Reiner Hähnle

Handbook of tableau methods

Preface. Introduction M. Fitting. Tableau Methods for Classical Propositional Logic M. DAgostino. First-Order Tableau Methods R. Letz. Equality and Other Theories B. Beckert. Tableaux for Intuitionistic Logics A. Waaler, L. Wallen. Tableau Methods for Modal and Temporal Logics R. Gore. Tableau Methods for Substructural Logics M. DAgostino, et al. Tableaux for Nonmonotonic Logics N. Olivetti. Tableaux for Many-valued Logics R. Hahnle. Implementing Semantic Tableaux J. Posegga, P. Schmitt. A Bibliography on Analytic Tableaux Theorem Proving G. Wrightson. Index.

The Liberalized delta-Rule in Free Variable Semantic Tableaux

In this paper we have a closer look at one of the rules of the tableau calculus presented by Fitting [4], called the δ-rule. We prove that a modification of this rule, called the δ+-rule, which uses fewer free variables, is also sound and complete. We examine the relationship between the δ+-rule and variations of the δ-rule presented by Smullyan [9]. This leads to a second proof of the soundness of the δ+-rule. An example shows the relevance of this modification for building tableau-based theorem provers.

Uniform notation of tableau rules for multiple-valued logics

A framework for axiomatizing arbitrary finitely valued logics with minimal overhead compared to the classical case is presented. The main idea is to work with tableaux using generalized signs, which makes it possible to express complex assertions regarding the possible truth values of a formula. The class of regular logical connectives which, together with a suitable restriction on queries (i.e. allowed signs) to the system, allow a uniform notation style representation of multiple-valued propositional and first-order logics is introduced. It has been demonstrated that various systems differing in their allowed classes of connectives and complexity, of rules may be formulated. This allows the use of tools and methods that are close to the ones used in classical logic, both on the theoretical (uniform notation in definitions and proofs) and practical (use of classical theorem provers with few modifications) sides. >A framework for axiomatizing arbitrary finitely valued logics with minimal overhead compared to the classical case is presented. The main idea is to work with tableaux using generalized signs, which makes it possible to express complex assertions regarding the possible truth values of a formula. The class of regular logical connectives which, together with a suitable restriction on queries (i.e. allowed signs) to the system, allow a uniform notation style representation of multiple-valued propositional and first-order logics is introduced. It has been demonstrated that various systems differing in their allowed classes of connectives and complexity, of rules may be formulated. This allows the use of tools and methods that are close to the ones used in classical logic, both on the theoretical (uniform notation in definitions and proofs) and practical (use of classical theorem provers with few modifications) sides.<<ETX>>

The Even More Liberalized delta-Rule in Free Variable Semantic Tableaux

In this paper we have a closer look at one of the rules of the tableau calculus presented in [3], called the δ-rule, and the modification of this rule, that has been proved to be sound and complete in [6], called the δ+-rule, which uses fewer free variables. We show 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. Examples show the relevance of this modification for building tableau-based theorem provers.

Many-valued logic and mixed integer programming

We generalize prepositional semantic tableaux for classical and many-valued logics toconstraint tableaux. We show that this technique is a generalization of the standard translation from CNF formulas into integer programming. The main advantages are (i) a relatively efficient satisfiability checking procedure for classical, finitely-valued and, for the first time, for a wide range of infinitely-valued propositional logics; (ii) easy NP-containment proofs for many-valued logics. The standard translation of two-valued CNF formulas into integer programs and Tseitins structure preserving clause form translation are obtained as a special case of our approach.

Towards an Efficient Tableau Proof Procedure for Multiple-Valued Logics

One of the obstacles against the use of tableau-based theorem provers for non-standard logics is the inefficiency of tableau systems in practical applications, though they are highly intuitive and extremely flexible from a proof theoretical point of view. We present a method for increasing the efficiency of tableau systems in the case of multiple-valued logics by introducing a generalized notion of signed formulas and give sound and complete tableau systems for arbitrary propositional finite-valued logics.

Transformations between signed and classical clause logic

In the last years two automated reasoning techniques for clause normal form arose in which the use of labels are prominently featured: signed logic and annotated logic programming, which can be embedded into the first. The underlying basic idea is to generalise the classical notion of a literal by adorning an atomic formula with a sign or label which in general consists of a possibly ordered set of truth values. In this paper we relate signed logic and classical logic more closely than before by defining two new transformations between them. As a byproduct we obtain a number of new complexity results and proof procedures for signed logics.

The SAT problem of signed CNF formulas

Signed conjunctive normal form (signed CNF) is a classical conjunctive clause form using a generalised notion of literal, called signed literal.A signed literal is an expression of the form S:p, where p is a classical atom and S, its sign, is a subset of a domain N.The informal meaning is “p takes one of the values in S”.Signed formulas are a logical language for knowledge representation that lies in the intersection of the areas constraint programming (CP) many-valued logic (MVL), and annotated logic programming (ALP). This central role of signed CNF justifies a detailed study of its subclasses including algorithms for and complexities of associated satisfiability problems (SAT problems). Although signed logic is used since the 1960s, there are only few systematic investigations of its properties. In contrast to work done in ALP and MVL, our present work is a more fine-grained study for the case of propositional CNF. We highlight the most interesting lines of current research: (i) signed versions of some main proponents of classical deduction systems including non-trivial refinements having no classical counterpart; (ii) incomplete local search methods for satisfiability checking of signed formulas; (iii) phase transition phenomena as known, for example, from classical SAT and the influence of the cardinality of N on the crossover point; (iv) the complexity of the SAT problem for signed CNF and its subclasses.

An Improved Method for Adding Equality to Free Variable Semantic Tableaux

Tableau-Based theorem provers can be extended to cover many of the nonclassical logics currently used in AI research. For both, classical and nonclassical first-order logic, equality is a crucial feature to increase expressivity of the object language. Unfortunately, all so far existing attempts of adding equality to semantic tableaux have been more or less experimental and turn out to be useless in practice. In the present work we introduce an approach that leads much further and sets the stage for more advanced developments. We identify the problems that stem specifically from choosing semantic tableaux as a framework and state soundness and completeness results for our method.

Proof theory of many-valued logic—linear optimization—logic design: connections and interactions

Abstractu2002In this paper proof theory of many-valued logic is connected with areas outside of logic, namely, linear optimization and computer aided logic design. By stating these not widely-known connections explicitly, I want to encourage interaction between the mentioned disciplines. Once familiar with the others’ terminology, I believe that the respective communities can greatly benefit from each other.