Ignacio Darío Viglizzo

On Some Semi-Intuitionistic Logics

Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which were defined by H. P. Sankappanavar as a generalization of Heyting algebras. We present a new, more streamlined set of axioms for semi-intuitionistic logic, which we prove translationally equivalent to the original one. We then study some formulas that define a semi-Heyting implication, and specialize this study to the case in which the formulas use only the lattice operators and the intuitionistic implication. We prove then that all the logics thus obtained are equivalent to intuitionistic logic, and give their Kripke semantics.

The foundations of DeLP: defeating relations, games and truth values

In this paper we examine the mechanism of DeLP (Defeasible Logic Programming). We first study the definition of the defeating relation in a formal setting that allows us to uncover some hidden assumptions, and suggest an alternative definition. Then we introduce a game-theoretic characterization of the system. We obtain a new set of truth values arising from games in which arguments for and against a given literal are played out. We study how additional constraints define protocols of admissible attacks. The DeLP protocol ensures the finiteness of the games, and therefore the existence of winning strategies for the corresponding games. The defeating relation among arguments determines the strategies that will win and consequently the truth values of queries. We find that the DeLP protocol also excludes the warranting of a literal and its negation.

Semi-Nelson Algebras

Generalizing the well known and exploited relation between Heyting and Nelson algebras to semi-Heyting algebras, we introduce the variety of semi-Nelson algebras. The main tool for its study is the construction given by Vakarelov. Using it, we characterize the lattice of congruences of a semi-Nelson algebra through some of its deductive systems, use this to find the subdirectly irreducible algebras, prove that the variety is arithmetical, has equationally definable principal congruences, has the congruence extension property and describe the semisimple subvarieties.

A labeled argumentation framework

To increase the expressivity of an argumentation formalism, we propose adding meta-level information to the arguments in the form of labels representing quantifiable data such as reliability degree, strength measure, skill, time availability, or any other feature about arguments. The extra information attached to an argument is then used in the acceptability determination process.We present a Labeled Argumentation Framework (LAF), combining the knowledge representation capabilities provided by the Argument Interchange Format (AIF) with an Algebra of Argumentation Labels, enabling us to handle the labels associated with the arguments. These labels are propagated through an argumentative graph according to the relations of support, conflict, and aggregation between arguments. Through this process we obtain final labels attached to the arguments that are useful to determine their acceptability.

An alternative foundation for DeLP: defeating relations and truth values

In this paper we recast the formalism of argumentation formalism known as DeLP (Defeasible Logic Programming) in game-theoretic terms. By considering a game between a Proponent and an Opponent, in which they present arguments for and against each literal we obtain a bigger gamut of truth values for those literals and their negations as they are defended and attacked. An important role in the determination of warranted literals is assigned to a defeating relation among arguments. We consider first an unrestricted version in which these games may be infinite and then we analyze the underlying assumptions commonly used to make them finite. Under these restrictions the games are always determined -one of the players has a winning strategy. We show how varying the defeating relation may alter the set of truth values reachable under this formalism. We also show how alternative characterizations of the defeating relation may lead to different assignations of truth values to the literals in a DeLP program.

Semi-intuitionistic Logic with Strong Negation

Motivated by the definition of semi-Nelson algebras, a propositional calculus called semi-intuitionistic logic with strong negation is introduced and proved to be complete with respect to that class of algebras. An axiomatic extension is proved to have as algebraic semantics the class of Nelson algebras.

A Labeled Abstract Bipolar Argumentation Framework

Argumentation is a form of reasoning where a claim is accepted or rejected according to the analysis of the arguments for and against it; furthermore, it provides a reasoning mechanism able to handle contradictory, incomplete and uncertain information in real-world situations. We combine Bipolar Argumentation Frameworks (an extension of Dung’s work) with an Algebra of Argumentation Labels modeling two independent types of interaction between arguments, representing meta-information associated with arguments, and introducing an acceptability notion that will give more information for arguments acceptability.