Fuzzy extension of interval-based temporal sub-algebras
S. Badaloni Dept.of Electronics and Computer Science University of Padova, Italy
[email protected]
Abstract
terval algebra is contained but the expressive power is lowered w.r.t. the Interval Algebra.
In a previous paper, we have proposed a fuzzy extension of Allen’s Interval Algebra, called IAf uz , which is able to model soft temporal constraints and uncertainty in a unified way. In this paper, we address the problem of finding tractable subalgebras of IAf uz , as it has been done in the classical case. In particular, we show that the fuzzy extensions of classical SAc and SA subalgebras are tractable sub-algebras of IAf uz .
In [2, 3], we have proposed a fuzzy extension of the Interval Algebra [1], called IAf uz , where degrees of preferences belonging to [0, 1] ⊆ R are attached to each atomic relation between temporal intervals, and possibility theory is used to model soft temporal constraints and uncertainty in a unified way. In particular, uncertainty is expressed by means of prioritized temporal constraints (their priority indicating the degree of necessity of their satisfaction), which can be represented in turn as IAf uz -relations.
Keywords: Temporal Reasoning, Fuzzy Constraints.
1
M. Giacomin Dept.of Electronics for Automation University of Brescia, Italy
[email protected]
Introduction
Dealing with temporal reasoning, a major difficulty is the tractability of the reasoning problem. For instance, it is well known that Allen’s Interval Algebra IA [1] is nontractable, as the two main interesting problems in this framework (finding a consistent scenario and computing the minimal network) are NP-complete. Since this fact seems to preclude general, large-scale applications, there has been a lot of work in identifying tractable sub-classes of IA (for example [14, 15, 19]). In particular, two tractable fragments of the interval language have been identified, i.e. the SAc and SA sub-algebras [15], that can be expressed using the point based language PA. In this way, the computational cost of full in-
While uncertainty issues have been discussed elsewhere [2, 3], in this paper we address the specific problem of finding tractable subalgebras of IAf uz , as it has been done in the classical case. Since IAf uz is defined as a generalization of the classical one, if the expressive power of its language is maintained then the intractability of the reasoning problem affects IAf uz too: in particular, the problems of finding the minimal network and of finding an optimal solution are NP-complete, as in the classical case. So, in this paper we generalize tractable sub-algebras of IA to our fuzzy framework. More specifically, first we show how point-algebras PAc and PA [19], and interval sub-algebras SAc and SA [15] can be fuzzy extended. Then, we show that the fuzzy extensions of SAc and SA, called SAfc uz and SAf uz respectively, are tractable sub-algebras of IAf uz .
2
The fuzzy Interval Algebra IAf uz
In classical Interval Algebra [1], temporal knowledge is represented as a graph where nodes represent intervals (i.e. elements of R2 ) and arcs are labelled by the set of equally possible atomic relations existing between pairs of intervals: I1 (rel 1 , rel 2 . . .)I2 where rel i are shown in Table 1 and correspond to 13 mutually exclusive atomic relations that may exist between two intervals (eq stands for equal, b stands for before, etc.). Table 1: Allen’s atomic relations and their inverse eq b d o m s f rel −1 rel eq a di oi mi si fi Given an IA-network N , a singleton labelling of N is an assignment of an atomic relation to each of its edges. A solution of N is a singleton labelling which satisfies all the constraints of the network and is consistent (i.e. it is possible to map each node of N to an element of R2 and have the atomic relations of the labelling hold). We denote the set of solutions of N as SOL(N ). In our approach (for a detailed description, see [2, 3]), we deal with relations between intervals I1 and I2 in this form: I1 (rel 1 [α1 ], rel 2 [α2 ] . . .)I2
union of fuzzy subsets corresponding to every rel i [αi ]. If αi belongs to {0, 1} we re-obtain the classical framework. So the Interval Algebra has been extended to a new fuzzy algebra IAf uz defined on the set: I = {b[α1 ], a[α2 ], m[α3 ], mi [α4 ], d[α5 ], di [α6 ], o[α7 ], oi [α8 ], s[α9 ], si [α10 ], f [α11 ], fi [α12 ], eq [α13 ]} where αi ∈ [0, 1], αi ∈ R, i = 1, . . . , 13 It is closed under the operations of inversion, conjunction, disjunction and composition, which corresponds to the usual operations on fuzzy relations adapted to our interval-based framework. More specifically, given the relation R = (rel 1 [α1 ], . . . , rel 13 [α13 ]), we define the unary inversion operator R−1 as R−1 = (rel 1 −1 [α1 ], . . . , rel 13 −1 [α13 ]), where rel i −1 is defined as in Table 1. Given any two relations R and R , where R = (rel 1 [α1 ], . . . , rel 13 [α13 ]) and R = (rel 1 [α 1 ], . . . , rel 13 [α 13 ]), we define the conjunction R = R ⊗ R as R = (rel 1 [α1 ], . . . , rel 13 [α13 ])
αi = min {α i , α i } i ∈ {1, . . . , 13}
the disjunction R = R ⊕ R as R = (rel 1 [α1 ], . . . , rel 13 [α13 ]) αi = max {α i , α i } i ∈ {1, . . . , 13} and the composition R = R ◦ R as
(1)
where αi is the preference degree of rel i (i = 1, . . . , 13), belonging to the interval [0, 1]. Given a relation R, we denote the preference degree assigned by R to a generic Allen’s atomic relation rel i as degR (rel i ). Interpreting the preference degree as membership degree leads to represent a soft constraint by a fuzzy relation according to the classical literature on FCSPs [7, 9, 5]. An atomic relation with a degree α is a fuzzy subset of R2 × R2 defined as follows: those pairs of intervals which satisfy “classically” the same atomic relation have membership degree α; all the others have membership degree 0. The semantics of (1) is the relation obtained by the
