code? 1. Assembly language 2. Assembly language + macros [ tigcc.ticalc.org ] .long 0 .macro sum from=0, to=5 .long 1 .long \from .long 2 .if \to-\from �− → .long 3 sum "(\from+1)",\to .long 4 .endif .long 5 .endm 3. High level language (C, C++, scala, ...) + compilation How much can we control the actual output in each case? GTTV’15, September 27, 2015 9/48

Our Approach ◮ A fully declarative approach where intended clauses are given first-order specifications in analogy to ASP . ◮ In the implementation, we harness state-of-the-art ASP grounders for instantiating terms variables in clauses. GTTV’15, September 27, 2015 10/48

Our Approach ◮ A fully declarative approach where intended clauses are given first-order specifications in analogy to ASP . ◮ In the implementation, we harness state-of-the-art ASP grounders for instantiating terms variables in clauses. ◮ The benefits of our approach: — Complex domain specifications supported — Database operations available — Uniform encodings enabled — Elaboration tolerance GTTV’15, September 27, 2015 11/48

Our Approach ◮ A fully declarative approach where intended clauses are given first-order specifications in analogy to ASP . ◮ In the implementation, we harness state-of-the-art ASP grounders for instantiating terms variables in clauses. ◮ The benefits of our approach: — Complex domain specifications supported — Database operations available — Uniform encodings enabled — Elaboration tolerance ◮ WYSIWYG: black ( X ) ∨ grey ( X ) ∨ white ( X ) ← node ( X ) . black ( a ) ∨ grey ( a ) ∨ white ( a ) . node ( a ) . node ( b ) . �− → black ( b ) ∨ grey ( b ) ∨ white ( b ) . black ( c ) ∨ grey ( c ) ∨ white ( c ) . node ( c ) . GTTV’15, September 27, 2015 12/48

Outline Clause Programs Modeling Methodology and Applications Implementation Discussion and Conclusion GTTV’15, September 27, 2015 13/48

Clause Programs: Syntax ◮ The signature P for predicate symbols is partitioned into domain predicates P d and varying predicates P v . GTTV’15, September 27, 2015 14/48

Clause Programs: Syntax ◮ The signature P for predicate symbols is partitioned into domain predicates P d and varying predicates P v . ◮ Domain rules in P d are normal rules of the form a ← c 1 , . . . , c m , ∼ d 1 , . . . , ∼ d n . GTTV’15, September 27, 2015 15/48

Clause Programs: Syntax ◮ The signature P for predicate symbols is partitioned into domain predicates P d and varying predicates P v . ◮ Domain rules in P d are normal rules of the form a ← c 1 , . . . , c m , ∼ d 1 , . . . , ∼ d n . ◮ The syntax for clause rules is a 1 ∨ · · · ∨ a k ∨ ¬ b 1 ∨ · · · ∨ ¬ b l ← c 1 , . . . , c m , ∼ d 1 , . . . , ∼ d n . where the head (resp. body) is expressed in P v (resp. P d ). GTTV’15, September 27, 2015 16/48

Clause Programs: Syntax ◮ The signature P for predicate symbols is partitioned into domain predicates P d and varying predicates P v . ◮ Domain rules in P d are normal rules of the form a ← c 1 , . . . , c m , ∼ d 1 , . . . , ∼ d n . ◮ The syntax for clause rules is a 1 ∨ · · · ∨ a k ∨ ¬ b 1 ∨ · · · ∨ ¬ b l ← c 1 , . . . , c m , ∼ d 1 , . . . , ∼ d n . where the head (resp. body) is expressed in P v (resp. P d ). ◮ For standard use cases, the domain part of a program should be stratified to enable evaluation by the grounder. GTTV’15, September 27, 2015 17/48

Example: Graph Colouring Domain rules node ( X ) ← edge ( X , Y ) . node ( Y ) ← edge ( X , Y ) . GTTV’15, September 27, 2015 18/48

Example: Graph Colouring Domain rules node ( X ) ← edge ( X , Y ) . node ( Y ) ← edge ( X , Y ) . Clause rules black ( X ) ∨ grey ( X ) ∨ white ( X ) ← node ( X ) . ¬ black ( X ) ∨ ¬ black ( Y ) ← edge ( X , Y ) . ¬ grey ( X ) ∨ ¬ grey ( Y ) ← edge ( X , Y ) . ¬ white ( X ) ∨ ¬ white ( Y ) ← edge ( X , Y ) . GTTV’15, September 27, 2015 19/48

Example: Graph Colouring Domain rules node ( X ) ← edge ( X , Y ) . node ( Y ) ← edge ( X , Y ) . Clause rules black ( X ) ∨ grey ( X ) ∨ white ( X ) ← node ( X ) . ¬ black ( X ) ∨ ¬ black ( Y ) ← edge ( X , Y ) . ¬ grey ( X ) ∨ ¬ grey ( Y ) ← edge ( X , Y ) . ¬ white ( X ) ∨ ¬ white ( Y ) ← edge ( X , Y ) . Uniform encoding that works for any graph instance! GTTV’15, September 27, 2015 20/48

Clause Programs: Semantics ◮ The Herbrand universe Hu ( P ) and the Herbrand base Hb ( P ) of a clause program P are defined as usual. ◮ The ground program Gnd ( P ) is the respective Herbrand instantiation of P over the universe Hu ( P ) . GTTV’15, September 27, 2015 21/48

Clause Programs: Semantics ◮ The Herbrand universe Hu ( P ) and the Herbrand base Hb ( P ) of a clause program P are defined as usual. ◮ The ground program Gnd ( P ) is the respective Herbrand instantiation of P over the universe Hu ( P ) . ◮ The domain reduct P I of P with respect to I contains the positive rule a ← c 1 , . . . , c m for every domain rule a ← c 1 , . . . , c m , ∼ d 1 , . . . , ∼ d n . such that { d 1 , . . . , d n } ∩ I d = ∅ . GTTV’15, September 27, 2015 22/48

Clause Programs: Semantics ◮ The Herbrand universe Hu ( P ) and the Herbrand base Hb ( P ) of a clause program P are defined as usual. ◮ The ground program Gnd ( P ) is the respective Herbrand instantiation of P over the universe Hu ( P ) . ◮ The domain reduct P I of P with respect to I contains the positive rule a ← c 1 , . . . , c m for every domain rule a ← c 1 , . . . , c m , ∼ d 1 , . . . , ∼ d n . such that { d 1 , . . . , d n } ∩ I d = ∅ . Definition An Herbrand interpretation I ⊆ Hb ( P ) is a domain stable model = Gnd ( P ) and I d is the least model of Gnd ( P ) I . of P iff I | GTTV’15, September 27, 2015 23/48

Example: Continued 1. Suppose the following facts as instance information: edge ( a , b ) , edge ( b , c ) , edge ( c , a ) . 2. Additional domain atoms from node ( X ; Y ) ← edge ( X , Y ) : node ( a ) , node ( b ) , node ( c ) . GTTV’15, September 27, 2015 24/48