Topic
Beth definability
About: Beth definability is a research topic. Over the lifetime, 60 publications have been published within this topic receiving 1334 citations.
Papers published on a yearly basis
Papers
TL;DR: The Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms, showing that the expressive power of each first-order system is rounded out, or the system is functionally complete.
Abstract: One task of metamathematics is to relate suggestive but nonelementary modeltheoretic concepts to more elementary proof-theoretic concepts, thereby opening up modeltheoretic problems to proof-theoretic methods of attack. Herbrand's Theorem (see [8] or also [9], vol. 2) or Gentzen's Extended Hauptsatz (see [5] or also [10]) was first used along these lines by Beth [1]. Using a modified version he showed that for all first-order systems a certain modeltheoretic notion of definability coincides with a certain proof theoretic notion. In the present paper the Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms.This may be interpreted as showing that (apart from some relatively minor exceptions which will be made apparent below) the expressive power of each first-order system is rounded out, or the system is functionally complete, in the following sense: Any functional relationship which obtains between concepts that are expressible in the system is itself expressible and provable in the system.A second application is concerned with the hierarchy of second-order formulas. A certain relationship is shown to hold between first-order formulas and those second-order formulas which are of the form (∃T1)…(∃Tk)A or (T1)…(Tk)A with A being a first-order formula. Modeltheoretically this can be regarded as a relationship between the class AC and the class PC⊿ of sets of models, investigated by Tarski in [12] and [13].
674 citations
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1 Jan 2007
TL;DR: This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics that uses an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual.
Abstract: This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum algebra construction, its generalization to a Lindenbaum category construction that includes proofs, and model cardinality spectra; these are used in some examples to show logics inequivalent. Generalizations of familiar results from first order to arbitrary logics are also discussed, including Craig interpolation and Beth definability.
66 citations
26 Aug 2013
TL;DR: This paper establishes that GNFO queries have distinctive features with respect to rewriting, and results include effective preservation theorems for GNFO, Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries withrespect to GNFO constraints within very restricted logics.
Abstract: The Guarded Negation Fragment (GNFO) is a fragment of first-order logic that contains all unions of conjunctive queries, a restricted form of negation that suffices for expressing some common uses of negation in SQL queries, and a large class of integrity constraints. At the same time, as was recently shown, the syntax of GNFO is restrictive enough so that static analysis problems such as query containment are still decidable. This suggests that, in spite of its expressive power, GNFO queries are amenable to novel optimizations. In this paper we provide further evidence for this, establishing that GNFO queries have distinctive features with respect to rewriting. Our results include effective preservation theorems for GNFO, Craig Interpolation and Beth Definability results, and the ability to express the certain answers of queries with respect to GNFO constraints within very restricted logics.
63 citations
TL;DR: A complete classification of Beth definability is provided for extensions of the basic description logic ALC with transitive roles, inverse roles, role hierarchies, and/or functionality restrictions, both on arbitrary and on finite structures.
Abstract: The Beth definability property, a well-known property from classical logic, is investigated in the context of description logics: if a general L-TBox implicitly defines an L-concept in terms of a given signature, where L is a description logic, then does there always exist over this signature an explicit definition in L for the concept? This property has been studied before and used to optimize reasoning in description logics. In this paper a complete classification of Beth definability is provided for extensions of the basic description logic ALC with transitive roles, inverse roles, role hierarchies, and/or functionality restrictions, both on arbitrary and on finite structures. Moreover, we present a tableau-based algorithm which computes explicit definitions of at most double exponential size. This algorithm is optimal because it is also shown that the smallest explicit definition of an implicitly defined concept may be double exponentially long in the size of the input TBox. Finally, if explicit definitions are allowed to be expressed in first-order logic, then we show how to compute them in single exponential time.
63 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 6 |
| 2020 | 12 |
| 2019 | 2 |
| 2018 | 3 |
| 2017 | 1 |