scholarly journals Generalized Completeness for SOS Resolution and its Application to a New Notion of Relevance

2021 ◽  
pp. 327-343
Author(s):  
Fajar Haifani ◽  
Sophie Tourret ◽  
Christoph Weidenbach
Keyword(s):  
Automated Reasoning ◽  
First Order ◽  
Completeness Result ◽  
Resolution Refutation ◽  
Unsatisfiable Core

AbstractWe prove the SOS strategy for first-order resolution to be refutationally complete on a clause set N and set-of-support S if and only if there exists a clause in S that occurs in a resolution refutation from $$N\cup S$$ N ∪ S . This strictly generalizes and sharpens the original completeness result requiring N to be satisfiable. The generalized SOS completeness result supports automated reasoning on a new notion of relevance aiming at capturing the support of a clause in the refutation of a clause set. A clause C is relevant for refuting a clause set N if C occurs in every refutation of N. The clause C is semi-relevant, if it occurs in some refutation, i.e., if there exists an SOS refutation with set-of-support $$S = \{C\}$$ S = { C } from $$N\setminus \{C\}$$ N \ { C } . A clause that does not occur in any refutation from N is irrelevant, i.e., it is not semi-relevant. Our new notion of relevance separates clauses in a proof that are ultimately needed from clauses that may be replaced by different clauses. In this way it provides insights towards proof explanation in refutations beyond existing notions such as that of an unsatisfiable core.

scholarly journals Model Completeness, Uniform Interpolants and Superposition Calculus

2021 ◽  
Author(s):  
Diego Calvanese ◽  
Silvio Ghilardi ◽  
Alessandro Gianola ◽  
Marco Montali ◽  
Andrey Rivkin
Keyword(s):  
Automated Reasoning ◽  
Quantifier Elimination ◽  
Effective Solution ◽  
Model Completeness ◽  
Present Contribution ◽  
First Order ◽  
Reduction Strategies ◽  
Infinite State Systems ◽  
Infinite State ◽  
Superposition Calculus

AbstractUniform interpolants have been largely studied in non-classical propositional logics since the nineties; a successive research line within the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. This further research line is motivated by the fact that uniform interpolants offer an effective solution to tackle quantifier elimination and symbol elimination problems, which are central in model checking infinite state systems. This was first pointed out in ESOP 2008 by Gulwani and Musuvathi, and then by the authors of the present contribution in the context of recent applications to the verification of data-aware processes. In this paper, we show how covers are strictly related to model completions, a well-known topic in model theory. We also investigate the computation of covers within the Superposition Calculus, by adopting a constrained version of the calculus and by defining appropriate settings and reduction strategies. In addition, we show that computing covers is computationally tractable for the fragment of the language used when tackling the verification of data-aware processes. This observation is confirmed by analyzing the preliminary results obtained using the mcmt tool to verify relevant examples of data-aware processes. These examples can be found in the last version of the tool distribution.

scholarly journals Exploring Steinitz-Rademacher Polyhedra: a Challenge for Automated Reasoning Tools

10.29007/d3ls ◽  
2018 ◽  
Author(s):  
Jesse Alama
Keyword(s):  
Automated Reasoning ◽  
Order Theory ◽  
Incidence Structures ◽  
First Order ◽  
First Order Theory

This note reports on some experiments, using a handful of standard automated reasoning tools, for exploring Steinitz-Rademacher polyhedra, which are models of a certain first-order theory of incidence structures. This theory and its models, even simple ones, presents significant, geometrically fascinating challenges for automated reasoning tools are.

scholarly journals Reasoning Inside The Box: Deduction in Herbrand Logics

10.29007/kx2m ◽  
2018 ◽  
Author(s):  
Liron Cohen ◽  
Yoni Zohar
Keyword(s):  
Automated Reasoning ◽  
Proof System ◽  
First Order ◽  
Proof Systems ◽  
The Rich ◽  
Order Structures ◽  
Effective Proof

Herbrand structures are a subclass of standard first-order structures commonly used in logic and automated reasoning due to their strong definitional character. This paper is devoted to the logics induced by them: Herbrand and semi-Herbrand logics, with and without equality. The rich expressiveness of these logics entails that there is no adequate effective proof system for them. We therefore introduce infinitary proof systems for Herbrand logics, and prove their completeness. Natural and sound finitary approximations of the infinitary systems are also presented.

A THREE-VALUED QUANTIFIED ARGUMENT CALCULUS: DOMAIN-FREE MODEL-THEORY, COMPLETENESS, AND EMBEDDING OF FOL

2017 ◽  
Vol 10 (3) ◽  
pp. 549-582 ◽  
Author(s):  
RAN LANZET
Keyword(s):  
Natural Language ◽  
Model Theory ◽  
Predicate Calculus ◽  
Extended Version ◽  
First Order ◽  
Completeness Result ◽  
Free Model ◽  
Simplifying Assumptions ◽  
Syntactic Restrictions ◽  
Order Predicate Calculus

AbstractThis paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible.

scholarly journals A proof of completeness for continuous first-order logic

2010 ◽  
Vol 75 (1) ◽  
pp. 168-190 ◽  
Author(s):  
Itaï Ben Yaacov ◽  
Arthur Paul Pedersen
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Complete Theory ◽  
Strong Completeness ◽  
First Order ◽  
Completeness Result ◽  
Metric Structures ◽  
Natural Classes

AbstractContinuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies anapproximatedform of strong completeness, whereby Σ⊧φ(if and) only if Σ⊢φ∸2−nfor alln < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth ofφ.Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theoryTisdecidableif for every sentenceφ, the valueφTis a recursive real, and moreover, uniformly computable fromφ. IfTis incomplete, we say it is decidable if for every sentenceφthe real numberφTois uniformly recursive fromφ, whereφTois the maximal value ofφconsistent withT. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

scholarly journals Automated reasoning-alternative methods

2004 ◽  
Vol 1 (3) ◽  
pp. 15-20
Author(s):  
Aleksandar Perovic ◽  
Nedeljko Stefanovic ◽  
Milos Milosevic ◽  
Dejan Ilic
Keyword(s):  
Automated Reasoning ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Alternative Methods ◽  
Formal Representation ◽  
First Order ◽  
First Order Theory ◽  
Interpretation Method

Our main goal is to describe a potential usage of the interpretation method (i.e. formal representation of one first order theory into another) together with quantifier elimination procedures developed in the GIS.

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