R = (rel 1 [α1 ], . . . , rel 13 [α13 ]) αi =
max
j,k:rel i ∈{rel j ◦rel k }
min {α j , α k }
i, j, k ∈ {1, . . . , 13} While in classical IA-networks a partial singleton labelling is locally consistent if it satisfies all the involved constraints, in our framework local consistency is graded. In particular, given an IAf uz -network N , we define the degree of local consistency of a partial singleton labelling s, denoted as degN (s), as follows: if the assignment is inconsistent then degN (s) = 0, otherwise degN (s) is equal to the preference degree of the least satisfied constraint. Accordingly, the concept of
k-consistency is generalized in this way: an IAf uz network is k-consistent if and only if, for every set of k−1 nodes, every assignment with a degree of local consistency α is extensible to any other k-th variable maintaining the same degree α. Path-consistency is k-consistency with k = 3. In this paper, we address a reasoning task that is of interest in our fuzzy framework, namely the computation of the minimal network equivalent to a given IAf uz -network N . Two networks N1 and N2 are equivalent if they involve the same variables and for every complete singleton labelling s degN1 (s) = degN2 (s). The minimal network of N is the ‘most explicit network’ among the equivalent ones. More specifically, an IAf uz -network N is minimal if and only if, for every relation Rij between a pair of intervals (Ii , Ij ), and ∀rel k [α] ∈ Rij , there is a complete singleton labelling s of N which assigns rel k to Rij and such that degN (s) = α. In the classical IA framework, the minimal network problem is usually faced by means of constraint propagation algorithms, used to render network constraints more explicit, enforcing consistency of sub-networks. In this paper, we will consider in particular the path-consistency algorithm developed for IA [17], which has been generalized to the fuzzy framework in [2]; its complexity is augmented at most by a factor k equal to the number of the levels of preference used to define the IAf uz network, yielding a worst-case complexity of O(k ∗ n3 ), where n is the number of nodes. Moreover, we will also consider the algorithm AAC proposed by van Beek and Cohen [15], which, given an IA-network, enforces minimality of all its 4subnetworks: AAC can be generalized as well to an O(k ∗n4 ) algorithm for IAf uz -networks.
3
From point-algebras to fuzzy point-algebras
The classical point-algebra PA [19] is based on the notion of time points. There are three basic relations that can hold between two
points, namely , therefore 8 possible relations between them can be expressed in point algebra, i.e. ∅, , ≥, =, ?. A particular subset of PA is the PAc algebra, that is PA without =, which has interesting computational properties, in that polynomial [O(n3 )] path-consistency algorithm achieves minimality [15]. In [10] it is proved that also PA is tractable, but, instead of path consistency, minimality of all the 4-subnetworks is required to ensure minimality of the whole network. As it will be shown in the following, the point algebras introduced above play a particular role in the study of tractable classes of IA. Since it turns out that the same role can be played in our fuzzy framework too, in this section we introduce the fuzzy extensions of PA and PAc , that we call PAf uz and PAfc uz , respectively. We define PAf uz in the same way as IAf uz , by considering points instead of intervals and PA relations instead of Allen’s relations. Therefore, PAf uz is defined on the set I = {< [α1 ], = [α2 ], > [α3 ]} where αi ∈ [0, 1], αi ∈ R, i = 1, 2, 3 PAfc uz is the subalgebra of PAf uz defined on the following set: I = {< [α1 ], = [α2 ], > [α3 ]} where αi ∈ [0, 1], αi ∈ R, i = 1, 2, 3 α2 ≥ min {α1 , α3 } The idea is to exclude the fuzzy counterpart of the = relation, which intuitively corresponds to the class of PAf uz relations (< [α1 ], = [α2 ], > [α3 ]) such that α2 < α1 and α2 < α3 . More formally, the relation between PAfc uz and PAc can be analyzed introducing the notion of α-cut. Definition 1 Given a PAf uz ( IAf uz ) relation Rf uz , its α-cut Rα is the PA (IA) relation made up of atomic relations rel i such that degRf uz (rel i ) ≥ α. Proposition 1 Given a relation R ∈ PAf uz , R ∈ PAfc uz iff ∀α ∈ [0, 1] Rα ∈ PAc .
In [11] we have proved that PAfc uz is an algebra (i.e. closed under inversion, conjunction and composition), and the proof given by van Beek and Cohen about the completeness of Path-Consistency algorithm for PAc networks [15] can be easily extended to PAfc uz networks. On the basis of these two results, it can be proved that the Path-Consistency algorithm applied to PAfc uz networks finds the equivalent minimal network. In fact, since the operations used by the application of the algorithm are only conjunctions and compositions of relations and the algebra is closed w.r.t. these operations, the network remains a PAfc uz network. Besides, since it is pathconsistent, it is also minimal.
4
Fuzzy extension of pointizable algebras
noted as {I1− , I1+ , I2− , I2+ }, and let us indicate their binary relations as −− −+ +− ++ −+ −+ , R12 , R12 , R12 , R11 , R22 . SPf uz R12 A is the class of the networks such that −+ −+ = {< [α1 ]} and R22 = {< [α2 ]}, R11 where α1 , α2 ∈ [0, 1]. In the following, a network N ∈ SPf uz A will be denoted by the 6-tuple of its PAf uz relations. It can be easily shown that N has at most 13 distinct solutions with degree of satisfaction strictly greater than 0, namely those which correspond to the possible relative dispositions of intervals I1 and I2 : let us call this set SOLIA ⊆ {}6 . Each element s ∈ SOLIA corresponds to an atomic relation of IA. Let fIA : SOLIA → IA be the relevant function (e.g. fIA (
