Bisimulation quantifiers and uniform interpolation for guarded first order logic

The idea that the good model-theoretic and algorithmic properties of Modal Logics are due to the guarded nature of their quantification was put forward by Andreka, van Benthem and Nemeti in a series of papers in the 1990s, exploiting the satisfiability problem, the tree model property, and other similar properties of the Guarded Fragment of First Order Logic(GF). Since then, further work on the Guarded Fragment has been done by various authors, in some cases reinforcing this idea, in some others not. At least at first sight, Craig interpolation is on the negative side: there are implications in GF without an interpolant in GF, while Modal Logic (and even the \u3bc-calculus, a powerful extension of Modal Logic) enjoys a much stronger form of interpolation, the uniform one, in which the interpolant of a valid implication not only exists, but only depends on the antecedent and on the common language of antecedent and consequent. However, Hoogland and Marx proved that Craig interpolation is restored in GFif we consider the modal character of GFwith more attention, that is, if relations appearing on guards are viewed as \u201cmodalities\u201d and the rest as \u201cpropositions\u201d, and only the latter enter in the common language. In this paper we strengthen this result by showing that GF enjoys a Modal Uniform Interpolation Theorem (in the sense of Hoogland and Marx)

Bisimulation Quantifiers and Uniform Interpolation for Guarded First Order Logic

The idea that the good model-theoretic and algorithmic properties of Modal Logics are due to the guarded nature of their quantification was put forward by Andreka, van Benthem and Nemeti in a series of papers in the 1990s, exploiting the satisfiability problem, the tree model property, and other similar properties of the Guarded Fragment of First Order Logic (GF). Since then, further work on the Guarded Fragment has been done by various authors, in some cases reinforcing this idea, in some others not. At least at first sight, Craig interpolation is on the negative side: there are implications in GF without an interpolant in GF, while Modal Logic (and even the μ-calculus, a powerful extension of Modal Logic) enjoys a much stronger form of interpolation, the uniform one, in which the interpolant of a valid implication not only exists, but only depends on the antecedent and on the common language of antecedent and consequent. However, Hoogland and Marx proved that Craig interpolation is restored in GF if we consider the modal character of GF with more attention, that is, if relations appearing on guards are viewed as “modalities” and the rest as “propositions”, and only the latter enter in the common language. In this paper we strengthen this result by showing that GF enjoys a Modal Uniform Interpolation Theorem (in the sense of Hoogland and Marx)

Uniform Interpolation for Coalgebraic Fixpoint Logic

We use the connection between automata and logic to prove that a wide class of coalgebraic fixpoint logics enjoys uniform interpolation. To this aim, first we generalize one of the central results in coalgebraic automata theory, namely closure under projection, which is known to hold for weak-pullback preserving functors, to a more general class of functors, i.e.; functors with quasi-functorial lax extensions. Then we will show that closure under projection implies definability of the bisimulation quantifier in the language of coalgebraic fixpoint logic, and finally we prove the uniform interpolation theorem

Querying the Guarded Fragment

Evaluating a Boolean conjunctive query Q against a guarded first-order theory F is equivalent to checking whether "F and not Q" is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since Q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment do not obviously carry over to conjunctive query answering over guarded theories, and had been left open in general. By investigating finite guarded bisimilar covers of hypergraphs and relational structures, and by substantially generalising Rosati's finite chase, we prove for guarded theories F and (unions of) conjunctive queries Q that (i) Q is true in each model of F iff Q is true in each finite model of F and (ii) determining whether F implies Q is 2EXPTIME-complete. We further show the following results: (iii) the existence of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof of the finite model property of the clique-guarded fragment; (v) the small model property of the guarded fragment with optimal bounds; (vi) a polynomial-time solution to the canonisation problem modulo guarded bisimulation, which yields (vii) a capturing result for guarded bisimulation invariant PTIME.Comment: This is an improved and extended version of the paper of the same title presented at LICS 201

Lindstrom theorems for fragments of first-order logic

Lindstr\"om theorems characterize logics in terms of model-theoretic conditions such as Compactness and the L\"owenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindstr\"om theorems for these languages can be challenging, as most known techniques rely on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindstr\"om theorems for several fragments of first-order logic, including the k-variable fragments for k>2, Tarski's relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindstr\"om proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems.Comment: Appears in Logical Methods in Computer Science (LMCS

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201