J. Lassez, M. J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification Revisited. Morgan Kaufmann, 1987. 24  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the justification of a high precision abstract unification algorithm. Following the approach of abstract interpretation [10] an abstract unification algorithm (the abstract operation) is constructed by mimicking the substitutions (the concrete data) which arise in a standard unification algorithm [17] (the concrete operation) with finite sharing, freeness and compoundness abstractions (the abstract data) The accuracy of the analysis depends, in part, on the substitution properties that the sharing abstractions capture. The popular sharing and freeness domain Share Theta Free [23] for ....

....that x is bound to a term with a principal functor f and an arity of 3. In contrast to other approaches [16] high precision does not come at the expense of gross inefficiency. The analysis exploits a confluence property of the unification algorithm (that all unifiers are equal up to renaming [17]) to split the analysis into two distinct phases. In the first phase compoundness information is tracked. In the second phase sharing and freeness is traced. The compoundness phase only operates on the compoundness component of the domain. Similarly, the sharing and freeness phase only operates on ....

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J. Lassez, M. J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification Revisited. Morgan Kaufmann, 1987. 24

.... i) v 62 var( n ) j = fv 7 n g mgu(n = v : E; i ffi j) if mgu(j(E) i) v 62 var( n ) j = fv 7 n g mgu(n = n : E; i) if mgu(t 1 = t n : E; i) f j f By induction it follows that dom(OE) cod(OE) if OE 2 mgu(E) or put another way, that the most general unifiers are idempotent [16]. Following [14] the semantics of a logic program is formulated in terms of a single unify operator. To construct unify, and specifically to rename apart program variables, an invertible substitution [16] Upsilon , is introduced. It is convenient to let Rvar Uvar denote a set of renaming ....

....= if OE 2 mgu(E) or put another way, that the most general unifiers are idempotent [16] Following [14] the semantics of a logic program is formulated in terms of a single unify operator. To construct unify, and specifically to rename apart program variables, an invertible substitution [16], Upsilon , is introduced. It is convenient to let Rvar Uvar denote a set of renaming variables that cannot occur in programs, that is Pvar Rvar = and suppose that Upsilon : Pvar Rvar. Definition2 unify. The partial mapping unify : Atom Theta Subst= Theta Atom Theta Subst= Subst= ....

J. Lassez, M. J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification Revisited. Morgan Kaufmann, 1987.

.... n g mgu(n = v : E; i ffi j) if mgu(j(E) i) v 62 var( n ) j = fv 7 n g mgu(n = 0 n : E; i) if mgu(t 1 = t 0 1 : t n = t 0 n : E; i) f j f 0 By induction it follows that dom(OE) cod(OE) if OE 2 mgu(E) or put another way, that the most general unifiers are idempotent [16]. Following [14] the semantics of a logic program is formulated in terms of a single unify operator. To construct unify, and specifically to rename apart program variables, an invertible substitution [16] Upsilon , is introduced. It is convenient to let Rvar Uvar denote a set of renaming ....

....cod(OE) if OE 2 mgu(E) or put another way, that the most general unifiers are idempotent [16] Following [14] the semantics of a logic program is formulated in terms of a single unify operator. To construct unify, and specifically to rename apart program variables, an invertible substitution [16], Upsilon , is introduced. It is convenient to let Rvar Uvar denote a set of renaming variables that cannot occur in programs, that is Pvar Rvar = and suppose that Upsilon : Pvar Rvar. Definition2 unify. The partial mapping unify : Atom Theta Subst= Theta Atom Theta Subst= Subst= ....

J. Lassez, M. J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification Revisited. Morgan Kaufmann, 1987.

...., V n is called the domain of q, and the set of variables appearing in t 1 , t n is called the range. We shall only consider substitutions such that domain range is empty, and this is understood unless otherwise stated in the rest of this paper. Such substitutions are idempotent [LMM88]. An element V i t i of a substitution is called a binding. We use the notation vars(o) to denote the set of all variables in the object o. We follow [LMM88] in this presentation. A unifier for two terms t 1 and t 2 is a substitution s such that t 1 s = t 2 s. We call a unifier s a most ....

