There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

An Introduction to MultiAgent Systems.

Publisher Summary This chapter discusses the mathematical theory of computation. Computation essentially explores how machines can be made to carry out intellectual processes. Any intellectual process that can be carried out mechanically can be performed by a general purpose digital computer. There are three established directions of mathematical research that are relevant to the science of computation—namely, numerical analysis, theory of computability, and theory of finite automata. The chapter explores what practical results can be expected from a suitable mathematical theory. Further, the chapter presents several descriptive formalisms with a few examples of their use and theories that enable to prove the equivalence of computations expressed in these formalisms. A few mathematical results about the properties of the formalisms are also presented.

A Basis for a Mathematical Theory of Computation

In this paper, we examine an argument-based semantics called semi-stable semantics. Semi-stable semantics is quite close to traditional stable semantics in the sense that every stable extension is also a semi-stable extension. One of the advantages of semi-stable semantics is that there exists at least one semi-stable extension. Furthermore, if there also exists at least one stable extension, then the semi-stable extensions coincide with the stable extensions. This, and other properties, make semi-stable semantics an attractive alternative for the more traditional stable semantics, which until now has been widely used in fields such as logic programming and answer set programming.

/pdf/semi-stable-semantics-2q6ow9khu8.pdf

Semi-Stable Semantics

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ACM Transactions on Database Systems

The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of our work. The outcome are paraconsistent logics with a lot of desirable properties.
 A preliminary version of this paper appears in Arieli and Avron (1994).

Reasoning with logical bilattices

The value of the four values

We define in precise terms the basic properties that an `ideal propositional paraconsistent logic' is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n-valued logics, each one of which is not equivalent to any k-valued logic with k < n.

/pdf/ideal-paraconsistent-logics-1kd2vee5ps.pdf

Ideal Paraconsistent Logics

The notion of a bilattice was first proposed by Ginsberg (1988) as a general framework for many applications. This notion was further investigated and applied for various goals by Fitting (1989, 1990, 1991, 1993). In this paper, we develop proof systems which correspond to bilattices in an essential way. We then show how to use those bilattices for efficient inferences from possibly inconsistent data. For this, we incorporate certain ideas of Kifer and Lozinskii (1992) concerning inconsistencies, which happen to well suit the framework of bilattices. The outcome is a paraconsistent logic with many desirable properties. >

Logical bilattices and inconsistent data

We introduce an abductive method for a coherent integration of independent datasources. The idea is to compute a list of data-facts that should be inserted to the amalgamated database or retracted from it in order to restore its consistency. This method is implemented by an abductive solver, called A system, that applies SLDNFA-resolution on a meta-theory that relates different, possibly contradicting, input databases. We also give a pure model-theoretic analysis of the possible ways to 'recover' consistent data from an inconsistent database in terms of those models of the database that exhibit as minimal inconsistent information as reasonably possible. This allows us to characterize the 'recovered databases' in terms of the 'preferred' (i.e., most consistent) models of the theory. The outcome is an abductive-based application that is sound and complete with respect to a corresponding model-based, preferential semantics, and - to the best of our knowledge - is more expressive (thus more general) than any other implementation of coherent integration of databases.

/pdf/coherent-integration-of-databases-by-abductive-logic-qya1r38xhf.pdf

Ofer Arieli

Papers

Reasoning with logical bilattices

The value of the four values

Ideal Paraconsistent Logics

Logical bilattices and inconsistent data

Coherent integration of databases by abductive logic programming