Wayne Snyder and Jean H. Gallier. Higher order unification revisited: Complete sets of transformations. Journal of Symbolic Computation, 8(1-2):101--140, 1989. 12  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as well as of satisfiability of one step rewrite constraints. 1 Introduction Context unification is a variant of second order unification and also a generalization of string unification. There are unification procedures for the more general problem of higher order unification (see e.g. [Pie73,Hue75,SG89,Pre95]) It is well known that higher order unification and second order unification are undecidable [Gol81,Far91,LV00] String unification was shown to be decidable by Makanin [Mak77] Recent upper complexity estimations are that it is in EXPSPACE [Gut98] in NEXPTIME [Pla99a] and even in PSPACE ....

Wayne Snyder and Jean Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989.

.... X(Xx 3 x 1 ) and x 1 ; x 2 ; x 3 Xx 1 x 1 x 2 are not. It is well known that higher order unification of patterns is decidable in polynomial time, and that there is a most general unifier (mgu) if any unifier exists at all [20] For convenience, we shall adopt Snyder and Gallier s convention [24] that s m abbreviates the sequence s 1 s 2 : s m , or s 1 ; s 2 ; s m depending on context. If is a one to one mapping from f1; kg to f1; mg, write s j the sequence s (1) s (2) s (k) To define higher order automata, we shall need patterns that are not too ....

W. Snyder and J. Gallier. Higher order unification revisited: Complete sets of tranformations. Journal of Symbolic Computation, 8(1 & 2):101--140, 1989. Special issue on unification. Part two. 19

.... X(Xx 3 x 1 ) and x 1 ; x 2 ; x 3 Xx 1 x 1 x 2 are not. It is well known that higher order unification of patterns is decidable in polynomial time, and that there is a most general unifier (mgu) if any unifier exists at all [14] For convenience, we shall adopt Snyder and Gallier s convention [15] that s m abbreviates the sequence s 1 s 2 : s m , or s 1 ; s 2 ; s m depending on context. If is a one to one mapping from f1; kg to f1; mg, write s j the sequence s (1) s (2) s (k) 3 Unification of Shallow Patterns To define higher order automata, we ....

W. Snyder and J. Gallier. Higher order unification revisited: Complete sets of tranformations. Journal of Symbolic Computation, 8(1 & 2):101--140, 1989. Special issue on unification. Part two. 15

....to a considerable extent based on an old technical report 2 where some proofs and more examples missing here (due to lack of space) are given in detail. 2 Second Order Term Languages The basic definitions, notions and lemmas about second order term languages are essentially taken from [12] and [24]. Moreover we assume familiarity with the basic notions and results of calculus (cf. 12] 2] Definition 1 (second order terms) For all i 0 let V i be a denumerable set of distinct (function) variables of arity i with V i V j = for i 6= j. The set of all variables is defined as V = S ....

....from the set of all bound variables. By j[A] x 1 Delta Delta Delta x ar(A) A(x 1 ; x ar(A) we denote the extensional normal form of V [ C (i.e. the normal form of A under the converse of the j reduction rule in calculus) Definition 2 (second order substitutions, cf. 12] [24]) A second order substitution oe is a finite set of (type respecting) variable assignments, i.e. pairs (X; t) with t 2 T also denoted by X t. It may be considered to be a total function oe : V t by defining oe(X) j[X] if there is no pair X t in oe. The domain of a substitution oe is ....

[Article contains additional citation context not shown here]

Wayne Snyder and Jean Gallier. Higher-order unification revisited: Complete sets of transformations. Research report, University of Pennsylvania, USA, March 1988.

....if the maximal depth of SO cycles does not grow too large. 1 Introduction Context unification is a variant of second order unification and also a generalization of string unification. There are unification procedures for the more general problem of higher order unification (see e.g. [Pie73,Hue75,SG89,Wol93,Pre95]) It is well known that general higher order unification and second order unification are undecidable [Gol81,Far91,LV99] and that string unification is decidable [Mak77] Recent upper complexity estimations for string unification are NEXPTIME [Pla99a] and PSPACE [Pla99b] Context unification ....

Wayne Snyder and Jean Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989.

