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The aim of this work is to present theoretically a new algebraic method for detecting nearly dangerous states in an expert system whose knowledge is represented by propositional Boolean logic. Given a dangerous state which does not happen at present, our method is able to detect if a dangerous situation would happen if an input variable of the Expert System changed. The method presented here is based on calculating just one Groebner basis of a polynomial ideal representing the system knowledge. In this way, although the Expert System is designed to notify only dangerous situations, we have developed an algebraic model for including in the expert system the ability to recommend watching carefully some Boolean variables such that if one of them changed, the system would fall in a dangerous situation. In this way, the same expert system not only warns about dangerous situations but also about nearly dangerous situations. It may be noted that our method does not require to change the Expert System for including this facility. As far as our knowledge goes, our work is completely new in the field of Expert Systems. [ABSTRACT FROM AUTHOR]

According to previous mathematical results, Expert Systems can be quickly and easily implemented on specialized mathematical software (‘Computer Algebra Systems’). These mathematical results provide two equivalent different approaches to implement algebraically Expert Systems: one based on propositional Boolean logic and another based on the ‘concept–attribute–value’ representation paradigm. Both approaches are equivalent in the sense that every Expert System implemented through one algebraic approach can be also implemented on the other algebraic approach. However, the performance of these algebraic approaches depend on the specific Expert System, and therefore, the Expert System must be implemented in both algebraic approaches in order to compare their performances. In this paper we have designed a natural language to implement algebraically Expert Systems which provides these three main advantages: we can implement directly any Expert System in both algebraic approach since our language is independent of the algebraic approach used; we can use any Computer Algebra System since our language is independent of the Computer Algebra System used; and we do not need to be acquainted with specialized abstract mathematical concepts like Groebner bases or normal forms since our language is natural and does not make use of these concepts. [ABSTRACT FROM AUTHOR]

The aim of this paper is to expound an original algebraic model for managing the knowledge provided by different expert humans when developing expert systems. This model is conceived as an extension of classical propositional logics in which each proposition is associated with a set of human experts who agree with it. In our model, the logical notions of tautological consequence and consistency of a set of formulae are reformulated taking into account the criteria and the knowledge of the different experts. The core of the paper is related to the discovery of a remarkable relation between these redefined logical concepts with the calculation of Groebner bases on an ideal of polynomials. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with expounding a new representation paradigm for modeling expert systems based on computing Groebner Bases. Previous research on Groebner Bases expert systems has so far been connected to modeling expert systems based on propositional logics. Our approach instead is based on the well-known Artificial Intelligence ‘Concept-Attribute-Value’ paradigm for representing knowledge. More precisely, our research is based on translating an already existent expert system described in terms of the ‘Concept-Attribute-Value’ paradigm to a new algebraic model which represents knowledge by means of polynomials. In this way, issues about consistence and inference within this expert system will be, through this new model, transformed into algebraic problems involving calculating Groebner Bases. By using this new model of ours, some interesting advantages ensue: on the one hand, knowledge representation may be performed in a more straightforward and intuitive way; on the other, calculating the Groebner Bases associated to our algebraic model is usually faster adopting this new ‘Concept-Attribute-Value’-based paradigm than it was in previous propositional logic-based expert systems. [Copyright &y& Elsevier]

Abstract: Railway interlocking systems are apparatuses that prevent conflicting movements of trains through an arrangement of tracks. A railway interlocking system takes into consideration the position of the switches (of the turnouts) and does not allow trains to be given clear signals unless the routes to be used by the trains do not intersect. A new model, based on Boolean Logic, and independent from the topology of the station is presented in this paper. According to this new model, any given proposed situation is safe if and only if a certain set of formulae (translating the position of trains and the movements allowed – the latter depend on the position of the switches and the colour of the semaphores) is consistent. The main procedure analyses the safety of a proposed situation and returns, if they exist, the sections where a collision could take place. The fact that trains could occupy more than one section is considered. The code of the corresponding Maple implementation is surprisingly brief. [Copyright &y& Elsevier]

Abstract: The present paper is concerned with expounding an altogether new method specifically designed to provide a straightforward proof that there is no possible solution for some family of systems of Diophantine linear equations which share the same dependent terms while differing in its independent terms. On the basis of this novel approach, a definition of testers is given and consequently used to shape this particular method, as illustrated with some suitable examples. [Copyright &y& Elsevier]

Abstract: This paper deals with a computer programme offering a valid simulation of cognitive human processes related to representation changes while problem solving. An introductory and brief recollection of preceding contributions by psychologists, cognitive scientists and AI researchers provides the necessary background and motivation for our work. A relevant trend of the present research concerns the formal mathematical study of representation phenomena, including precisions about isomorphic and homomorphic representation changes. In order to simplify problem resolution processes, the implemented system performs certain representation changes, by making use of specific procedures which assign and modify a certain relevance value to every attribute involved in the problem representation, on behalf of their respective importance so as to actually solve the problem. The paper includes a full account of mathematical definitions and propositions involved in this system. [Copyright &y& Elsevier]

Abstract: We have recently developed a package in Maple that allows to perform logical computations in any existing or proposed many-valued logic (that can be defined using truth tables). It has applications in logic engineering (e.g. when creating a new logic adapted to some requirements), in theoretical logic (e.g. checking if two axiomatizations of a logic are equivalent), or simply in checking the properties of a well known logic. Now, such a package has been tuned so that it can simultaneously deal with the logic proposed by the end user plus the standard Boolean logic, in order to conveniently check properties. This new approach is used to easily develop some disquisitions on why Belnap's four-valued logic is defined the way it is. Finally, it is shown how a new four valued logic that mixes ideas from Łukasiewicz's and Kleene's logics behaves similarly to Belnap's logic. [Copyright &y& Elsevier]