A. Colmerauer. Prolog and infinite trees. In K. L. Clark and S.-A. Tarnlund, editors, Logic Programming, volume 16, pages 231--251. Academic Press, London, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....share qd, rather than that. 9] proposes Collapse Axiom in order to fill this gap, i.e. A and B share that A knows T iff A and B share that qo . However, this axiom is not valid but contingent. If it is valid, then the following two dialogues must be equivalent in terms of the shared content. 6) A: You know I know the USSR collapsed. B: Uh huh. 7) A: The USSR collapsed. B: Uh huh. For, the content A and B share that A knows that B knows that A knows that the USSR collapsed is equivalent to the content A and B share that A knows the USSR collpased , if the axiom is valid. This is ....

.... satisfying: r e TYPE is of one of the foliowing forms: 1) Collapse, ussr; 1) for short [Collapse ussr] or (2) Know, T, p; 1) for short, KTp] or ( Know, T, p; 0) for short, KTI, or ( Share, T, U , p; 1) for short, T,op] or (5) Exist, ussr; 1) for short, E ussr] or (6) USSR, ussr; 1) for short, U R ussr] or (7) X, p; 1) for short, X p] where T, U, e A, B) and p C PROP. is the dual of z, vice versa. A basic type z e BTYPE is either (I) 5) or (6) p E PROP (an atomic proposition) is a tuple ( s, 1) for short (s: r) where s E SIT, ....

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A. Colmerauer. Prolog and Infinite Trees, in: LK. L. Clark and S. -A. Tirnlund eds. Logic Prograraming, London: Academic Press, 1982, 231-252.

.... FT is complete, all three models are elementarily equivalent (i.e. satisfy exactly the same first order formulae) While the feature graph model captures intuitions common in linguistically motivated investigations, the feature tree model provides the connection to the tree constraint systems [9, 10, 16, 17] employed in logic programming. Our proof of FT s completeness will exhibit a simplification algorithm that computes for every feature description an equivalent solved form from which the solutions of the description can be read of easily. For a closed feature description the solved form is ....

A. Colmerauer. Prolog and infinite trees. In K. Clark and S.- A. Tarnlund, editors, Logic Programming, pages 153--172. Academic Press, 1982.

....In other papers on Oz tasks are called actors. store contains constraints x = y and x = f(y) in a normal form. The constraint store grows monotonically. The constraints are interpreted in a fixed first order structure, called the universe. The universe contains rational trees (as in Prolog II [4, 5]) an extension to records is straightforward [21] Suppose that OE is the conjunction of all constraints in the store. We say that the store entails a constraint , if OE is valid in the universe. The procedure store contains the bindings of names to procedures (to be explained later) The ....

Alain Colmerauer. Prolog and infinite trees. In K.L. Clark and S.-A. Tarnlund, editors, Logic Programming, pages 153--172. Academic Press, 1982.

....substitution s such that s = s # e. The most remarkable characteristic of hyperset theory is that hyperset theory admits circular sets such as x x. This paper concentrates on the formalization using equational system of sets which will be introduced in the next subsection, in detail. [5] call it as a regular tree. 2.3 Equational Systems of Sets A circular set such as x = x, a is specified by the notion of equational systems of sets, defined as follows. Definition 1 (Barwise Moss [4] 1. A flat equational system of sets is a triple = X, A, e) where X and A are sets ....

Alan Colmerauer. Prolog and infinite trees. In K. L. Clark and S. A. Tarnlund, editors, Logic Programming, pages 231--252. Academic Press, London, 1982.

....denotes a constraint, i.e. a logic formula involving only the equality predicate. Any logic theory of the type just defined has a clear and well understood semantics. It can be seen as a constraint logic programming program in CLP (RT ) JL87] It can also be seen as a generalization of Prolog II [Col82]. For instance by using the Hohfeld Smolka construction [HS88] 6 One benefit of this choice is that we can benefit from the extensive theoretical work and implementation techniques which were developed for Constraint Logic P rogramming. In Section 5 we will expand on this issue by giving the ....

Colmerauer, Alain, 1982. Prolog and infinite trees. In: Logic Programming, S. A. Tarnlund (ed.), Academic Press, New York, 231-251.

