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

One of the most significant drawbacks of classical logic is its being useless in the presence of an inconsistency. Nevertheless, the classical calculus is a very convenient framework to work with. In this work we propose means for drawing conclusions from systems that are based on classical logic, although the information might be inconsistent. The idea is to detect those parts of the knowledge base that ‘cause’ the inconsistency, and isolate the parts that are ‘recoverable’. We do this by temporarily switching into Ginsberg/Fitting multivalued framework of bilattices (which is a common framework for logic programming and nonmonotonic reasoning). Our method is conservative in the sense that it considers the contradictory data as useless and regards all the remaining information unaffected. The resulting logic is nonmonotonic, paraconsistent, and a plausibility logic in the sense of Lehmann.

A Model-Theoretic Approach for Recovering Consistent Data from Inconsistent Knowledge Bases

Reasoning with the maximally consistent subsets (MCS) of the premises is a well-known approach for handling contradictory information. In this paper we consider several variations of this kind of reasoning, for each one we introduce two complementary computational methods that are based on logical argumentation theory. The difference between the two approaches is in their ways of making consequences: one approach is of a declarative nature and is related to Dung-style semantics for abstract argumentation, while the other approach has a more proof-theoretical flavor, extending Gentzen-style sequent calculi. The outcome of this work is a new perspective on reasoning with MCS, which shows a strong link between the latter and argumentation systems, and which can be generalized to some related formalisms. As a by-product of this we obtain soundness and completeness results for the dynamic proof systems with respect to several of Dung’s semantics. In a broader context, we believe that this work helps to better understand and evaluate the role of logic-based instantiations of argumentation frameworks.

Reasoning with maximal consistency by argumentative approaches

Current approaches for giving semantics to abstract argumentation frameworks dismiss altogether any possibility of having conflicts among accepted arguments by requiring that the latter should be ‘conflict free’. In reality, however, contradictory phenomena coexist, or it may happen that one cannot make a choice between conflicting indications but still would like to keep track to all of them. For this purpose we introduce in this paper a new kind of argumentation semantics, called ‘conflict-tolerant’, in which all the accepted arguments must be justified (in the sense that each one of them can be defended), but some of them may still attack each other. In terms of graphical representation of argumentation systems, where attacks are represented by directed edges, this means that the possibility of accepting ‘loops’ of arguments is not automatically ruled out without any further considerations. To provide conflict-tolerant semantics we enhance the two standard approaches for defining coherent (conflict-free) semantics for argumentation frameworks. The extension-based approach is generalized by relaxing the ‘conflict-freeness’ requirement of the chosen sets of arguments, and the three-valued labeling approach is replaced by a four-valued labeling system that allows to capture mutual attacks among accepted arguments. We show that our setting is not a substitute of standard (conflict-free) semantics, but rather a generalized framework that accommodates both conflict-free and conflict-tolerant semantics. Moreover, the one-to-one relationship between extensions and labelings of conflict-free semantics is carried on to a similar correspondence between the extended approaches for providing conflict-tolerant semantics. Thus, in our setting as well, these are essentially two points of views for the same thing.

http://www2.mta.ac.il/~oarieli/Papers/jlc14.pdf

On the acceptance of loops in argumentation frameworks

In this paper we integrate priorities in sequent-based argumentation. The former is a useful and extensively investigated tool in the context of non-monotonic reasoning, and the latter is a modular and general way of handling logical argumentation. Their combination offers a platform for representing and reasoning with maximally consistent subsets of prioritized knowledge bases. Moreover, many frameworks of the resulting formalisms satisfy common rationality postulates and other desirable properties, like conflict preservation.

Prioritized Sequent-Based Argumentation

We introduce a general method for paraconsistent reasoning in knowledge systems by classical second-order formulae. A standard technique for paraconsistent reasoning on inconsistent classical theories is by shifting to multiple-valued logics. We show how these multiple-valued theories can be "shifted back" to two-valued classical theories (through a polynomial transformation), and how preferential reasoning based on multiple-valued logic can be represented by classical circumscription-like axioms. By applying this process we manage to overcome the shortcoming of classical logic in properly handling inconsistent data, and provide new ways of implementing multiple-valued paraconsistent reasoning in knowledge systems. Standard multiple-valued reasoning can thus be performed through theorem provers for classical logic, and multiple-valued preferential reasoning can be implemented using algorithms for processing circumscriptive theories (such as DLS and SCAN).

Ofer Arieli

Papers

A Model-Theoretic Approach for Recovering Consistent Data from Inconsistent Knowledge Bases

Reasoning with maximal consistency by argumentative approaches

On the acceptance of loops in argumentation frameworks

Prioritized Sequent-Based Argumentation

Modeling Paraconsistent Reasoning by Classical Logic