E. Schwalb and R. Dechter. Processing disjunctions in temporal constraint networks. Artificial Intelligence. To appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....only if the (non conditional) temporal problem V, E is consistent. See Appendix for proof) The implication of the above theorem is that we can perform Strong Consistency checking by using specialized algorithms for non conditional temporal problems such as IDCP [4] for STPs, known techniques [18] for TCSPs, and Epilitis [22] for DTPs. 5.1. USES OF THE STRONG CONSISTENCY CONCEPT FOR PLANNING A plan that is represented as a strongly consistent CTP can be executed according to a fixed schedule, in the sense that every action it includes has an assigned, specific time. Not all of the actions ....

Schwalb, E. and R. Dechter: 1997, 'Processing Disjunctions in Temporal Constraint Networks'. Artificial Intelligence 93, 29-61.

....classes of problems. In effect, it has been shown to be equivalent to global consistency for convex binary problems. This means that if a binary convex CSP is PC, its solutions can be derived backtrack free. The PC property has also received particular attention in the area of temporal reasoning [ Schwalb and Dechter, 1997 ] where lower forms of consistency prove to be of less interest. In this paper we show that Montanari s theorem extends to triangulated constraint graphs. Triangulated constraint graphs can be made PC by ensuring that every path of length two is PC. In this case there is no need for additional ....

E. Schwalb and R. Dechter. Processing disjunctions in temporal constraint networks. Artificial Intelligence, 93:29--61, 1997.

....5.1. A CTP V; L; E; OV;O;P is Strongly Consistent consistent. See Appendix for proof) The implication of the above theorem is that we can perform Strong Consistency checking by using specialized algorithms for non conditional temporal problems such as IDCP [4] for STPs, known techniques [18] for TCSPs, and Epilitis [22] for DTPs. 5.1. Uses of the Strong Consistency Concept for Planning A plan that is represented as a strongly consistent CTP can be executed according to a xed schedule, in the sense that every action it includes has an assigned, speci c time. Not all of the actions ....

Schwalb, E. and R. Dechter: 1997, `Processing Disjunctions in Temporal Constraint Networks'. Arti cial Intelligence 93, 29-61.

....reactions to sequences of temporally related events. 2. 2 Possible evolutions as chronicles In the temporal system IxTeT [ Dousson et al. 1993 ] used for dynamic situation assessment, temporal evolutions are taken into account in the shape of chronicles, that are Temporal Constraint Networks [ Schwalb and Dechter, 1997 ] on which classical constraint propagation techniques can be run: time points represent instantaneous changes or begin end points of intervals of time over which a uent is true, and constraints between them are precedences labelled by arithmetic intervals of possible values, allowing to express ....

E. Schwalb and R Dechter. Processing disjunctions in temporal constraint networks. Articial Intelligence, 93:29-61, 1997.

....t ) are still processed at the logical level, but reasoning upon the mere temporal relations can be done through a purely temporal algebraic model. One usually gets a constraint network that is mainly used to check the temporal consistency of the given problem. Temporal Constraint Networks (TCN) [Schwalb and Dechter, 1997] rely on qualitative (or symbolic) constraint algebras [Allen, 1983, Vilain et al. 1989] but more speci cally tackle quantitative (or numerical) constraints. They are now at the heart of many application domains such as scheduling [Dubois et al. 1993] supervision [Dousson et al. 1993] ....

....control community, that behaves in a continuous game manner, and we prove that it solves our problem. The paper ends with a discussion about relevance of the approach and eciency issues. 2 CONTINGENT CONSTRAINTS AND CONTROLLABILITY We rst recall the basics of Temporal Constraint Networks [Schwalb and Dechter, 1997]. At the qualitative level, we rely on the time point continuous algebra [Vilain et al. 1989] where time points are related by a number of relations, that can be represented through a graph where nodes are time points and edges correspond to precedence ( relations. We can use the same ....

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Schwalb, E. and Dechter, R. (1997). Processing disjunctions in temporal constraint networks. Articial Intelligence, 93:29-61.

....would most often involve a number of disjunctions exponential in n. In general, this explosion cannot be avoided, since even simple TCSP problems may incur such fragmentation which renders the number of disjunctions in one relation exponential to the numbers of relations in the network 6 (cf. [37]) However, very often relations overlap, contain each other or there are only a nite number of them such as in networks based on IO . Thus, having computed composition, we optimize the resulting representation before we proceed with further iteration. Optimization may be done in a naive way, ....

....structures I Q and IO , respectively. STP and TCSP stand for non disjunctive and disjunctive quantitative constraint systems, respectively, as described by Dechter et al. 9] 10 The term integration stands for the integration of TCSPs with Allen s model [27] TCSP LPC (cf. Schwalb Dechter [37]) is not really a representation schema on its own. Viewed from a representational perspective, it is equivalent to TCSPs, but it propagates only a limited number of disjunctions in each step such that propagation, as a whole, remains polynomial in the number of relations. Constraint ....

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E. Schwalb and R. Dechter. Processing disjunctions in temporal constraint networks. Articial Intelligence, 93(1-2):29-61, 1997.

