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Probabilistic Approach for Representing and Reasoning with Repetitive Events  Rasiah Loganantharaj Automated Reasoning Laboratory USL, Lafayette, Louisiana [email protected]

Steve Giambrone Dept. of Philosophy USL Lafayette, Louisiana [email protected]

Abstract

Fortunately, many interesting repetitive events are either periodic or stochastic. Consider the statement \Faculty meeting starts at 4pm on every Friday during Fall of 1994." It is a periodic repetitive event. In reality, the meeting may not start sharply at 4pm on every Friday, instead it may start at 4.05pm on many Fridays and at 4pm or 4.10pm on some Fridays. This kinds of uncertainty could be modeled by associating probabilities with the starting time. Suppose the review of the faculty meeting in the last semester (Spring of 94) revealed that the meeting started at 4pm, 4.05pm, and 4.10pm, respectively, for 15%, 80% and 5% of the total events. These kinds of uncertainty should not surprise anyone since we all operate under uncertainty of one kind or other. If we assume the pattern of the previous semester repeats for the Fall of 94 then the information about the faculty meeting could be restated as "meeting starts on every Friday in Fall of 1994 at 4pm with the probability of 15%, at 4.5pm with the probability of 80%, and at 4.10pm with the probability of 5%. These probabilistic information are on quantitative relations. Similarly, we can have probabilistic information on qualitative relation of repetitive events. We will look at qualitative relations among repetitive events. For example, consider the following information: John drinks coee during his breakfast with the probability of 90%. Sometimes, he drinks his coee before the breakfast and drinks orange juice during his breakfast. The probability of John drinking coee before the breakfast is 8%. There were few occasions he drank water during his breakfast and drank coee after the breakfast. The probability of John drinking coee after the breakfast is 2%. In this information the probabilities are associated with the qualitative relationship between drinking coee and having breakfast. Assume the

In this paper we study repetitive events with qualitative constraints. We provide a formalism for representing and reasoning with repetitive events using probabilistic approach.

1 Introduction

In this paper we introduce a probabilistic approach for representing and reasoning with repetitive events. If an event repeats over a speci ed period then it is called a repetitive event. The event \ John attends Math 101 class at 6pm on every Monday in Fall 1994" is an example of repetitive event. Very often, we come across repetitive events in every day activities. In spite of the urgent need for formally studying the problem of representing and reasoning with repetitive events, very little has been done in this area. In this paper we focus on formalizing some aspects of representing and reasoning with repetitive events, and providing a practical solution to solve the problem using probabilistic theory. There are two types of repetitive events: periodic and aperiodic. Periodic event repeats at regular intervals. For example, Mondays repeats after every 7 days. All repetitive events except the periodic events are called aperiodic repetitive events. Aperiodic events can be further divided into stochastic and random events. In random repetitive events there is no regularity among the temporal distances or periods, thus it is extremely dicult to study. \Raining at Melbone Beach in Florida in Winter 1994" is an example for random repetitive event.

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paper appeared at the FLAIRS95 conference pp 26-

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following event happened. Mike talked to John over the phone while John was having coee. Suppose, we are interested in nding how Mike's telephone conversation was related to John's breakfast. In this example we have two types of information: temporal assertion of an actual event and information about a stochastic repetitive event. In this paper, we will provide the details of how to use these information. This paper is organized as the following. We introduce interval-based logic in Section 2. In Section 3, we describe the representation of uncertain temporal knowledge and its propagation. This paper is concluded by a summary and discussion in section 4.

2 Background on Interval Based System Allen [1] has proposed an interval logic that uses time intervals as primitives. In his logic, the following seven relations and their inverses are de ned to express the temporal relations between two intervals: before(after), meets(met-by), overlaps(overlapped-by), starts(started-by), during(contains), ends(ended-by), and equals. Here, the inverse relations are indicated within parentheses. Since the inverse of equal is same as itself, there are, in fact, only thirteen relations. Temporal inferencing is performed by manipulating the network corresponding to the intervals. Each interval maps onto a node of a network called temporal constraint network (TCN). A temporal relation, say R, from an interval, say Ii , to another interval, say Ij , is indicated by the label Rij on the directed arc from Ii to Ij . Obviously, the label Rji of the directed arc from Ij to Ii is the inverse relation of Rij . If we have de nite information about the relation Ii to Ij then Rij will be a primitive interval relation, otherwise it will be disjunctions of two or more primitive interval relations. Suppose the relation Ii to Ij (Rij ) and the relation Ij to Ik (Rjk ) are given. The relation between Ii and Ik , constrained by Rij and Rjk, is given by composing Rij and Rjk. In general, Rij and Rjk can be disjunctions of primitive relations, they are represented as: Rij = frij1 ; rij2 ; : : :; rijm g and Rjk = 1 ; r2 ; : : :; rn g where r is one of the primitive frjk ij jk jk relations de ned in the system. The interval Ii is related to the interval Ik by the temporal relation

given by the following expression:

[ ( [

p) Rij  Rjk = (rijl  rjk l=1:::m p=1:::n

)

p ) is a composition (transitive relawhere (rijl  rjk p , and is obtained from the entry tion) of rijl and rjk j of the transitivity table [1] at the riji row and the rjk column. Alternatively this could be written as

