Possibilistic Logic Research Articles

Inférence dans les réseaux possibilistes basés sur le conditionnement ordinal

Junction tree inference algorithms in possibilistic networks are efficient tools to deal with uncertain pieces of information. In presence of important clusters in terms of variables number, these algorithms are inadequate to resolve such instances. In this paper, we present two alternatives for standard algorithms: - An exact approach which consists ofcompactly representing uncertainty at the level of nodes and clusters using possibilistic logic. - An approximative approach which allows to avoid the graphical transformation from the initial graph to the junction tree structure using the idempotency of the minimum operator.

Formalizing argumentative reasoning in a possibilistic logic programming setting with fuzzy unification

Possibilistic Defeasible Logic Programming (P-DeLP) is a logic programming language which combines features from argumentation theory and logic programming, incorporating the treatment of possibilistic uncertainty at the object-language level. In spite of its expressive power, an important limitation in P-DeLP is that imprecise, fuzzy information cannot be expressed in the object language. One interesting alternative for solving this limitation is the use of PGL +, a possibilistic logic over Gödel logic extended with fuzzy constants. Fuzzy constants in PGL + allow expressing disjunctive information about the unknown value of a variable, in the sense of a magnitude, modelled as a (unary) predicate. The aim of this article is twofold: firstly, we formalize DePGL +, a possibilistic defeasible logic programming language that extends P-DeLP through the use of PGL + in order to incorporate fuzzy constants and a fuzzy unification mechanism for them. Secondly, we propose a way to handle conflicting arguments in the context of the extended framework.

Introducing possibilistic logic in ILP for dealing with exceptions

In this paper we propose a new formalization of the inductive logic programming (ILP) problem for a better handling of exceptions. It is now encoded in first-order possibilistic logic. This allows us to handle exceptions by means of prioritized rules, thus taking lessons from non-monotonic reasoning. Indeed, in classical first-order logic, the exceptions of the rules that constitute a hypothesis accumulate and classifying an example in two different classes, even if one is the right one, is not correct. The possibilistic formalization provides a sound encoding of non-monotonic reasoning that copes with rules with exceptions and prevents an example to be classified in more than one class. The benefits of our approach with respect to the use of first-order decision lists are pointed out. The possibilistic logic view of ILP problem leads to an optimization problem at the algorithmic level. An algorithm based on simulated annealing that in one turn computes the set of rules together with their priority levels is proposed. The reported experiments show that the algorithm is competitive to standard ILP approaches on benchmark examples.

Combining multiple prioritized knowledge bases by negotiation

Recently, several belief negotiation models have been introduced to deal with the problem of belief merging. A negotiation model usually consists of two functions: a negotiation function and a weakening function. A negotiation function is defined to choose the weakest sources and these sources will weaken their point of view using a weakening function. However, the currently available belief negotiation models are based on classical logic, which makes them difficult to define weakening functions. In this paper, we define a prioritized belief negotiation model in the framework of possibilistic logic. The priority between formulae provides us with important information to decide which beliefs should be discarded. The problem of merging uncertain information from different sources is then solved by two steps. First, beliefs in the original knowledge bases will be weakened to resolve inconsistencies among them. This step is based on a prioritized belief negotiation model. Second, the knowledge bases obtained by the first step are combined using a conjunctive operator which may have a reinforcement effect in possibilistic logic.

Hybrid possibilistic networks

Possibilistic networks and possibilistic logic are two standard frameworks of interest for representing uncertain pieces of knowledge. Possibilistic networks exhibit relationships between variables while possibilistic logic ranks logical formulas according to their level of certainty. For multiply connected networks, it is well-known that the inference process is a hard problem. This paper studies a new representation of possibilistic networks called hybrid possibilistic networks. It results from combining the two semantically equivalent types of standard representation. We first present a propagation algorithm through hybrid possibilistic networks. This inference algorithm on hybrid networks is strictly more efficient (and confirmed by experimental studies) than the one of standard propagation algorithm.

A split-combination approach to merging knowledge bases in possibilistic logic

We propose an adaptive approach to merging possibilistic knowledge bases that deploys multiple operators instead of a single operator in the merging process. The merging approach consists of two steps: the splitting step and the combination step. The splitting step splits each knowledge base into two subbases and then in the second step, different classes of subbases are combined using different operators. Our merging approach is applied to knowledge bases which are self-consistent and results in a knowledge base which is also consistent. Two operators are proposed based on two different splitting methods. Both operators result in a possibilistic knowledge base which contains more information than that obtained by the t-conorm (such as the maximum) based merging methods. In the flat case, one of the operators provides a good alternative to syntax-based merging operators in classical logic.

A logic of soft constraints based on partially ordered preferences

Representing and reasoning with an agent's preferences is important in many applications of constraints formalisms. Such preferences are often only partially ordered. One class of soft constraints formalisms, semiring-based CSPs, allows a partially ordered set of preference degrees, but this set must form a distributive lattice; whilst this is convenient computationally, it considerably restricts the representational power. This paper constructs a logic of soft constraints where it is only assumed that the set of preference degrees is a partially ordered set, with a maximum element 1 and a minimum element 0. When the partially ordered set is a distributive lattice, this reduces to the idempotent semiring-based CSP approach, and the lattice operations can be used to define a sound and complete proof theory. A generalised possibilistic logic, based on partially ordered values of possibility, is also constructed, and shown to be formally very strongly related to the logic of soft constraints. It is shown how the machinery that exists for the distributive lattice case can be used to perform sound and complete deduction, using a compact embedding of the partially ordered set in a distributive lattice.

