Jonsson, P., Backstrom, C.: A unifying approach to temporal constraint reasoning.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(scheduling, causal reasoning, etc. need to manage disjunctive assertions of temporal constraints, as well as conjunctive and hypothetical queries, one to many constraints, etc. This gives rise to the need to manage nonbinary constraints. Although there are some models for non binary CSP [10] [8], we intend to show the high expressiveness provided by the Labelled TCSP model [4] which allows us to specify and reason about a great variety of constraints, such as disjunctive assertions, other complex and non binary time point constraints, and costs associated to constraints. This model is ....

P. Jonsson and C. Bckstrm. A unifying approach to temporal constraint reasoning. Artif. Intel., 102:143--155, 1998.

....supported by TMR project CHOROCHRONOS funded by ESPRIT IV. 1 Two very important operations in CLP and CDB systems are deciding consistency of a set of constraints, and performing variable elimination. Subclasses of linear constraints over the reals have also been considered in temporal reasoning [5, 18, 19, 22, 23, 31, 14, 15]. Important operations in temporal reasoning applications are the following: i) deciding consistency of a set of binary temporal constraints, ii) performing variable elimination, and (iii) computing the strongest feasible constraints between every pair of variables. Disjunctions of linear ....

....are the following: i) deciding consistency of a set of binary temporal constraints, ii) performing variable elimination, and (iii) computing the strongest feasible constraints between every pair of variables. Disjunctions of linear constraints over the reals are important in many applications [13, 5, 18, 19, 20, 22, 23, 31, 14, 15]. The problem of deciding consistency for an arbitrary set of disjunctions of linear constraints is NP complete [34] It is therefore interesting to discover classes of disjunctions of linear constraints for which consistency can be decided in PTIME. In [30] Lassez and McAloon have studied the ....

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P. Jonsson and C. Backstrom. A unifying approach to temporal constraint reasoning. Artificial Intelligence, 102:143--155, 1998.

....maintaining the derived constraints and maintaining only the input ones. Currently, we are working on nonpropagation techniques which allow us to retract constraints in a very efficient way. We are also studying the possibility of adding more expressive temporal constraints to the scheduler [15]. ACKNOWLEDGEMENTS This work is proposed in the Intelligent Planning Scheduling Group of the Polytechnic University of Valencia (http: www.dsic.upv.es users ia gps) and partially supported by the grant CICYT TAP98 0345 from the Spanish government. ....

P. Jonsson and C. Bckstrm, `A unifying approach to temporal constraint reasoning', Artificial Intelligence, 102, 143-155, (1998).

.... Marin [31] Wetprasit Sattar [46] and Pujari 29 et al. 33,32] extend the time point algebra (cf. 45] by comparisons on distances, which our approach does not allow for. However, they are complementary to GTNs, because they cannot express non binary relations. Jonsson B ackstr om [20,21] and Koubarakis [24] use networks where each relation is Horn, meaning that at most one positive literal must exist per conjunction. This way, a scale up is achieved from subclasses of Allen s calculus to interval relations with quantities where consistency can be determined in polynomial time. ....

....a before b then c before d . Naturally, this line of research neglects the actual algebraic operations on higher arity temporal relations, like composition and intersection, and their implications which are given in our proposal (along the same lines cf. 10] To sum up, none of these approaches [31,33,20,21,24,12,4,35,10] uses abstraction neither for eciency nor for understandability purposes such as we do. There exist few approaches to temporal abstraction, e.g. cf. Shahar Cheng [38] However, their abstraction does not move along the heterarchy of temporal reasoning mechanisms, such as our proposal does. ....

P. Jonsson and C. Backstrom. A unifying approach to temporal constraint reasoning. Articial Intelligence, 102:143-155, 1998.

.... (linear inequalities and inequations) also named Disjunctive Linear Relations (DLRs) These expressions are a unifying approach to manage disjunctive constraints on points, intervals and durations, such that these expressions subsume most of the formalism for temporal constraint reasoning (Jonsson Bckstrm, 1998). Moreover, DLRs are able to represent disjunctions of non disjunctive metric constraints (x 1 y 1 c 1 x 2 y 2 c 2 . x n y n c n ) where x i and y are time points, c i real numbers and n1 (Stergiou Koubarakis, 1998) Obviously, the satisfiability problem for an arbitrary set of ....