Rij  Rjk = fT(rij ; rjk)jrij 2 Rij ^ rjk 2 Rjkg where T(rij ; rjk) is the value of Allen's look up composition table of row rij and column rjk . Temporal constraints are propagated to the rest of the network to obtain the minimal temporal network in which each label between a pair of intervals is minimal with respect to the given constraints. Vilain et al. [9] have shown that the problem of obtaining minimal labels for an interval-based temporal constraint network is NP-complete. Approximation algorithms, however, are available for temporal constraint propagation. Allen proposed an approximate algorithm that has an asymptotic time complexity of O(N 3 ) where N is the number of intervals. Allen's algorithm is not guaranteed to obtain the minimum relations, but it always nd the superset of the minimal label. Since any set is a super set of a null set it is not very comforting because it is possible that global inconsistency may be hiding under 3consistency.

3 Representation of Uncertainty of Repetitive Events Representing and reasoning with uncertainty is not new to AI problem solving. In many expert system applications, uncertainty have been studied under approximate reasoning. Mycin system [8, 2] has used certainty factor whose value varies from -1 to 1 through 0 to represent the con dence on an evidence or on a rule. The value 1 indicates the assertion is true while the value -1 indicates the evidence is false. 0 indicates no opinion on the evidence. The other values correspond to some mapping of the belief on the evidence onto the scale of -1 to 1. Prospector model [5, 6] uses probabilistic theory with Bayes' theorem and other approximation

techniques to propagate evidences over causal network. Fuzzy logic [10] has also been used in expert systems to capture knowledge with fuzzy quanti ers such as `very much', `somewhat' etc. Other techniques have also been used to handle uncertainty in expert systems. Let us formally represent uncertain temporal relations between intervals. The relation Rij , the relation Ii to Ij , is represented as frij1 (w1); rij2 (w2); :::; rijm(wm )g where rijl (wl ) is a primitive relation with its probability or the relative weight of wl . Since each primitive relation between a pair of intervals is associated with a weight to represent the probability or the relative strength, the summation of the weights must be equal to 1 which we P call probabilistic condition for the weights. That is, mi=1 wi = 1. The boundary value of the weight 1 indicates that the relation is true while the value 0 indicates that the relation is false. Further, the inverse relation of Rij will be the inverse of all the primitive relations of Rij with the same weights. In the presence of new evidences, the probability values of the relations are modi ed to take account of the new evidences. That is, when the relations between a pair of intervals are modi ed or re ned because of other constraining relations, the probability or the weights of the relations are adjusted to satisfy the probabilistic condition. This process is called normalization. Let us explain this with an example: Suppose (1) R12 = fb(0:6); O(0:3); d(0:1)g, (2) R13  R32 = fb(0:4); o(0:5); m(0:1)g. The relation R12 is re ned to fb(w1); o(w2)g. The weights w1 and w2 are computed using the intersection operation and then the weights are normalized such that w1 + w2 = 1. When propagating constraints with probabilistic weights, we may need to de ne such as union, intersection, composition and normalization operations which are used in propagation. 0

0

Union operation: Rij [p Rij = fr(w)jr(wij ) 2 Ri ^ r(wij ) 2 Rj ^ w = max(wij ; wij )g Intersection Operation: Rij \p Rij = fr(w)jr(wij ) 2 Ri ^ r(wij ) 2 Rj ^ w = min(wij ; wij )g Composition Operation: m (wm ) 2 Rjk Rij Rjk = f[p r(w)jrijl (wl ) 2 Rij ^ rjk m l ^ r = T(rij ; rjk) ^ w = min(wl ; wm )g Normalization operation: Suppose the label Rij 0

0

0

0

0

0

takes the following form after re nement

frij1 (w1); rij2 (w2); :::; rijm(wm )g. Let w =

Pm w . i=1 i

After normalization operation the label becomes frij1 (w1); rij2 (w2); :::;Prijm(wm )g where wi = wi=w which ensures that mi=1 wi = 1 0

0

0

0

0

In this approach we use possibilistic ways of combining the constraints as has been used in many expert systems under uncertainty.

3.1 Temporal Propagation A temporal constraint network (TCN) is constructed from the given temporal assertions such that each node of the constraint network represents each interval of the temporal assertions. The labels on each arc of a TCN corresponds to the relations between the corresponding intervals. Further the summation of the weights of each component in each arc should be equal to 1. If no constraint is speci ed between a pair of intervals it will take a universal constraint1 as label in the TCN and it will not be used for the purpose of propagation. When propagating a constraint, other labels of the network may get updated to a subset of their original labels. The process of updating of a label as a result of propagating a constraint is called label re nement. The label re nement takes the following forms: (1) The weights of the labels of an arc get changed, or (2) the primitive relations of a label get reduced. In either case normalization operation is performed to ensure the summation of the weights adds to 1. Suppose we are propagating the label Rij of the arc < I; J > to the arc < I; K > of the triangle IJK. Let the labels of the arcs < J; K > and < I; K > are respectively Rjk and Rik . The new label of the arc < I; K > is computed as Rik \ fRij  Rjk g. When the new relation is not null, the weights on the label are normalized.

