G. Huet, R'esolution d' ' Equations dans les Langages d'Ordre 1,2, ..., !, Th`ese de Doctorat d' ' Etat, Universit'e de Paris 7, 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we have discussed how coinductively defined sets can be turned into inductively defined ones using the fixpoint rule. 5.2. Related work 5.2.1. Recursive type equality and subtyping The present work was inspired by an observation that meant that the standard unification closure algorithm [HK71, Hue76] (see [ASU86, Section 6.7] for a presentation) can be turned into an axiomatization of recursive type equality simply by adding the type equation to be proved to the assumptions in the premises; see Figure 4. Unification closure works by building a bisimulation between two recursive types. This ....

G. Huet. R'esolution d'equations dans des langages d'ordre 1, 2, : : : , omega (th`ese de Doctorat d'Etat). PhD thesis, Univ. Paris VII, September 1976.

....by applying two algorithms: ffl unification between pattern trees, ffl lub of pattern trees. Both algorithms are described in detail in [MM93] as well as an algorithm to simplify pattern trees. The unification between pattern trees behaves similarly as unification between regular trees 3 (see [Hu76, Co82a, Co84]) Regular trees have been included in the language Prolog II ( Co82b] The main difference with usual unification is that pattern trees cannot be syntactically equal. The equivalence between two pattern trees is reduced to the equivalence between their sons. The process finish by unification of ....

G. Huet, Resolution d"equations dans les languages d'ordre 1, 2, : : : ; !, Th`ese de doctorat d'etat, Universit'e Paris VI.

....the entailment x = f(x; y) y = f(y; y) j= CFT x = y 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 new notions providing for an elegant presentation of our results. Huet [11] uses the related notion of equivalence simplifiable in his study of rational tree unification. We improve on Colmerauer s [10] results for rational trees since our constraints are closed under existential quantification. For instance, our algorithm is complete for quantified negative ....

G. Huet. R'esolution d'equations dans des langages d'ordre 1; 2; \Delta \Delta \Delta ; !. Th`ese de Doctorat d'Etat, l'Universit'e Paris VII, Sept. 1976.

....and introduction of existential quantifiers are mixed and full unification is required, but in proof checking and semi automated theorem proving, these rules can be applied separately and thus pattern matching can be used instead of unification. Higher order matching is conjectured decidable in [6] and the problem is still open. In [5] 6] 7] Huet has given a semi decision algorithm and shown that in the particular case in which the variables occurring in the term a are at most second order this algorithm terminates, and thus that second order matching is decidable. In [10] Statman has ....

....are mixed and full unification is required, but in proof checking and semi automated theorem proving, these rules can be applied separately and thus pattern matching can be used instead of unification. Higher order matching is conjectured decidable in [6] and the problem is still open. In [5] [6] [7] Huet has given a semi decision algorithm and shown that in the particular case in which the variables occurring in the term a are at most second order this algorithm terminates, and thus that second order matching is decidable. In [10] Statman has reduced the conjecture to the definability ....

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G. Huet, R'esolution d' ' Equations dans les Langages d'Ordre 1,2, ..., !, Th`ese de Doctorat d' ' Etat, Universit'e de Paris VII, 1976. 18

....that the left members of matching equations are composed only of first order terms (i.e. not containing sets, arrows or applications) The syntactic matching substitution from t to t 0 , when it exists, is unique and can be computed by a simple recursive algorithm given for example by G. Huet [Hue76] It can also be computed by the following set of rules SyntacticMatching where the symbol is assumed to be associative and commutative and where T is denoted ; Decomposition (f(t 1 ; t n ) f(t 0 1 ; t 0 n ) P 7 7 V i=1; n t i ; t 0 i P ....

G. Huet. R'esolution d'equations dans les langages d'ordre 1,2, ...,!. Th`ese de Doctorat d'Etat, Universit'e de Paris 7 (France), 1976.

....the inequation (ff e 0 ; ff) ff e ; ff) in Figure 7 is replaced by the equality ff e 0 = ff e . For the Curry Hindley Calculus all extracted constraints are equational and the resulting problem is unification, which can be solved efficiently by any number of unification algorithms [Hue76,PW78,MM82]. Since the additional inequational constraints for the Damas Milner Calculus seemed rather innocuous at first sight, it was long believed that Damas Milner typability was equally efficient (e.g. Lei83,MH88] before it was refuted by Kanellakis, Mairson and Mitchell [KM89,Mai90,KMM91] and ....

