J-L. Lassez & K. McAloon, A Canonical Form for Generalized Linear Constraints, IBM Research Report, T.J. Watson Research Center, RC 15004, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a set of size constraints in MS, DS, or SD. The consistency of [ can be determined in polynomial time. Proof. If follows from the fact that the constraints of [ are a special case of Lassez and McAloon s Generalized Linear Constraints for which satis ability can be solved in polynomial time [44]. Theorem 30 RSAT for B 7 [ MS is polynomial. Proof. Let be the input set of topological constraints forming a scenario of RCC 7 relations, and the input set of metric size constraints. We show that the algorithm of Figure 8 decides the consistency of RSAT for [ in polynomial time. If ....

....decision procedure. If s [ is consistent for some scenario, then [ is consistent, otherwise it is inconsistent. Enforcing path consistency (Step 1) requires cubic time. Steps 2, 3 and 5 can be accomplished by running know polynomial algorithms for solving a set of linear inequalities (e.g. [18,37,44]) Finally, the propagation of the containment constraints from to of Step 5 can be accomplished in polynomial time using Steps 3 5 of Algorithm B7 MetricSize (see Figure 8) To conclude we observe that RCC 7 MetricSize can be used also as an approximate procedure for RCC 8 and metric size ....

J.L. Lassez and K. McAloon. A canonical form for generalized linear constraints. Journal of Symbolic Computation, 13:1-24, 1992.

....below we shall demonstrate that several important constraint types have the 1 independence property. A more restricted notion of 1 independence has been widely studied in the literature of constraint programming, where it has been called simply independence (see [Lassez and McAloon 1989; Lassez and McAloon 1992; Lassez and McAloon 1991] for example) The earlier property applies to an individual constraint class containing positive constraints and negative constraints, and has been used in the development of consistency checking algorithms and canonical forms [Lassez and McAloon 1991; Lassez and ....

.... and McAloon 1992; Lassez and McAloon 1991] for example) The earlier property applies to an individual constraint class containing positive constraints and negative constraints, and has been used in the development of consistency checking algorithms and canonical forms [Lassez and McAloon 1991; Lassez and McAloon 1992]. However, we will show below that the more general notion of 1 independence of one class with respect to another, introduced here, can be used to prove the tractability of a wide variety of disjunctive constraint classes for which the earlier notion of independence does not hold. Consider the ....

Lassez, J.-L. and McAloon, K. 1992. A canonical form for generalized linear constraints. Journal of Symbolic Computation 13, 1-24.

....from . In the examples below we shall demonstrate that several important constraint types have the 1 independence property. A more restricted notion of 1 independence has been widely studied in the literature of constraint programming, where it has been called simply independence (see [Lassez and McAloon 1989; Lassez and McAloon 1992; Lassez and McAloon 1991] for example) The earlier property applies to an individual constraint class containing positive constraints and negative constraints, and has been used in the development of consistency checking algorithms and canonical forms [Lassez and ....

....P.Jonsson, and M. Koubarakis Note that is the set of constraints speci ed by a disjunction of disequalities, and the problem CSP( corresponds to deciding whether a convex polyhedron, possibly minus the union of a nite number of hyperplanes, is the empty set. It was shown in [Lassez and McAloon 1989] that the set [ is independent (using their more restrictive notion of independence referred to in Section 3.2, above) and hence that this problem is tractable. However, the set of constraints speci ed by linear Horn constraints corresponds to the much larger set , and this set ....

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Lassez, J.-L. and McAloon, K. 1989. A canonical form for generalized linear constraints. In TAPSOFT '89, Advanced Seminar on Foundations of Innovative Software Development, Volume 351 of Lecture Notes in Computer Science (1989), pp. 19-27. Springer-Verlag.

....both x 1 and x 2 . Thus, each H 0 i can have at most one point in common with L, so the rest of L cannot be a subset of S m i=1 H 0 i which contradicts that L S S m i=1 H 0 i . Hence, the lemma follows. 2 A more complicated proof of the previous lemma appears in Lassez and McAloon [22]. We can now tie together the results and end up with a sufficient condition for satisfiability of Horn DLRs. 7 Lemma 17 Let Gamma be a set of arbitrary Horn DLRs. Let C Gamma be the set of convex DLRs in Gamma and let D = fD 1 ; D k g Gamma be the set of DLRs that are not ....

