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ALGORITHMIC CORRESPONDENCE AND COMPLETENESS IN MODAL LOGIC. I. The Core Algorithm SQEMA
- CONSIDERED FOR PUBLICATION IN LOGICAL METHODS IN COMPUTER SCIENCE
, 2006
"... In terms of frame validity, modal formulae express universal monadic secondo-rder properties, but in many important cases these have first-order equivalents. This is important for both logical and computational reasons, since first-order logic is much better studied and behaved than monadic second- ..."
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Cited by 18 (2 self)
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In terms of frame validity, modal formulae express universal monadic secondo-rder properties, but in many important cases these have first-order equivalents. This is important for both logical and computational reasons, since first-order logic is much better studied and behaved than monadic second-order logic. Furthermore, first-order definability often goes together with canonicity, which in turn implies frame-completeness of logics axiomatized with such formulae. Sahlqvist’s theorem is a general result on first-order definability and canonicity of a large syntactic class of modal formulae. Sahlqvist’s approach was paralleled and further developed by van Benthem into the substitution method. Establishing first-order definability of modal formulae amounts to elimination of secondorder quantifiers. Two algorithms have been developed and implemented for elimination of predicate quantifiers in second-order logic: SCAN, based on a constraint resolution procedure, and DLS, based on a logical equivalence established by Ackermann. In this paper we introduce a new algorithm, SQEMA, for computing first-order equivalents and proving canonicity of modal formulae. Like DLS, it uses (a modal version of) Ackermann’s lemma, but unlike both SCAN and DLS it works directly on modal formulae,
SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC
"... Abstract. We define Sahlqvist fixed point formulas. By extending the technique of Sambin and Vaccaro we show that (1) for each Sahlqvist fixed point formula ϕ there exists an LFPformula χ(ϕ), with no free first-order variable or predicate symbol, such that a descriptive µ-frame (an order-topological ..."
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Abstract. We define Sahlqvist fixed point formulas. By extending the technique of Sambin and Vaccaro we show that (1) for each Sahlqvist fixed point formula ϕ there exists an LFPformula χ(ϕ), with no free first-order variable or predicate symbol, such that a descriptive µ-frame (an order-topological structure that admits topological interpretations of least fixed point operators as intersections of clopen pre-fixed points) validates ϕ iff χ(ϕ) is true in this structure, and (2) every modal fixed point logic axiomatized by a set Φ of Sahlqvist fixed point formulas is sound and complete with respect to the class of descriptive µ-frames satisfying {χ(ϕ) : ϕ ∈ Φ}. We also give some concrete examples of Sahlqvist fixed point logics and classes of descriptive µ-frames for which these logics are sound and complete. 1.
Game Solution, Epistemic Dynamics and Fixed-Point Logics
, 2010
"... Current methods for solving games embody a form of “procedural rationality” that invites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a number of anal ..."
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Cited by 6 (3 self)
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Current methods for solving games embody a form of “procedural rationality” that invites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a number of analysis from recent years that look different conceptually, and find that they are all mathematically equivalent. This shows how an abstract logical perspective can bring out basic invariant structure in games. We then generalize this to an exploration of fixed-point logics on finite trees that best fit game-theoretic equilibria. We end with some open questions that suggest a broader program for merging current computational logics with notions and results from game theory. This paper is largely a program for opening up an area: an extended version of the technical results will be found in the forthcoming dissertation [26].
Sahlqvist correspondence for modal mu-calculus, Studia Logica
"... Abstract. We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them. ..."
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Abstract. We define analogues of modal Sahlqvist formulas for the modal mu-calculus, and prove a correspondence theorem for them.
Annotation Theories over Finite Graphs
, 2009
"... Abstract. In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph's nodes. Such theories, which we call annotatio ..."
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Abstract. In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph's nodes. Such theories, which we call annotation theories, can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPTime-or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of
Man Muss Immer Umkehren!
"... The 19th century geometrist Jacobi famously said that one should always try to invert every geometrical theorem. But his advice applies much more widely! Choose any class of relational frames, and you can study its valid modal axioms. But now turn the perspective around, and fix some modal ..."
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The 19th century geometrist Jacobi famously said that one should always try to invert every geometrical theorem. But his advice applies much more widely! Choose any class of relational frames, and you can study its valid modal axioms. But now turn the perspective around, and fix some modal
Computational Complexity
"... Abstract. Automated deduction is not just application or implementation of logical systems. The field of computational logic also poses deep challenges to our understanding of logic itself. I will discuss some key issues. This text is just an appetizer that will be elaborated in the lecture. ..."
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Abstract. Automated deduction is not just application or implementation of logical systems. The field of computational logic also poses deep challenges to our understanding of logic itself. I will discuss some key issues. This text is just an appetizer that will be elaborated in the lecture.
IOS Press Game Solution, Epistemic Dynamics and Fixed-Point Logics
"... Abstract. Current methods for solving games embody a form of “procedural rationality ” that in-vites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a num ..."
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Abstract. Current methods for solving games embody a form of “procedural rationality ” that in-vites logical analysis in its own right. This paper is a brief case study of Backward Induction for extensive games, replacing earlier static logical definitions by stepwise dynamic ones. We consider a number of analysis from recent years that look different conceptually, and find that they are all mathematically equivalent. This shows how an abstract logical perspective can bring out basic in-variant structure in games. We then generalize this to an exploration of fixed-point logics on finite trees that best fit game-theoretic equilibria. We end with some open questions that suggest a broader program for merging current computational logics with notions and results from game theory. This paper is largely a program for opening up an area: an extended version of the technical results will be found in the dissertation [26].