....a derivation that would lead us to claim, incorrectly, that I 1 has no semi unifiers. If we consider system I 0 = ff(y; y) x; x yg it is easy to see that it is unsolvable. This is caught by the extended occurs check, Rule 9. 5. 2 Arrow Graph Rewriting System Fast unification algorithms [Hue76,PW78,MM82,ASU86] use term graphs, a data structure that supports sharing of subexpressions, to eliminate the potentially exponential cost of copying terms and applying substitutions. We present arrow graphs, which are term graphs with additional structure to represent equations and inequations between terms, and ....

G. Huet. R'esolution d'equations dans des langages d'ordre 1, 2, . . . , omega (th`ese de Doctorat d'Etat). PhD thesis, Univ. Paris VII, Sept. 1976.

....backtracking to an alternative unifier of the same pair of terms may occur and the search for a higher order unifier may go on forever. Higher order matching, the special case of higher order unification we need, was conjectured to be decidable in the simply typed case (no polymorphism) by Huet [11], but this is still an open problem. The third order case was recently shown to be decidable by Dowek [5] On the other hand, Dowek also showed that strongly polymorphic higher order matching is undecidable [6] Prolog supports ML style polymorphism, so we included it in our notion of higher order ....

G. Huet, R'esolution d"equations dans les langages d'ordre 1; 2; : : : ; !, Th`ese de Doctorat d'Etat, Universit'e de Paris-VII, 1976.

....and unknown beyond. All the results above hold for the languages with explicit typing, which corresponds to the Church style simply typed lambda calculus with a single constant type o and no variable types. The higher order matching problem in the simply typed lambda calculus due to G. Huet [Hue76] is formally stated as follows: given an equation s(x 1 ; x n ) fij t, where s, t are terms of fixed simple types, and t does not contain free variables, do there exist simply typed terms s 1 ; s n of appropriate types such that s[s 1 =x 1 ; s n =x n ] t modulo ....

G. Huet. R'esolution d' ' Equations dans les Langages d'Ordre 1, 2, . . . , !. Th`ese de Doctorat d' ' Etat, Universit'e de Paris VII, 1976.

....bound did not follow from Statman s result [15] Second, we settle the new F (1; cn= log(n) exp 1 (cn= log(n) 2 2 Delta Delta Delta 2 oe cn= log(n) lower bound for a long standing, today still open, higher order matching problem in the simply typed lambda calculus due to G. Huet [9]. The problem consists in deciding, given a term t of type oe 1 : oe n and a term u of type (both in normal forms) whether there exist terms s i of types oe i (1 i n) such that ts 1 : s n = fij u. This gives an example asked for by Compton and Henson in Problem 10.11 from [2] ....

....in simply typed calculus, cf. Section 12. In fact, if one could normalize faster than in time F (1; cn) it would be possible to decide fi(j) equality faster than in F (1; cn) in contradiction with our Theorem 19. 15. The exp 1 (cn=log(n) Lower Bound for Higher Order Matching G. Huet [9] posed the following, today still open, decidability problem for the simply typed lambda calculus, called the higher order matching problem: Given a term t of type oe 1 : oe n and a term u of type (both in normal forms) do there exist terms s i of types oe i (for 1 i n) such that ....

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G. Huet. R'esolution d' ' Equations dans les Langages d'Ordre 1, 2, . . . , !. Th`ese de Doctorat d' ' Etat, Universit 'e de Paris VII, 1976.

....Finally, we have discussed how coinductively defined sets can be turned into inductively defined ones using the fixpoint rule. 5.2. Related work 5.2.1. Recursive type equality and subtyping The present work was inspired by an observation that meant that the standard unification closure algorithm [HK71, Hue76] (see [ASU86, Section 6.7] for a presentation) can be turned into an axiomatization of recursive type equality simply by adding the type equation to be proved to the assumptions in the premises; see Figure 4. Unification closure works by building a bisimulation between two recursive types. This ....

G. Huet. R'esolution d'equations dans des langages d'ordre 1, 2, . . . , omega (th`ese de Doctorat d'Etat). PhD thesis, Univ. Paris VII, September 1976.

....Plotkin [5] 6] This conjecture has been studied by Plotkin and Statman. At last section 7 presents the higher order matching conjecture and the proof that the definability conjecture implies the higher order matching conjecture. The decidability of higher order matching is conjectured in Huet [4], the equivalence of the higher order matching problem and the higher order matching problem with closed terms if proved in [7] and the proof that the definability conjecture implies the higher order matching conjecture is from [8] this proof is also discussed in Wolfram [10] 1 Typed ....

G. Huet, R'esolution d' ' Equations dans les Langages d'Ordre 1,2, ..., !, Th`ese de Doctorat d' ' Etat, Universit'e de Paris 7, 1976.

