We prove completeness and decidability results for the temporal logic CTL in Coq/Ssreflect. Our basic result is a constructive proof that for every formula one can obtain either a finite model satisfying the formula or a proof in a Hilbert system certifying the unsatisfiability of the formula. The proof is based on a history-augmented tableau system obtained as the dual of Brünnler and Lange’s ...

Tableau calculi are the state-of-the-art for reasoning in description logics (DL). Despite recent improvements, tableau-based reasoners still cannot process certain knowledge bases (KBs), mainly because they end up building very large models. To address this, we propose a tableau calculus with individual reuse: to satisfy an existential assertion, our calculus nondeterministically tries to reus...

Redundancy is a substantial concept when avoiding endless computation in theorem proving methods. In this paper, we are targeting a special class of tableau calculi, namely rigid clausal tableaux. These tableaux are (1) clausal in the sense that they consist solely of literals, and are (2) rigid in the sense that they employ rigid variables. Such calculi are e.g. rigid hyper tableaux [7] and mu...

1 Introduction: In [1] Fitch proposed a new proof proceedure for several standard modal logics. The chief characteristic of this was the inclusion in the object language, of symbols representing worlds in Kripke models. In this paper we incorporate the device into a tableau proof system and it is seen that the resulting (propositional) proof system is highly analogous to a classical first order...

In this paper we classify the irreducible representations of the symmetric group Sn and give a proof of the hook formula for the dimension of each irreducible. 1. The Irreducible Representations of Sn We construct the irreducible representations of the symmetric group. The number of irreducible representations of Sn is the number of conjugacy classes of Sn, which is the number of partitions of ...

Partitions can be graphically represented by Young frames, which are Young tableaux with empty boxes. The i-th part λi corresponds to the i-th row of the frame, consisting of λi boxes. Conversely, the Young frames of n boxes can be uniquely labelled by a partition λ ` n. We will therefore identify a Young frame with the partition labelling it. A Young tableau (YT) of N objects and of shape λ ` ...

Due to the wide range of modelling constructs supported by the expressive DLSROIQ, the typically used tableau algorithms in competitive reasoning systems such as FaCT++ [16], HermiT,3 or Pellet [14] have a very high worst-case complexity. The development of tableau optimisations that help to achieve practical efficiency is, therefore, a longstanding challenge in DL research (see, e.g., [11, 17]...

Interval temporal logics are difficult to deal with in many respects. In the last years, various meaningful fragments of Halpern and Shoham’s modal logic of time intervals have been shown to be decidable with complexities that range from NP-complete to non-primitive recursive. However, even restricting the attention to finite interval structures, the step from modeltheoretic decidability result...

Abstract We describe , an automated theorem prover for first-order logic that relies on a concurrent search procedure to find tableau proofs, with processes corresponding individual branches of the tableau. Since branch closure may require instantiating free variables shared across branches, communicate via channels exchange information about substitutions used closure. present proof and its im...

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.