On Minimal Corrections in ASP s Janota 1 Joao Marques-Silva 2 Mikol - PowerPoint PPT Presentation

On Minimal Corrections in ASP s Janota 1 Joao Marques-Silva 2 Mikol´ aˇ RCRA 2017, Bari 1 INESC-ID/IST, University of Lisbon, Portugal 2 LaSIGE, Faculty of Science, University of Lisbon, Portugal Janota and Silva On Minimal Corrections in ASP 1 / 15

Context and History • Number of interesting problems about propositional formulae: Janota and Silva On Minimal Corrections in ASP 2 / 15

Context and History • Number of interesting problems about propositional formulae: • Minimally Unsatisfiable Set (MUS), diagnostics, debugging, SMT Janota and Silva On Minimal Corrections in ASP 2 / 15

Context and History • Number of interesting problems about propositional formulae: • Minimally Unsatisfiable Set (MUS), diagnostics, debugging, SMT • Minimal Correction Set (MCS), diagnostics, debugging Janota and Silva On Minimal Corrections in ASP 2 / 15

Context and History • Number of interesting problems about propositional formulae: • Minimally Unsatisfiable Set (MUS), diagnostics, debugging, SMT • Minimal Correction Set (MCS), diagnostics, debugging • Prime Implicant/Implicate, model checking Janota and Silva On Minimal Corrections in ASP 2 / 15

Context and History • Number of interesting problems about propositional formulae: • Minimally Unsatisfiable Set (MUS), diagnostics, debugging, SMT • Minimal Correction Set (MCS), diagnostics, debugging • Prime Implicant/Implicate, model checking • Minimal model, circumscription Janota and Silva On Minimal Corrections in ASP 2 / 15

Context and History • Number of interesting problems about propositional formulae: • Minimally Unsatisfiable Set (MUS), diagnostics, debugging, SMT • Minimal Correction Set (MCS), diagnostics, debugging • Prime Implicant/Implicate, model checking • Minimal model, circumscription • Backbone, fault-localization Janota and Silva On Minimal Corrections in ASP 2 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ • φ ⊆ ψ ⇒ � SAT( ψ ) ⇒ SAT( φ ) � Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ • φ ⊆ ψ ⇒ � SAT( ψ ) ⇒ SAT( φ ) � � � • φ ⊆ ψ ⇒ UNSAT( φ ) ⇒ UNSAT( ψ ) Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ • φ ⊆ ψ ⇒ � SAT( ψ ) ⇒ SAT( φ ) � � � • φ ⊆ ψ ⇒ UNSAT( φ ) ⇒ UNSAT( ψ ) • MUS — subset minimum for the UNSAT predicate. Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ • φ ⊆ ψ ⇒ � SAT( ψ ) ⇒ SAT( φ ) � � � • φ ⊆ ψ ⇒ UNSAT( φ ) ⇒ UNSAT( ψ ) • MUS — subset minimum for the UNSAT predicate. • MSS — subset maximum for the SAT predicate. Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ • φ ⊆ ψ ⇒ � SAT( ψ ) ⇒ SAT( φ ) � � � • φ ⊆ ψ ⇒ UNSAT( φ ) ⇒ UNSAT( ψ ) • MUS — subset minimum for the UNSAT predicate. • MSS — subset maximum for the SAT predicate. • L 1 , L 2 sets of literals: Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ • φ ⊆ ψ ⇒ � SAT( ψ ) ⇒ SAT( φ ) � � � • φ ⊆ ψ ⇒ UNSAT( φ ) ⇒ UNSAT( ψ ) • MUS — subset minimum for the UNSAT predicate. • MSS — subset maximum for the SAT predicate. • L 1 , L 2 sets of literals: � � • L 1 ⊆ L 2 ⇒ L 1 | = ϕ ⇒ L 2 | = ϕ Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ • φ ⊆ ψ ⇒ � SAT( ψ ) ⇒ SAT( φ ) � � � • φ ⊆ ψ ⇒ UNSAT( φ ) ⇒ UNSAT( ψ ) • MUS — subset minimum for the UNSAT predicate. • MSS — subset maximum for the SAT predicate. • L 1 , L 2 sets of literals: � � • L 1 ⊆ L 2 ⇒ L 1 | = ϕ ⇒ L 2 | = ϕ • prime implicant — subset minimum for the · | = ϕ predicate. Janota and Silva On Minimal Corrections in ASP 3 / 15

Monotone Predicates • These problems are instances of monotone predicates. [Marques-Silva et al., 2013] • Example for sets of clauses φ , ψ • φ ⊆ ψ ⇒ � SAT( ψ ) ⇒ SAT( φ ) � � � • φ ⊆ ψ ⇒ UNSAT( φ ) ⇒ UNSAT( ψ ) • MUS — subset minimum for the UNSAT predicate. • MSS — subset maximum for the SAT predicate. • L 1 , L 2 sets of literals: � � • L 1 ⊆ L 2 ⇒ L 1 | = ϕ ⇒ L 2 | = ϕ • prime implicant — subset minimum for the · | = ϕ predicate. • Literal a backbone if φ | = l L 1 ⊆ L 2 ⇒ � � l ∈L 2 φ | = l ⇒ � l ∈L 1 φ | � = l Janota and Silva On Minimal Corrections in ASP 3 / 15

What about ASP? • Unlike propositional logic, ASP is not monotone. Janota and Silva On Minimal Corrections in ASP 4 / 15

What about ASP? • Unlike propositional logic, ASP is not monotone. • How to define minimality? Janota and Silva On Minimal Corrections in ASP 4 / 15

What about ASP? • Unlike propositional logic, ASP is not monotone. • How to define minimality? • Can the algorithms from propositional logic be adapted? (or at least some) Janota and Silva On Minimal Corrections in ASP 4 / 15

What about ASP? • Unlike propositional logic, ASP is not monotone. • How to define minimality? • Can the algorithms from propositional logic be adapted? (or at least some) Example ← not move ( a ) . % program move ( a ) ← stone ( b ) , not stone ( c ) . % program stone ( c ) ← . % fact (input) Janota and Silva On Minimal Corrections in ASP 4 / 15

What about ASP? • Unlike propositional logic, ASP is not monotone. • How to define minimality? • Can the algorithms from propositional logic be adapted? (or at least some) Example ← not move ( a ) . % program move ( a ) ← stone ( b ) , not stone ( c ) . % program stone ( c ) ← . % fact (input) Possible fix: add stone ( b ), remove stone ( c ) Janota and Silva On Minimal Corrections in ASP 4 / 15

Maximal Consistent Set in ASP Definition Janota and Silva On Minimal Corrections in ASP 5 / 15

Maximal Consistent Set in ASP Definition • Let P be a consistent ASP program and S be a set of atoms. Janota and Silva On Minimal Corrections in ASP 5 / 15

Maximal Consistent Set in ASP Definition • Let P be a consistent ASP program and S be a set of atoms. • A set L ⊆ S is a maximal consistent subset of S w.r.t. P Janota and Silva On Minimal Corrections in ASP 5 / 15

Maximal Consistent Set in ASP Definition • Let P be a consistent ASP program and S be a set of atoms. • A set L ⊆ S is a maximal consistent subset of S w.r.t. P • if the program P ∪ { s . | s ∈ L} is consistent Janota and Silva On Minimal Corrections in ASP 5 / 15

Maximal Consistent Set in ASP Definition • Let P be a consistent ASP program and S be a set of atoms. • A set L ⊆ S is a maximal consistent subset of S w.r.t. P • if the program P ∪ { s . | s ∈ L} is consistent • and for any L ′ , such that L � L ′ ⊆ S , the program � s . | s ∈ L ′ � P ∪ is inconsistent. Observe: In monotone case L is maximally consistent iff L ∪{ s } is inconsistent for any s ∈ S \ L . Does not hold in non-monotone. Janota and Silva On Minimal Corrections in ASP 5 / 15

Choice Rules Notation • Consider set of atoms: S = { s 1 , . . . , s k } . Janota and Silva On Minimal Corrections in ASP 6 / 15

Choice Rules Notation • Consider set of atoms: S = { s 1 , . . . , s k } . • Let choice ( S ) denote the choice rule 0 ≤ { s 1 , . . . , s k } Janota and Silva On Minimal Corrections in ASP 6 / 15

Choice Rules Notation • Consider set of atoms: S = { s 1 , . . . , s k } . • Let choice ( S ) denote the choice rule 0 ≤ { s 1 , . . . , s k } • Let atleast1 ( { s 1 , . . . , s k } ) denote the choice rule 1 ≤ { s 1 , . . . , s k } . Janota and Silva On Minimal Corrections in ASP 6 / 15

Choice Rules Notation • Consider set of atoms: S = { s 1 , . . . , s k } . • Let choice ( S ) denote the choice rule 0 ≤ { s 1 , . . . , s k } • Let atleast1 ( { s 1 , . . . , s k } ) denote the choice rule 1 ≤ { s 1 , . . . , s k } . Idea Janota and Silva On Minimal Corrections in ASP 6 / 15