A complete transformation system for polymorphic higher-order unification

Polymorphic higher-order unification is a method for unifying terms in the poly\-mor\-phi\-cally typed $\lambda$-calculus, that is, given a set of pairs of terms $S_0 = \{s_1 = t_2,\ldots,s_n = t_n\}$, called a unification problem, finding a substitution $\sigma$ such that $\sigma(s_i)$ and $\sigma(t_i)$ are equivalent under the conversion rules of the calculus for all $i$, $1\leq i\leq n$. I present the method as a transformation system, i.e.\ as a set of schematic rules $U \Rightarrow U'$ such that any unification problem $\delta({U})$ can be transformed into $\delta({U'})$ where $\delta$ is an instantiation of the meta-level variables in $U$ and $U'$. By successive use of transformation rules one possibly obtains a solved unification problem with obvious unifier. I show that the transformation system is correct and complete, i.e.\if $\delta({U}) \Rightarrow \delta({U'})$ is an instance of a transformation rule, then the set of all unifiers of $\delta({U'})$ is a subset of the set of all unifiers of $\delta({U})$ and if $\cal U$ is the set of all unification problems that can be obtained from successive applications of transformation rules from an unification problem $U$, then the union of the set of all unifiers of all unification problems in $\cal U$ is the set of all unifiers of $U$. The transformation rules presented here are essentially different from those in Gallier and Snyder (1989) or Nipkow (1990). The correctness and completeness proofs are in lines with those of Gallier and Snyder (1989)

Motel user manual

MOTEL is a logic-based knowledge representation languages of the KL-ONE family. It contains as a kernel the KRIS language which is a decidable sublanguage of first-order predicate logic. Whereas KRIS is a single-agent knowledge representation system, i.e. KRIS is only able to represent general world knowledge or the knowledge of one agent about the world, MOTEL is a multi-agent knowledge representation system. The MOTEL language allows modal contexts and modal concept forming operators which allow to represent and reason about the believes and wishes of multiple agents. Furthermore it is possible to represent defaults and stereotypes. Beside the basic resoning facilities for consistency checking, classification, and realization, MOTEL provides an abductive inference mechanism. Furthermore it is able to give explanations for its inferences

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

Comparative concept similarity over Minspaces: Axiomatisation and Tableaux Calculus

We study the logic of comparative concept similarity \CSL introduced by Sheremet, Tishkovsky, Wolter and Zakharyaschev to capture a form of qualitative similarity comparison. In this logic we can formulate assertions of the form " objects A are more similar to B than to C". The semantics of this logic is defined by structures equipped by distance functions evaluating the similarity degree of objects. We consider here the particular case of the semantics induced by \emph{minspaces}, the latter being distance spaces where the minimum of a set of distances always exists. It turns out that the semantics over arbitrary minspaces can be equivalently specified in terms of preferential structures, typical of conditional logics. We first give a direct axiomatisation of this logic over Minspaces. We next define a decision procedure in the form of a tableaux calculus. Both the calculus and the axiomatisation take advantage of the reformulation of the semantics in terms of preferential structures.Comment: 25 page

Ontology-based data access with databases: a short course

Ontology-based data access (OBDA) is regarded as a key ingredient of the new generation of information systems. In the OBDA paradigm, an ontology defines a high-level global schema of (already existing) data sources and provides a vocabulary for user queries. An OBDA system rewrites such queries and ontologies into the vocabulary of the data sources and then delegates the actual query evaluation to a suitable query answering system such as a relational database management system or a datalog engine. In this chapter, we mainly focus on OBDA with the ontology language OWL 2QL, one of the three profiles of the W3C standard Web Ontology Language OWL 2, and relational databases, although other possible languages will also be discussed. We consider different types of conjunctive query rewriting and their succinctness, different architectures of OBDA systems, and give an overview of the OBDA system Ontop

A Foundational View on Integration Problems

The integration of reasoning and computation services across system and language boundaries is a challenging problem of computer science. In this paper, we use integration for the scenario where we have two systems that we integrate by moving problems and solutions between them. While this scenario is often approached from an engineering perspective, we take a foundational view. Based on the generic declarative language MMT, we develop a theoretical framework for system integration using theories and partial theory morphisms. Because MMT permits representations of the meta-logical foundations themselves, this includes integration across logics. We discuss safe and unsafe integration schemes and devise a general form of safe integration

SAT-based Explicit LTL Reasoning

We present here a new explicit reasoning framework for linear temporal logic (LTL), which is built on top of propositional satisfiability (SAT) solving. As a proof-of-concept of this framework, we describe a new LTL satisfiability tool, Aalta\_v2.0, which is built on top of the MiniSAT SAT solver. We test the effectiveness of this approach by demonnstrating that Aalta\_v2.0 significantly outperforms all existing LTL satisfiability solvers. Furthermore, we show that the framework can be extended from propositional LTL to assertional LTL (where we allow theory atoms), by replacing MiniSAT with the Z3 SMT solver, and demonstrating that this can yield an exponential improvement in performance

Modal satisfiability via SMT solving

Modal logics extend classical propositional logic, and they are robustly decidable. Whereas most existing decision procedures for modal logics are based on tableau constructions, we propose a framework for obtaining decision procedures by adding instantiation rules to standard SAT and SMT solvers. Soundness, completeness, and termination of the procedures can be proved in a uniform and elementary way for the basic modal logic and some extensions.Fil: Areces, Carlos Eduardo. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina.Fil: Areces, Carlos Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina.Fil: Fontaine, Pascal. Université de Lorraine; Francia.Fil: Fontaine, Pascal. National Institute for Research in Digital Science and Technology; Francia.Fil: Merz, Stephan. Université de Lorraine; Francia.Fil: Merz, Stephan. National Institute for Research in Digital Science and Technology; Francia.Ciencias de la Computació