Jean-Louis Lassez and Ken McAloon. A constraint sequent calculus. In Proceedings of the 5th IEEE Symposium on Logic in Computer Science, LICS'90, pages 52--61, Philadelphia, June 1990. IEEE Computer Society Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....it considerably reduces the non determinism of the residue rule, allowing more e#cient implementations of the calculus. The notion of theory convexity we adopt here is based on one due to Nelson and Oppen [NO79] It is also related to the notion of independence of negative constraints (see [LM90] for a general treatment) from the constraint programming literature. Definition 5.1 (# Convex Theory) Let # be a signature. A theory of signature # is # convex i# for every set # of # literals and every finite non empty set # of positive # literals, p i# # # . We prove in the ....

Jean-Louis Lassez and Ken McAloon. A constraint sequent calculus. In Proceedings of the 5th IEEE Symposium on Logic in Computer Science, LICS'90, pages 52--61, Philadelphia, June 1990. IEEE Computer Society Press.

....of other semantics (compositional, open program, unfolding, etc. studied by Levi et al. 8, 40, l, 43] In addition, an interesting follow up to the work presented here would be a categorical analysis of the Jaffar Lassez criteria [29] for constraint domains and of the Lassez MacAloon [35] constraint calculi. Our categorical study should also be extended to the case of higher order logic, L Jk, and dependent type theory. It would seem that a categorical formulation of abstract interpretation [5, 14] would yield some interesting insights as well. Appendix Lemma A.1 (Lemma 4.8) ....

J. L. Lassez and K. MacAloon. A constraint sequent calculus. In Proceedings of tthe 5th Symp. on LICS, 1990.

....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 McAloon 1992] However, we will ....

.... 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 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 ....

Lassez, J.-L. and McAloon, K. 1991. A constraint sequent calculus. In Constraint Logic Programming, Selected Research, pp. 33-43. MIT Press.

.... Delta Delta :cn iff D j= 9 c :c i for i = 1; n. or, equivalently, D j= c W n i=1 c i iff for some j, D j= c c j . INC was first formulated and proved by Colmerauer [6] for equations over infinite trees. A more general formulation and its consequences were developed in [29]. Both [29] and [39] have useful weakenings of INC. The advantages of this property are that: ffl testing the satisfiability of a conjunction of constraints and negated constraints reduces to a series of tests involving a single negated constraint ffl problems involving disjunction are simpler, ....

.... Delta :cn iff D j= 9 c :c i for i = 1; n. or, equivalently, D j= c W n i=1 c i iff for some j, D j= c c j . INC was first formulated and proved by Colmerauer [6] for equations over infinite trees. A more general formulation and its consequences were developed in [29] Both [29] and [39] have useful weakenings of INC. The advantages of this property are that: ffl testing the satisfiability of a conjunction of constraints and negated constraints reduces to a series of tests involving a single negated constraint ffl problems involving disjunction are simpler, since the ....

J-L. Lassez & K. McAloon, A Constraint Sequent Calculus, Proc. of 5th. Symp. on Logic in Computer Science, 52--62, 1990.

....the 3 two element Boolean algebra is denoted by BOOL and the free Boolean algebra generated by an infinite set of generators is denoted by BOOL1 . An important property of constraint domains is generally phrased as a form of independence. This property has been investigated in some generality in [25]. The significance of the property for the optimization of bottom up execution of CDBs and CLP programs is discussed in [26] Definition 2.2 A constraint domain (D; L) has the independence of negative constraints property if, for all constraints c; c 1 ; c n 2 L, D j= 9 c :c 1 ....

J-L. Lassez & K. McAloon, A Constraint Sequent Calculus, Proc. of Symp. on Logic in Computer Science, 52--62, 1990.

....only would that give a strengthening of the results, but the proofs might be simpler, by avoiding the clutter of substitutions. However, in some cases it is not possible to directly lift results from logic programming to CLP. In particular, there is a form of independence of negative constraints [5, 19] which holds for the Herbrand universe (when presented as a constraint domain) but not for many other domains. This turns out to be a major reason why some results cannot be lifted from LP to CLP. We show that it affects completeness for top down execution, termination for bottom up evaluation, ....