....is NP complete. Interesting tractable subclasses of DLRs and conditions on tractability are identified in (Cohen et al. 1996; Jonsson Bckstrm, 1996; and Stergiou Koubarakis, 1996) The two main tractable subclasses are Horn linear and Ord Horn linear constraints (Stergiou Koubarakis, 1996; Jonsson Bckstrm, 1998). However, these subclasses subsume temporal algebras whose management is also polynomial. The management of a set of disjunctions of linear constraints is mainly based on general methods from linear programming, although some specific methods have been defined for tractable subclasses (Stergiou ....

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Jonsson, P., & Bckstrm, C. (1998). A unifying approach to temporal constraint reasoning. Artificial Intelligence, 102, 143-155.

....be expressed as A sat(F) with additional constraints on the lengths of the intervals. Moreover, in [3] it was suggested that many important forms of constraints on lengths can be expressed in the form of Horn disjunctive linear relations. This class of relations is known to be tractable [23, 24] and at the same time it allows us to express all elementary constraints such as xing the length, or bounding the length of an interval by a given number, or comparing the lengths of two intervals. It was proved in [3] that only three out of the 18 maximal tractable fragments for A sat(F) ....

....nite sets D of DLRs, denoted DLRsat, is that of checking whether there exists an assignment f of variables in V to real numbers such that all DLRs in D are satis ed. Such an f is said to be a model of D. The satis ability problem for nite sets of Horn DLRs is denoted hornDLRsat. Theorem 2 ([23, 24]) The problem DLRsat is NP complete and hornDLRsat is solvable in polynomial time. We are interested in how the complexity of a problem depends on the value of parameter F which, in our case, is a set of qualitative relations. Therefore we shall allow only those constraints on lengths which can ....

Jonsson, P., Backstrom, C.: A unifying approach to temporal constraint reasoning. Arti cial Intelligence 102 (1998) 143-155

....instance, the point algebra [16] is only useful for time points and Allen s interval algebra [1] is only useful for time intervals. Such restricted languages may not be sufficient for modelling realworld problems so several formalisms for multisorted temporal reasoning have been proposed [2, 7, 9, 13, 15]. However, the basic temporal formalisms are much easier to analyse from a complexity theoretic standpoint; all tractable subclasses of Allen s interval algebra are known, for instance [10] The goal of this paper is to study the computational complexity of Meiri s [13] Qualitative Algebra which ....

P. Jonsson and C. Bckstrm. A unifying approach to temporal constraint reasoning. Artificial Intelligence, 102(1):143--155, 1998.

....Allen s algebra combined with some forms of disjunctive linear constraints, a well known framework which subsumes many di erent types of temporal reasoning problems. Some examples of these problems, including scheduling, planning, and inde nite temporal constraint databases, can be found in [26, 31, 49] (see also [10] for more information on tractable disjunctive constraints) De nition 2 Let V = fx 1 ; x n g be a set of real valued variables, and ; linear polynomials (polynomials of degree one) over V with rational coecients. A linear relation over V is an expression of the form R ....

....; r 6 (pm) 1 ) o) 1 r 9 = A = fr j r 6= rg Table 3: The 18 maximal tractable subalgebras of Allen s algebra. 7 is a disjunctive linear relation, and (x 2y 3z 42:3) x z 6= 4y 8) x 6= 3 12 ) is a Horn disjunctive linear relation. Proposition 1 ([26, 31]) The problem DLRsat is NP complete and hornDLRsat is solvable in polynomial time. We can now de ne the general interval satis ability problem with metric information. De nition 3 Let I be an instance of A sat(X) over a set V of variables and let H be a nite set of DLRs over the set fv ; v ....

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P. Jonsson and C. Backstrom. A unifying approach to temporal constraint reasoning. Articial Intelligence, 102(1):143-155, 1998.

....qualitative and metric time has proven this problem to be computationally hard. However, recent results show that tractable reasoning is possible in certain subclasses of Allen s algebra augmented with quite advanced metric information. The linear programming approach by Jonsson and Backstrom [9] and Koubarakis [11] offers a straightforward method for extending the ORD Horn subclass with metric constraints. Several other subclasses of Allen s algebra with this property are exhibited in Drakengren and Jonsson [5] Almost certainly, these methods can be adapted to the point interval algebra ....

