We present a sound, complete, modular and lean labelled tableau calculus for many propositional modal logics where the labels contain \free" and \universal" variables. Our \lean" Prolog implementation is not only surprisingly short, but compares favourably with other considerably more complex implementations for modal deduction.

In this paper, we present a further liberalization of the-rule in free variable semantic tableaux. It is eeective in that (1) it is both a natural and intuitive liberalization, and (2) can reduce the proof size non elementarily as compared to previous versions of the-rule.

We apply the framework of the generic δ-rule presented in [4] to show how to map the δ-rule, which uses ǫ-terms as syntactical objects to expand existentially quantified formulae, in the context of standard δ-rules based on Skolem terms. Structural results coming out from such mapping process are discussed.

In this paper we present a three valued many sorted logic for dealing with preorders, incorporating subsort relations into the syntax of the language, and where formulas taking the third boolean value as interpretation contain a term or a predicate not well-sorted w.r.t. the signature. For this logic a ground tableau-based deduction method and a free variable extension version are proposed, pro...

We present a sound, complete, modular and lean labelled tableau calculus for many propositional modal logics where the labels contain \free" and \universal" variables. Our \lean" Prolog implementation is not only surprisingly short, but compares favourably with other considerably more complex implementations for modal deduction.

Labelled tableaux are extensions of semantic tableaux with annotations (labels, indices) whose main function is to enrich the modal object language with semantic elements. This paper consists of three parts. In the first part we consider some options for labels: simple constant labels vs labels with free variables, logic depended inference rules vs labels manipulation based on a label algebra. ...

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 δ + –rule, that in addition reduces the number of different Skolem–function symbols that...

Different theorem provers tend to produce proof objects in different formats and this is especially the case for modal logics, where several deductive formalisms (and provers based on them) have been presented. This work falls within the general project of establishing a common specification language in order to certify proofs given in a wide range of deductive formalisms. In particular, by usi...

In this paper we present a logic for dealing with preorders, where functions and predicates behave monotonically or antimonotonically in their arguments, and incorporating order-sorted relations into the syntax of the language. For this logic a ground tableau-based deduction method and a free variable extension version are proposed, proving their soundness and completeness. Finally, an implemen...

We present a calculus that integrates equality handling by superposition into a free variable tableau calculus. We prove completeness of this calculus by an adaptation of the model generation [1, 15] technique commonly used for completeness proofs of resolution calculi. The calculi and the completeness proof are compared to earlier results of Degtyarev and Voronkov [7].