....representation by a finite graph for finite trees. In the most common use, one node corresponds to each path of the finite tree. A regular tree t is a tree such that the number of distinct subtrees of t is finite. Such a tree can be infinite, but it can still be represented by a finite graph [6], see Fig. 1 for an example. t = f 0 a 1(01) g can be represented by Sigma Sigmafl fl fl 1 0 0 a g ll Fig. 1. An infinite regular tree 2.2 Best Representation The naive representation, which consists in using any graph representing the tree [6] is very easy to ....

....by a finite graph [6] see Fig. 1 for an example. t = f 0 a 1(01) g can be represented by Sigma Sigmafl fl fl 1 0 0 a g ll Fig. 1. An infinite regular tree 2. 2 Best Representation The naive representation, which consists in using any graph representing the tree [6], is very easy to deal with and quite widely used for small problems. But we can do far better if we observe that some nodes can represent different paths of the tree, as long as the subtrees at these paths are the same. This is called sharing the subtrees (see e.g. 1] In fact, the best we can ....

Colmerauer, A. PROLOG and infinite trees. In Logic Programming (1982), K. L. Clark and S.-A. Tarnlund, Eds., vol. 16 of APIC Studies in Data Processing, Academic Press, pp. 231--251.

....and are an essential part of the standard unification procedure in SLD resolution. An alternative approach used in some implementations of logic programming systems, such as Prolog II, SICStus and Oz, does not require the occurs check axioms. This approach is based on the theory of rational trees [14, 15], denoted TdT. It assumes the congruence axioms and the identity axioms together with a uniqueness axiom for each substitution in rational solved form. Informally speaking these state that, after assigning a ground rational tree to each parameter variable, the substitution uniquely defines a ....

A. Colmerauer. Prolog and infinite trees. In K. L. Clark and S. ]k. T&rnlund, editors, Logic Programming, APIC Studies in Data Processing, volume 16, pages 231-251. Academic Press, New York, 1982.

....during visits of the second author to Leeds, funded by EPSRC under grant M05645. 1. INTRODUCTION The intended computation domain of most logic based languages includes the algebra (or structure) of finite trees. Other (constraint) logic based languages, such as Prolog II and its successors [18, 20], SICStus Prolog [63] and Oz [59] refer to a computation domain of rational trees. A rational tree is a possibly infinite tree with a finite number of distinct subtrees and where each node has a finite number of immediate descendants. These properties ensure that rational trees, even though ....

....representation makes use of connected, rooted, directed and possibly cyclic graphs where nodes are labeled with variable and function symbols as is the case of finite trees. Applications of rational trees in logic programming include graphics [32] parser generation and grammar manipulation [18, 35], and computing with finite state automata [18] Other applications are described in [34] and [37] Going from Prolog to CLP, 53] combines constraints on rational trees and record structures, while the logic based language Oz allows constraints over rational and feature trees [59] The expressive ....

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A. Colmerauer. Prolog and infinite trees. In K. L. Clark and S. A. Tarnlund, editors, Logic Programming, APIC Studies in Data Processing, volume 16, pages 231--251. Academic Press, New York, 1982.

....by the schema (2.7) are called the occur check axioms and are an essential part of the standard unification procedure in SLD resolution. An alternative approach used in some implementations of Prolog, does not require the occur check axioms. This approach is based on the theory of rational trees [5, 6]. It assumes the basic axioms and the identity axioms together with a set of uniqueness axioms [10, 11] These state that each equation in rational solved form uniquely defines a set of trees. Thus, an equation z = t where z 2 vars(t) and t 2 (T Vars n Vars) denotes the axiom (expressed in terms ....

....algorithm based on that of Robinson [17] which includes the occur check. With such algorithms, the resulting unifier is both unique and idempotent. Unfortunately, this is not what is implemented by most Prolog systems. In particular, if the algorithm is as described in [11] and used in Prolog III [5], then the resulting unifier is in rational solved form. This algorithm does not generate idempotent or even variable idempotent substitutions even when the occurcheck would never have succeeded. However, it has been shown that the substitution obtained in this way uniquely defines a system of ....

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A. Colmerauer. Prolog and Infinite Trees. In K. L. Clark and S. A. Tarnlund, editors, Logic Programming, APIC Studies in Data Processing, volume 16, pages 231--251. Academic Press, New York, 1982.