....as well as of satisfiability of one step rewrite constraints. 1 Introduction Context unification is a variant of second order unification and also a generalization of string unification. There are unification procedures for the more general problem of higher order unification (see e.g. [Pie73, Hue75, SG89, Pre95]) It is well known that general higher order unification and second order unification are undecidable [Gol81, Far91, LV99] String unification was shown to be decidable by Makanin [Mak77] Recent upper complexity estimations are that it is in NEXPTIME [Pla99a] and even in PSPACE [Pla99b] ....

Wayne Snyder and Jean Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101-- 140, 1989.

....We denote by T F ;V = the quotient set of T F ;V . Let us for simplicity use the word term for formula as well and var for vars or fvars. We now define the unification of first order abstract terms as a set of non deterministic rules of transformation. This elegant approach is due to [10, 19]. A pair of terms (for short a pair) is a multiset of two terms, denoted by hs; ti; we call a substitution an abstract unifier of a pair hs; ti if s = t . A system of terms (for short a system) is a multiset of pairs; a substitution is an abstract unifier of a system if it unifies each ....

W. Snyder and J. Gallier. Higher-order unification revisited: Complete sets of transformations. Journal of Symbolic Computation, 8:101--140, 1989.

....the context for this study as an approach to a problem of modularity in environments for machine assisted proof. Unification algorithms for Pi have been discussed before [9, 10, 22, 23] our contribution is to present the search space for the unification algorithm as a reduction system following [27]. This presentation has a certain simplicity and it is based on a formal theory of type similarity. 1 However, our main purpose for re presenting unification in this particular way is to support the second and third aims above. The rest of this introduction is devoted to a discussion of the ....

....a space characterised as the inductive closure of a set of rules the inference system. Even in the case when the problem of interest is solvable in polynomial time, such as with certain unification problems, the clarity of exposition obtainable by this decomposition makes the method a common one [27]. 4 This is not as great a restriction as it might seem at first sight. Refinements of resolution, for example, are often formulated as distinct inference systems that take advantage of the structure of a search space to prioritise certain constructions over others. Viewed from the point of view ....

[Article contains additional citation context not shown here]

Wayne Snyder and Jean Gallier. Higher-order unification revisited : Complete sets of transformations. Journal of Symbolic Computation, 8:101--140, 1989. 20

....First order unification has the characteristic that variables in the codomain of the unifier is a subset of variables in the domain of unification, up to renaming. In higher order unification, however, arbitrarily many new logic variables may appear in the codomain of unifier substitutions (see [22]) Since these new variables are not originally associated with constant symbols, setsub will not be able to instantiate them to proper object level variables. At the end of this chapter will discuss how the setsub technique can be extended beyond first order unification. The cut ( in the first ....

....algorithm adds no complication when performing first order unification. It performs the same operations as standard first order unification; it always terminates and returns a most general unifier if it succeeds. This should be clear from the presentation of higherorder unification given in [22]. One other crucial property of Prolog s meta level unification is that only idempotent substitutions are constructed as unifiers. The proof follows the operational account of the unification program given above. All substitutions here are idempotent (toeoe = toe for all terms t) Let v 1 ; ....

[Article contains additional citation context not shown here]

Wayne Snyder and Jean H. Gallier. Higher order unification revisited: Complete sets of transformations. Journal of Symbolic Computation, 8(1-2):101--140, 1989. 88

....first order acceptable problem into a real first order problem (Def. 6 7) and solve it by the given E unification algorithm (Def. 8) 2. Variable binding: Bind free variables occurring as heads of disagreement pairs (heads of alien subterms not touched in the step above) by partial bindings as in [7]. This is done by imitating the corresponding head (rule E Imitation) or by projecting on one of its arguments (rule Projection) First, we define the set of all abstractions that cover a given subterm, e.g. x; y cover the subterm f(a; y) in x Delta y Delta v(x; f(a; y) A subterm is given ....

Snyder, W. and Gallier, J.: Higher-Order Unification Revisited: Complete Sets of Transformations. J. Symbolic Computation 8 (1989), 101-140.

....set of all types in T 1 . Define a type substitution as a total map Theta : Tvars Types such that Theta(X ) 6= X for finitely many X 2 Tvars. Extend the type substitution domain to all types, Theta : Types Types in the usual way. See, for example, the presentation in Snyder and Gallier [SG89] Let Theta denote the application of substitution Theta to type and let Theta 2 Theta 1 be the composition of substitutions Theta 2 and Theta 1 defined by ( Theta 2 Theta 1 ) x) Theta 2 ( Theta 1 (x) 5 The support of a substitution Theta, denoted su( Theta) is the set of type ....

