G'erard Huet. R'esolution d"equations dans des langages d'ordre 1; 2; . . . ; !. PhD thesis, Universit 'e Paris VII, September 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be ( reduced in two steps to x 1 ; x 2 ; y 1 : x 1 ( y 2 : x 2 y 2 ) y 1 ) If a term cannot be further reduced by (resp. then it is in normal form (resp. normal form) It is well known that the reduction relation de ned by ; is strongly terminating and Church Rosser [Wol93,Hue76,Bar84]. Hence for every term t, there is a normal form t# , which is unique up to = Remark 2.7. Let t be a term in normal form. Let t result from t by a series of reductions. Then t is in normal form. Remark 2.8. Let t be a term of type in normal form. Let m = ar( Then t ....

Gerard Huet. Resolution d'equations dans des langages d'ordre 1,2,. . . !. These de doctorat d`etat, Universite Paris VII, 1976. In French.

.... [Her30] is given credit for the rst description of a uni cation algorithm in a footnote of his thesis, but it was not until 1965 that it was introduced into automated deduction through the seminal work by Alan Robinson [Rob65, Rob71] The rst algorithms were exponential, and later almost linear [Hue76, MM82] and linear algorithms [MM76, PW78] were discovered. In the practice of theorem proving, generally variants of Robinson s algorithm are still used, due to its low constant overhead on the kind of problems encountered in practice. For further discussion and a survey of uni cation, see [Kni89] We ....

Gerard Huet. Resolution d'equations dans des langages d'ordre 1; 2; : : : ; !. PhD thesis, Universite Paris VII, September 1976.

.... x: x:fx ) f Similar problems arise if we try to enrich the calculus with extra rewrite rules which may be confluent by themselves, but which when taken in conjunction with j contraction fail to be confluent [16] Recently several researchers [2,15,20,19,22, 49] have adopted older proposals [41,62,68] that j conversion be interpreted as an expansion: t ) x:tx if t : A B and the resulting rewrite relation has been shown confluent. In these works infinite reduction sequences such as: f ) x:fx ) x: y:fy)x ) 1.2) are prohibited by imposing syntactic restrictions to limit the ....

....fragment and is proved strongly normalising and confluent by modifying the tradi tional proofs so as to cope with the presence of expansions and the non congruent nature of reduction. Finally we show that the normal forms of the restricted rewrite relation are exactly Huet s long fij normal forms [41] and that these normal forms may be calculated by first contracting all fi redexes and then performing any remaining j expansions or, vice versa, by performing all j expansions and then contracting any remaining fi redexes. Historically the use of j expansions, as opposed to j contractions, can ....

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

....be ( reduced in two steps to x 1 ; x 2 ; y 1 : x 1 ( y 2 : x 2 y 2 ) y 1 ) If a term cannot be further reduced by (resp. then it is in normal form (resp. normal form) It is well known that the reduction relation de ned by ; is strongly terminating and Church Rosser [Wol93,Hue76,Bar84]. Hence for every term t, there is a normal form t# , which is unique up to = Remark 2.7. Let t be a term in normal form. Let t 0 result from t by a series of reductions. Then t 0 is in normal form. Remark 2.8. Let t be a term of type in normal form. Let m = ar( Then ....

Gerard Huet. Resolution d'equations dans des langages d'ordre 1,2,. . . !. These de doctorat d`etat, Universite Paris VII, 1976. In French.

....between logic and programming need to be refined, maybe even beyond the Curry Howard correspondence. Griffin s paper [10] emphasized the interactions between continuations (in Scheme) and classical reasoning. This starting point led to a convincing typed system, recently introduced by Parigot [14], which showed how Curry Howard correspondence could be extrapolated to capture more programming constructs. Thus, we are suggested to approach the relations between reasoning and programming in a different way. Same kind of connections have to be found with regard to such programming tools as ....