....of the basic axioms together with the following occur check axioms : 8z 2 Vars : 8t 2 (T Vars n Vars) z 2 vars(t) z = t) However, an alternative approach used in some implementations of Prolog, does not require these occur check axioms. This is based on the theory of rational trees [7, 8]. It assumes the basic axioms together with a set of uniqueness axioms [11, 12] These state that each equation in rational solved form uniquely defines a set of trees. Thus, an equation z = t where z 2 vars(t) and t 2 (T Vars n Vars) denotes the axiom (expressed in terms of the usual first order ....

....identity is removed. Let oe = Phi z 7 f(z; y) x 7 z Psi , be the computed substitution. Then, we have vars(xoe) vars(z) fzg vars(xoe 2 ) vars Gamma f(z; y) Delta = fy; zg: Hence oe is not variable idempotent. If the algorithm is as described in [12] and used in Prolog III [7], then the resulting unifier is in rational solved form. This algorithm does not generate idempotent or even variable idempotent substitutions even when the occur check would never have succeeded. Since, it has been shown that the substitution obtained in this way is unique (up to variable ....

A. Colmerauer. Prolog and Infinite Trees. In K. L. Clark and S. A. Tarnlund, editors, Logic Programming, APIC Studies in Data Processing, volume 16, pages 231--251. Academic Press, New York, 1982.

....however: The mode values of corresponding paths must be unified as well as the graph structures, taking into account both the polarities of the paths and the labels of the nodes at the ends of the paths. A rational term is a possibly infinite term which has a finite set of subterms [3]. 16 In particular, nodes labeled ground must be dealt with properly. First, unification of two one node submode graphs with ground nodes simply succeeds or fails, depending on the polarities of the paths leading to those ground nodes. Second, a one node submode graph G 1 with a ....

Colmerauer, A., Prolog and Infinite Trees. In Logic Programming, Clark, K. L. and Tarnlund, S. - A. (eds.), Academic Press, London, 1982, pp. 231--251.

....and are an essential part of the standard unification procedure in SLD resolution. An alternative approach used in some implementations of logic programming systems, such as Prolog II, SICStus and Oz, does not require the occurs check axioms. This approach is based on the theory of rational trees [14, 15], denoted RT . It assumes the congruence axioms and the identity axioms together with a uniqueness axiom for each substitution in rational solved form. Informally speaking these state that, after assigning a ground rational tree to each parameter variable, the substitution uniquely defines a ....

A. Colmerauer. Prolog and infinite trees. In K. L. Clark and S. A. Tarnlund, editors, Logic Programming, APIC Studies in Data Processing, volume 16, pages 231--251. Academic Press, New York, 1982.

.... decided in almost linear time (linear time if no infinite terms are allowed [21] For entailment problems much less is known, and the few existing results give intractable lower bounds for the constraint languages they study, except for equality constraints where polynomial time algorithms exist [3, 4]. In this paper, we consider the entailment problem for conditional equality constraints. Conditional equality constraints extend the usual equality constraints with an additional kind of constraint # # # , which is satisfied if # = # or # = # . Conditional equality constraints have been used ....

....is decidable. Thus for these di#erent forms of constraints, the problems are intractable or may even be undecidable. In the constraint logic programming community, the entailment problems over equality constraints have been considered by Colmerauer and shown to be polynomial time decidable [3, 4, 15, 24]. Previous work leaves open the question of whether there are other constraint languages with e#ciently decidable entailment problems besides equality constraints over trees (finite or infinite) 1.1 Contributions We consider two forms of the entailment problem: simple entailment and restricted ....

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A. Colmerauer. Prolog and Infinite Trees. In K. L. Clark and S.-A. Tarnlund, editors, Logic Programming, pages 231--251. Academic Press, London, 1982.

....names true and false, or the integers from 1 to 4. 2.4 Unification Unification in Berlioz uses an algorithm similar to that described in [Mehl99] which is a variation of rational tree unification algorithm for cyclic trees. The details of rational tree unification algorithms can be found in [Col82] and [HS84] Some slight differences compared to what is mentioned in [Mehl99] In case unification for any two nodes fails, an exception is raised immediately in Berlioz, whereas in [Mehl99] only a fail flag is set in termination status, but the unification process continues until there is no ....