J.-L. Lassez and K. McAloon. A canonical form for generalized linear constraints. J. Symb. Logic 13 (1992) 1--14.

....of equations then INC holds iff Sigma is infinite [37] While several constraint domains have INC, many interesting domains do not. Linear equations over the reals have INC, but when inequalities or multiplication are introduced, INC is lost (although a much weaker form of INC does hold [30]) Similarly, constraints over the integers do not have INC unless the language of constraints is severely restricted, for example, to lower bounds on variables. Since logic formalisms typically only include function symbols that are treated in the Herbrand style, the problem of adding ....

J-L. Lassez & K. McAloon, A Canonical Form for Generalized Linear Constraints, Journal of Symbolic Computation 13, 1--24, 1992.

.... y) c 0 ( x; z) then the inference rules are sound and complete for inferring f from F . In the important case of linear arithmetic constraints, independence of negative constraints does not hold. However we can obtain a weaker version of the theorem using a weaker notion of independence [24]. A constraint domain (D; L) has the independence of inequations property if, for all constraints c 2 L, and all conjunctions of equations e 1 ; e n 2 L, D j= 9 c :e 1 Delta Delta Delta :en iff D j= 9 c :e i for i = 1; n. Both Lin and Q Lin have this property. ....

....weak independence of inequations if, for some such L 0 and every c 2 L 0 , D j= 9(c V i :e i ) iff, for every i, D j= 9(c :e i ) where each e i is a conjunction of equations. For example, Lin has weak independence since we can take L 0 to be L, and use the independence result of [24], but Lin does not have independence of negative constraints. Similarly, the constraint domain H 9 ( Sigma) does not have independence of negative constraints when Sigma is finite, but it does have weak independence. In this case we take L 0 to be the quantifier free subset of L, and the ....

J-L. Lassez & K. McAloon, A Canonical Form for Generalized Linear Constraints, Journal of Symbolic Computation, 1992.

....used in bottom up computation, and also (d) simplifies the detection that the answers of one query are contained in the answers to another, as discussed in the next subsection. Because of the closure requirements on L, most works on the independence of negative constraints for example, 5] and [18] are not applicable. However there are still several constraint domains known to have this property, including the algebras of finite, rational and infinite trees with equational constraints, when there are infinitely many function symbols [17, 25] feature trees with infinitely many sorts and ....

J-L. Lassez & K. McAloon, A Canonical Form for Generalized Linear Constraints, Journal of Symbolic Computation to appear.

.... in the context of a CLP( language for the following reasons ffl Rewriting inequalities as equalities simplifies subsequent constraint solving steps [PJSY92a, Stu91] ffl It is an essential part of a procedure for the search of the minimal representation of a set of linear constraints [LM92b]. ffl Detection of implicit equalities simplifies the output of a CLP( system [JMSY92] However, the detection of implicit equalities (and therefore null slack variables) and the way they are handled in CLP(R) can cause problems to the failure analysis procedure. In the example below it is ....

J.-L. Lassez and K. McAloon. A canonical form for generalized linear constraints. Journal of Symbolic Computation, 13(1):1--24, January 1992.

.... disconnected from the only other region x k contained in z, and that are all non tangential proper part of z) Despite deciding the consistency of a set of containment constraints can be accomplished in polynomial time by applying known algorithms for solving a set of linear inequalities (e.g. [26, 32]) these implicit constraints make RSAT intractable when topological information are combined with metric size information in either MS, DS or SD. As we will show, in order to have RSAT polynomial, one or more particular basic relations need to be removed from the available set of topological ....

....tractability more ecient specialized algorithms for solving these problems could be designed. 27 Proof. If follows from the fact that the constraints of [ are a special case of Lassez and McAloon s Generalized Linear Constraints for which satis ability can be solved in polynomial time [32]. Theorem 27 RSAT for B n ffPOgg [ MS is polynomial. Proof. Let be the input set of topological constraints, and the input set of metric size constraints. Consider the following algorithm: 1. Check whether is consistent; if is inconsistent, then return no ; 2. Add to the strongest ....

