Jean H. Gallier and Wayne Snyder. Designing unification procedures using transformations: a survey. EATCS Bulletin, 40:273--326, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Therefore, in this section, we do not rely on the syntactic conventions in any way. Subtype satisfaction is a generalization of the well known problem of unification, and the techniques we use here are based on those used to solve unification. For more details, consult a survey on unification [19, 20, 30, 10, 21 6, 31, 1]. One difference between unification and our satisfaction problems is that we work with types that go beyond simple types, but our substitutions involve only simple types. This is not the typical case with unification, and it makes our problem easier to solve. If S 1 ; S 2 are substitutions and V ....

J.H. Gallier and W. Snyder. Designing unification procedures using transformations: A survey. In Y.N. Moschovakis, editor, Logic from Computer Science, volume 21 of Mathematical Sciences Research Institute Publications, pages 153--215. Springer-Verlag, 1992.

....Therefore, in this section, we do not rely on the syntactic conventions in any way. Subtype satisfaction is a generalization of the well known problem of unification, and the techniques we use here are based on those used to solve 21 unification. For more details, consult a survey on unification [19, 20, 30, 10, 6, 31, 1]. One difference between unification and our satisfaction problems is that we work with types that go beyond simple types, but our substitutions involve only simple types. This is not the typical case with unification, and it makes our problem easier to solve. If S 1 ; S 2 are substitutions and V ....

J.H. Gallier and W. Snyder. Designing unification procedures using transformations: A survey. In Y.N. Moschovakis, editor, Logic from Computer Science, volume 21 of Mathematical Sciences Research Institute Publications, pages 153--215. Springer-Verlag, 1992.

....algorithm (section 6) we decided to design a a rule based subsumption algorithm and apply the same techniques to its verification. We think that this rulebased approach turns out to be more suitable for mechanical verification, since the proof e#ort was reduced by the following facts (see [2]) Termination aspects of the algorithm are clearly separated. We can reason about the algorithm before proving its termination. Properties of the algorithm are identified as invariants that are preserved in each step of transformation. A family of algorithms can be verified with the ....

Gallier, J., and Snyder, W. Designing unification procedures using transformations: A survey. Bulletin of the EATCS, 40 (1990), 273--326.

....problem f(x f) f (f a) g has solucions x = z(z (f a) and x = z(f (z a) neither of which is more general than the other. It s known that third order matching is decidable; though it s not known whether the problem in general is decidable, the conjecture is that it is ( 7] see also [8] 6] [4] and [2] Decidability is not enough however: in processing we are interested in effectively computing all possible unifiers, and in general this is not possible because with higher order terms we can have an infinite number of unifiers for a given equation that cannot be expressed with a finite ....

....when its head is a free variable. A higher order unification equation is a pair of terms to be unified, OE = and a Higher Order Unification System (HOUS) is a set of such equations, that must be satisfied simultaneously. A common method for solution of a HOUS is that of transformations ( 19] [4]) there are a set of transformation rules that transform a system S to a system S 0 with the same set of unifiers. The transformation rules we will use not only transform the HOUS but will also compute a unifier; any state of the process is represented by the unifier oe computed up to the ....

Jean Gallier and Wayne Snyder. Designing unification procedures using transformations: a survey. In Workshop Logic For Computer Science. MSRI Berkeley, 1989.

....one. Given two expressions, if they are unifiable, then the algorithm finishes and returns an unifier. However, we have to generate a complete set of critical pairs, thus we need an algorithm which generates a complete set of (minimum) unifiers. Based on Huet s algorithm, Gallier Snyder [GS90] describe an algorithm fulfilling this requirement in terms of transformation systems. Unfortunately, the search tree may be infinitely branching at certain nodes (flexible flexible steps) The only 10 The soundness of the higher order rule can be proved rewriting G(X [ Y ) G(X [ Y ) into G(X ....

....conditions to ensure that fresh variables are not introduced during the rewriting process. 4 A higher order unification algorithm In this section we describe the unification algorithm used to compute extended critical pairs. We use the notation on transformations introduced by Gallier Snyder [GS90]. The expression u oe denotes the sequence u oe 1 ; u oe n , where oe = foe 1 ; oe n g is a set of indexes, n] denotes the set f1; ng, and we write u n instead of u [n] when no confusion is possible. t) denotes the type of the expression t. We suppose that any ....