....IEEE, 1987. 12] Parigot M. 1988) Programming with proofs: a second order type theory. Proceedings of ESOP 88 (ed. H. Ganziger) Lectures Notes in Comp. Science 300. Springer Verlag. 1988. 13] Parigot M. 1992) Recursive Programming with Proofs. Theoretical Computer Science 94 (1992) p.335 356. [14] Parigot M. 1992) calculus: an algorithmic interpretation of classical natural deduction. In Proceedings International Conference on Logic Programming and Automated Deduction, St Petersburg. LNCS 624. Springer Verlag. 1992. 15] Paulin Mohring C. 1989) Extracting F programs from proofs in the ....

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

....induction on the structure of the type inference derivation. 2 The small terms property allows reasoning about sharing between sub terms, and is a key requirement in our formulation of minimization (see section 4. 5) It is already to be found, for instance, in the theory of uni cation [Hue76]. Among works more closely related to ours, Aiken and Wimmers [AW92] and Palsberg [Pal95] use a similar convention. 11 3 Simplifying equality constraints We are done introducing our type inference system, which speci es how to associate a constrained type scheme with a given program. We shall ....

Gerard Huet. Resolution d'equations dans des langages d'ordre 1, 2, : : : , !. PhD thesis, Universite Paris 7, September 1976. 36

....First of all, it is an off the shelf mechanism, well studied and well understood, which has both a logical and a procedural interpretation. This permits the development of equational, declarative analyses while also enabling computational implementations via Huet s classic HOU algorithm [Hue76]. Also interesting is the fact that basic HOU has given birth to a number of off springs, which as we shall see, also are of linguistic interest. For example Higher Order Coloured Unification can be shown to constrain over generation in linguistically plausible ways whilst HOU with entailment ....

G'erard P. Huet. R'esolution d' ' Equations dans des Langages d'ordre 1,2,...,w. Th`ese d` ' Etat, Universit'e de Paris VII, 1976.

.... complete set of T unifiers CSU T (s; t) satisfying the minimality condition: 8oe 2 CSU T (s; t) oe 6= oe 6 W T ) where W = V ar(s) V ar(t) Note that since CSU T (s; t) exists and it is finite, for any s; t 2 ( Sigma [ V) then CSU T (s; t) exists and it is unique up to j V T [11]. Similar definitions can be given for unification problems consisting of some finite system of equations E , instead of a single equation s = t. When the context is clear, we will omit the prefix T before the word unifier. Sometimes we will speak of the set of most general unifiers when referring ....

Huet, G. R'esolution d"equations dans des langages d'ordre 1,2,: : : ; !. Th`ese d' ' Etat, Univ. de Paris, VII, 1976.

....definition 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 ....

G.P. Huet. R'esolution d"equations dans des langages d'ordre 1,2,...,!. PhD thesis, Universit'e de Paris VII, 1976. Th`ese d' ' Etat (in French).

.... where W = V ar(s) V ar(t) The set of most general T unifiers of s and t, denoted by S T (s; t) is a minimal complete set of T unifiers if (8oe 2 S T (s; t) oe W T ) oe j W T ) where W = V ar(s) V ar(t) Note that since S T (s; t) exists, it is unique up to j V T [7]. Similar definitions can be given starting from a system of equations E instead of the unification problem s = t. When the context is clear, we will omit the prefix T before the word unifier. For any satisfiable Herbrand system E involving terms from ( Sigma[V ) a unification algorithm should ....

Huet, G. R'esolution d"equations dans des langages d'ordre 1,2,: : : ; !. Th`ese d' ' Etat, Univ. de Paris, VII, 1976.

....of rules directly corresponds to the rules of higherorder pre unification as they can be found in [Koh94] which generalize Huet s pre unification transformations (see for instance [Sny91] for variable conditions. With these rules we use the tableau mechanism to construct Huet s unification tree [Hue76]. We call a branch Theta in a higher order tableau T closed, iff Theta ends in a flex flex pair or a trivial pair A 6= A. Note that the HT (subst) rule immediately closes the branch Theta that ends in a solved pair. A tableau h Gamma: Ri T is called closed, iff each branch of h Gamma: Ri T ....

G'erard P. Huet. R'esolution d' ' Equations dans des Langages d'ordre 1,2,...,w. Th`ese d` ' Etat, Universit'e de Paris VII, 1976.

