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 describe how a combination of the classical “universal” E-unification and “rigid”E-unification, called “mixed” E-unification, can be used to efficiently handle equality in universal formula semantic tableaux, that are an extension of free variable tableaux.

Even though higher-order calculi for automated theorem proving are rather old, tableau calculi have not been investigated yet. This paper presents two free variable tableau calculi for higher-order logic that use higher-order uniication as the key inference procedure. These calculi diier in the treatment of the substitutional properties of equivalences. The rst calculus is equivalent in deducti...

implements a rst-order theorem prover based on free-variable semantic tableaux. It is complete, sound, and eecient.

A recent development in the theory of M-sets has paved a new means of an approach to CH. Here we further clarify the theory and form part two of our previous paper[P]. The set B i will be used to represent binary numbers. A set of positions associated with a one symbol we will denote by pos Elements of pos we will denote by p(1) j. Sets of transformations T i. A knowable set H is denoted by K(H...

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 usele...

“prove((E,F),A,B,C,D) :!, prove(E,[F|A],B,C,D). prove((E;F),A,B,C,D) :!, prove(E,A,B,C,D), prove(F,A,B,C,D). prove(all(H,I),A,B,C,D) :!, \+length(C,D), copy_term((H,I,C),(G,F,C)), append(A,[all(H,I)],E), prove(F,E,B,[G|C],D). prove(A,_,[C|D],_,_) :((A= -(B); -(A)=B)) -> (unify(B,C); prove(A,[],D,_,_)). prove(A,[E|F],B,C,D) :prove(E,F,[A|B],C,D).” implements a first-order theorem prover based on...

We present several calculi that integrate equality handling by superposition and ordered paramodulation into a free variable tableau calculus. We prove completeness of this calculus by an adaptation of themodel generation [1, 13] technique commonly used for completeness proofs of resolution calculi. The calculi and the completeness proof are compared to earlier results of Degtyarev and Voronkov...