....that domain range is empty, and this is understood unless otherwise stated in the rest of this paper. Such substitutions are idempotent [LMM88] An element V i t i of a substitution is called a binding. We use the notation vars(o) to denote the set of all variables in the object o. We follow [LMM88] in this presentation. A unifier for two terms t 1 and t 2 is a substitution s such that t 1 s = t 2 s. We call a unifier s a most general unifier (mgu) of t 1 and t 2 if it has the following property: Let q be any unifier for t 1 3 and t 2 . Then, there is some substitution a such ....

J-L. Lassez, M.J. Maher, and K. Marriott, "Unification Revisited,*In Foundations of Deductive Databases and Logic Programming, Ed. J. Minker, Morgan Kaufman, 1988.

....give the type of each node. 1.2. Unication and anti unication A unication algorithm is an algorithm solving sets of equations over a set of terms. Although it is a well known concept, we recall some denitions in order to precise notations and stay self contained. For a good introduction, see [17]. 5 Denition 4. Terms, substitutions) Given a set V of variables and a set F = S i F i of function symbols with an associated number called arity, the set T of terms is recursively dened as : V T 8f i 2 F; 8t i 2 T ; f i (t 1 ; t i ) 2 T : A substitution is a nite mapping from V to ....

J.-L. Lassez, M.J. Maher, and K. Mariott. Foundations of deductive databases and logic programming, chapter Unication revisited, pages 587625. Morgan Kaufman, Los Altos, CA, 1987.

....achieved thanks to an algebraic based approach. If the definition of the supremum and infimum operators can be supported by the set union and intersection in the propositional calculus frame, first order logic needs more sophisticated tools, especially for unification: we adopt the approach of [ Lassez et al. 1987 ] which allows a lattice on the terms algebra to be defined properly, thanks to the anti unification operator. Example: 1) p(x; g(y; b) is the anti unified literal of p(a; g(a; b) and p(1; g(b; b) 2) The anti unification of fa(x) b(x)g and fa(1) b(2)g is fa(x) b(y)g. In fact, ....

J.-L. Lassez, M.J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification revisited. 1987.

....of the C base properties, which uses particular types as atomic basic notions. In principle, views, are obtained using any types, which are constructed using finite 4 As a theoretical observation, one could show that the C base is the program base wrt an appropriate pre interpretation (see Maher [6] for motivations) Also, the program base, as it is defined in [7] is obtained applying our definition in the context of the Herbrand pre interpretation. 4 pre interpretations (Proposition 2.1 holds for an arbitrary pre interpretation) Following the above theory, the toolkit supports a view ....

Maher, M. Equivalence of Logic Programs,Foundations of Deductive Databases and Logic Programming. Morgan-Kaufmann, 1988, 627-658

....This can always be achieved by renaming the bound type variables. Let id be the identity mapping and [ ff] the replacement of ff by . Juxtaposition RS of substitutions R and S denote the composition of mappings. We define S T R iff TS = R and as short form S R iff 9T:S T R. In [LMM87] it is stated that the set of substitutions with the relation is a complete lower semi lattice. Give two types 1 ; 2 a unifier is a substitution S with S 1 = S 2 . A most general unifier S has property S S 0 for every other unifier S 0 . We denote this as mgu( 1 ; 2 ) S. ....

J-L. Lassez, M.J. Maher, and K. Marriott. Unification revisited. Foundations of Deductive Databases and Logic Programming, pages 587 -- 625, 1987.

....of view, 2 Each is a special case of the latter layer. 3 Although is defined in terms of clauses instead of programs, it can be extended to the latter case as: For two programs P 1 and P 2 , P 1 P 2 iff for every clause D i in P 2 there exists a clause C j in P 1 such that C j D i . [M88] studied the equivalence relations between logic programs. Similarly, some orderings between programs can be defined as follows: Definition 1 For any two programs P 1 and P 2 , 1. P 1 TP P 2 if TP1 (X) TP2 (X) for every X HB. 2. P 1 P 2 if [ P 1 ] X) P 2 ] X) for every X HB. 3. P ....

Maher, M.J., Equivalence of logic programs, Foundations of Deductive Databases and Logic Programming, Morgan Kaufmann, 1988.