....major advantage of Independence of Negated Constraints is that the language of constraints L can be extended to allow the conjunction of negated constraints with a (relatively) incremental cost in testing for satisfiability. A more general form of Independence of Negated Constraints is studied in [19], where a subclass of constraints the negatable constraints are allowed to appear negated. The form that we consider here considers every constraint in L to be negatable. The startling thing about Independence of Negated Constraints is that it has a fundamental effect on several semantic ....

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J-L. Lassez & K. McAloon, A Constraint Sequent Calculus, Proc. of Logic in Computer Science, 52--62, 1990.

....y 6 0 are not independent from each other: is not satisfiable although both f(x; y) 0 x 6 0 and f(x; y) 0 y 6 0 are. Please write titlerunninghead (Shortened) Article Title in file 3 A general study of the property shows its importance in various symbolic computation areas [43]. In constraint data bases, the property allows the efficient containment test between constraint relations [38] The property is necessary for the inference of constrained functional dependencies in polynomial time [45] Examples of constraint systems with the independence property are: universal ....

.... test between constraint relations [38] The property is necessary for the inference of constrained functional dependencies in polynomial time [45] Examples of constraint systems with the independence property are: universal closures of the definite Horn clauses of a logic programming language [43], term equations over finite or infinite trees [20] or in shallow theories [22] linear equations over the real numbers [42] various constraint classes over feature trees [8, 7, 56] infinite Boolean algebras with positive constraints [36] and various simple subclasses of constraints (e.g. ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

....to T is the same as entailment with respect to FT. The two major technical contributions of this paper are (1) an incremental entailment simplification system that is proved to be sound and complete, and (2) a proof showing that FT satisfies the so called independence of negative constraints [7, 14, 15]. The incremental entailment simplification system is the prerequisite for FT s use with either of the constraint programming frameworks ALPS, CCP or KAP mentioned at the beginning of this section. The indepence property means among other things that negative constraints can essentially be handled ....

....X V(fl) Then: 1. if A j= fl OE fl , then fl j= A 9XOE 9X ; 2. fl j= A :9XOE iff fl OE is unsatisfiable in A. That is, the conjunction fl OE is satisfiable if and only if fl either entails 9XOE, or it does not determine 9XOE. The so called independence of negative constraints [7, 14, 15] is an important property of constraint systems. If it holds, simplification of conjunctions of 13 positive and negative constraints can be reduced to entailment simplification of conjunctions of positive constraints. To define the independence property, we assume that a constraint system is a ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

....many sorts A 1 ; A n we have j= T A 1 x . A n x; and for finitely many features f 1 ; f n we have j= T xfg 9y(xf 1 y) 9y(xf n y) Since we allow for existential quantification, our Independence Theorem is stronger than what is usually stated in the literature [10, 15, 16]. In fact, independence for existentially quantified constraints over finite or rational constructor trees does not hold if the alphabet of constructors is finite. To see this, note that the disjunction 9y 1 (x = f 1 (y 1 ) 9y n (x = f n (y n ) is valid if there are no other constructors ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

....trees. The independence of inequations states: if a conjunction of positive and negative equational constraints are inconsistent then one of the negative constraints is inconsistent with the positive constraints. Independence 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 ....

.... 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 Constraints property if, for all constraints c; c 1 ; c n 2 L, D j= 9 c :c 1 Delta Delta Delta :cn iff D j= 9 c :c i for i = ....

J-L. Lassez & K. McAloon, A Constraint Sequent Calculus, Proc. of Symp. on Logic in Computer Science, 52--62, 1990.

....case. Recently there has been interest in weak independence, usually called independence in the literature, as a general means of lifting satisfiability and canonicity results from conjunctions of positive constraints to conjunctions with negative constraints. In particular Lassez and McAloon [15] studied canonical forms and Colmerauer [5] has investigated sufficient conditions for weak independence of equations and inequations in a general algebraic setting. However, Colmerauer s results do not apply in the Boolean domain as positive Boolean constraints do not admit eliminable variables ....

J.-L. Lassez and K. McAloon. A constraint sequent calculus. In IEEE Symposium on Logic in Computer Science, pages 52--61. IEEE Press, 1990.

....(those constraints which determine which properties variables do not have) usually does not affect the ability to place other negative constraints. I may not say both x = 3 and x = 4; but I may very well say x 6= 3 and x 6= 4. This property is explored further by Lassez and McAloon in [19] and [16]. It is worth noting that the property holds true for other constraint systems as well; in particular it holds for all of the primitive constraint systems described here. 2.3 Arithmetic 2.3.1 Declarative Arithmetics In Prolog, when one wishes to perform an arithmetical operation, one has to use ....