.... algorithm presented by Ait Kaci is a node merging process using the UNION FIND method (originally used for testing the equivalence of finite automata [Hopcroft Karp 71] It has its analogue in the unification algorithm for rational terms based on a fast procedure for congruence closure [Huet 76] Node merging is a destructive operation Since actual merging of nodes to build new node equivalence classes modifies the argument DGs, they must be copied before unification is invoked if the argument DGs need to be preserved. For example, during parsing there are two kinds of representations ....

G'erard Huet. R'esolution d'Equations dans des Langages d'Ordre 1, 2, : : : , !. Th`ese de Doctorat d'Etat, Universit'e de Paris VII, France. 1976.

....by analogy. Curien followed the similar line to use second order matching for computing similarities between formulas in automatic analogical proofs [4] Second order matching is also used by Kolbe and Walther [12] for reusing proofs. The standard second order matching algorithm was given by Huet [10] (see also [11] Let us use s = t to denote a matching problem, where s is a second order simply typed term (short: term) possibly containing free variables and t a term containing no free variables. An essential case that a second order matching algorithm should consider is the so called ....

....there exist no sequences of transformations that terminate with a pair of the above form. In the case of failure, S 0 is not matchable. In this section a second order matching algorithm is described via the transformations. The transformations are a reformulation of the standard algorithm by Huet [10] (see also [11] They are in fact a restriction of the corresponding transformations by Snyder and Gallier [19] The first transformation is the usual simplification rule, which simplifies a matching pair with identical heading symbols on both sides. hfx:a(s n ) x:a(t n )g [ S; oei = ....

G. Huet. R'esolution d'Equations dans les langages d'Ordre 1,2, ...,!. Th`ese de Doctorat d'Etat, Universit'e de Paris 7 (France), 1976.

....type , but not a greatest type . The problems arising in constructing a structure for Sigma 3 are dual to those encountered for Sigma 2 , and we leave to the reader to construct an appropriate structure. The result is the first type inference algorithm for this system. 6 Related work Huet [16] gave the first unification algorithm for recursive types; see also the papers by Cardone and Coppo [7, 8] Mitchell [23, 24] gave the first inference algorithm for atomic subtyping, without recursive types. With no further assumptions about the partial order, this problem is PSPACE complete [35, ....

G. Huet. R'esolution d"equations dans les langages d'ordre 1, 2, . . . , !. Th`ese de Doctorat d'Etat, Universit'e de Paris VII, 1976.

.... syntactic operator have exactly one inference rule scheme (without side conditions) it follows that type checking for an annotated term is (RAM )linear time equivalent to solving a unification problem [Wan87,Hen88a] and thus can be done efficiently in linear [PW78,MM82] or almostlinear time [Hue76,ASU86] We will show that type inference for unannotated (or partially annotated) terms and computation of minimal completions can actually be done in essentially the same time. 3 Type constraint characterization Recall that the universe of type expressions is the class of terms T (A; V ) ....

....constraint normalization algorithm algorithm can be refined to accommodate constraints of the form ff b ff 0 . Term graphs with equivalence classes have been used for fast implementations of unification. We shall not go into details, but refer the reader to the literature; e.g. HK71,AHU74,Hue76, PW78,MM82,ASU86] The equivalence classes are represented by a system of equivalence class representatives (ecr s) and there are two operations available on ecr s: find(n) for node n is a function that returns the ecr of the equivalence class to which n belongs; union(n; n 0 ) which can only ....

G. Huet. R'esolution d'equations dans des langages d'ordre 1, 2, . . . , omega (th`ese de Doctorat d'Etat). PhD thesis, Univ. Paris VII, Sept. 1976.

....for Sigma 3 are dual to those encountered for Sigma 2 , so we will not bother to give the appropriate structure here. However, a historical note is in order. Cardone and Coppo [7, 8] have claimed that a type inference algorithm for Sigma 3 can be based on Huet s unification algorithm [17] (see p. 56 of [8] and they refer to Courcelle [9] in support. However, Huet s algorithm does not apply to types with , and Courcelle does not make that claim. Moreover, Courcelle s ordering on types is not the same as the ordering of Cardone and Coppo. Courcelle s ordering is the the Bohm ....

....are always treated covariantly, so, for example, Omega Omega is less than nat nat in the Bohm order. Cardone and Coppo order function types contravariantly in the argument type, as do we. Hence, we believe we have the first type inference algorithm for this system. 6 Related work Huet [17] gave the first unification algorithm for recursive types. Mitchell [24, 25] gave the first inference algorithm for atomic subtyping, without recursive types. With no further assumptions about the partial order, this problem is PSPACEcomplete [36, 16, 13] and if the partial order is a disjoint ....

G. Huet. R'esolution d"equations dans les langages d'ordre 1, 2, . . . , !. Th`ese de Doctorat d'Etat, Universit'e de Paris VII, 1976.