....by fX : g. Two substitutions 1 and 2 are equal if and only if their supports are equal and 8X 2 su( 1 ) 1 (X) 2 (X) Every substitution : Tvars Types can be extended to a map from types to types, Types Types, using standard methods. See, for example, Snyder and Gallier [SG89] We normally assume the appropriate 58 59 domain of the substitution map, either Tvars or Types, from the context of any discussion. The composition of substitutions and 0 is denoted 0 ffi , or simply 0 by juxtaposition, and defined ( 0 ffi ) def = 0 ( 1 If ....

[Article contains additional citation context not shown here]

Wayne Snyder and Jean Gallier. Higher order unification revisited: Complete sets of transformations. Journal of Symbolic Computation, 8:101--140, 1989.

....as in the last case. ffl If S = x:case y:X(t) of : G, then X(t) must be of one of the following forms: x:c(t 0 ) such that Case Select applies on S. In this case, Imitation is applicable with a binding oe such that 9 0 : 0 oe as in proof of higher order unification (see [36, 33]) Since 0 is R normalized and dummy free, the induction hypothesis applies with a smaller solution, decreasing B. x:x(t 0 ) such that Case Select applies on S. This case proceeds as the above with Projection instead of Imitation. x:f(t 0 ) such that Case Eval applies on S. Here, ....

Wayne Snyder and Jean Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989. 33

....; oe k n ] # fi where the y i 2 V oe i are new variables. For notational convenience, if y i 2 V i , i = 1; n, and x 2 V n for a base type , then the j adjustment y n :x(y n ) may also be written as x. The intention of j adjustment is twofold: to put a term into j expanded form as in [26] and to transform a term typed with respect to into a fij equivalent term of the same type, but now typable with respect to v. Note that if =v, where all types under the subtype relation v have the same corresponding domain types, j adjustment degenerates into j expansion. Consider the subtype ....

....on the structure of t 1 and t 2 . Since j[u i # fi ; ff j[t i ; we have j[u 1 # fi ; ff j[u 2 # fi ; 2 5.1 A Pre Unification Algorithm This section presents a unification algorithm for an arbitrary subtype relation . The presentation is inspired by the work of Snyder and Gallier [26]. Following Huet [5] we only investigate pre unification where a solution is a substitution together with a set of solvable constraints, the so called flex flex pairs. The first step in this algorithm is to j adjust all terms to be unified: instead of some terms s and t with s; t : for some ....

[Article contains additional citation context not shown here]

W. Snyder and J. Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989.

....equations are well typed in the context Gamma, for every variable X occurring in the problem, Gamma X = Gamma. We can conclude by Proposition 5.5 that this problem is the the pre cooking of a problem in calculus. We call solved form any solved systems of equation in calculus in the sense of [46]. Corollary 5.13. Let a = fij b a higher order unification problem such that aF = oe b F can be rewritten by the system Unif to a disjunction of systems that has one of its constitutive systems P solved. Let Q be the system resulting of the normalisation with the strategy Back of the ....

....of systems that has one of its constitutive systems P solved. Let Q be the system resulting of the normalisation with the strategy Back of the system P and R = F Gamma1 (Q) Then R is a solved form and the solutions of R are solutions of a = fij b. If we apply the trivial solution of [46] to solve the flexible flexible equations, we get back the solution built in the proof of the Proposition 5.6. Proof. Let be a substitution in calculus. Applying the previous results, if is a solution of R then F is a solution of RF , F is a solution of Q, F is a solution of P , F is ....

W. Snyder and J. Gallier, Higher order unification revisited: Complete sets of tranformations, Journal of Symbolic Computation, 8 (1989), pp. 101--140. Special issue on unification. Part two.

.... simply typed terms [11] allows for a usable definition of the unification of terms ( unification) Huet gives an extensive presentation of unification [19] Paulson gives the main lines of it in the context of theorem proving [32] and Snyder and Gallier revisit it in a deduction rule setting [39]. Nipkow extends the deduction rule setting towards simple types with type variables [31] which is in fact the domain of Prolog. Miller studies the correspondence between logical quantifications (8 and 9) and quantification [26] One can see the extension of Prolog with simply typed terms ....

