Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion Finite Model Reasoning in Expressive Fragments of First-Order Logic Lidia Tendera Institute of Mathematics and Informatics Opole University, Poland M4M Kanpur 8.–9. January 2017
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion O UTLINE Introduction/Motivation Standard translation of ML Base Fragments Definitions Properties and Complexity Extensions of Base Fragments More operators Special classes of structures Deciding (Fin)Sat More or less natural reductions Finitary unravellings Linear/Integer Programming Conclusion
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion C LASSICAL D ECISION P ROBLEM L – any logic FO – first-order logic ◮ Sat ( L ): given a formula ϕ ∈ L , does ϕ admit a model ? ◮ FinSat ( L ): given a formula ϕ ∈ L , does ϕ admit a finite model ? Theorem (Church, Turing, Trahtenbrot) Sat( FO ) and FinSat( FO ) are undecidable and recursively inseparable. Possible response: � devise incomplete algorithms � identify decidable fragments
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion W HY F IN S AT ? Databases, systems etc. are often considered to be finite . ◮ L has the finite model property (FMP) iff every satisfiable ϕ ∈ L has a finite model. Observation ◮ If L has FMP then Sat ( L ) and FinSat ( L ) coincide. ◮ If L is a fragment of FO and L has FMP then Sat ( L ) is decidable.
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion D ECIDABLE F RAGMENTS OF FO Note: one cannot study all possible fragments! ◮ defined by restrictions on signatures e.g. [Löwenheim-Skolem 1915] monadic theories (FMP) ◮ prenex classes defined by quantifier prefix ∃ ∗ ∀ ∗ , ∃ ∗ ∀∃ ∗ , ∃ ∗ ∀∀∃ ∗ (equality free) ◮ defined by other syntactic restrictions and suitably motivated Also: we want to identify reasons for a logic to be (un)decidable, (in)tractable etc. ◮ Can we decide whether a formula is satisfiable without actually seeing a model? ◮ Some formulas have only infinite models. Can we decide whether they are satisfiable?
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion M OTIVATION : M ODAL L OGIC [V ARDI 1996]: W HY IS MODAL LOGIC SO ROBUSTLY DECIDABLE ? ◮ Propositional modal logic: Boolean logic + operators ♦ (possibly) and � (necessary) ◮ Good model-theoretical and algorithmic properties, robustly decidable ◮ Variants and extensions of modal logics have applications in various areas of computer science: ◮ verification of hardware and software ◮ artificial intelligence ◮ distributed systems ◮ knowledge representation
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion S TANDARD T RANSLATION OF M ODAL L OGIC (1) ◮ Modal logic can be translated into FO : P ∧ ♦ ( Q ∨ � ¬ P ) Px ∧ ∃ y ( Rxy ∧ ( Qy ∨ ∀ z ( Ryz → ¬ Pz ))) � ◮ FO 3 undecidable [Kahr, Moore, Wang, 1959] [Gabbay, 1981] TWO variables suffice! Observation ◮ ML can be embedded in the two-variable fragment FO 2
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion S TANDARD T RANSLATION OF M ODAL L OGIC (1) ◮ Modal logic can be translated into FO : P ∧ ♦ ( Q ∨ � ¬ P ) Px ∧ ∃ y ( Rxy ∧ ( Qy ∨ ∀ x ( Ryx → ¬ Px ))) � ◮ FO 3 undecidable [Kahr, Moore, Wang, 1959] [Gabbay, 1981] TWO variables suffice! Observation ◮ ML can be embedded in the two-variable fragment FO 2
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion S TANDARD T RANSLATION OF M ODAL L OGIC (2) P ∧ ♦ ( Q ∨ � ¬ P ) Px ∧ ∃ y ( Rxy ∧ ( Qy ∨ ∀ z ( Ryz → ¬ Pz ))) � The translation suggests other restrictions of FO : ◮ fluted fragment FL : variables appear in some fixed order and no quantifier-rescoping occurs; order of quantification of variables matches order of appearance in predicates. ◮ guarded fragment GF : quantifiers are relativized by atomic formulas ◮ unary negation fragment UNF : negation is applied only to subformulas with a single free variable.
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion P ROBLEMS R EDUCING TO (F IN )S AT E XAMPLE : Q UERY A NSWERING A knowledge base � D , O� : database D (a set of facts, i.e. ground atoms), ontology O (i.e. a logical formula). ◮ Query Answering: given a knowledge base � D , O� and a query Q : does � D , O� entail Q , i.e. D ∧ O | = Q ? Observation iff D ∧ O ∧ ¬ Q is unsatisfiable D ∧ O | = Q
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion B ASE F RAGMENTS FRAGMENTS EMBEDDING MODAL LOGIC ◮ two-variable fragment FO 2 ◮ fluted fragment FL ◮ guarded fragment GF ◮ unary negation fragment UNF Theorem All four base fragments enjoy the finite model property. ◮ FMP often gives a bound on the size of minimal models. Hence: ◮ FMP often gives an upper bound for the computational complexity of Sat ( L )= FinSat ( L ).
