Logic Journal of IGPL - recent issues

A defeasible logic for modelling policy-based intentions and motivational attitudes

Governatori, G., Padmanabhan, V., Rotolo, A., Sattar, A. — 2009-05-24

In this paper we show how defeasible logic could formally account for the non-monotonic properties involved in motivational attitudes like intention and obligation. Usually, normal modal operators are used to represent such attitudes wherein classical logical consequence and the rule of necessitation comes into play, i.e., A/ A, that is from A derive A. This means that such formalisms are affected by the Logical Omniscience problem. We show that policy-based intentions exhibit non-monotonic behaviour which could be captured through a non-monotonic system like defeasible logic. To this end we outline a defeasible logic of intention that specifies how modalities can be introduced and manipulated in a non-monotonic setting without giving rise to the problem of logical omniscience. In a similar way we show how to add deontic modalities defeasibly and how to integrate them with other motivational attitudes like beliefs and goals. Finally we show that the basic aspect of the BOID architecture is captured by this extended framework.

On complete representations of algebras of logic

Khaled, M., Sayed-Ahmed, T. — 2009-05-24

We show that there exists an atomic polyadic equality algebra of dimension n that is elementary equivalent to a completely representable algebra, but its diagonal free reduct (obtained by deleting diagonals and substitutions) is not completely representable.

Algorithms for finding coalitions exploiting a new reciprocity condition

Boella, G., Sauro, L., van der Torre, L. — 2009-05-24

We introduce a reciprocity criterion for coalition formation among goal-directed agents, which we call the indecomposable do-ut-des property. It refines an older reciprocity property, called the do-ut-des or give-to-get property by considering the fact that agents prefer to form coalitions whose components cannot be formed independently. A formal description of this property is provided as well as an analysis of algorithms and their complexity. We provide an algorithm to decide whether a coalition has the desired property, and we show that the problem to verify whether a single coalition satisfies the property is tractable. Moreover, we provide an algorithm to search all the sub-coalitions of a given coalition satisfying the new property. Even if this problem is not computationally tractable, we show that in several cases, also the complexity of this problem may decrease considerably.

Logics for Qualitative Coalitional Games

Agotnes, T., Van Der Hoek, W., Wooldridge, M. — 2009-05-24

Qualitative Coalitional Games (QCGs) are a variant of coalitional games in which an agent's desires are represented as goals that are either satisfied or unsatisfied, and each choice available to a coalition is a set of goals, which would be jointly satisfied if the coalition made that choice. A coalition in a QCG will typically form in order to bring about a set of goals that will satisfy all members of the coalition. Our goal in this paper is to develop and study logics for reasoning about QCGs. We begin by introducing a logic for reasoning about "static" QCGs, where participants play a single game, and we then introduce and study Temporal QCGs (TQCGs), i.e., games in which a sequence of QCGs is played. In order to represent and reason about such games, we introduce a linear time temporal logic of QCGs, called L(TQCG). We give a complete axiomatisation of L(TQCG), use it to investigate the properties of TQCGs, identify its expressive power, establish its complexity, characterise classes of TQGCs with formulas from our logical language, and use it to formulate several (temporal) solution concepts for TQCGs.

Acknowledgements

2009-05-24

Unifiability in extensions of K4

Gencer, C., De Jongh, D. — 2009-04-07

We extend and generalize the work on unifiability of [8]. We give a semantic characterization for unifiability and non-unifiability in the extensions of K4. We apply this in particular to extensions of KD4, GL and K4.3 to obtain a syntactic characterization and give a concrete decision procedure for unifiability for those logics. For that purpose we use universal models.

Indecomposability of negative dense subsets of R in Constructive Reverse Mathematics

Loeb, I. — 2009-04-07

In 1970 Vesley proposed a substitute of Kripke's Scheme. In this paper it is shown that —over Bishop's constructive mathematics— the indecomposability of negative dense subsets of R is equivalent to a weakening of Vesley's proposal. This result supports the idea that full Kripke's Scheme might not be necessary for most of intuitionistic mathematics. At the same time it contributes to the programme of Constructive Reverse Mathematics and gives a new answer to a 1997 question of Van Dalen.

Economic theory and the Alternative Set Theory AFA-+AD+DC

Tohme, F. — 2009-04-07

Many authors in the discipline as well as outsiders have claimed that the main results from Mathematical Economics are far removed from real world phenomena. A more precise version of this position is that one of the main reasons for this unrealistic stance is the use of the wrong formal tools. So, for example, it has been pointed out that the computability of choice functions as well as the existence of economic equilibria and of states of the world may not be ensured in general if the assumed set theory is ZFC. We will show that there exists a very natural set theory that overcomes some formal limitations of contemporary economic theory. A switch to an alternative set theory helps to obtain in a more natural way results widely accepted by mathematical economists. Moreover, alternative set-theoretical frameworks convey different intuitions about how agents behave when solving problems. We claim that AFA^–+AD+DC is the adequate alternative set-theoretical universe for economic theory.

Factorization of residuated lattices

Krupka, M. — 2009-04-07

We discuss the problem of factorization of residuated lattices by similarity relations. As the main result, we introduce a natural structure of residuated lattice on factorized residuated lattice. Some consequences are also discussed: the problem of representatives and factor projections, sequential factorization, application to fuzzy sets, application to factorization of concept lattices of data with fuzzy attributes.