3.2 Temporal Constraints Propagation Algorithm for Uncertain Constraints This is an extension of Allen's propagation algorithm. The algorithm uses a rst in rst out (FIFO) queue to maintain the constraints that need to be a universal constraint is the weakest constraint and thus it is a disjunctions of all the primitive relations of the time model 1

propagated. The queue is initialized with all the pairs of constrained intervals. The propagation of constraints is initiated by removing an arc, say < Ii ; Ij >, from the queue and checking whether the relation between Ii to Ij can constrain the relations on all the arcs incident to either Ii or Ij except the arc < Ii Ij >. When a new relation is constrained, that is, the old label is modi ed, the arc (the pair of intervals related by this relation) is placed in the queue. The main propagation algorithm is described in Figure 1. In this algorithm, we use the notation Rij to denote the label of the arc i to j. When we omit the weights on the labels and use the union, the intersection, and the composition operations of temporal logic, the algorithm becomes identical to the one of Allen's [1]. One may expect the asymptotic complexity to be O(N 3) where N is the number of nodes of the TCN. Intuitively one may make a conclusion that this algorithm will also converge in O(N 3 ) time. This may be misleading because we have not yet considered the instability eect of the algorithm due to it normalization operation. An arc is placed back in the queue when either the number of primitive relations of the label is reduced or the weight of the primitive relation of the label is modi ed even when the label remains unchanged. An arc may be placed in the queue at most 12 times as a result of the disjunctive relations being re ned one at a time till it becomes a singleton relation. On the other hand, the number of times an arc is placed in the queue due to the change of weight depends on the following parameters: (1) the resolution of the weights (the number of decimal places that counts) and (2) the error bound we are prepared to tolerate. Let us consider an example. The probability of `having coee' before breakfast is 0.15, during breakfast is 0.8 and after breakfast is 0.05. The probability of `having a coee' overlapping `reading morning newspaper' is 0.8 and `having coee' meets `reading the newspaper' is 0.2. Suppose we want to nd the relation between having breakfast and reading newspaper. Let us propagate the constraints and nd out how the labels gets re ned. Suppose Ib ; Ic ; Ir respectively represent the intervals of having breakfast, coee and reading newspaper. Using the notations de ned in this paper we can de ne the labels of the initial TCN as following.

Initial TCN Rcb = fb(0:15); d(0:8); a(0:05)g Rcr = fo(0:8); m(0:2)g Rbr = to be computed TCN after propagating Rbc to < Ib; Ir > Rcb = fb(0:15); d(0:8); a(0:05)g Rcr = fo(0:8); m(0:2)g Rbr = fo(:8); oi(:15); d(:15);di(:8);f(:15); fi(:8); b(:05); a(:15)g After normalizing Rbr Rbr = fo(:26); oi(:049); d(:049); di(:26);f(:049); fi(:26); b(:024); a(:049)g This is also the 3-consistent TCN labels. From the label (Rbr ) on the arc Ib to Ir it is clear that breakfast overlapping, containing or nished by reading is highly probable than the other relations.

4 Summary and Discussion In this paper we have proposed a formalism for representing and reasoning with repetitive events. We have shown a way to combine repetitive events with propositional assertions. The uncertainty of the repetitive events are modeled as probabilistic weights of the label of a temporal constraint arc. A 3-consistency algorithm is used to propagate the constraints. We have extended Allen's algorithm to handle probabilistic relations among intervals. The operations we have de ned for the labels with weights are applicable to both the points and the intervals. Therefore, our formalism will be applicable to both the points and the intervals. Many works done in the area of repetitive events were focused on periodic repetitive events. Ladkin [3] proposed non convex intervals to represent repetitive events. He maps the repetitive events onto a time line and hence the non convex intervals extends to in nity in the future. The possible relations among the non convex intervals are in nite; therefore, it may not be very attractive to use such representation for reasoning with repetitive events. Poesio et al. [4] had proposed an approach to use periodic repetitive events for maintaining appointments. Morris et al. [7] recently extended non convex intervals for representing and reasoning with repetitive events. To our knowledge this is the rst approach to use probabilistic to handle uncertainty in stochastic repetitive events.

procedure propagate1() 1 While queue is not empty Do 2 f get next < i; j > from the queue /* propagation begins here */ 3 For ( ( k 2 set of intervals ) ^k 6= i ^ k 6= j )Do 4 f temp Rik \p (Rij Rjk ) 5 If temp is null Then signal contradiction and Exit 6 Normalize temp 7 If Rik =6 temp Then 8 f place < i; k > on queue 9 Rik temp g 10 temp Rkj \p (Rki Rij ) 11 If temp is null Then signal contradiction and Exit 12 Normalize temp 13 If Rkj =6 temp Then 14 f place < k; j > on queue 15 Rkj temp g 16 g 17 g Figure 1: propagation algorithm

Acknowledgment This research was supported by a grant from Louisiana Education Quality Support Fund (LEQSF 1993-96RD-A-36).

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