Clausal Resolution for Modal Logics of Confluence

We present a clausal resolution-based method for normal multimodal logics of confluence, whose Kripke semantics are based on frames characterised by appropriate instances of the Church-Rosser property. Here we restrict attention to eight families of such logics. We show how the inference rules related to the normal logics of confluence can be systematically obtained from the parametrised axioms that characterise such systems. We discuss soundness, completeness, and termination of the method. In particular, completeness can be modularly proved by showing that the conclusions of each newly added inference rule ensures that the corresponding conditions on frames hold. Some examples are given in order to illustrate the use of the method.Comment: 15 pages, 1 figure. Preprint of the paper accepted to IJCAR 201

Modal Resolution: Proofs, Layers and Refinements

Resolution-based provers for multimodal normal logics require pruning of the search space for a proof in order to ameliorate the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal level at which clauses occur in order to reduce the number of possible inferences. We show that the calculus is complete for the logics being considered. We also show that the calculus can be combined with other strategies. In particular, we discuss the completeness of combining modal layering with negative and ordered resolution and provide experimental results comparing the different refinements

On Sub-Propositional Fragments of Modal Logic

In this paper, we consider the well-known modal logics $\mathbf{K}$, $\mathbf{T}$, $\mathbf{K4}$, and $\mathbf{S4}$, and we study some of their sub-propositional fragments, namely the classical Horn fragment, the Krom fragment, the so-called core fragment, defined as the intersection of the Horn and the Krom fragments, plus their sub-fragments obtained by limiting the use of boxes and diamonds in clauses. We focus, first, on the relative expressive power of such languages: we introduce a suitable measure of expressive power, and we obtain a complex hierarchy that encompasses all fragments of the considered logics. Then, after observing the low expressive power, in particular, of the Horn fragments without diamonds, we study the computational complexity of their satisfiability problem, proving that, in general, it becomes polynomial

Many-valued logics. A mathematical and computational introduction.

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics

Hyperresolution for guarded formulae

AbstractThis paper investigates the use of hyperresolution as a decision procedure and model builder for guarded formulae. In general, hyperresolution is not a decision procedure for the entire guarded fragment. However we show that there are natural fragments of the guarded fragment which can be decided by hyperresolution. In particular, we prove decidability of hyperresolution with or without splitting for the fragment GF1− and point out several ways of extending this fragment without losing decidability. As hyperresolution is closely related to various tableaux methods the present work is also relevant for tableaux methods. We compare our approach to hypertableaux, and mention the relationship to other clausal classes which are decidable by hyperresolution