On the Restraining Power of Guards

Grädel, Erich

doi:10.2307/2586808

Cited by 246 publications

(275 citation statements)

References 21 publications

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“…Decidability of GF now follows because we can test satisfiability for arbitrary (loosely) guarded formulas φ by testing for the existence of a quasi-model for φ whose size is effectively bounded by the length of φ. ■ This decision procedure can be adapted easily to give an optimal complexity result (Grädel 1999B). Satisfiability is 2EXPTIME-complete for guarded formulas, and it is EXPTIME-complete for GF with a fixed bound on the arities of predicates.…”

Section: Lemmamentioning

confidence: 99%

Crs and Guarded Logics: A Fruitful Contact

Benthem¹

2013

Bolyai Society Mathematical Studies

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1Introduction: back and forth between algebra and model theory Algebra and model theory are complementary stances in the history of logic, and their interaction continues to spawn new ideas, witness the interface of First-Order Logic and Cylindric Algebra. This chapter is about a more specialized contact: the flow of ideas between algebra and modal logic through 'guarded fragments' restricting the range of quantification over objects. Here is some general background for this topic. For a start, the connection between algebra and model theory is rather tight, since we can view universal algebra as the equational logic part of standard first-order model theory. This extension of the set of models leaves several base laws of relational algebra valid, while others become invalid: Associativity (R S) ; T = R ; (S ; T)) is a typical example.Crucially, in this contraction process, the set of algebraic validities becomes decidable.But if we impose additional conditions on the relation U, then undecidability may reappears. For instance, if we require transitivity, relational algebra is undecidable again: in logical modeling. Intuitively, the core calculus of action embodied in relational algebra seems simple, and undecidability comes as a surprise. Thus, we want to find a semantics that gives just the bare bones of action, while additional effects of 'standard set-theoretic modeling' are separated out as negotiable decisions of formulation that engender the undecidability. This theme underlies the systems presented in this chapter.Fragments But there is also a quite different technical way of viewing relativization as a general logical device. Already Wadge 1975 showed how relational algebra can be axiomatized smoothly by using explicit pair notation (x, y) : R, making transitions explicit as objects, which suggests viewing it as a fragment of first-order logic. Now it is a well-known result of Tarski's that standard relational algebra translates into the undecidable 3-variable fragment of first-order logic, through clauses such aswhich typically use existential quantification over objects in the domain. But the clause in our earlier description replaces this by another syntactic format, namelyThus, we end up inside a sub-language of the 3-variable fragment, where patterns of quantification are restricted or 'guarded' in some way by atomic formulas. Similar points hold for CRS and first-order dependence logics, and the result there is that we end up in a sub-language of full first-order logic known as the Guarded Fragment.In this paper, we will develop this fragment view as well, and eventually, we will also address the following fundamental question about our presentation so far. What is the relation between the two lines of (a) taking a logical language and extending its class of models, and (b) retaining the original model class while restricting the language? 4 Arrow logic in a nutshellMotivation: core content versus wrappings Relational algebra is a calculus of transition

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Section: Lemmamentioning

confidence: 99%

Crs and Guarded Logics: A Fruitful Contact

Benthem¹

2013

Bolyai Society Mathematical Studies

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“…The tree model property [8] for modal logics and the generalised tree model property [11] of guarded logics account for various decidability and complexity results for modal and guarded logics; they stand behind the applicability of tree automata to their algorithmic model theory; they support modeltheoretic characterisations of modal and guarded fragments of first-order and second-order logics; and, even though the unfoldings themselves are typically infinite, they also point us in the right direction for understanding certain approaches to the finite model property and the combinatorial challenges posed by the finite model theory of modal and guarded logics [10,19,26,23,11,13].…”

Section: Introductionmentioning

confidence: 99%

Bisimulation and Coverings for Graphs and Hypergraphs

Otto

2013

Logic and Its Applications

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Abstract. We survey notions of bisimulation and of bisimilar coverings both in the world of graph-like structures (Kripke structures, transition systems) and in the world of hypergraph-like general relational structures. The provision of finite analogues for full infinite tree-like unfoldings, in particular, raises interesting combinatorial challenges and is the key to a number of interesting model-theoretic applications.

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“…The satisfiability problem for FHL \↓2 is decidable. This holds because the standard translation ST of FHL into first order classical logic [1,13] maps formulae in the considered fragment into universally guarded formulae [13], that have a decidable satisfiability problem [8].…”

Section: Introductionmentioning

confidence: 99%

“…However, in practice, the overhead coming from the translation cannot be completely ingnored. In fact, the standard translation of F is a universally guarded formula, which has to be rewritten into an equisatisfiable guarded one [8]. Moreover, decision procedures for guarded logic such as the above mentioned ones apply to constant-free formulae.…”

Section: Introductionmentioning

confidence: 99%

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A Tableaux Based Decision Procedure for a Broad Class of Hybrid Formulae with Binders

Cerrito¹,

Mayer²

2011

Lecture Notes in Computer Science

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In this paper we provide the first (as far as we know) direct calculus deciding satisfiability of formulae in negation normal form in the fragment of hybrid logic with the satisfaction operator and the binder, where no occurrence of the 2 operator is in the scope of a binder. A preprocessing step, rewriting formulae into equisatisfiable ones, turns the calculus into a satisfiability decision procedure for the fragment HL(@, ↓) \2↓2, i.e. formulae in negation normal form where no occurrence of the binder is both in the scope of and contains in its scope a 2 operator.The calculus is based on tableaux, where nominal equalities are treated by means of substitution, and termination is achieved by means of a form of anywhere blocking with indirect blocking. Direct blocking is a relation between nodes in a tableau branch, holding whenever the respective labels (formulae) are equal up to (a proper form of) nominal renaming. Indirect blocking is based on a partial order on the nodes of a tableau branch, which arranges them into a tree-like structure.

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On the Restraining Power of Guards

Cited by 246 publications

References 21 publications

Crs and Guarded Logics: A Fruitful Contact

Crs and Guarded Logics: A Fruitful Contact

Bisimulation and Coverings for Graphs and Hypergraphs

A Tableaux Based Decision Procedure for a Broad Class of Hybrid Formulae with Binders

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