Jean H. Gallier and Wayne Snyder. Designing unification procedures using transformations: A survey. Bulletin of the EATCS, 40:273--326, 1990.

....Theorem 8 (strong normalization) Polymorphic selective calculus is strongly normalizing. 7.4 Type unification The key for type synthesis is unification. We give here a unification algorithm for the label selective monotypes defined above. It can be expressed as a simple E unification problem [11], where the equational theory is that deciding equality of record types. Then, our type substitution operation using record type concatenation constitutes a complete set of reduction for this theory. We next give this unification procedure as a complete set of equivalence preserving ....

Jean Gallier and Wayne Snyder. Designing unification procedures using transformations: a survey. In Y. N. Moschovakis, editor, Logic from Computer Science, pages153--215. SpringerVerlag (1989).

....constant, a second order free variable or a first order bound variable (in this later case n = 0) and t i are also second order terms in normal form. 3 2. 1 Word Unification It is easy to describe a complete 1 (non terminating) procedure for word unification in terms of transformation rules [GS90]. Any state of the process is represented by a pair hS; oei, where S is the problem and oe the substitution computed until that moment. We proceed by applying a substitution ae, that transforms the pair into a new one hae(S) aeffioei where ae(S) can be later simplified. At some point, more than a ....

J. H. Gallier and W. Snyder. Designing unification procedures using transformations: A survey. Bulletin of the EATCS, 40:273--326, 1990.

....represented by oe(t) Notation tj p represents subterm at position p of t, and t[u] p represents term t where subterm at position p has been replaced by u. 2. 1 Word unification It is easy to describe a complete 1 (non terminating) procedure for word unification in terms of transformation rules (Gallier and Snyder, 1990). Any state of the process is represented by a pair hS; oei, where S is the problem and oe the substitution computed until that moment. We proceed by applying a substitution ae, that transforms the pair into a new one hae(S) aeffioei where ae(S) can be later simplified. At some point, more than ....

Gallier, J. H. and Snyder, W. (1990). Designing unification procedures using transformations: A survey. Bulletin of the EATCS, 40:273--326.

....Higher order unification was pioneered by Darlington [8] and was used first to extend resolution to second order [31] and higher order logic [32] It was also used in the method of matings [2] and in higher order logic programming [27] for instance. A more complete survey can be found in [14]. It consists in, given two simply typed terms t and t 0 of the same type, finding complete sets of substitutions oe such that toe = fi t 0 oe (resp. toe = fij t 0 oe) where = fi is fi equivalence, and = fij is fij equivalence. It was made practical by Huet, who invented a much more ....

J. Gallier and W. Snyder. Designing unification procedures using transformations: A survey. In I. Moschovakis, editor, Workshop on Logic From Computer Science, MSRI, Berkeley, CA, USA, 1989.

....expression a(b P (c Q P ) denotes a(b p 1 (c q 1 1 : c q m 1 1 ) b pn (c q 1 n : c q mn n ) Notice that capital letters denote set of indexes whereas lower case letters denote concrete indexes. We use the notation on transformations introduced by Gallier and Snyder (Gallier and Snyder, 1990) for describing unification processes. Any state of the process is represented by a pair hS; oei where S = ft 1 = u 1 ; t n = u n g is the set of unification problems still to be solved and oe is the substitution leading from the initial problem to the actual one. The algorithm is ....

Gallier, J. H. and Snyder, W. (1990). Designing unification procedures using transformations: A survey. Bulletin of the EATCS, 40:273--326.

....Therefore, in this section, we do not rely on the syntactic conventions in any way. Subtype satisfaction is a generalization of the well known problem of unification, and the techniques we use here are based on those used to solve unification. For more details, consult a survey on unification [19, 20, 30, 10, 6, 31, 1]. One difference between unification and our satisfaction problems is that we work with types that go beyond simple types, but our substitutions involve only simple types. This is not the typical case with unification, and it makes our problem easier to solve. If S 1 ; S 2 are substitutions and V ....

J.H. Gallier and W. Snyder. Designing unification procedures using transformations: A survey. In Y.N. Moschovakis, editor, Logic from Computer Science, volume 21 of Mathematical Sciences Research Institute Publications, pages 153--215. Springer-Verlag, 1992.

