On the existence of polynomial time algorithms for interpolation problems in propositional logic (original) (raw)

From Strong Amalgamability to Modularity of Quantifier-Free Interpolation

arXiv (Cornell University), 2012

The use of interpolants in verification is gaining more and more importance. Since theories used in applications are usually obtained as (disjoint) combinations of simpler theories, it is important to modularly re-use interpolation algorithms for the component theories. We show that a sufficient and necessary condition to do this for quantifierfree interpolation is that the component theories have the 'strong (sub-)amalgamation' property. Then, we provide an equivalent syntactic characterization, identify a sufficient condition, and design a combined quantifier-free interpolation algorithm capable of handling both convex and non-convex theories, that subsumes and extends most existing work on combined interpolation.

A Note on the Interpolation Theorem in First Order Logic

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1982

1. The purpose of this paper is to present a model theoretic argument for the interpolation theorem. Model theoretic arguments in the literature are of three basic types : BUCHI-CRAIG, ROBINSON and HENKIN. BUCHI-CRAIG arguments formulate the interpolation theorem in terms of pseudoelementary classes ([3]). Here there are both algebraic and topological arguments. Algebraic arguments include those using ultrapowers ([7]) and ultralimits ([2]), and topological arguments include those which put a topology on the space of the models and use the normality of this topology to establish the interpolation theorem ([14]). ROBIXSON arguments prove the interpolation theorem from ROBINSON'S joint consistency lemma. Arguments here are distinguished one from another by the techniques used to prove ROBINSON'S lemma: elementary chains ([l], [ 5 ] ) ; saturated and special models ([4]. [ 5 ] ) ; and recursively saturated models ([l], [9]). HENKIN arguments are based on refinements of the HENKIN style proof of the compactness theorem. These include those using separable pairs of theories ([6], [5]), and thoye using consistency properties ([12], [ll], [8], [El).

Failure of interpolation in relevant logics

Journal of Philosophical Logic, Vol. 22, 449-479, 1993

Craig's interpolation theorem fails for the propositional logics E of entailment, R of relevant implication and T of ticket entailment, as well as in a large class of related logics. This result is proved by a geometrical construction, using the fact that a non-Arguesian projective plane cannot be imbedded in a three-dimensional projective space. The same construction shows failure of the amalgamation property in many varieties of distributive lattice-ordered monoids.

Complexity Results for Quantified Boolean Formulae Based on Complete Propositional Languages ∗ Sylvie Coste-Marquis

2005

Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.

A non-classical refinement of the interpolation property for classical propositional logic

2015

We refine the interpolation property of the {∧,∨,¬}-fragment of classical propositional logic, showing that if 2 ¬φ, 2 ψ and φ ψ then there is an interpolant χ, constructed using at most atomic formulas occurring in both φ and ψ and negation, conjunction and disjunction, such that (i) φ entails χ in Kleene’s strong three-valued logic and (ii) χ entails ψ in Priest’s Logic of Paradox.

Quantifier-free interpolation in combinations of equality interpolating theories

ACM Transactions on Computational Logic, 2014

The use of interpolants in verification is gaining more and more importance. Since theories used in applications are usually obtained as (disjoint) combinations of simpler theories, it is important to modularly reuse interpolation algorithms for the component theories. We show that a sufficient and necessary condition to do this for quantifier-free interpolation is that the component theories have the strong ( sub -) amalgamation property. Then, we provide an equivalent syntactic characterization and show that such characterization covers most theories commonly employed in verification. Finally, we design a combined quantifier-free interpolation algorithm capable of handling both convex and nonconvex theories; this algorithm subsumes and extends most existing work on combined interpolation.

Combination of Uniform Interpolants via Beth Definability

Journal of Automated Reasoning

Uniform interpolants were largely studied in non-classical propositional logics since the nineties, and their connection to model completeness was pointed out in the literature. A successive parallel research line inside the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. In this paper, we investigate cover transfer to theory combinations in the disjoint signatures case. We prove that, for convex theories, cover algorithms can be transferred to theory combinations under the same hypothesis needed to transfer quantifier-free interpolation (i.e., the equality interpolating property, aka strong amalgamation property). The key feature of our algorithm relies on the extensive usage of the Beth definability property for primitive fragments to convert implicitly defined variables into their explicitly defining terms. In the non-convex case, we show by a counterexample that covers may not exist in t...

Complexity results for quantified boolean formulae based on complete propositional languages

2006

L O ] 1 0 A ug 2 01 8 Interpolation in extensions of first-order logic

2018

We provide a constructive proof of the interpolation theorem for extensions of classical first order logic with a special type of geometric axioms, called singular geometric axioms. As a corollary, we obtain a direct proof of interpolation for first-order logic with identity. Interpolation is a central result in first-order logic. It asserts that for any theorem A → B there exists a formula C, called interpolant, such that A → C and C → B are also theorems and C only contains non-logical symbols that are contained in both A and B. The aim of this paper is to extend interpolation beyond first-order logic. In particular, we show how to prove interpolation in Gentzen’s sequent calculi with singular geometric rules, a special case of geometric rules investigated in [5]. Interestingly, singular geometric rules include the rules for first-order logic with identity (as well as those for the theory of strict partial orders and other order theories). Thus, from interpolation for singular geo...

An interpolation theorem in equational logic

1988

The Centre for Mathematics and Computer Science is a research institute of the Stichting Mathematisch Centrum, which was founded on February 11 , 1946, as a nonprofit institution aiming at the promotion of mathematics, computer science, and their applications. It is sponsored by the Dutch Government through the Netherlands Organization for the Advancement of Pure Research (Z.W.0.).

On the existence of polynomial time algorithms for interpolation problems in propositional logic (original) (raw)

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