Emilio Muñoz-Velasco

Relational approach for a logic for order of magnitude qualitative reasoning with negligibility non-closeness and distance

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for orderof-magnitude reasoning and its dual tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object.

An ATP of a Relational Proof System for Order of Magnitude Reasoning with Negligibility, Non-closeness and Distance

We introduce an Automatic Theorem Prover (ATP) of a dual tableau system for a relational logic for order of magnitude qualitative reasoning, which allows us to deal with relations such as negligibility, non-closeness and distance. Dual tableau systems are validity checkers that can serve as a tool for verification of a variety of tasks in order of magnitude reasoning, such as the use of qualitative sum of some classes of numbers. In the design of our ATP, we have introduced some heuristics, such as the so called phantom variables , which improve the efficiency of the selection of variables used un the proof.

Implementing a relational theorem prover for modal logic [image omitted]

An automatic theorem prover for a proof system in the style of dual tableaux for the relational logic associated with modal logic has been introduced. Although there are many well-known implementations of provers for modal logic, as far as we know, it is the first implementation of a specific relational prover for a standard modal logic. There are two main contributions in this paper. First, the implementation of new rules, called ( ) and ( ), which substitute the classical relational rules for composition and negation of composition in order to guarantee not only that every proof tree is finite but also to decrease the number of applied rules in dual tableaux. Second, the implementation of an order of application of the rules which ensures that the proof tree obtained is unique. As a consequence, we have implemented a decision procedure for modal logic . Moreover, this work would be the basis for successive extensions of this logic, such as , and .

Dual tableau for a multimodal logic for order of magnitude qualitative reasoning with bidirectional negligibility

We present a relational proof system in the style of dual tableaux for the relational logic associated with a multimodal propositional logic for order of magnitude qualitative reasoning with a bidirectional relation of negligibility. We study soundness and completeness of the proof system and we show how it can be used for verification of validity of formulas of the logic.

A Logic for Order of Magnitude Reasoning with Negligibility, Non-closeness and Distance

This paper continues the research line on the multimodal logic of qualitative reasoning; specifically, it deals with the introduction of the notions non-closeness and distance. These concepts allow us to consider qualitative sum of medium and large numbers. We present a sound and complete axiomatization for this logic, together with some of its advantages by means of an example.

Sub-propositional Fragments of the Interval Temporal Logic of Allen's Relations

Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. The most influential propositional interval-based logic is probably Halpern and Shohams Modal Logic of Time Intervals, a.k.a. HS. While most studies focused on the computational properties of the syntactic fragments that arise by considering only a subset of the set of modalities, the fragments that are obtained by weakening the propositional side have received very scarce attention. Here, we approach this problem by considering various sub-propositional fragments of HS, such as the so-called Horn, Krom, and core fragment. We prove that the Horn fragment of HS is undecidable on every interesting class of linearly ordered sets, and we briefly discuss the difficulties that arise when considering the other fragments.

Relational dual tableau decision procedures and their applications to modal and intuitionistic logics

Abstract We study a class DL of certain decidable relational logics of binary relations with a single relational constant and restricted composition. The logics in DL are defined in terms of semantic restrictions on the models. The main contribution of the present article is the construction of relational dual tableau decision procedures for the logics in DL . The systems are constructed in the framework of the original methodology of relational proof systems, determined only by axioms and inference rules, without any external techniques. All necessary bookkeeping is contained in the proof tree itself and used according to the explicit rules. All the systems are deterministic, producing exactly one proof tree for every formula. Furthermore, we show how the systems for logics in DL can be used as deterministic decision procedures for some modal and intuitionistic logics.

A PDL APPROACH FOR QUALITATIVE VELOCITY

We introduce the syntax, semantics, and an axiom system for a PDL-based extension of the logic for order of magnitude qualitative reasoning, developed in order to deal with the concept of qualitative velocity, which together with qualitative distance and orientation, are important notions in order to represent spatial reasoning for moving objects, such as robots. The main advantages of using a PDL-based approach are, on the one hand, all the well-known advantages of using logic in AI, and, on the other hand, the possibility of constructing complex relations from simpler ones, the flexibility for using different levels of granularity, its possible extension by adding other spatial components, and the use of a language close to programming languages.

A new deduction system for deciding validity in modal logic K

A new deduction system for deciding validity for the minimal decidable normal modal logic K is presented in this paper. Modal logics could be very helpful in modeling dynamic and reactive systems such as bio-inspired systems and process algebras. In fact, recently it has been presented the Connectionist Modal Logics which combines the strengths of modal logics and neural networks. Thus, modal logic K is the basis for these approaches. Soundness, completeness, and the fact that the system itself is a decision procedure are proved in this paper. The main advantages of this approach are: first, the system is deterministic, that is, it generates one proof tree for a given formula; second, the system is a validity-checker, hence it generates a proof of a formula (if such exists); third, the language of deduction and the language of a logic coincide. Some of these advantages are compared to other classical approaches.

Reasoning with qualitative velocity: towards a hybrid approach

Qualitative description of the movement of objects can be very important when there are large quantity of data or incomplete information, such as in positioning technologies and movement of robots. We present a first step in the combination of fuzzy qualitative reasoning and quantitative data obtained by human interaction and external devices as GPS, in order to update and correct the qualitative information. We consider a Propositional Dynamic Logic which deals with qualitative velocity and enables us to represent some reasoning tasks about qualitative properties. The use of logic provides a general framework which improves the capacity of reasoning. This way, we can infer additional information by using axioms and the logic apparatus. In this paper we present sound and complete relational dual tableau that can be used for verification of validity of formulas of the logic in question.