P. Jonsson and C. Backstrom, A unifying approach to temporal constraint reasoning, Artif. Intell. 102(1) (1998) 143--155.

.... [Jeavons and Cooper 1995] the connected row convex constraints rst identi ed in [Deville et al. 1997] see also [Jeavons et al. 1998] the ORD Horn constraints over temporal intervals described in [Nebel and Burckert 1995] the disjunctive linear constraints over the real numbers described in [Jonsson and B ackstr om 1998; Koubarakis 1996] the extended Horn clauses described in [Chandru and Hooker 1991] and the tractable set constraints described in [Drakengren 1997; Drakengren and Jonsson 1998; Drakengren and Jonsson 1997] In all of these cases our results lead to simpli cations of earlier proofs, and in ....

.... in [Nebel and Burckert 1995] the disjunctive linear constraints over the real numbers described in [Jonsson and B ackstr om 1998; Koubarakis 1996] the extended Horn clauses described in [Chandru and Hooker 1991] and the tractable set constraints described in [Drakengren 1997; Drakengren and Jonsson 1998; Drakengren and Jonsson 1997] In all of these cases our results lead to simpli cations of earlier proofs, and in many cases we are able to generalise the earlier results to Building Tractable Disjunctive Constraints 3 obtain larger families of tractable constraint classes. We also describe ....

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Jonsson, P. and B ackstr om, C. 1998. A unifying approach to temporal constraint reasoning. Articial Intelligence 102, 143-155.

....Whether these subclasses are interesting or not is of course a matter of taste but we can provide some evidence that they are worth studying. First, simple constraint languages extended with disjunctions have historically proved to have interesting properties. For instance, the Horn DLRs (Jonsson Bckstrm 1998) which subsumes almost all previously presented temporal languages for total ordered time can be viewed as a point algebra extended with disjunctions. Several other similar examples are given in Cohen et al. 1997) Secondly, disjunctions can compactly describe complex relations. Consider for ....

....and define X 1 = PA to Theta f6=g and X 2 = Delta where Delta = fr 2 PA to j r = or f=g rg. As we will see later on, X 1 and X 2 are the only maximal tractable subclasses of SAT to . Lemma 16 SAT to (X i ) 1 i 2, are tractable problems. Proof: Tractability of X 1 has been proved by Jonsson and Bckstrm (1998) and Koubarakis (1996) while the tractability of X 2 is shown in Theorem 10. 2 The NP completeness results for SAT to are all based on the previously presented NP completeness results; interestingly, many of these results hold even when restricted to total orders. Lemma 17 SAT to (N i ) 1 i 5, ....

Jonsson, P., and Bckstrm, C. 1998. A unifying approach to temporal constraint reasoning. Artificial Intelligence 102(1):143--155.

....M . Such an M is said to be a model of D. The satisfiability problem for finite sets of Horn DLRs is denoted hornDLRsat. Proposition 1. There is an algorithm for solving hornDLRsat which runs in polynomial time wrt the size of the instance. Proof. See Koubarakis [Kou96] or Jonsson Backstrom [JB98]. The Alg hornDLRsat algorithm is, while still polynomial, rather expensive and relies on a linear programming algorithm, so when there is no need for specifying metric information, the following subclass of Horn DLRs gives us an algorithm of lower complexity. Definition 2 (Point algebra) The ....

Peter Jonsson and Christer Bckstrm. A unifying approach to temporal constraint reasoning. Artificial Intelligence, 102(1):143--155, 1998.

....the framework of DLRs makes it unnecessary to distinguish between qualitative and metric information. Nevertheless, when it comes to identifying tractable subclasses, the distinction is still convenient. 2.2 Complexity Results Proposition 2.3 The problem DLRsat is NP complete. Proof: See [Jonsson and Backstrom, To appear] 2 Proposition 2.4 hornDLRsat(H) is solvable in polynomial time. Proof: See Jonsson and Backstrom [To appear] first published as [Jonsson and Backstrom, 1996] or Koubarakis [1996] 2 See Algorithm 2.11 for a polynomial time algorithm. 2.3 Polynomial time Algorithm Some auxiliary concepts are ....

Peter Jonsson and Christer Backstrom. A unifying approach to temporal constraint reasoning. Artificial Intelligence, To appear.

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Jonsson, P., Backstrom, C.: A unifying approach to temporal constraint reasoning.

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