....consistency: we only provide the definition of operational entailment, and the actual rule does not fit comfortably into our sequent calculus. We do not provide a definition for the operational assertion of negative constraints because asserting negation is a very expensive process (see [19] and [16]) The definition of operational entailment: Not N a M [ N 62 C M not(a) The definitions of not and con are a little strange. In the former case, the rule reads whatever proves a is inconsistent with M , whilst in the latter, it reads whatever proves a is consistent with M . In both ....

J-L. Lassez. A constraint sequent calculus. In Logic in Computer Science, pages 52--61, 1990.

....with respect to T is the same as entailment with respect to FT. The two major technical contributions of this paper are (1) an incremental simplification system for entailment that is proven to be sound and complete, and (2) a proof showing that the independence of negative constraints property [9, 16, 17] holds for FT. The incremental entailment simplification system is the prerequisite for FT s use with either of the constraint programming frameworks ALPS, CCP, KAP or LIFE mentioned at the beginning of this section. Roughly, these systems are concurrent thanks to a new effective discipline for ....

....Let X V(fl) Then: 1. if A j= fl OE fl , then fl j= A 9XOE 9X ; 2. fl j= A :9XOE if and only if fl OE is unsatisfiable in A. That is, the conjunction fl OE is satisfiable if and only if fl either entails 9XOE, or it does not determine 9XOE. The independence of negative constraints [9, 16, 17] is an important property of constraint systems. If it holds, simplification of conjunctions of positive and negative constraints can be reduced to entailment simplification of conjunctions of positive constraints. In order to see why, observe that fl :OE 1 . OE n is unsatisfiable if and ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

....to to T is the same as entailment with respect to FT. The two major technical contributions of this paper are (1) an incremental entailment simplification system that is proved to be sound and complete, and (2) a proof showing that FT satisfies the so called independence of negative constraints [6, 13, 14]. The incremental entailment simplification system is the prerequisite for FT s use with either of the constraint programming frameworks ALPS, CCP or KAP mentioned at the beginning of this section. The indepence property means among other things that negative constraints can essentially be handled ....

....X V(fl) Then: 1. if A j= fl OE fl , then fl j= A XOE X ; 2. fl j= A XOE iff fl OE is unsatisfiable in A. That is, the conjunction fl OE is satisfiable if and only if fl either entails XOE, or it does not determine XOE. The so called independence of negative constraints [6, 13, 14] is an important property of constraint systems. If it holds, simplification of conjunctions of positive and negative constraints can be reduced to entailment simplification of conjunctions of positive constraints. To define the independence property, we assume that a constraint system is a pair ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

....with respect to T is the same as entailment with respect to FT. The two major technical contributions of this paper are (1) an incremental simplification system for entailment that is proven to be sound and complete, and (2) a proof showing that the independence of negative constraints property [9, 16, 17] holds for FT. The incremental entailment simplification system is the prerequisite for FT s use with either of the constraint programming frameworks ALPS, CCP, KAP or LIFE mentioned at the beginning of this section. Roughly, these systems are concurrent thanks to a new effective discipline for ....

....6 Let X V(fl) Then: 1. if A j= fl OE fl , then fl j=A 9XOE 9X ; 2. fl j=A :9XOE if and only if fl OE is unsatisfiable in A. That is, the conjunction fl OE is satisfiable if and only if fl either entails 9XOE, or it does not determine 9XOE. The independence of negative constraints [9, 16, 17] is an important property of constraint systems. If it holds, simplification of conjunctions of positive and negative constraints can be reduced to entailment simplification of conjunctions of positive constraints. In order to see why, observe that fl :OE 1 : OE n is unsatisfiable if and ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

....In the constraint f(x; y) 0 x 6 0 y 6 0, the conjuncts x 6 0 and y 6 0 are not independent from each other. izes (and is characterized by) the semantics of bottomup and top down computations [32] A general study of the property shows its importance in various symbolic computation areas [31]. In constraint data bases, the property allows the efficient containment test between constraint relations [26] The property is necessary for the inference of constrained functional dependencies in polynomial time [33] Examples of constraint systems with the independence property are: universal ....