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion FMP AND C OMPLEXITY OF FO 2 Theorem (Mortimer, 75) FO 2 has doubly exponential model property: every satisfiable ϕ ∈ FO 2 has a model of size at most doubly exponential in | ϕ | . Theorem (Grädel, Kolaitis, Vardi, 97) FO 2 has exponential model property: every satisfiable ϕ ∈ FO 2 has a model of size at most exponential in | ϕ | . Corollary Sat ( FO 2 ) is NE XP T IME -complete.
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion F LUTED F RAGMENT FL ◮ First identified by W.V.Quine in 1968: ◮ homogeneous m -adic formulas (generalization of monadic fragment) ◮ later generalized to fluted fragment ◮ Examples of fluted formulas: No student admires every professor ∀ x 1 ( student ( x 1 ) → ¬∀ x 2 ( prof ( x 2 ) → admires ( x 1 , x 2 ))) No lecturer introduces any professor to every student ∀ x 1 ( lecturer ( x 1 ) → ¬∃ x 2 ( prof ( x 2 ) ∧ ∀ x 3 ( student ( x 3 ) → intro ( x 1 , x 2 , x 3 )))) . ◮ Order of quantification of variables matches order of appearance in predicates.
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion F LUTED F RAGMENT FL ◮ First identified by W.V.Quine in 1968: ◮ homogeneous m -adic formulas (generalization of monadic fragment) ◮ later generalized to fluted fragment ◮ Examples of fluted formulas: No student admires every professor ∀ x 1 ( student ( x 1 ) → ¬∀ x 2 ( prof ( x 2 ) → admires ( x 1 , x 2 ))) No lecturer introduces any professor to every student ∀ x 1 ( lecturer ( x 1 ) → ¬∃ x 2 ( prof ( x 2 ) ∧ ∀ x 3 ( student ( x 3 ) → intro ( x 1 , x 2 , x 3 )))) . ◮ Order of quantification of variables matches order of appearance in predicates.
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion F LUTED F RAGMENT FL ◮ First identified by W.V.Quine in 1968: ◮ homogeneous m -adic formulas (generalization of monadic fragment) ◮ later generalized to fluted fragment ◮ Examples of fluted formulas: No student admires every professor ∀ x 1 ( student ( x 1 ) → ¬∀ x 2 ( prof ( x 2 ) → admires ( x 1 , x 2 ))) No lecturer introduces any professor to every student ∀ x 1 ( lecturer ( x 1 ) → ¬∃ x 2 ( prof ( x 2 ) ∧ ∀ x 3 ( student ( x 3 ) → intro ( x 1 , x 2 , x 3 )))) . ◮ Order of quantification of variables matches order of appearance in predicates.
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion F LUTED F RAGMENT FL ◮ First identified by W.V.Quine in 1968: ◮ homogeneous m -adic formulas (generalization of monadic fragment) ◮ later generalized to fluted fragment ◮ Examples of fluted formulas: No student admires every professor ∀ x 1 ( student ( x 1 ) → ¬∀ x 2 ( prof ( x 2 ) → admires ( x 1 , x 2 ))) No lecturer introduces any professor to every student ∀ x 1 ( lecturer ( x 1 ) → ¬∃ x 2 ( prof ( x 2 ) ∧ ∀ x 3 ( student ( x 3 ) → intro ( x 1 , x 2 , x 3 )))) . ◮ Order of quantification of variables matches order of appearance in predicates.
Introduction/Motivation Base Fragments Extensions of Base Fragments Deciding (Fin)Sat Conclusion FL FORMAL DEFINITION ◮ Let x 1 , x 2 , . . . be a fixed sequence of variables. ◮ The fluted fragment with k free variables, FL [ k ] , is defined by simultaneous induction for all k : - any atom p ( x ℓ , . . . , x k ) is in FL [ k ] ; - FL [ k ] is closed under Boolean operations; - FL [ k ] contains ∃ x k + 1 ϕ and ∀ x k + 1 ϕ for any ϕ ∈ FL [ k + 1 ] . ◮ The fluted fragment, FL [ k ] is the union: � FL [ k ] . FL = k ≥ 0 ◮ For all m > 0, we define FL m , to be the set of fluted formulas containing at most the variables x 1 , . . . , x m , free or bound.
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