....allows a most general unifier. Basically, the construction of this most general unifier mimics the algorithm of Martelli Montanari, extended with limit procedures. The unification result in this paper is incomparable with unification results for infinite sets of equations based on lattices by Huet [9] and Eder [5] where substitutions always have a finite domain. In [5] an artificial top element 1 is added to the collection of idempotent substitutions, and each set of equations between terms that cannot be unified by a substitution with finite support is awarded the same most general unifier, ....

....is, for the in this example there does not exist a natural number M such that (x) M 1 j (x) M , so we cannot construct its limit . 5 Related and Future Work The unification result in this paper is incomparable with unification results for infinite sets of equations based on lattices [9, 5] where substitutions have a finite domain. In [5] an artificial top element 1 is added to the collection of idempotent substitutions, in order to obtain a complete lattice, and each set of equations between finite terms which cannot be unified by a substitution (with finite support) is awarded the ....

G. Huet, R'esolution d' ' Equations dans des Langages d'Ordre 1,2,...,!, PhD thesis, University of Paris VII, 1976.

....must hold, whereas equations must be solved. 2. Syntactic unification As mentioned above, syntactic unification of first order terms was introduced by Post and Herbrand in the early part of this century Various researchers have studied the problem further [Champeaux 1986, Corbin and Bidoit 1983, Huet 1976, Martelli and Montanari 1982, Paterson and Wegman 1978, Robinson 1971, VenturiniZilli 1975] and, among other results, it was shown that linear time algorithms for unification exist [Martelli and Montanari 1982, Paterson and Wegman 1978] In this section we review the major results by presenting a ....

....this results in a quadratic algorithm. To develop an asymptotically faster algorithm, however, it is necessary to abandon the recursive descent approach, and recast the problem of unification as the construction of a certain kind of equivalence relation on graphs. This second approach is due to Huet [1976]. 2.3.1. Term dags and substitution Concerning example 2.8, it should be remarked that the explosion in the size of the terms occurred precisely because there were duplicate occurrences of the same variables, which cause a duplication of ever larger and larger terms. In order to fix this problem, ....

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Huet G. [1976], R'esolution d' ' Equations dans des Langages d'ordre 1,2,...,!, Th`ese d` ' Etat, Universit 'e de Paris VII.

....induction over the structure of its type, and the definition of the j long form of a normal term is by induction over the structure of the term itself. The j long form appeared in [17] under the name of long reduced form and in [15] under the name of j normal form, and was further investigated in [16], under the name of extensional form. In systems with dependent types the corresponding definition is more complicated. First when t = x : U ]u we have to take also the j long form of the term U and when t = x c 1 : c p ) we have to take also the j long form of the terms P 1 ; P n . So ....

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

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G'erard Huet. R'esolution d"equations dans des langages d'ordre 1; 2; . . . ; !. PhD thesis, Universit 'e Paris VII, September 1976.

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

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

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Gerard Huet. Resolution d'equations dans des langages d'ordre 1; 2; : : : ; !. PhD thesis, Universite Paris VII, September 1976. BIBLIOGRAPHY 103

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Gerard Huet. Resolution d'equations dans des langages d'ordre 1; 2; : : : ; !. PhD thesis, Universite Paris VII, September 1976.

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Gerard Huet. Resolution d'equations dans des langages d'ordre 1; 2; : : : ; !. PhD thesis, Universite Paris VII, September 1976.

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Gerard Huet. Resolution d'equations dans des langages d'ordre 1; 2; : : : ; !. PhD thesis, Universite Paris VII, September 1976.

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Gerard Huet. Resolution d'equations dans des langages d'ordre 1; 2; : : : ; !. PhD thesis, Universite Paris VII, September 1976.

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Gerard Huet. Resolution d'equations dans des langages d'ordre 1; 2; : : : ; !. PhD thesis, Universite Paris VII, September 1976.

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G.P. Huet. R'esolution d"equations dans des langages d'ordre 1,2,...,!. DSc thesis, Univ. Paris VII, 1976.

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