Non-monotone Set Research Articles

Handling and measuring inconsistency in non-monotonic logics

We address the issue of quantitatively assessing the severity of inconsistencies in non-monotonic frameworks. While measuring inconsistency in classical logics has been investigated for some time now, taking the non-monotonicity into account poses new challenges. In order to tackle them, we focus on the structure of minimal strongly K-inconsistent subsets of a knowledge base K—a sound generalization of minimal inconsistent subsets to arbitrary, possibly non-monotonic, frameworks which induces a generalization of Reiter's famous hitting set duality between minimal inconsistent and maximal consistent subsets of a knowledge base. We propose measures based on this notion and investigate their behavior in a non-monotonic setting by revisiting existing rationality postulates, analyzing the compliance of the proposed measures with these postulates, and by investigating their computational complexity. Motivated by the observation that a knowledge base of a non-monotonic logic can also be repaired by adding formulas – whereas Reiter's duality is only concerned about removing –, we also investigate situations where we are given potential additional assumptions to repair a knowledge base. For this, we characterize the minimal modifications to a knowledge base in terms of a hitting set duality

Revision of defeasible preferences

There are several contexts of non-monotonic reasoning where a priority between rules is established with the purpose of solving conflicts. We investigate how to modify such a priority (preference) relation in a non-monotonic logic in order to change the conclusions of the theory itself. We shall argue that the approach we adopt has a natural counterpart in legal reasoning and argumentation, where users cannot typically change the facts or the rules, but can propose their preferences about the relative strength of the rules.The main result of the present work is the proof that the problem of revising a non-monotonic theory by changing only the superiority order between conflicting rules is, in general, computationally hard.After such an analysis, we identify three contraction/revision/update operations and study them against the AGM postulates for belief revision, to discover that only a (small) part of these postulates are satisfied in the specific non-monotonic setting.

Measuring Strong Inconsistency

We address the issue of quantitatively assessing the severity of inconsistencies in nonmonotonic frameworks. While measuring inconsistency in classical logics has been investigated for some time now, taking the nonmonotonicity into account poses new challenges. In order to tackle them, we focus on the structure of minimal strongly kb-inconsistent subsets of a knowledge base kb---a generalization of minimal inconsistency to arbitrary, possibly nonmonotonic, frameworks. We propose measures based on this notion and investigate their behavior in a nonmonotonic setting by revisiting existing rationality postulates, analyzing the compliance of the proposed measures with these postulates, and by investigating their computational complexity.

Backdoors to tractable answer set programming

Answer Set Programming (ASP) is an increasingly popular framework for declarative programming that admits the description of problems by means of rules and constraints that form a disjunctive logic program. In particular, many AI problems such as reasoning in a nonmonotonic setting can be directly formulated in ASP. Although the main problems of ASP are of high computational complexity, complete for the second level of the Polynomial Hierarchy, several restrictions of ASP have been identified in the literature, under which ASP problems become tractable.In this paper we use the concept of backdoors to identify new restrictions that make ASP problems tractable. Small backdoors are sets of atoms that represent “clever reasoning shortcuts” through the search space and represent a hidden structure in the problem input. The concept of backdoors is widely used in theoretical investigations in the areas of propositional satisfiability and constraint satisfaction. We show that it can be fruitfully adapted to ASP. We demonstrate how backdoors can serve as a unifying framework that accommodates several tractable restrictions of ASP known from the literature. Furthermore, we show how backdoors allow us to deploy recent algorithmic results from parameterized complexity theory to the domain of answer set programming.

Backdoors to Tractability of Answer-Set Programming

The practical results of answer-set programming indicate that classical complexity theory is insufficient as a theoretical framework to explain why modern answer-set programming solvers work fast on industrial applications. Complexity analysis by means of parameterized complexity theory seems to be promising, because we think that the reason for the gap between theory and practice is the presence of a "hidden structure" in real-world instances. The application of parameterized complexity theory to answer-set programming would give a crucial understanding of how solver heuristics work. This profound understanding can be used to improve the decision heuristics of modern solvers and yields new efficient algorithms for decision problems in the nonmonotonic setting. My research aims to explain the gap between theoretical upper bounds and the effort to solve real-world instances. I will further develop by means of parameterized complexity exact algorithms which work efficiently for real-world instances. The approach is based on backdoors which are small sets of atoms that represent "clever reasoning shortcuts" through the search space. The concept of backdoors is widely used in the areas of propositional satisfiability and constraint satisfaction. I will show how this concept can be adapted to the nonmonotonic setting and how it can be utilized to improve common algorithms.

Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations.

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

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.

Handling uncertainty and defeasibility in a possibilistic logic setting

Default rules express concise pieces of knowledge having implicit exceptions, which is appropriate for reasoning under incomplete information. Specific rules that explicitly refer to exceptions of more general default rules can then be handled in a non-monotonic setting. However, there is no assessment of the certainty with which the conclusion of a default rule holds when it applies. We propose a formalism in which uncertain default rules can be expressed, but still preserving the distinction between the defeasibility and uncertainty semantics by means of a two steps processing. Possibility theory is used for representing both uncertainty and defeasibility. The approach is illustrated in persistence modeling problems.

Nonmonotonic Logics and Semantics

Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas iff a holds in all models in which each of the elements of A holds. A more liberal semantics has been considered: a formula a may be deduced from a set A of formulas iff a holds in all of the preferred models in which all the elements of A hold. Shoham proposed that the notion of preferred models be defined by a partial ordering on the models of the underlying language. A more general semantics is described in this paper, based on a set of natural properties of choice functions. This semantics is here shown to be equivalent to a semantics based on comparing the relative importance of sets of models, by what amounts to a qualitative probability measure. The consequence operations defined by the equivalent semantics are then characterized by a weakening of Tarski's properties in which the monotonicity requirement is replaced by three weaker conditions. Classical propositional connectives are characterized by natural introduction-elimination rules in a nonmonotonic setting. Even in the nonmonotonic setting, one obtains classical propositional logic, thus showing that monotonicity is not required to justify classical propositional connectives.

Self-reference and incompleteness in a non-monotonic setting

Le premier theoreme d'incompletude de Godel etablit que pour toute theorie axiomatique T, d'un type assez normal, il y a des propositions indecidables dans T. L'A. se demande si ce resultat peut etre generalise, et dans quelles conditions, a des analogues de ces theories axiomatiques comme celle qu'il nomme «structure non-monotone», et qui inclut des inferences non-deductives

Non-monotonic set theoretic operations

We introduce a new operation called the non-monotonic intersection. This operation has significant applications in default and other commonsense reasoning systems. In addition to being non-monotonic this operator is not a pointwise operator. Furthermore it is neither commutative nor associative but rather supports a concept of priority amongst its arguments. We also introduce a related non-monotonic union. We generalize these ideas with the aid of t-norms.

Connectives and quantifiers in fuzzy sets

We discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets. We discuss the extension principles used in extending operators to fuzzy subsets. We provide a structure for aggregating linguistic variables in the theory of approximate reasoning. We look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation. We discuss non-monotone set operators. We introduce the concept of linguistic quantifiers and look at some applications of these structures. We are particularly concerned with their application to linguistic summarizers, syllogistic reasoning and multi-criteria decision making.