A sound and complete fuzzy temporal constraint logic

In this work, we define an extended fuzzy temporal constraint logic (EFTCL) based on possibilistic logic. EFTCL allows us to handle fuzzy temporal constraints between temporal variables and, therefore, enables us to express interrelated events through fuzzy temporal constraints. EFTCL is compatible with a theoretical temporal reasoning model: the fuzzy temporal constraint networks (FTCN). The syntax, the semantics and the deduction and refutation theorems for EFTCL are similar to those defined for the sound and noncomplete fuzzy temporal constraint logic (FTCL). In this paper, a resolution principle for performing inferences which take these constraints into account is proposed for EFTCL. Moreover, we prove the soundness and the completeness of the refutation by resolution in EFTCL.

PRODUCT-BASED CAUSAL NETWORKS AND QUANTITATIVE POSSIBILISTIC BASES

In possibility theory, there are two kinds of possibilistic causal networks depending if possibilistic conditioning is based on the minimum or on the product operator. Similarly there are also two kinds of possibilistic logic: standard (min-based) possibilistic logic and quantitative (product-based) possibilistic logic. Recently, several equivalent transformations between standard possibilistic logic and min-based causal networks have been proposed. This paper goes one step further and shows that product-based causal networks can be encoded into product-based knowledge bases. The converse transformation is also provided.

Bipolar possibility theory in preference modeling: Representation, fusion and optimal solutions

The bipolar view in preference modeling distinguishes between negative and positive preferences. Negative preferences correspond to what is rejected, considered unacceptable, while positive preferences correspond to what is desired. But what is tolerated (i.e., not rejected) is not necessarily desired. Both negative and positive preferences can be a matter of degree. Bipolar preferences can be represented in possibilistic logic by two separate sets of formulas: prioritized constraints, which describe what is more or less tolerated, and weighted positive preferences, expressing what is particularly desirable. The problem of merging multiple-agent preferences in this bipolar framework is then discussed. Negative and positive preferences are handled separately and are combined in distinct ways. Since negative and positive preferences are stated separately, they may be inconsistent, especially in this context of preference fusion. Consistency can be enforced by restricting what is desirable to what is tolerated. After merging, and once the bipolar consistency is restored, the set of preferred solutions can be logically characterized. Preferred solutions should have the highest possible degree of feasibility, and only constraints with low priority may have to be discarded in case of inconsistency inside negative preferences. Moreover, preferred solutions should satisfy important positive preferences when feasible (positive preferences may be also inconsistent). Two types of preferred solutions can be characterized, either in terms of a disjunctive combination of the weighted positive preferences, or in terms of a cardinality-based evaluation.

Reasoning with multiple-source information in a possibilistic logic framework

This paper addresses the problem of merging uncertain information in the framework of possibilistic logic. It presents several syntactic combination rules to merge possibilistic knowledge bases, provided by different sources, into a new possibilistic knowledge base. These combination rules are first described at the meta-level outside the language of possibilistic logic. Next, an extension of possibilistic logic, where the combination rules are inside the language, is proposed. A proof system in a sequent form is given, sound and complete with respect to the possibilistic logic semantics. Possibilistic fusion modes are illustrated on an example inspired from an application of a localization of a robot in an uncertain environment.

Reasoning with partially ordered information in a possibilistic logic framework

In many applications, the reliability relation associated with available information is only partially defined, while most of existing uncertainty frameworks deal with totally ordered pieces of knowledge. Partial pre-orders offer more flexibility than total pre-orders to represent incomplete knowledge. Possibilistic logic, which is an extension of classical logic, deals with totally ordered information. It offers a natural qualitative framework for handling uncertain information. Priorities are encoded by means of weighted formulas, where weights are lower bounds of necessity measures. This paper proposes an extension of possibilistic logic for dealing with partially ordered pieces of knowledge. We show that there are two different ways to define a possibilistic logic machinery which both extend the standard one.

An annotated logic theorem prover for an extended possibilistic logic

In this paper we present a theorem prover for possibilistic logic extended with fuzzy constants and fuzzily restricted quantifiers. First of all, we propose a representation for this logic in terms of Horn clauses. We then show how to transform this Horn clause formalism into the generalized annotated logic formalism proposed by Kifer and Subrahmanian in 1992. Specifically, the valuation in a possibilistic clause generates the annotation which will be attached to the head of the annotated clause. We also show how the inference rules of this possibilistic logic can be translated in terms of the mechanisms provided by this annotated logic. Finally, we discuss the implementation of a theorem prover for this possibilistic logic, now translated to annotated logic, in system KOMET, a large-purpose system which has generalized annotated logic as its underlying framework. In this paper we also present proofs relative to the theoretical issues and some examples implemented in KOMET.

