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

Simple contrapositive assumption-based frameworks are a general setting for structured argumentation, providing a robust approach to reasoning with arguments and counter-arguments. In this paper we extend these frameworks with priorities and introduce some new results concerning the semantics of the resulting formalisms.

Prioritized Simple Contrapositive Assumption-Based Frameworks.

We identify three classes of four-state paraconsistent logics according to their different approaches towards the disjunctive syllogism, and investigate three representatives of these approaches: Quasi-classical logic, which always accepts this principle, Belnap's logic, that rejects the disjunctive syllogism altogether, and a logic of inconsistency minimization that restricts its application to consistent fragments only. These logics are defined in a syntactic and a semantic style, which are linked by a simple transformation. It is shown that the three formalisms accommodate knowledge minimization, and that the most liberal formalism towards the disjunctive syllogism is also the strongest among the three, while the most cautious logic is the weakest one.

http://www2.mta.ac.il/~oarieli/Papers/kr10a.pdf

On the application of the disjunctive syllogism in paraconsistent logics based on four states of information

In this paper we introduce hypersequent-based frameworks for the modeling of defeasible reasoning by means of logic-based argumentation. These frameworks are an extension of sequent-based argumentation frameworks, in which arguments are represented not only by sequents, but by more general expressions, called hypersequents . This generalization allows to incorporate, as the deductive-base of our formalism, some well-studied logics like the modal logic S5, the relavent logic RM, and Godel-Dummett logic LC, to which no cut-free sequent calculi are known. In this paper we take S5 as the core logic and show that the hypersequent-based argumentation frameworks that are obtained in this case yield a robust defeasible variant of S5 with several desirable properties.

Hypersequential Argumentation Frameworks: An Instantiation in the Modal Logic S5

We introduce a modular framework for formalizing reasoning with incomplete and inconsistent information. This framework is composed of non-deterministic semantic structures and distance-based considerations. The combination of these two principles leads to a variety of entailment relations that can be used for reasoning about non-deterministic phenomena and are inconsistency-tolerant. We investigate the basic properties of these entailments and demonstrate their usefulness in the context of model-based diagnostic systems.

/pdf/non-deterministic-distance-semantics-for-handling-incomplete-52ayr2cxaz.pdf

Non-deterministic Distance Semantics for Handling Incomplete and Inconsistent Data

We introduce a family of preferential logics that are useful for handling information with different levels of uncertainty. The corresponding consequence relations are non-monotonic, paraconsistent, adaptive, and rational. It is also shown that any formalism in this family that is based on a well-founded ordering of the different types of uncertainty, can be embedded in a corresponding four-valued logic with at most three uncertainty levels.

Ofer Arieli

Papers

Prioritized Simple Contrapositive Assumption-Based Frameworks.

On the application of the disjunctive syllogism in paraconsistent logics based on four states of information

Hypersequential Argumentation Frameworks: An Instantiation in the Modal Logic S5

Non-deterministic Distance Semantics for Handling Incomplete and Inconsistent Data

Preferential Logics for Reasoning with Graded Uncertainty