J.L. Lassez and K. McAloon. A canonical form for generalized linear constraints. Journal of Symbolic Computation, 13:1-24, 1992.

.... of negative constraints has been investigated in greater generality in [163] The property has been shown to hold for several classes of constraints including equations on finite, rational and infinite trees [161, 160, 174] linear real arithmetic constraints (where only equations may be negated) [162], sort and feature constraints on feature trees [12] and infinite Boolean algebras with positive constraints [106] among others [163] We consider a restricted form of independence of negative constraints [177] Definition 2.3. A constraint domain (D; L) has the Independence of Negated ....

.... a cost for communication between the two kinds of algorithms [133] Some elements of the CHIP solver are described in [114] Disequality constraints can be handled using entailment of the corresponding equation (discussed in Section 10.4) since an independence of negative constraints holds [162]. For the domain of word equations WE , an algorithm is known [179] but no efficient algorithm is known. In fact, the general problem, though easily provable to be NP hard, is not known to be in NP. The most efficient algorithm known still has the basic structure of the Makanin algorithm but uses ....

J-L. Lassez & K. McAloon, A Canonical Form for Generalized Linear Constraints, Journal of Symbolic Computation, to appear.

....solvers, such as linear constraint solvers [Golub and Van Loan 1989] non linear constraint solvers [Dennis, Jr. and Schnabel 1983; Witkin et al. 1990; Witkin and Welch 1990; Gleicher and Witkin 1991; Gleicher and Witkin 1992] and linear equality and inequality solvers [Jaffar et al. 1992; Lassez and McAloon 1992; Lassez and Lassez 1991; Helm et al. 1992; Huynh et al. 1992] Domain specific algorithms are capable of satisfying more expressive constraints within their domain of knowledge, but they have the drawbacks cited in Section 2 including less coverage, less efficiency, and less usability relative to ....

Lassez, J.-L. and McAloon, K. 1992. A canonical form for generalized linear constraints.

.... importance in the context of a CLP( language for the following reasons ffl Rewriting inequalities as equalities simplifies subsequent constraint solving steps [JJY92, Stu91] ffl It is an essential part of a procedure for the search of the minimal representation of a set of linear constraints [LM92a]. ffl In the context of a CLP( derivation, detection of implicit equalities may produce bindings for variables which must be communicated to the inference engine [JJY92, Stu91] see also figure 2) ffl Detection of implicit equalities simplifies the output of a CLP( system [JMSY92] 11 Of ....

J.-L. Lassez and K. McAloon. A canonical form for generalized linear constraints. Journal of Symbolic Computation, 13(1):1--24, January 1992.

....capabilities of Prolog are far from optimal and can still be improved. For instance, in Prolog, there is no way to assert that X Y , except in so far as one assigns values to X and Y which satisfy the relation (e.g. fX = 5; Y = 3g X Y ) However, Davenport et al. [5] and Lassez [13] 17] and [18] show how asserting explicit inequalities can be incorporated usefully and efficiently into declarative languages. To sum up, quoting Saraswat, the study of logic programming languages is the study of how to introduce incompleteness usefully in inference engines, and yet retain predictability , ....

....naive arithmetic system (see [28] we may see that the two systems are equivalent in expressive power. Consequently, we should not be surprised that we cannot do very much with our arithmetic we must wait until someone implements a better system (such as those described by Lassez and McAloon [18] and those by Davenport et al. in [5] 5.1 The Towers of Hanoi We start off with a program that solves the Towers of Hanoi problem. This problem is somewhat clich ed amongst examples demonstrating the power of programming languages, since it is both very simple, and beautifully illustrates the ....

J-L. Lassez and K. McAloon. A canonical form for generalized linear constraints. to appear in Journal of Symbolic Computation.

.... subspace containing f(H) It can be calculated by determining a maximal set, R 0 e , of independent equality restrictions verified by the points in f(H) l j m X i=1 ff i :y i = fi (16) This determination is not very complicated and can be done by standard linear programming techniques [35]. For each one of these independent equality restrictions we reduce a dimension of f(H) This operation is very simple: we only have to reduce a component y j of f for which ff j 6= 0. This reduction is repeated for each independent equality restriction. So if there is k of these restrictions we ....

J.L. Lassez and K. McAloon. A canonical form for generalized linear constraints. Journal of Symbolic Computation, 13:1--24, 1992.

....efficiency [Jaffar and Lassez, 1987] In the CLP class of languages [Jaffar and Lassez, 1987, Lassez, 1990] it is possible to compute over universes other than that of Herbrand. The user can more easily express his problem and can expect a quicker computation of the solution or its approximation [Lassez and Lassez, 1991, Lassez and McAloon, 1991, for example] The notion of derivation sequence is similar to our bc resolution. It is more restrictive in the sense that it applies only to Horn clauses, but in another sense it is less restrictive because not limited to Herbrand terms, as in bc resolution. The ....

....1987] In the CLP class of languages [Jaffar and Lassez, 1987, Lassez, 1990] it is possible to compute over universes other than that of Herbrand. The user can more easily express his problem and can expect a quicker computation of the solution or its approximation [Lassez and Lassez, 1991, Lassez and McAloon, 1991, for example] The notion of derivation sequence is similar to our bc resolution. It is more restrictive in the sense that it applies only to Horn clauses, but in another sense it is less restrictive because not limited to Herbrand terms, as in bc resolution. The constraints used in [Jaffar and ....

Lassez and McAloon, 1991 Lassez, J.-L. and McAloon, K. (1991). A canonical form for generalized linear constraints. Technical Report RC 15004, IBM Research Division, T.J. Watson Research Center. To appear in the Journal of Symbolic Computation.

....[49, 25, 2] 8. 2 Domain Specific Constraint Solvers Multi way, local propagation solvers may be used either individually or in concert with domainspecific solvers, such as linear constraint solvers [17] non linear constraint solvers [10, 50, 51, 16, 8] and linear equality and inequality solvers [29, 32, 31, 30, 19, 28]. Domain specific algorithms are capable of satisfying more expressive constraints within their domain of knowledge, but they have the drawbacks cited in Section 2 including less coverage, less efficiency, and less usability relative to local propagation techniques. Some systems, such as ThingLab ....

J.-L. Lassez and K. McAloon. "A Canonical Form for Generalized Linear Constraints". Journal of Symbolic Computation 13, 1 (Jan 1992).

....of x and y are not in S i . But since we intersect only a finite number of sets, only finitely many convex combinations are excluded from S, and the result follows. 2 The following is a generalisation of Lemma 13 of Jonsson and Backstrom (1996) which is a simplified version of the proof by Lassez and McAloon, 1992), from convex to almost convex sets. Drakengren Jonsson Lemma 5.6 Let S R n be an almost convex set, and let H 1 ; H k be hyperplanes. If S S k i=1 H i , then there exists a j, 1 j k such that S H j . Proof: Induction on k, letting H k = S k i=1 H i . For k = 1, the ....

Lassez, J.-L. and McAloon, K. (1992). A canonical form for generalized linear constraints.

....In the latter case, we consider also periodicity constraints. Linear arithmetic constraints have been extensively studied in Operations Research [Sch86] They provided one of the first constraint domains to which the CLP scheme of [JL87] was applied [JMSY92] leading to further theoretical studies [Las90, LM92]. Constraint databases with linear arithmetic constraints were studied in [ACGK94, BJM93, BK95, BLLM95, RS93, Sri93] Periodicity constraints were used in [KSW90, TCR94] to represent periodic data in temporal databases. While in the real number case, our results are developed in the standard ....

J-L. Lassez and K. McAloon. A Canonical Form for Generalized Linear Constraints. Journal of Symbolic Computation, 13:1--24, 1992.

....by considering those constraints one at a time, even in the presence of arbitrary additional constraints from Gamma . In the examples below we shall demonstrate that several important constraint types have this independence property. A more restrictive notion of independence was described in [18]. This earlier notion of independence was defined for a single set of relations, and has been used to establish the tractability of a number of constraint types. However, we will show below that the more general notion of independence introduced here can be used to prove the tractability of a wide ....

Lassez, J-L., and McAloon, K., "A Canonical Form for Generalized Linear Constraints ", Lecture Notes in Computer Science, 351, Springer-Verlag, Berlin/New York, (1989), pp. 19--27.

....(e.g. x 1 4x 2 x 3 6= 0) Note that Delta is the set of relations defined by a disjunction of disequalities, and the problem CSP( Gamma [ Delta ) corresponds to deciding whether a convex polyhedron minus the union of a finite number of hyperplanes is the empty set. It was shown in [17] that the set Gamma [ Delta is independent (using their more restrictive notion of independence referred to above) and hence that this problem is tractable. However, the set of relations specified by linear Horn constraints corresponds to the much larger set Gamma Theta Delta , and ....

....their more restrictive notion of independence referred to above) and hence that this problem is tractable. However, the set of relations specified by linear Horn constraints corresponds to the much larger set Gamma Theta Delta , and this set is not independent in the sense defined in [17] (see [15] In order to establish that this larger set of constraints is tractable we shall use the more general notion of independence introduced in this paper. Consider any set of constraints C in CSP( Gamma [ Delta) Let C 0 be the subset of C which is specified by weak linear ....

Lassez, J-L., and McAloon, K., "A Canonical Form for Generalized Linear Constraints ", Technical Report RC15004 (#67009), IBM Research Division, T.J. Watson Research Center, (1989).

.... defining : a tell(c) agent that checks for satisfiability of oe [ c when executed in store oe and updates the store accordingly 9 Let us just mention that handling the FG subsystem requires maintenance of two extra variable sets J and K and incremental detection of implicit equalities [12, 8], and that disequations may be treated one by one [7] an ask(c) agent that checks wether oe CT c where c ranges over equations (ax = b) and inequations (ax b) Furthermore, ask(c) and tell(c) should be atomic operations. Most of the notations used in our code come directly from the ....

J.L.Lassez and K.McAloon. A canonical form for generalized linear constraints. Journal of Symbolic Computation, 1989.

....feasible way whether the store is satisfiable with respect to integer solutions. Therefore the integral constraint becomes non primitive and satisfiability is checked over the rational numbers, which can be done in polynomial time, although the solution set need not be convex anymore [27]. The extension of the notion of primitive constraints allows us on the one hand to combine symbolic constraints of CP(FD) e.g. alldifferent, with symbolic constraints of extended ILP, e.g. assign. On the other hand, inference algorithms in existing nonprimitive constraints may be improved and ....

J.-L. Lassez and K. McAloon. A canonical form for generalized linear constraints. Journal of Symbolic Computation, 13:1--24, 1992.

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J-L. Lassez & K. McAloon, A Canonical Form for Generalized Linear Constraints, IBM Research Report, T.J. Watson Research Center, RC 15004, 1989.

....presence of inequations was made easier by the drawing of an analogy between terms algebras and linear algebra. For instance we introduced in [20] the notion of dimension of the solution space of a system of equations, which was successfully exploited to derive interesting concepts and proofs. In [23] we studied linear constraints from an affine geometry point of view. Finding again analogies with the situation in term algebras led us to a systematic study of negative constraints in an abstract setting. The axiomatization proposed in [22] is sufficiently general to account for a variety of ....

J-L. Lassez and K. McAloon, A Canonical Form for Generalized Linear Constraints, Journal of Symbolic Computation, to appear.

.... Z = Under[i] Over[i] Then call min: Z; Z = wp(Z) At this point, the feasible region is included in the union of the family of hyperplanes Under[i] Over[i] wp(Z) If a polyhedron is covered by a finite set of hyperplanes, it must be completely contained in one of them, e.g. [28]. Therefore for at least one value of i, the minimum of Under[i] Over[i] is fixed at wp(Z) Now the process can be iterated on a smaller set of indices. Writing things out in detail, the meetgoals procedure becomes newmeetgoals(continuous R,P[ MODES] KINDS] continuous ....

J.-L. Lassez and K. McAloon, A canonical form for generalized linear constraints, Journal of Symbolic Computation 13 (1992) pp 1-24.

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