. Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established. 1 Introduction The logics that have appeared in artificial intelligence form a rich and varied collection. While classical (and maybe intuitionistic) logic su#ces for the formal development of mathematics, artificial intelligence has found uses for modal, temporal, relevant, and many-valued logics, among others. Indeed, I take it as a basic principle that an application should find (or create) an appropriate logic, if it needs one, rather than reshape the application to fit some narrow class of `established' logics. In this paper I want to enlarge the variety of logics...

Abstract. 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.

Abstract. We define a collection of mappings that transform many-valued clausal forms into satisfiability equivalent Boolean clausal forms, analyze their complexity and evaluate them empirically on a set of benchmarks with state-of-the-art SAT solvers. Our results provide empirical evidence that encoding combinatorial problems with the mappings defined here can lead to substantial performance improvements in complete SAT solvers. 1

1 Introduction Computing the probability of a complex event from the probability of the primitive events constituting it depends upon the dependencies (if any) known to exist between the events being composed. For example, consider two events e1; e2. The probability, P(e1 ^ e2) of the occurrence of both is events is 0 if the events are mutually exclusive. However, if the events are independent, then P(e1 ^ e2) = P(e1) \Theta P(e2). If we are ignorant of the relationship between these two events, then, as stated by Boole[1], the best we can say about P(e1 ^ e2) is that it lies in the interval [max(0; P(e1) + P(e2) \Gamma 1); min(P(e1); P(e2)]. In short, computing the probability of a complex event depends fundamentally upon our knowledge about the dependences between the events involved. In [2] we proposed a language called Hybrid Probabilistic (Logic)

This paper gives an overview of the system with a special focus on the new features of 3 T A P Version 4.0, including: efficient completion-based equality reasoning, methods for handling redundant axiom sets, utilization of pragmatic information contained in axioms to rearrange the search space, and a graphical user interface for controlling 3 T A P and visualizing its output.

this article, there is considerable activity in MVL deduction which is why we felt justified in writing this survey. Needless to say, we cannot give a general introduction to MVL in the present context. For this, we have to refer to general treatments such as [153, 53, 93]. 2 A classification of many-valued logics according to their intended application

This paper tries to identify the basic problems encountered in automated theorem proving in manyvalued logics and demonstrates to which extent they can be currently solved. To this end a number of recently developed techniques are reviewed. We list the avenues of research in many-valued theorem proving that are in our eyes the most promising. 1 Introduction The purpose of this note is to review a number of techniques that lead to a computationally adequate representation of the search space of many-valued logics and to identify the avenues of research in many-valued theorem proving that are in our eyes the most promising. We do not mention the large number of possible applications of many-valued theorem proving, but refer to [15] for an extensive list of applications and to [18] for a case study. If one is doing many-valued deduction, typically a number of problems that are not as much prominent in classical deduction have to be addressed: 1. The number of case distinctions is much la...

We describe the automated theorem prover "Deep Thought" ( d DT ). The prover can be used for arbitrary multiple-valued first-order logics, provided the connectives can be defined by truth tables and the quantifiers are generalizations of the classical universal resp. existential quantifiers. d DT has been tested with many interesting multiple-valued logics as well as classical first-order predicate logic. d DT uses a free-variable semantic tableau calculus with generalized signs. For the existential tableau-rules two liberalized versions are implemented. The system utilizes a static index to control the application of axioms as wells as the search for applicable rules. A dynamic lemma generation strategy and various heuristics to control the tableau expansion and branch closure are integrated into d DT . Theoretically, contradiction sets of arbitrary size can be discovered to close a branch. 1 Introduction Deep Thought ( d DT ) is an automated theorem prover which is able to cope wi...

This paper gives an overview of the system with a special focus on the new features of 3 T A P Version 4.0, including: efficient completion-based equality reasoning, methods for handling redundant axiom sets, utilization of pragmatic information contained in axioms to rearrange the search space, and a graphical user interface for controlling 3 T A P and visualizing its output.

Abstract We report on the results of a long-term project to formalize the semantics of OCL 2.0 in Higher-order Logic (HOL). The ultimate goal of the project is to provide a formalized, machine-checked semantic basis for a theorem proving environment for OCL (as an example for an objectoriented specification formalism) which is as faithful as possible to the original informal semantics. We report on various (minor) inconsistencies of the OCL semantics, discuss the more recent attempt to align the OCL semantics with UML 2.0 and suggest several extensions which make, in our view, OCL semantics more fit for future extensions towards program verifications and specification refinement, which are, in our view, necessary to make OCL more fit for future extensions. 1