....normalizing. 5.3 Type unification The base of a type synthesis algorithm is unification. We give here a unification algorithm for monotypes defined above. The reason we have to design a new algorithm is that we work modulo type normalization (cf 4. 2) That is, we have a good form of E unification [8], where the equivalence relation on terms can be expressed by an oriented rewriting system. Base type OE u = v (u 6= v; u; v base types) Redundancy OE = OE Non recurrent OE ff = 6= ff; ff 2 V ar( Type structure OE u = p 1 ) 1 ; u base type) ....

Jean H. Gallier and Wayne Snyder. Designing unification procedures using transformations : a survey. In Piergiorgio Odiffredi, editor, Logic and Computer Science. Academic Press, 1990.

....normalizing. 7.4 Type unification The key for type synthesis is unification. We give here a unification algorithm for the label selective monotypes defined above. It can be expressed as a simple E unification Research Report No. 35 October 16 Jacques Garrigue and Hassan At Kaci problem [11], where the equational theory is that deciding equality of record types. Then, our type substitution operation using record type concatenation constitutes a complete set of reduction for this theory. We next give this unification procedure as a complete set of equivalence preserving ....

Jean Gallier and Wayne Snyder. Designing unification procedures using transformations: a survey. In Y. N. Moschovakis, editor, Logic from Computer Science, pages 153--215. Springer-Verlag (1989).

.... Q P ) denotes a(b p 1 (c q 1 1 : c q m 1 1 ) b pn (c q 1 n : c q mn n ) As usual, small letters like p denotes correlative lists of indexes [1: p] so a(b p ) denotes a(b 1 ; b p ) We also use the notation on transformations introduced by Gallier and Snyder [2] for describing unification processes. Any state of the process is represented by a pair hS; oei where S = ft 1 = u 1 ; t n = u n g is the set of unification problems still to be solved and oe is the substitution computed until that moment, i.e. the substitution leading from the ....

....and H and I are unary variables. The following are examples of stratified terms w.r.t. the position p = of) this stratified forest. F(f(G(a; b) G(I(a) b) H(a) F (a; H(b) The following are examples of stratified terms G(I(a) b) f(H(a) g(H(b) g(I(a) w.r.t. positions [1] [2] and [1; 1] respectively, of the same forest. The unification problem fG(I(a) b) g(I(a) g is not stratified because, although both terms are stratified, they are stratified w.r.t. two different positions. Notice that any term without free variables is a stratified term w.r.t. any position ....

J. H. Gallier and W. Snyder. Designing unification procedures using transformations: A survey. Bulletin of the EATCS, 40:273--326, 1990.

....Y n 7 oe(u m ) Notation tj p represents subterm at position p of t, and t[u] p represents term t where subterm at position p has been replaced by u. 2. 1 Word Unification It is easy to describe a complete 3 (nonterminating) procedure for word unification in terms of transformation rules [GS90]. Any state of the process is represented by a pair hS ; oei, 1 Goldfarb [Gol81] also makes this assumption when he proves undecidability of SOU. He uses an special notation for second order terms, called L languages. However, we will use the usual notation for them. 2 In such case, two ....

J. H. Gallier and W. Snyder. Designing unification procedures using transformations: A survey. Bulletin of the EATCS, 40:273--326, 1990.

....unsorted ones. The preferred method is to describe a set of transformations that can be applied to a given set of equations with little or no control necessary, resulting in a solved form equation set. The algorithms presented here retain this transformational approach which, as pointed out by Gallier and Snyder [ 1990 ] has the following advantages: ffl Logic and control are cleanly separated, which allows for a large number of different implementations. ffl Implementing the algorithm efficiently can be thought of as optimizing the application of the transformations. ffl The algorithm often brings a ....

Jean Gallier and Wayne Snyder. Designing unification procedures using transformations: A survey. Technical Report BU-CS TR 90-005, Boston University, September 1990.

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Jean H. Gallier and Wayne Snyder. Designing unification procedures using transformations: a survey. EATCS Bulletin, 40:273--326, 1990.

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J. H. Gallier and W. Snyder. Designing unification procedures using transformations: A survey. Bulletin of the EATCS, 40:273--326, 1990.

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Gallier, J. and W. Synder, Designing unification procedures using transformations: A survey, in: Proc. Workshop on Logic from Computer Science, Berkeley, CA (1990).

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