Kanger’s Choices in Automated Reasoning

Degtyarev, Anatoli; Воронков, Андрей

doi:10.1007/978-94-010-0630-9_4

Cited by 3 publications

(3 citation statements)

References 41 publications

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“…The strategy was first introduced as dummy instantiation in the seminal work of Kanger [8] (in 1963, i.e., even before the introduction of unification), and later studied under the names subterm instantiation and minus-normalisation [4,5]; the relationship to general Simultaneous Rigid E -unification (SREU) was observed in [3]. The present paper addresses the topic of solving a problem using the restricted strategy in an efficient way and makes the following main contributions:…”

Section: Background and Motivating Examplementioning

confidence: 99%

Efficient Algorithms for Bounded Rigid E-unification

Backeman

Rümmer

2015

Lecture Notes in Computer Science

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Abstract. Rigid E -unification is the problem of unifying two expressions modulo a set of equations, with the assumption that every variable denotes exactly one term (rigid semantics). This form of unification was originally developed as an approach to integrate equational reasoning in tableau-like proof procedures, and studied extensively in the late 80s and 90s. However, the fact that simultaneous rigid E -unification is undecidable has limited the practical relevance of the method, and to the best of our knowledge there is no tableau-based theorem prover that uses rigid E -unification. We recently introduced a new decidable variant of (simultaneous) rigid E -unification, bounded rigid E -unification (BREU), in which variables only represent terms from finite domains, and used it to define a first-order logic calculus. In this paper, we study the problem of computing solutions of (individual or simultaneous) BREU problems. Two new unification procedures for BREU are introduced, and compared theoretically and experimentally.

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Section: Background and Motivating Examplementioning

confidence: 99%

Efficient Algorithms for Bounded Rigid E-unification

Backeman

Rümmer

2015

Lecture Notes in Computer Science

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“…An early solution in the class of "terminating, but incomplete" algorithms for SREU was introduced as dummy instantiation in the seminal work of Kanger [11] (in 1963, i.e., even before the introduction of unification), and later studied under the names subterm instantiation and minus-normalisation [5,6]; the relationship to SREU was observed in [4]. In contrast to full SREU, subterm instantiation only considers E -unifiers where substituted terms are taken from some predefined finite set, which directly implies decidability.…”

Section: Subterm Instantiation and Bounded Rigid E -Unificationmentioning

confidence: 99%

“…In contrast to full SREU, subterm instantiation only considers E -unifiers where substituted terms are taken from some predefined finite set, which directly implies decidability. The impact of subterm instantiation on practical theorem proving was again limited, however, among others because no efficient search procedures for dummy instantiation were available [6].…”

Section: Subterm Instantiation and Bounded Rigid E -Unificationmentioning

confidence: 99%

Free Variables and Theories: Revisiting Rigid E-unification

Backeman

Rümmer

2015

Frontiers of Combining Systems

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Abstract. The efficient integration of theory reasoning in first-order calculi with free variables (such as sequent calculi or tableaux) is a longstanding challenge. For the case of the theory of equality, an approach that has been extensively studied in the 90s is rigid E -unification, a variant of equational unification in which the assumption is made that every variable denotes exactly one term (rigid semantics). The fact that simultaneous rigid E -unification is undecidable, however, has hampered practical adoption of the method, and today there are few theorem provers that use rigid E -unification. One solution is to consider incomplete algorithms for computing (simultaneous) rigid E -unifiers, which can still be sufficient to create sound and complete theorem provers for first-order logic with equality; such algorithms include rigid basic superposition proposed by Degtyarev and Voronkov, but also the much older subterm instantiation approach introduced by Kanger in 1963 (later also termed minus-normalisation). We introduce bounded rigid E -unification (BREU) as a new variant of Eunification corresponding to subterm instantiation. In contrast to general rigid E -unification, BREU is NP-complete for individual and simultaneous unification problems, and can be solved efficiently with the help of SAT; BREU can be combined with techniques like congruence closure for ground reasoning, and be used to construct theorem provers that are competitive with state-of-the-art tableau systems. We outline ongoing research how BREU can be generalised to other theories than equality.

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Herbrand Proofs and Expansion Proofs as Decomposed Proofs

Ralph

2020

Journal of Logic and Computation

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The reduction of undecidable first-order logic to decidable propositional logic via Herbrand’s theorem has long been of interest to theoretical computer science, with the notion of a Herbrand proof motivating the definition of expansion proofs. In this paper we construct simple deep inference systems for first-order logic, both with and without cut, such that ‘decomposed’ proofs—proofs where the contractive and non-contractive behaviour of the proof is separated—in each system correspond to either expansion proofs or Herbrand proofs. Translations between proofs in this system, expansion proofs and Herbrand proofs are given, retaining much of the structure in each direction.

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Kanger’s Choices in Automated Reasoning

Cited by 3 publications

References 41 publications

Efficient Algorithms for Bounded Rigid E-unification

Efficient Algorithms for Bounded Rigid E-unification

Free Variables and Theories: Revisiting Rigid E-unification

Herbrand Proofs and Expansion Proofs as Decomposed Proofs

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