Adding similarity-based reasoning capabilities to a Horn fragment of possibilistic logic with fuzzy constants

PLFC is a first-order possibilistic logic dealing with fuzzy constants and fuzzily restricted quantifiers. The refutation proof method in PLFC is mainly based on a generalized resolution rule which allows an implicit graded unification among fuzzy constants. However, unification for precise object constants is classical. In order to use PLFC for similarity-based reasoning, in this paper we extend a Horn-rule sublogic of PLFC with similarity-based unification of object constants. The Horn-rule sublogic of PLFC we consider deals only with disjunctive fuzzy constants and it is equipped with a simple and efficient version of PLFC proof method. At the semantic level, it is extended by equipping each sort with a fuzzy similarity relation, and at the syntactic level, by fuzzily “enlarging” each non-fuzzy object constant in the antecedent of a Horn-rule by means of a fuzzy similarity relation.

A note on the model-theoretic semantics of fuzzy logic programming for dealing with inconsistency

Inconsistency is likely to exist in a knowledge base built up from various sources of information, especially in the presence of uncertainty and fuzziness. In such a case, inconsistency is often local and thus should not cause the entire knowledge base to collapse. For a fuzzy logic system to be able to deal with local inconsistency, an appropriate model-theoretic semantics is required. This paper analyses and shows that the model-theoretic semantics approach of possibilistic logic, which has recently become popular to be applied to the development of fuzzy set logic whose formulas can have fuzzy set constants, cannot deal with local inconsistency. We then introduce the annotated logic-based approach as a solution, and analyse the difference and relation between the two approaches.

Possibilistic logic: a retrospective and prospective view

Possibilistic logic is a weighted logic introduced and developed since the mid-1980s, in the setting of artificial intelligence, with a view to develop a simple and rigorous approach to automated reasoning from uncertain or prioritized incomplete information. Standard possibilistic logic expressions are classical logic formulas associated with weights, interpreted in the framework of possibility theory as lower bounds of necessity degrees. Possibilistic logic handles partial inconsistency since an inconsistency level can be computed for each possibilistic logic base. Logical formulas with a weight strictly greater than this level are immune to inconsistency and can be safely used in deductive reasoning. This paper first recalls the basic features of possibilistic logic, including information fusion operations. Then, several extensions that mainly deal with the nature and the handling of the weights attached to formulas, are suggested or surveyed: the leximin-based comparison of proofs, the use of partially ordered scales for the weights, or the management of fuzzily restricted variables. Inference principles that are more powerful than the basic possibilistic inference in case of inconsistency are also briefly considered. The interest of a companion logic, based on the notion of guaranteed possibility functions, and working in a way opposite to the one of usual logic, is also emphasized. Its joint use with standard possibilistic logic is briefly discussed. This position paper stresses the main ideas only and refers to previous published literature for technical details.

A BIG-STEPPED PROBABILITY APPROACH FOR DISCOVERING DEFAULT RULES

This paper deals with the extraction of default rules from a database of examples. The proposed approach is based on a special kind of probability distributions, called "big-stepped probabilities", which are known to provide a semantics for non-monotonic reasoning. The rules which are learnt are genuine default rules, which could be used (under some conditions) in a non-monotonic reasoning system and can be encoded in possibilistic logic.

Logical representation and fusion of prioritized information based on guaranteed possibility measures: Application to the distance-based merging of classical bases

In the possibility theory framework, prioritized information can be logically expressed in different formats. The most usual way, used in standard possibilistic logic, is to associate necessity degrees with propositional formulas. This paper considers another representation and fusion of prioritized information using guaranteed possibility measures. Prioritized pieces of information are then represented by sets of weighted formulas, called Δ-knowledge bases, where weights are lower bounds of guaranteed possibility degrees of formulas.We first show that the basic notions of standard possibilistic logic have natural counterparts when dealing with Δ-knowledge bases. In particular we present the inference machinery, and provide syntactic, but semantically meaningful, merging of Δ-knowledge bases. In the second part of the paper, we show that distance-based merging propositional knowledge bases can be naturally encoded using Δ-knowledge bases. Moreover, this encoding is more efficient than the necessity-based encoding of distance-based merging operator.

Temporal scenario modelling and recognition based on possibilistic logic

We propose in this paper a new approach for the modelling and recognition of temporal scenarios. A scenario is represented by three different structures. The first one models the logical dependency between the elements of the scenario, using possibilistic logic, while the second one is the minimal temporal graph representing all temporal constraints between the events. The third structure explains the way the matching between observations and scenarios has to be done. The consistency between the three structures is ensured.

Fusion of possibilistic knowledge bases from a postulate point of view

This paper proposes a postulate-based analysis of the fusion of possibilistic logic bases which are made of pieces of information expressing knowledge associated with certainty degrees. We propose two main sets of postulates: one only focuses on plausible conclusions, while the other considers both plausible conclusions and the certainty degrees associated with them. For each rational postulate, the class of operators that it satisfies is identified. The existence of weights associated with the pieces of information considerably enriches the postulates-based analysis, and leads to a refined classification of combination operators.