# Semantic Tableaux with Equality

@article{Beckert1997SemanticTW,
title={Semantic Tableaux with Equality},
author={Bernhard Beckert},
journal={J. Log. Comput.},
year={1997},
volume={7},
pages={39-58}
}
This paper tries to identify the basic problems encountered in handling equality in the semantic tableau framework, and to describe the state of the art in solving these problems. The two main paradigms for handling equality are compared: adding new tableau expansion rules and using E-unification algorithms.
20 Citations

#### Figures, Tables, and Topics from this paper

Equality and Other Theories
The general purpose foreground reasoner calls a special purpose background reasoner to handle problems from a certain theory to increase the efficiency of automated deduction systems. Expand
The Hyper Tableaux Calculus with Equality
In most theorem proving applications, a proper treatment of equational theories or equality is mandatory. In this paper we show how to integrate a modern treatment of equality in the hyper tableauExpand
Hyper Tableaux with Equality
• Mathematics, Computer Science
• 2007
This paper shows how to integrate a modern treatment of equality in the hyper tableau calculus based on splitting of positive clauses and an adapted version of the superposition inference rule, where equations used for paramodulation are drawn from a set of positive unit clauses, the candidate model. Expand
Modal Tableau Systems with Blocking and Congruence Closure
• Mathematics, Computer Science
• TABLEAUX
• 2015
Congruence closure is integrated into a modal tableau system with equality-based blocking for an extension of modal logic characterised by frames in which the accessibility relation is transitive and every world has a distinct immediate predecessor. Expand
Simplification: A General Constraint Propagation Technique for Propositional and Modal Tableaux
The rule of simplification is proposed, which plays for tableaux the role of subsumption for resolution and of unit for the Davis-Putnam procedure, and gives an unifying view of a number of tableaux-like calculi such as DPLL, KE, HARP, hyper-tableaux, BCP, KSAT. Expand
An algorithm based on equivalence classes for tableau with equality
• Mathematics, Medicine
• Proceedings of 2004 International Conference on Machine Learning and Cybernetics (IEEE Cat. No.04EX826)
• 2004
This paper introduces an algorithm based on equivalence classes for tableau with equality - CECA, which use the method based on the transformation into disjunctions of inequalities and theExpand
The Hyper Tableaux Calculus with Equality and an Application to Finite Model Computation
• Mathematics, Computer Science
• J. Log. Comput.
• 2010
This article shows how to integrate a modern treatment of equality in the hyper tableau calculus based on splitting of positive clauses and an adapted version of the superposition inference rule, where equations used for superposition are drawn from a set of positive unit clauses, and superposition inferences into positive literals is restricted into (positive) unit clauses only. Expand
Simpli cation with renaming: A general proof technique for tableau and sequent-based provers
Tableau and sequent calculi are the basis for most popular interactive theorem provers for hardware and software veri cation. Yet, when it comes to decision procedures or automatic proof search,Expand
A Taxonomy of Theorem-Proving Strategies
• M. Bonacina
• Computer Science
• Artificial Intelligence Today
• 1999
This article presents a taxonomy of strategies for fully-automated general-purpose first-order theorem proving. It covers forward-reasoning ordering-based strategies and backward-reasoningExpand
Equational Reasoning in Cocktail
This thesis describes an extension of the tableau method to allow reasoning with equalities, and introduces sets of constraints to rewrite rigid E-unification problems into more simple ones. Expand

#### References

SHOWING 1-10 OF 32 REFERENCES
An Improved Method for Adding Equality to Free Variable Semantic Tableaux
• Mathematics, Computer Science
• 1992
This work identifies the problems that stem specifically from choosing semantic tableaux as a framework and state soundness and completeness results for the method. Expand
An algorithm for reasoning about equality
A simple technique for reasoning about equalities that is fast and complete for ground formulas with function symbols and equality is presented. A proof of correctness is given as well.
Equality-based binary resolution
• Computer Science
• JACM
• 1986
Equality-based binary resolution is complete without making use of paramodulation and leads to refutations that are less than half as long as standard refutations with the equality axioms. Expand
The Even More Liberalized delta-Rule in Free Variable Semantic Tableaux
• Mathematics, Computer Science
• Kurt Gödel Colloquium
• 1993
It is shown that, an even more liberalized version, the $$\delta ^{ + ^ + }$$-rule, that in addition reduces the number of different Skolem-function symbols that have to be used, is also sound and complete. Expand
Simultaneous Regid E-Unification Is Undecidable
• Mathematics, Computer Science
• CSL
• 1995
The undecidability of simultaneous rigid E-unification is proved using reduction of Hubert's tenth problem, and the ∃*-fragment of intuitionistic logic with equality and representability of recursively enumerable sets is obtained. Expand
Proving Theorems with the Modification Method
• D. Brand
• Mathematics, Computer Science
• SIAM J. Comput.
• 1975
Completeness of this “Modification Method” implies completeness of Paramodulation without the functionally reflexive axioms, thus proving a conjecture of Wos and Robinson (1969), and holds with some other restrictions, such as limiting paramodulation into variables. Expand
A Completion-Based Method for Mixed Universal and Rigid E-Unification
A completion-based method for handling a new version of E- unification, called “mixed” E-unification, that is a combination of the classical “universal” and “rigid” versions, and state soundness and completeness results, and describe experiments with an implementation of the method. Expand
Completion without Failure 1
• 1989
We present an \unfailing" extension of the standard Knuth-Bendix completion procedure that is guaranteed to produce a desired canon-ical system, provided certain conditions are met. We prove thatExpand
Basic Narrowing Revisited
• Computer Science, Mathematics
• J. Symb. Comput.
• 1989
This paper presents an equation solving calculus that features three classes of rules, whose application is don't care nondeterministic, that enhance the power of the failure rules and reduce the number of necessary don't know steps. Expand
Rigid Unification by Completion and Rigid Paramodulation
• Computer Science
• KI
• 1994
This paper addresses the problem of computing complete sets of rigid-E-unifiers, initially introduced by Gallier, Narendran, Plaisted and Snyder, with a proof of its NP-completeness and proves soundness, completeness and termination of the algorithm. Expand