A fuzzy extension of interval-based constraint networks

The aim of this work is to integrate the ideas of flexibility and uncertainty into Allen’s interval-based temporal logic [1], defining a new formalism (IA) which extends classical Interval Algebra (IA). We refer to Dubois, Fargier & Prade approach to Fuzzy Constraint Satisfaction Problem (FCSP ) [3], in which constraints are satisfied to a degree; this framework is based on Fuzzy Sets and Possibility Theory, which is a more robust approach to uncertainty with respect to probabilistic one [2]. While FCSP has been applied in job-shop scheduling problems to deal with quantitative temporal information (such as job release date and due date) [4], in the present work we deal exclusively with the qualitative aspect of temporal knowledge since it may be convenient for the solution of planning problems. A new fuzzy interval algebra IA is defined. Classical concepts of consistency and minimality are generalized to deal with IA networks as well as path-consistency and branch & bound algorithms. 1 The Fuzzy Interval Algebra Classical Interval Algebra can be viewed as a special case of CSP, since an interval can be interpreted as an element of R2 and a relation between a pair of intervals as a subset of R2 ×R2. According to the definition of “soft constraint” given in [3] as the constraint which directly expresses preferences among solutions, this subset becomes fuzzy, its membership function indicates the degree of preference of an assignment among the others. Allen’s relations are qualitative, expressed as disjunctions of the thirteen atomic relations of Figure 1: then an assignment is a choice of an atomic relation among the allowed ones. In a fuzzy approach we assign to every atomic relation reli a preference degree. So we deal with relations between intervals I1 and I2 in the form I1(rel1[α1], rel2[α2] . . .)I2 (1) where αi is the preference degree of reli (i = 1, . . . , 13). An atomic relation with a degree α is a fuzzy subset of R2 × R2 defined in this way: the pairs of intervals which satisfy “classically” the same atomic relation have membership degree α, all the rel eq b a d di o oi m mi s si f fi rel eq a b di d oi o mi m si s fi f Figure 1: Allen’s atomic relations and their inverse