.... test between constraint relations [26] The property is necessary for the inference of constrained functional dependencies in polynomial time [33] Examples of constraint systems with the independence property are: universal closures of the definite Horn clauses of a logic programming language [31], term equations over finite or infinite trees [14] linear equations over the real numbers [30] various constraint classes over feature trees [7, 6, 39] infinite Boolean algebras with positive constraints [24] and various simple subclasses of constraints (e.g. inequations over numbers where ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

....be presented as a rewriting system which must be sound with respect to the declarative semantics and preferably complete. A sufficient condition for completeness is well foundness of the solver. This notion of constraint programming is a generalization of the approach of Steele [Ste80] and Lassez [LM89]. Here the constraint system is assumed to be built in and therefore, programming simply means imposing constraints. When designing a constraint programming language it is essential to develop a systematic way to define constraints and a generic way to construct a solver for each defined ....

J-L. Lassez and K. McAloon. A constraint sequent calculus. Technical report, IBM T.J. Watson Research Center, 1989.

....introduce the expression 9y c(y) The dependencies in [2, 16] do not introduce existential quantifications. An important property of constraint domains is generally phrased as a form of independence. This property, Independence of Negative Constraints, has been investigated in some generality in [13]. The significance of the property for the optimization of CDBs and CLP programs is discussed in [15] The property is used in [16] to identify tractable classes of constrained functional dependencies and constraint generating dependencies. We use 9 p to denote the existential closure of the ....

J-L. Lassez & K. McAloon, A Constraint Sequent Calculus, Proc. of Symp. on Logic in Computer Science, 52--62, 1990.

....to T is the same as entailment with respect to FT. The two major technical contributions of this paper are (1) an incremental entailment simplification system that is proved to be sound and complete, and (2) a proof showing that FT satisfies the so called independence of negative constraints [7, 14, 15]. The incremental entailment simplification system is the prerequisite for FT s use with either of the constraint programming frameworks ALPS, CCP or KAP mentioned at the beginning of this section. The indepence property means among other things that negative constraints can essentially be handled ....

....Let X V(fl) Then: 1. if A j= fl OE fl , then fl j= A 9XOE 9X ; 2. fl j= A :9XOE iff fl OE is unsatisfiable in A. That is, the conjunction fl OE is satisfiable if and only if fl either entails 9XOE, or it does not determine 9XOE. The so called independence of negative constraints [7, 14, 15] is an important property of constraint systems. If it holds, simplification of conjunctions of positive and negative constraints can be reduced to entailment simplification of conjunctions of positive constraints. To define the independence property, we assume that a constraint system is a pair ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

.... of FT s relative simplification method [7] which can be seen from the fact that the entailment x = f(x; y) y = f(y; y) j= CFT x = y 3 Since we allow for existential quantification in simple constraints, our independence result is in fact stronger than what is usually stated in the literature [13, 23, 24]. See also the discussion at the end of Section 5.4. holds in CFT. It of course also holds in Colmerauer s rational tree system. Our operational investigations are based on congruences and normalizers of constraints, two straightforward notions providing for an elegant presentation of the ....

.... many sorts A 1 ; A n we have j= T A 1 x : A n x; and for finitely many features f 1 ; f n we have j= T xfg 9y(xf 1 y) 9y(xf n y) Since we allow for existential quantification, our Independence Theorem is stronger than what is usually stated in the literature [13, 23, 24]. Independence of existentially quantified constraints has been shown for a class of Boolean constraint systems in [16] and for finite and rational constructor trees over an infinite signature in [26] In fact, independence for existentially quantified constraints over finite or rational ....

J. L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

.... logic programming (CLP) it has made possible the manipulation of inequations [9] The property also characterizes (and is characterized by) the semantics of bottom up and top down computations [22] A general study of the property shows its importance in various symbolic computation areas [20]. In constraint data bases, the property allows the efficient containment test between constraint relations [17] The property is necessary for the inference of constrained functional dependencies in polynomial time [23] Examples of constraint systems with the independence property are: universal ....

.... test between constraint relations [17] The property is necessary for the inference of constrained functional dependencies in polynomial time [23] Examples of constraint systems with the independence property are: universal closures of the definite Horn clauses of a logic programming language [20], term equations over finite or infinite trees [9] linear equations over the real numbers [19] various constraint classes over feature trees [5, 4, 28] infinite Boolean algebras with positive constraints [15] and various simple subclasses of constraints (e.g. inequations over numbers where ....

J.-L. Lassez and K. McAloon. A constraint sequent calculus. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 52--61, June 1990.

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