A-ordered tableaux

In resolution proof procedures refinements based on A-orderings of literals have a long tradition and are well investigated. In tableau proof procedures such refinements were only recently introduced by the authors of the present paper. In this paper we prove the following results: we give a completeness proof of A-ordered ground clause tableaux which is a lot easier to follow than the previous one. The technique used in the proof is extended to the non-clausal case as well as to the non-ground case and we introduce an ordered version of Hintikka sets that shares the model existence property of standard Hintikks sets. We show that A-ordered tableaux are a proof confluent refinement of tableaux and that A-ordered tableaux together with the connection refinement yield an incomplete proof procedure. We introduce A-ordered first-order NNF tableaux, prove their completeness, and we briefly discuss implementation issues

A-ordered tableaux

In resolution proof procedures refinements based on A-orderings of literals have a long tradition and are well investigated. In tableau proof procedures such refinements were only recently introduced by the authors of the present paper. In this paper we prove the following results: we give a completeness proof of A-ordered ground clause tableaux which is a lot easier to follow than the previous one. The technique used in the proof is extended to the non-clausal case as well as to the non-ground case and we introduce an ordered version of Hintikka sets that shares the model existence property of standard Hintikks sets. We show that A-ordered tableaux are a proof confluent refinement of tableaux and that A-ordered tableaux together with the connection refinement yield an incomplete proof procedure. We introduce A-ordered first-order NNF tableaux, prove their completeness, and we briefly discuss implementation issues

Tableau calculi for description logics revision

Focusing on the Ontology Change problem, we consider an environment where Description Logics (DLs) are the logical formalization to express knowledge bases, and the integration of distributed ontologies is developed under new extensions and modifications of the Belief Revision theories yielded originally in [2]. When using tableaux algorithms to reason about DLs, new information is yielded from the models considered in order to achieve knowledge satisfiability. Here a whole new theory have to be reinforced in order to adapt belief revision definitions and postulates to properly react over beliefs on extensions generated from these DL’s reasoning services. In this text we give a brief background of these formalisms and comment the research lines to be taken in our way to this goal.Eje: Agentes y Sistemas InteligentesRed de Universidades con Carreras en Informática (RedUNCI

Positive Unit Hyperresolution Tableaux and Their Application to Minimal Model Generation

Minimal Herbrand models of sets of first-order clauses are useful in several areas of computer science, e.g. automated theorem proving, program verification, logic programming, databases, and artificial intelligence. In most cases, the conventional model generation algorithms are inappropriate because they generate nonminimal Herbrand models and can be inefficient. This article describes an approach for generating the minimal Herbrand models of sets of first-order clauses. The approach builds upon positive unit hyperresolution (PUHR) tableaux, that are in general smaller than conventional tableaux. PUHR tableaux formalize the approach initially introduced with the theorem prover SATCHMO. Two minimal model generation procedures are described. The first one expands PUHR tableaux depth-first relying on a complement splitting expansion rule and on a form of backtracking involving constraints. A Prolog implementation, named MM-SATCHMO, of this procedure is given and its performance on benchmark suites is reported. The second minimal model generation procedure performs a breadth-first, constrained expansion of PUHR (complement) tableaux. Both procedures are optimal in the sense that each minimal model is constructed only once, and the construction of nonminimal models is interrupted as soon as possible. They are complete in the following sense The depth-first minimal model generation procedure computes all minimal Herbrand models of the considered clauses provided these models are all finite. The breadth-first minimal model generation procedure computes all finite minimal Herbrand models of the set of clauses under consideration. The proposed procedures are compared with related work in terms of both principles and performance on benchmark problems

Deductive Systems in Traditional and Modern Logic

The book provides a contemporary view on different aspects of the deductive systems in various types of logics including term logics, propositional logics, logics of refutation, non-Fregean logics, higher order logics and arithmetic

Finitary proof systems for Kozen’s μ.

We present three finitary cut-free sequent calculi for the modal [my]-calculus. Two of these derive annotated sequents in the style of Stirling’s ‘tableau proof system with names’ (4236) and feature special inferences that discharge open assumptions. The third system is a variant of Kozen’s axiomatisation in which cut is replaced by a strengthening of the v-induction inference rule. Soundness and completeness for the three systems is proved by establishing a sequence of embeddings between the calculi, starting at Stirling’s tableau-proofs and ending at the original axiomatisation of the [my]-calculus due to Kozen. As a corollary we obtain a completeness proof for Kozen’s axiomatisation which avoids the usual detour through automata or games

Action Logic Programs: How to Specify Strategic Behavior in Dynamic Domains Using Logical Rules

We discuss a new concept of agent programs that combines logic programming with reasoning about actions. These agent logic programs are characterized by a clear separation between the specification of the agent’s strategic behavior and the underlying theory about the agent’s actions and their effects. This makes it a generic, declarative agent programming language, which can be combined with an action representation formalism of one’s choice. We present a declarative semantics for agent logic programs along with (two versions of) a sound and complete operational semantics, which combines the standard inference mechanisms for (constraint) logic programs with reasoning about actions