.... X Sigma L Gamma Delta oe Phi Gamma A OE ; 4 This situation can be considered to be a variation on the programs as worlds view of the semantics of logic programs [107, 151] 5 Pym s work higher order unification for dependent types ( Pi) builds directly on Huet s seminal work in [81, 80]. See also [45] corresponding to the sequent (X ) Delta L OE, with proof object Phi, in L. Reduction steps in OR can then be seen to correspond to reduction steps in L. Such a calculus can be interpreted in structures. 4 Proof search and Programming A proof object is not only a ....

G. Huet. R'esolution d"equations dans les languages d'ordre 1, 2 , : : : , !. Th`ese de Doctorat d' ' Etat, Universit'e de Paris VII, 1976.

....Before a rule is applied to an equation of P the equation is first transformed to an equation of representatives. If this new equation is of the form x = E t x or at 1 = E at 2 it is placed into S. This method ensures termination of the algorithm. The function rep is first described in [Hue76]. Colmerauer [Col84a] uses a simpler version of this function in an algorithm for the unification of syntactic trees that may be infinite. Definition 3.12 The representative rep(t; S) of a term t in normal form with respect to a reduced set S of equations is defined as: rep(t; S) Nf (rep(t 1 ; ....

Huet, G.: Resolution d'equations dans les langages d'ordre1,2,...,\Omega\Gamma Th`ese de Doctorat d' ' Etat, Universit'e Paris VI. 1976.

....processing positive and negative constraints incrementally. 2 The Feature Tree Structure This section gives a formal definition of CFT s standard model T . T is a first order structure whose universe consists of all feature trees obtainable from given alphabets of sorts and features. 4 Huet [17] uses the related notion of equivalence simplifiable in his study of rational tree unification. From now on we assume that an infinite alphabet SOR of symbols called sorts and an infinite alphabet FEA of symbols called features are given. For several results of this paper (e.g. ....

....with classical trees in that it allows for finer grained descriptions. The resulting constraint system CFT is a conservative extension of both Prolog II s rational tree system [12, 13] and the feature tree system FT [9, 7] Thus CFT brings together the work on classical tree constraints (e.g. [17, 12, 13, 23, 26]) and the work on feature descriptions (e.g. 21, 20, 1, 2, 4, 5, 6, 29, 9, 7, 11] two lines of research that seemed to be rather far apart in the past. The declarative semantics of CFT was specified both algebraicly (the feature tree structure T ) and logically (the first order theory CFT ....

G. Huet. R'esolution d'equations dans des langages d'ordre 1; 2; \Delta \Delta \Delta ; !. Th`ese de Doctorat d'Etat, l'Universit'e Paris VII, Sept. 1976.

....the solution: Gamma Gamma Gamma Gamma Gamma oe L [ or any cons In [MM93] algorithms to compute the lub and glb of a set of DDPs are provided. In fact, DDPs are a complex variant of regular trees, and these algorithms behave similarly to those described for them, see [Hu76, Co84, Co83]. The computation of the glb behaves as the unification of DDPs, while the lub is the union and simplification of DDPs. MM93] also provides a method to reduce a system of equations to solved form. The equation solver proceeds bottom up from simple equations to more complicated ones applying the ....

G. Huet, Resolution d"equations dans les languages d'ordre 1, 2, : : : ; !, Th`ese de doctorat d'etat, Universit'e Paris VI.

....second order then the equation has the form (f a 1 : an ) b B.P. 105, 78153 Le Chesnay CEDEX, France. Gilles.Dowek inria.fr where f is a second order variable and all the variables occurring in the terms a i and b are first order. We consider a variant of the unification algorithm of [2] [3] in which when we consider an equation on the form x = b such that all the variables occurring in the term b are first order, if the variable x does not occurs in b then one substitutes b for x and otherwise one fails. When restricted to second order superficial unification problems, the ....

....select the success nodes such that all the closed terms substituted to the variables f 1 ; fn taking place of a variable f of the initial problem are equal, else fail to give any information. Remark: The algorithm above generalizes both first order unification [7] and second order matching [3] [4] as for problems in these classes it does not fail and gives a complete set of unifiers. ....

G. Huet, R'esolution d' ' Equations dans les Langages d'Ordre 1,2, ..., !, Th`ese de Doctorat d' ' Etat, Universit'e de Paris VII, 1976.

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G. Huet, R'esolution d' ' Equations dans les Langages d'Ordre 1,2,..., !, Th`ese de Doctorat d' ' Etat, Universit'e de Paris VII, 1976.

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G. Huet, R'esolution d' ' Equations dans les Langages d'Ordre 1,2, ..., !, Th`ese de Doctorat d' ' Etat, Universit'e de Paris 7, 1976.

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