....structures IQ I and IO , respectively. STP and TCSP stand for non disjunctive and disjunctive quantitative constraint systems, respectively, as described by Dechter et al. 1991] 2 The term integration stands for the integration of TCSPs with Allen s model [Meiri, 1996] TCSP LPC (cf. Schwalb and Dechter [1997]) is not really a representation schema on its own. Viewed from a representational perspective, it is equivalent to TCSPs, but it propagates only a limited number of disjunctions in each step such that propagation, as a whole, remains polynomial in the number of relations. This heterarchy mirrors ....

....at least 15 hours. With this information and with (9) one may conclude, finally, that Mr. George will arrive last. Though for most temporal reasoning mechanisms the two families of abstraction operators, and , play the major role, one may think of alternative operators, too. For instance, Schwalb and Dechter [1997] encountered the TCSP fragmentation problem by restricting propagation to (almost) convex constraints. An operator that abstracts from general nonconvex relations into a limited number of convex disjunctions may render constraint propagation more efficient. However, the disadvantage remains that ....

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E. Schwalb and R. Dechter. Processing disjunctions in temporal constraint networks. Artificial Intelligence, 93(1-2):29--61, 1997.

....a significant tractable extension of simple temporal problems that allows disjunctions involving an arbitrary number of linear disequations (e.g. 3x 5y 7z 6= 8) but at most one linear inequality. Naturally, the applications of this latter class go beyond temporal reasoning. Finally, SD97] studied the performance of local consistency algorithms for processing TCSPs on hard problems in the transition region. This paper continues the work on quantitative temporal constraints and makes the following contributions: 1. We extend the framework of simple temporal problems studied ....

E. Schwalb and R. Dechter. Processing Disjunctions in Temporal Constraint Networks. Artificial Intelligence, 93:29--61, 1997.

....would most often involve a number of disjunctions exponential in n. In general, this explosion cannot be avoided, since even simple TCSP problems may incur such fragmentation which renders the number of disjunctions in one relation exponential to the numbers of relations in the network 8 (cf. [15]) However, very often relations overlap, contain each other or there are only a finite number of them such as in networks based on IO . Thus, having computed composition, I optimize the resulting representation before I proceed with further iteration. Lemma 10 A locally optimal ....

....in preliminary practical experiences optimization improved performance a lot. Even without any costs for optimization the range factor in the computational complexity of the propagation algorithm will prove too hard to live with for many or most applications. Similarly to Schwalb Dechter [15], I propose to use more efficient, but less accurate approximation techniques. They encounter the fragmentation problem for TCSPs by using only the convex closure of intervals (in several variations) for constraint propagation and, thus, avoid exponential numbers of disjunctions. Accordingly, I ....

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E. Schwalb and R. Dechter, Processing disjunctions in temporal constraint networks', Artificial Intelligence, 93(1-2), 29--61, (1997).

....using a finite set of intervals, or using the intervals implicitly denoted by a known periodic granularity, so that only the instants within these intervals can instantiate the constrained variable. A similar result (limited to a finite set of intervals) has been recently found independently [SD]. Other researchers [Lad86,MR93] have investigated temporal CSPs involving non convex intervals and periodic intervals, however the considered CSPs are substantially different since their variables must be instantiated with intervals (as opposed to instants in our framework) and only qualitative ....

E. Schwalb and R. Dechter. Processing disjunctions in temporal constraint networks. Artificial Intelligence. To appear.

.... C ij = Gamma7; Gamma1] 1; 20] The PC operation C ij : C ij C ik Omega C kj leads to C ij = Gamma6; Gamma4] 1; 3] 8; 14] 15; 20] The number of intervals of constraint (X j Gamma X i ) 2 C ij , which was 2, becomes 4 : this is the so called fragmentation problem (Schwalb and Dechter 1997). Applying path consistency (henceforth PC) to a TCSP consists of repeating, until stability, the process of considering each triangle (X i ; X k ; X j ) and replacing the label C ij on edge (X i ; X j ) by its intersection with the composition, C ik Omega C kj , of the labels on the other two ....

.... replacing the label C ij on edge (X i ; X j ) by its intersection with the composition, C ik Omega C kj , of the labels on the other two edges: C ij : C ij (C ik Omega C kj ) As illustrated in Figure 1, applying PC even to a scheduling TCSP may lead to the so called fragmentation problem (Schwalb and Dechter 1997): the constraints, which are initially of length lower than or equal to 2, may be fragmented . See (Schwalb and Dechter 1997) for another example illustrating the fragmentation problem, and for more details. Schwalb and Dechter in (Schwalb and Dechter 1997) gave a number of weak versions of PC, ....

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Schwalb, E. and Dechter, R.: 1997, Processing disjunctions in temporal constraint networks, Artificial Intelligence 93, 29--61.

....distances, D, as being bounded by metaintervals to which our mechanism can be applied, too. Second, though our approach requires exponential time at worst, it remains efficient when the input is restricted to input conditions that describe convex relations. Similarly to the way Schwalb Dechter [12] proceeded for TCSP relations, one may also employ approximating defragmentation procedures. But in addition to their approach, we may exploit the interdependencies between distance constraints from the set of arrays between two intervals in order to derive plausible, approximating distance ....