....the justification of a high precision abstract unification algorithm. Following the approach of abstract interpretation [10] an abstract unification algorithm (the abstract operation) is constructed by mimicking the substitutions (the concrete data) which arise in a standard unification algorithm [17] (the concrete operation) with finite sharing, freeness and compoundness abstractions (the abstract data) The accuracy of the analysis depends, in part, on the substitution properties that the sharing abstractions capture. The popular sharing and freeness domain Share Theta Free [23] for ....

....that x is bound to a term with a principal functor f and an arity of 3. In contrast to other approaches [16] high precision does not come at the expense of gross inefficiency. The analysis exploits a confluence property of the unification algorithm (that all unifiers are equal up to renaming [17]) to split the analysis into two distinct phases. In the first phase compoundness information is tracked. In the second phase sharing and freeness is traced. The compoundness phase only operates on the compoundness component of the domain. Similarly, the sharing and freeness phase only operates on ....

[Article contains additional citation context not shown here]

J. Lassez, M. J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification Revisited. Morgan Kaufmann, 1987.

....pre interpretation I V1 as the set of all terms of L . Proposition 3.5 Let L = hF; P; Vi be a language with the most precise pre interpretation I V1 . Let P be a definite logic program on L: then C(P ) fl I V1 (M I V1 (P ) Proof immediately follows from the well know result of Maher [20] (see also Apt [1] for a simpler proof) 2 Proposition 3.5 says that D should be a countable set. However, sometimes it is enough to have only one constant in D. For instance, taking the set fAE g as D and computing the least non Herbrand models of the above programs P and P with respect to I ....

Maher, M. Equivalence of Logic Programs, Foundations of Deductive Databases and Logic Programming, ed. J.Minker, Morgan-Kaufmann, 1988, 627-658.

....safe approximations of correct answers. However, it is important to see that domain fixpoint operator can be used to compute the set of correct atomic answers precisely. This can be achieved with pre interpretations containing non term domain elements. Let L = hF; P; Vi be a language. Lassez [18] says that given a tuple t = of Terms = L) the set of ground instances of t = uniquely determines the tuple t = itself provided that F is not a single constant. This motivates to the following definition: Definition 3.4 Sufficiently Precise Pre interpretation Let L be a ....

Lassez, J.-L., Maher, M., Marriott, K. Unification Revisited, Foundations of Deductive Databases and Logic Programming, ed. J.Minker, Morgan-Kaufmann, 1988, 587-625.

....similar to the standard backtracking procedures usually used for solving constraint problems. However, what really counted was the observation that term equations are just constraints of a special type and that thus the unification algorithm is just a special kind of constraint solving algorithm [81]. This has led to the definition of a general framework, called Constraint Logic Programming (CLP) 68] which has all the features of logic programming but is parametric with respect to the kind of constraints used within the language. Moreover, it has also brought fundamental changes in areas ....

J-.L. Lassez, M.J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification Revisited. Morgan Kaufmann Publishers, Inc, 1988.

....of the supremum and the infimum operators can be supported by the set union and intersection in the propositional calculus frame, first order logic needs more sophisticated tools. We use the subsumption and reduction operators defined by Plotkin [Plo70] but also adopt the approach of [Hue76] and [LMM87] which allows a lattice on the terms algebra to be defined properly thanks to the anti unification operator. Example: p(x; g(y; b) is the anti unified literal of p(a; g(a; b) and p(1; g(b; b) In fact, anti unification allows the infimum to generalize the terms so as 4 contrary to what ....

J-L. Lassez, M.J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification Revisited. J. Minker, 1987.

....set of a term. The type descriptions must be mutually exclusive 3 Any definite program has the least model wrt any pre interpretation (see Lloyd [8] 4 As a theoretical observation, one could show that the C base is the program base wrt an appropriate pre interpretation (see Maher [9] for motivations) Also, the program base, as it is defined in [10] is obtained applying our definition in the context of the Herbrand pre interpretation. conditions, which split the set of terms of the language into disjoint subsets, called types. A type has a unique representation set and ....

Maher, M. Equivalence of Logic Programs,Foundations of Deductive Databases and Logic Programming. Morgan-Kaufmann, 1988, 627-658

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J.-L. Lassez, M.J. Maher, and K. Marriott. Foundations of Deductive Databases and Logic Programming, chapter Unification revisited. 1987.

No context found.

Maher, M.J., Equivalence of logic programs, Foundations of Deductive Databases and Logic Programming, Morgan Kaufmann, 1988.

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