Interval-Based Possibilistic Logic Salem Benferhat 1 , Julien Hu´ e 1 , Sylvain Lagrue 1 , Julien Rossit 2 1 Universit´ e Lille – Nord de France CRIL – CNRS UMR 8188 Artois, F-62307 Lens {benferhat,hue,lagrue}@cril.fr 2 Universit´ e Paris Descartes LIPADE – France julien.rossit@parisdescartes.fr Abstract Possibilistic logic is a well-known framework for dealing with uncertainty and reasoning under incon- sistent knowledge bases. Standard possibilistic logic expressions are propositional logic formulas associ- ated with positive real degrees belonging to [0,1]. However, in practice it may be difficult for an expert to provide exact degrees associated with formulas of a knowledge base. This paper proposes a flexible representation of un- certain information where the weights associated with formulas are in the form of intervals. We first study a framework for reasoning with interval-based possibilistic knowledge bases by extending main concepts of possibilistic logic such as the ones of necessity and possibility measures. We then provide a characterization of an interval-based possibilistic logic base by means of a concept of compatible stan- dard possibilistic logic bases. We show that interval- based possibilistic logic extends possibilistic logic in the case where all intervals are singletons. Lastly, we provide computational complexity results of deriv- ing plausible conclusions from interval-based possi- bilistic bases and we show that the flexibility in rep- resenting uncertain information is handled without extra computational costs. 1 Introduction Possibilistic logic (e.g. [Dubois et al., 1994; Dubois and Prade, 2004]) is an important framework for representing and reason- ing with uncertain and inconsistent pieces of information. Un- certainty is syntactically represented by a set of weighted for- mulas of the form K = {〈ϕ i ,α i 〉 : i =1, .., n} where ϕ i ’s are propositional formulas and α i ’s are real numbers belonging to [0,1]. The pair 〈ϕ i ,α i 〉 means that ϕ i is certain (or important) to at least a degree α i . An inference machinery has been pro- posed in [Lang, 2000] to derive plausible conclusions from a possibilistic knowledge base, which needs log 2 (m) calls to the satisfiability test of a set of propositional clauses (SAT), where m is the number of different degrees used in K. Uncertainty is also represented at the semantic level by associating a possi- bility degree with each possible world (or interpretation). Several extensions of possibilistic logic have been proposed to replace the unit interval [0, 1] by some complete lattice or even by a partial pre-order. In [Lafage et al., 1999], a set of assumptions which supports a formula is used in- stead of a real positive number. In [Dubois et al., 1992], the degrees are replaced by a set of positive values (not nec- essarily in [0, 1]) representing a time frame where the for- mulas are known to be true. In [Benferhat et al., 2004; Benferhat and Prade, 2006], a partially ordered extension of possibilistic logic has been proposed. However, these exten- sions either increases the computational complexity (e.g. when dealing with partially pre-ordered information) or fail to gen- eralize possibilistic logic. For instance, the so-called timed possibilistic logic proposed in [Dubois et al., 1992] does not recover standard possibilistic logic when sets of times assigned to formulas are singletons belonging to [0, 1]. This paper is in the spirit of these extensions of possibilis- tic logic. It studies theoretical foundations, with an analysis of computational issues, of interval-based possibilistic logic. The question addressed in this paper is whether one can extends and increases the expressive power of standard possibilistic logic, by representing imprecision regarding uncertainty associated with formulas, without increasing the computational complex- ity of the reasoning process. The framework considered in this paper is the one of interval-based possibilistic logic. At the syntactic level, pieces of information are represented by an interval-based possibilis- tic knowledge base, of the form IK = {〈ϕ i ,I i 〉 : i =1, .., n} where I i =[α i ,β i ] is a closed sub-interval of ]0, 1]. The pair 〈ϕ i ,I i 〉, called an interval-based weighted formula, means that the weight associated with ϕ i is one of the elements in I i . This disjunctive interpretation of 〈ϕ i ,I i 〉 should not be confused with the conjunctive interpretation (used in [Dubois et al., 1992]), which corresponds to the fact that ∀α i ∈ I i , 〈ϕ i ,α i 〉 is true. The conjunctive interpretation of intervals makes sense when considering temporal information, where 〈ϕ i ,I i 〉 means that ϕ i is true in all the interval time I i . Similarly, the semantic of interval-based possibilistic logic is an interval-based possibility distribution, where a sub- interval of [0,1] is assigned to each interpretation. Unlike stan- dard possibilistic logic, an interval-based possibility distribu- tion only induces a partial pre-order over the set of interpreta- tions. On the basis of a disjunctive interpretation of intervals, we 750 Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence