Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric $d_T$ in the general case of a real linear space $X$ ordered by a cone $K$. We show that $d_T$ has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of $d_T$ and some results concerning the topology of $d_T$, including a brief study of the $d_T$-convergence of monotone sequences. It is shown most of the results are true without any assumption of an Archimedean-type property for $K$. One considers various completeness properties and one studies the relations between them. Since $d_T$ is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering, with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. The Thompson metric $d_T$ and order-unit (semi)norms $|\cdot|_u$ are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although $d_T$ and $|\cdot|_u$ are only topological (and not metrical) equivalent on $K_u$, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.Comment: 36 page

On completeness results for predicate lukasiewicz, product, gödel and nilpotent minimum logics expanded with truth-constants

In this paper we deal with generic expansions of first-order predicate logics of some left-continuous t-norms with a countable set of truth-constants. Besides already known results for the case of Lukasiewicz logic, we obtain new conservativeness and completeness results for some other expansions. Namely, we prove that the expansions of predicate Product, Gödel and Nilpotent Minimum logics with truth-constants are conservative, which already implies the failure of standard completeness for the case of Product logic. In contrast, the expansions of predicate Gödel and Nilpotent Minimum logics are proved to be strong standard complete but, when the semantics is restricted to the canonical algebra, they are proved to be complete only for tautologies. Moreover, when the language is restricted to evaluated formulae we prove canonical completeness for deductions from finite sets of premises.Peer Reviewe

Strong Completeness of Coalgebraic Modal Logics

Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics often present subtle difficulties - up to the point that canonical models may fail to exist, as is the case e.g. in most probabilistic logics. Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee strong completeness. We apply this method to reconstruct canonical model theorems that are either known or folklore, and moreover instantiate our method to obtain new strong completeness results. In particular, we prove strong completeness of graded modal logic with finite multiplicities, and of the modal logic of exact probabilities

Verification in ACL2 of a Generic Framework to Synthesize SAT–Provers

We present in this paper an application of the ACL2 system to reason about propositional satisfiability provers. For that purpose, we present a framework where we define a generic transformation based SAT–prover, and we show how this generic framework can be formalized in the ACL2 logic, making a formal proof of its termination, soundness and completeness. This generic framework can be instantiated to obtain a number of verified and executable SAT–provers in ACL2, and this can be done in an automatized way. Three case studies are considered: semantic tableaux, sequent and Davis–Putnam methods.Ministerio de Ciencia y Tecnología TIC2000-1368-C03-0

Building validation tools for knowledge-based systems

The Expert Systems Validation Associate (EVA), a validation system under development at the Lockheed Artificial Intelligence Center for more than a year, provides a wide range of validation tools to check the correctness, consistency and completeness of a knowledge-based system. A declarative meta-language (higher-order language), is used to create a generic version of EVA to validate applications written in arbitrary expert system shells. The architecture and functionality of EVA are presented. The functionality includes Structure Check, Logic Check, Extended Structure Check (using semantic information), Extended Logic Check, Semantic Check, Omission Check, Rule Refinement, Control Check, Test Case Generation, Error Localization, and Behavior Verification