Acknowledgements

2009-04-07

Defeasible inheritance systems and reactive diagrams

Gabbay, D. M, Schlechta, K. — 2009-02-02

Inheritance diagrams are directed acyclic graphs with two types of connections between nodes: x -> y (read x is a y) and x y (read as x is not a y). Given a diagram D, one can ask the formal question of "is there a valid (winning) path between node x and node y?" Depending on the existence of a valid path we can answer the question "x is a y" or "x is not a y".

The answer to the above question is determined through a complex inductive algorithm on paths between arbitrary pairs of points in the graph.

This paper aims to simplify and interpret such diagrams and their algorithms. We approach the area on two fronts.

(1)

Suggest reactive arrows to simplify the algorithms for the winning paths.

(2)

We give a conceptual analysis of (defeasible or nonmonotonic) inheritance diagrams, and compare our analysis to the "small" and "big sets" of preferential and related reasoning.

In our analysis, we consider nodes as information sources and truth values, direct links as information, and valid paths as information channels and comparisons of truth values. This results in an upward chaining, split validity, off-path preclusion inheritance formalism of a particularly simple type.

We show that the small and big sets of preferential reasoning have to be relativized if we want them to conform to inheritance theory, resulting in a more cautious approach, perhaps closer to actual human reasoning.

We will also interpret inheritance diagrams as theories of prototypical reasoning, based on two distances: set difference, and information difference.

We will see that some of the major distinctions between inheritance formalisms are consequences of deeper and more general problems of treating conflicting information.

It is easily seen that inheritance diagrams can also be analysed in terms of reactive diagrams - as can all argumentation systems.

AMS Classification: 68T27, 68T30

An Indian logic-based argument representation formalism for knowledge-sharing

Mahalakshmi, G.S., Geetha, T.V. — 2009-02-02

Knowledge-sharing is the fundamental aspect of learning from ancient days. By exchanging questions and answers in a debate fashion, knowledge is explored. Argumentation can be thought of as a knowledge-sharing mechanism where construction of arguments and counter-arguments towards reaching mutually agreed upon conclusions is modelled after the rational discussion of knowledge-sharing. The procedural approach of generation and exchange of arguments is with the intention of reaching a definite conclusion at the end of discussion irrespective of the winning or losing of the individual arguments. A procedural argumentation system for knowledge-sharing, similar to the one discussed above will be of immense use only when the representational facts of world knowledge are well captured and represented as identical with the representation of natural intelligence. Indian Philosophy suggests various rules for classifying and representing the world knowledge to enhance the procedure of argumentation in reaching new inferences. This paper discusses a new mathematical model that caters to the special requirements of classification and hypothetical argumentative reasoning (tarka) of Indian philosophy. The main focus is the emphasis given to the existence of relations between concepts and foundation of invariable connection between concepts/relations of the arguments that is indicated by the philosophy. The extracted knowledge from the arguments exchanged is captured using ‘Indianised logics’, an Indian Logic based argument representation formalism. The arguments are deeply analysed for the presence of conceptual and relational conflicts. This is achieved by referring to the presence and/or absence of invariable concomitance, so that, the decision of justifying the represented invalid knowledge with or without proof shall be made clear in knowledge-sharing through rational discussion.

An argument game for stable semantics

Caminada, M., Wu, Y. — 2009-02-02

In this paper, we present a discussion game for argumentation under stable semantics. Our work is inspired by Vreeswijk and Prakken, who have defined a similar game for preferred semantics. In the current paper, we restate Vreeswijk and Prakken's work using the approach of argument labellings and then show how it can be adjusted for stable semantics. The nature of the resulting argument game is somewhat unusual, since stable semantics does not satisfy the property of relevance.

Equivalence and quantifier rules for logic with imperfect information

Caicedo, X., Dechesne, F., Janssen, T. M.V. — 2009-02-02

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot be so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore (restricted versions of) those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games.

Approximate formulae for a logic that capture classes of computational complexity

Arratia, A., Ortiz, C. E. — 2009-02-02

This paper presents a syntax of approximate formulae suited for the logic with counting quantifiers SOLP. This logic was formalised by us in [1] where, among other properties, we showed the following facts: (i) In the presence of a built–in (linear) order, SOLP can describe NP–complete problems and some of its fragments capture the classes P and NL; (ii) weakening the ordering relation to an almost order we can separate meaningful fragments, using a combinatorial tool adapted to these languages.

The purpose of our approximate formulae is to provide a syntactic approximation to the logic SOLP, enhanced with a built-in order, that should be complementary of the semantic approximation based on almost orders, by means of producing logics where problems are syntactically described within a small counting error. We introduce a concept of strong expressibility based on approximate formulae, and show that for many fragments of SOLP with built-in order, including ones that capture P and NL, expressibility and strong expressibility are equivalent. We state and prove a Bridge Theorem that links expressibility in fragments of SOLP over almost-ordered structures to strong expressibility with respect to approximate formulae for the corresponding fragments over ordered structures. A consequence of these results is that proving inexpressibility over fragments of SOLP with built-in order could be done by proving inexpressibility over the corresponding fragments with built-in almost order, where separation proofs are allegedly easier.

Erratum

2009-02-02

Acknowledgements

2009-02-02