Alain Colmerauer, Prolog and infinite trees. In K. Clark and S. Tarnlund, ed., Logic Programming, pp. 231-251, Academic Press, NY 1982.

....of both its practical and theoretical interests, extensive research [Rob79, Hue76, Kir90, Bau93, BS98, BO99] has been focused on the definition of good unification algorithms. Linear and almost linear, in both space and time, sequential algorithms were defined [Rob76, Hue76, Bax76, PW78, MM82, Col82, CB83, Muk83, Jaf84, RP90] The advent of parallel processing led to the study of unification on parallel machines [Rob85, MO84] In [Yas84, DKM84] unification was proved to be LogSpace complete for P, which means that there are inputs where parallel unification runs not significantly faster than ....

....the unifier induces a relation on V which can be extended into a valid relation for D V;E . 2 All the above mentioned sequential algorithms for mgu are also procedures for MGU . However, in order to solve MGU , step 5 of Algorithm 1 can be omitted, and if we are interested in infinite unification [Col82, Jaf84, KR89] step 4 must also be removed (see Section 6) 3 Unification on sets Unification, as originally defined [Rob65] is also called binary unification since it applies to unordered pairs of terms. However, in some contexts, for instance factoring [CL74] unification applies to finite ....

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A. Colmerauer. Prolog and infinite trees. In K.L. Clark and SA. Tarnlund, editors, Logic Programming, pages 231--251. Academic Press, 1982.

....## t # # (s = t) # e # # RSubst . In the rest of the paper, we will often write a substitution # # RSubst to denote a set of equations in rational solved form (and vice versa) Some logic based languages, such as Prolog II, SICStus and Oz, are based on RT , the theory of rational trees [9, 10]. This is a syntactic equality theory (i.e. a theory where the function symbols are uninterpreted) augmented with a uniqueness axiom for each substitution in rational solved form. It is worth noting that any set of equations in rational solved form is, by definition, satisfiable in RT . ....

A. Colmerauer. Prolog and infinite trees. In K. L. Clark and S. A. Tarnlund, editors, Logic Programming, APIC Studies in Data Processing, volume 16, pages 231--251. Academic Press, New York, 1982.

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A. Colmerauer. Prolog and infinite trees. In K. L. Clark and S.-A. Tarnlund, editors, Logic Programming, volume 16, pages 231--251. Academic Press, London, 1982.

No context found.

A. Colmerauer, Prolog and infinite trees, in: K. L. Clark, S. A. Tarnlund (Eds.), Logic Programming, APIC Studies in Data Processing, Vol. 16, Academic Press, New York, 1982, pp. 231--251.

No context found.

A. Colmerauer, Prolog and infinite trees, in: K. L. Clark, S. A. Tarnlund (Eds.), Logic Programming, APIC Studies in Data Processing, Vol. 16, Academic Press, New York, 1982, pp. 231--251.

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A. Colmerauer, Prolog and infinite trees, in: K. L. Clark, S. A. Tarnlund (Eds.), Logic Programming, APIC Studies in Data Processing, Vol. 16, Academic Press, New York, 1982, pp. 231--251.

No context found.

A. Colmerauer, Prolog and infinite trees, in: K. L. Clark, S. A. Tarnlund (Eds.), Logic Programming, APIC Studies in Data Processing, Vol. 16, Academic Press, New York, 1982, pp. 231--251.

No context found.

A. Colmerauer. Prolog and infinite trees. In K. L. Clark and S.-A. Tarnlund, editors, Logic Programming, volume 16, pages 231--251. Academic Press, London, 1982.

No context found.

A. Colmerauer. Prolog and infinite trees. In K. L. Clark and S. A. Tarnlund, editors, Logic Programming, APIC Studies in Data Processing, vol. 16, pages 231--

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A. Colmerauer, Prolog and Infinite Trees, in K.L. Clark and S.A. Tarnlund (eds) Logic Programming, Academic Press, New York, 1982, pp. 231-251.

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A. Colmerauer. Prolog and Infinite Trees, in: LK. L. Clark and S. -A. Tarnlund eds. Logic Programming, London: Academic Press, 1982, 231-252.

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