Eddie Schwalb and Rina Dechter. Processing disjunctions in temporal constraint networks. Artificial Intelligence, 93:29--61, 1997.

....constraints, removal of redundant non conjunctive set) 5] These two rules are devoted to scheduling or assignment problems. They are based on the constraints modeling only. In the case of assignment problems, theses two rules correspond to algorithm Upper Lower Tightening proposed in [9]. R3. Simplification of resource sharing constraints for yet assigned tasks using energetic adjustments [6] This rule is dedicated to the disjunctive scheduling problem. It is based on the analysis of sets of tasks competing for the same resource to be performed. R4. Simplification of the ....

....have been made by rules R1 and R2; Rule R4 expresses these deductions in terms of resource assignment. For instance, let consider a task i which can be processed either on resource 1 m with duration [2,4] or on resource 2 m with duration [7,10] If the deduced duration (by applying R1) is [6,9] then by applying R2 on the non conjunctive set ( 4 ft st i i 7 st ft i ) the constraint 4 ft st i i is inconsistent. Rule R4 then removes the assignment 1 m for task i and update the feasible duration for task i on resource 2 m : 7,9] R5. Propagation of resource ....

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E. SCHWALB, R. DECHTER, Processing disjunctions in temporal constraint networks, Artificial Intelligence, 93, pp. 29-61, 1997.

....yet. As they prove, if C 2 ij = k (C 2 ik fi C 2 kj ) then C 2 ij is redundant prone; after applying ULT to a constraint problem P 1 , one can check if this condition is fulfilled in order to remove some constraints which are not path consistent yet. In a subsequent paper, [46], they improve ULT, introducing a new algorithm, called LPC, and some of its variants to better approximate path consistency simply modifying the intersection operation. Definition 2.8.1 If T : I 1 [ In and S : J 1 [ Jm are two constraints, their loose intersection, T loose S, is ....

....particular, it is true when S = T ik fi T kj . The operation of loose intersection is not commutative: for instance, if T : f[0; 2] 1; 4]g and S : f[0; 3]g, then T loose S is the set f[0; 2] 1; 3]g, while S loose T is the set f[0; 3]g. However, contrary to what it is remarked in [46], it may be that T loose S = T loose S; take for instance S = f[0; 2]g and T = f[1; 3]g; then T loose S = f[1; 2]g and so it is S loose T . 12 The range of the constraints is the difference between the lowest and the highest numbers specified, 13] 1: input P 1 2: P 3 ....

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Eddie Schwalb and Rina Dechter. Processing disjunctions in temporal constraint networks. AI, 93:29--61, 1997.

....the domain of a variable to a given set of (disjoint) intervals, or binary, restricting the distance X i Gamma X j between two variables to a given set of (disjoint) intervals. They show that the problem is easy when the set of intervals is restricted to a single interval. Schwalb and Dechter [9] show that the complexity of enforcing path consistency on TCSPs is exponential, due to the fragmentation of intervals into subintervals. They present algorithms that bound the fragmentation by computing looser constraints that subsume the subintervals (at the cost of losing precision) Koubarakis ....

E. Schwalb and R. Dechter. Processing disjunctions in temporal constraint networks. Artificial Intelligence, 93:29--61, 1997.

....stochastic and incomplete. Likewise, consistency inference has complete solution algorithms (e.g. variable elimination) or incomplete versions in the form of local consistency algorithms. Consistency inference. Local consistency (or consistency enforcing, or constraint propagation) algorithms [15, 13, 6] are polynomial algorithms that transform a given constraint network into an equivalent, yet more explicit network by deducing new constraints to be added onto the network. Intuitively, a consistency enforcing algorithm will make any partial solution of a small subnetwork extensible to some ....

....satisfaction problems. Artificial Intelligence, 14:263 313, 1980. 11] L. M. Kirousis. Fast parallel constraint satisfaction. Artificial Intelligence, 64:147 160, 1993. 12] G. Kondrak and P. van Beek. A theoretical valuation of selected algorithms. Artificial Intelligence, 89:365 387, 1997. [13] A. K. Mackworth. Consistency in networks of relations. Artificial Intelligence, 8(1) 99 118, 1977. 14] A. K. Mackworth and E. C. Freuder. The complexity of some polynomial network consistency algorithms for constraint satisfaction problems. Artificial Intelligence, 25, 1985. 15] U. ....

E. Schwalb, and R. Dechter. Processing disjunctions in temporal constraint networks. Artificial Intelligence, 93(1-2), 29--61, 1997.

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E. Schwalb and R. Dechter. Processing disjunctions in temporal constraint networks. Artificial Intelligence. To appear.

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Schwalb, E. and Dechter, R. (1997). Processing disjunctions in temporal constraint networks, AI, 93, 29-61.

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Schwalb, E., Dechter, R.: Processing Disjunctions in Temporal Constraint Networks. Artificial Intelligence 93 (1997) 29-61

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