W. Snyder and J. Gallier. Higher-order unification revisited: complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989.

.... fij b a higher order unification problem such that a F = b F can be rewritten by the system Unif to a disjunction of systems that has one of its constitutive systems P solved. Let Q be the normal form of P by the Anti rules and R = F Gamma1 (Q) Then R is a solved systems in the sense of [SG89] the solutions of R are solutions of a = fij b. If we apply the trivial solution of [SG89] to solve the flexible flexible equations, we get back the solution of the previous section. Proof: Let be a substitution in calculus, if is a solution of R then F is a solution of R F , F is a ....

....Unif to a disjunction of systems that has one of its constitutive systems P solved. Let Q be the normal form of P by the Anti rules and R = F Gamma1 (Q) Then R is a solved systems in the sense of [SG89] the solutions of R are solutions of a = fij b. If we apply the trivial solution of [SG89] to solve the flexible flexible equations, we get back the solution of the previous section. Proof: Let be a substitution in calculus, if is a solution of R then F is a solution of R F , F is a solution of Q, F is a solution of P , F is a solution of a F = b F and is a solution of ....

[Article contains additional citation context not shown here]

W. Snyder and J. Gallier. Higher order unification revisited: Complete sets of tranformations. Journal of Symbolic Computation, 8(1 & 2):101--140, 1989. Special issue on unification. Part two.

.... Gamma 1. This solution oe 0 is a solution to the problem Psi. Remark This method, in which an interpolation problem Phi( Psi; oe) is constructed from a pair Psi; oe where Psi is an arbitrary problem and oe a solution to Psi, can be compared to the one used in the completeness proof of [9] in which a problem in solved form is constructed from such a pair. 4 A Decision Procedure Theorem Third Order Matching is Decidable Proof A decision procedure is obtained by considering the problem Phi and enumerating all the ground substitutions such that the term substituted for x has a ....

W. Snyder, J. Gallier, Higher-Order Unification Revisited: Complete Sets of Transformations, Journal of Symbolic Computation, 8, 1989, pp. 101-140.

....U n ff n ) For ff , fi , j , fij conversion and the definition of fij and head normal form (hnf ) for a term T we refer to [Bar84] as well as for the definition of free variables, closed formulas, and substitutions. Unification and sets of partial bindings AB h fl are well explained in [SG89]. An example for a pre clause, i.e. not in proper clause normal form, consisting of a positive literal, a negative literal, and a special negative equation literal (also called unification constraint) is C : P o T ) T [h fl o Y n fl n ] F [Q a = Y b ] F . The ....

.... No ] F Equiv C [M fl = N fl ] F C [8P fl o P M) P N] F Leib C [F fl n fl U n = H ffi m fl V m ] F G 2 AB h fl n fl for a constant h C [F U n = H V m ] F [F = G] F F lexF lex AB h fl specifies the set of partial bindings of type fl for head h as defined in [SG89] Fig. 2. Extensional HO Resolution Calculus ER We now sketch the main results on ER as discussed in detail in [Ben99] Definition 3 (Extensional HO Resolution) We define three calculi: ER : fCnf; Res; F ac; P rimg [ UNI [ fEquiv; Leibg employs all rules (except FlexFlex ) displayed in Fig. ....

W. Snyder and J. H. Gallier. Higher-Order Unification Revisited: Complete Sets of Transformations. Journal of Symbolic Computation, 8:101--140, 1989.

No context found.

Wayne Snyder and Jean H. Gallier. Higher order unification revisited: Complete sets of transformations. Journal of Symbolic Computation, 8(1-2):101--140, 1989. 12

No context found.

W. Snyder and J. H. Gallier. Higher order unification revisited: Complete sets of transformations. Journal of Symbolic Computation, 8(1-2):101--140, 1989. 12

No context found.

Wayne Snyder and Jean Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989. 20

No context found.

Wayne Snyder and Jean Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989.

No context found.

Snyder, Wayne and Gallier, Jean. (July/August 1989). Higherorder unification revisited: Complete sets of transformations. Journal of Symbolic Computation, 8(2):101--140.

No context found.

Wayne Snyder and Jean Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989.

No context found.

W. Snyder and J. Gallier. Higher-order unification revisited: Complete sets of transformations. J. Symbolic Computation, 8:101--140, 1989.

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC