G. Huet, A unification algorithm for typed lambda calculus, Theoretical Computer Science 1(1) (1973) 27--57.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....This idea goes back to Huet [1972] who defines a complete calculus for higher order logic. Recently, Miller and Nadathur [1987] have recasted this idea in a logic programming language called Prolog. To be complete, such a system must incorporate a higher order unification algorithm [Huet, 1975]. Unfortunately, higher order unification is undecidable [Huet, 1973; Goldfarb, 1981] and many possible solutions have to be taken into account if two terms are to be unified whose initial symbols are function variables. Besides an ongoing discussion of whether higher order extensions of ....

G. Huet. A unification algorithm for typed lambda calculus. Journal of Theoretical Computer Science, 1:27--57, 1975.

....additional input. The work on general purpose unification algorithms derives from work on term rewriting (see 4) since the axioms must usually be represented as a confluent rewrite rule set. One of the major achievements of built in unification was Huet s higher order unification algorithm, Huet, 1975; Jouannaud and Kirchner, 1991 ] which builds in the #, # and optionally the # rules of #calculus. This algorithm makes automated higher order theorem proving possible. Higher order unification is a badly behaved problem: there can be infinitely many unifiers and the problem is undecidable. ....

G. Huet. A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1:27--57, 1975.

....We investigate how these ideas can be developed within the Calculus of Constructions (CC) The adaptation provides an conservative extension, denoted CC . Strong normalisation for fi reductions is preserved. We recover the alternate recursive coding for integers introduced in AF2 by Parigot [12, 13]. Thus, the computational behaviour for terms coding integers is improved. Moreover, as expected, all partial recursive functions are now definable. Relationships with primitive coding through Church integers within the pure Calculus is studied, giving some insights into logical expressiveness ....

....and Pitts, based on Moggi s categorical monads. However, our approach is quite different, since the kind of extension we are about to work with, requires the consideration of lambda terms no longer normalisable. Therefore, it is much like to be compared with Constable and Mendler [8] or Parigot [13]. The starting point is the thesis that recursive structures can be coded by fixpoints. So, this paper is concerned with the study of an extension of a typed framework with a fixpoint constructor, allowing us to reason with non terminating programs or other recursive program schemes. Our study, ....

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G. Huet. A Unification Algorithm for Typed Lambda Calculus. Theoretical Computer Science, 1, 1, 1975 pp. 27--57.

....program matches the source scheme S 1 , and if the hypotheses are satisfied, then the output program is the result of replacing t by the instantiation of the output scheme S 2 . The matching process has to instantiate functions, therefore this is at least a second order matching as described in [9, 10, 7]. A way to avoid this is to use combinators, such as the composition operator, to express programs. If we use the composition, identity, pro ections and product to express programs, the axioms of the algebra of programs are those of categorical products. Then the matching process becomes ....

....sel; sel ffi y = P g or fx = P a ffi ( ffi sel) sel ffi y = P Idg. These sets are selector solved forms for the match. For application to program transformation, it is sufficient to have the selector solved form, although it is weaker than the fully solved form. For the same purpose, G. Huet in [9] also introduced a weaker form of solved form called presolved forms in higher order unification processes. Let A be the axiom of the associativity of ffi. We define a selector solved form by: e o A selector solved form for P is any finite set of equations fs 1 = P t 1 ; 1 1 1 ; s n = P t n g ....

G. Huet. A Unification algorithm for typed lambda calculus. In Theoritical Computer Science, 1(1):27:57, 1973.

....dolina, 842 15 Bratislava, Slovakia. E mail: galbavy dcs.fmph.uniba.sk 1 frequently by terms, e.g. in infinite models. We can find them as substitutions produced by a unification procedure for an infinitary equational theory, e.g. for associativity [Plo72] or in higher order unification [Hue75]. There exist equational theories that are not finitely presentable, i.e. their generating set of equational axioms is infinite [McN92] If we consider theories also with other predicates than equality, we can think of theories generated by an infinite set of disequations or inequations. The ....

G. Huet. A unification algorithm for typed lambda-calculus. Theoretical Computer Science, 1(1):27--57, 1975.

....E unification, i.e. unification modulo = fij and =E . The problem of combining calculi with first order equational theories was initiated in [BT88] and the higherorder E unification problem has been already successfully studied [QW96, NQ91, Sny90] by extending the techniques developed [Hue75] for unification of simply typed terms. On the side of functional programming languages implementations, the operation of substitution (issued from fi reduction) is expensive and explicit substitutions aim at controlling this operation. The internalization of the substitution calculus was ....

G. Huet. A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1(1):27--57, 1975.

....of two terms, unification has to search for convertible instances of those terms. In other words, the conversion steps of the early proof searching methods are mixed with the unification process [And71] this is the so called higher order unification, or unification in simply typed calculus [Hue75]. A similar transformation has been applied successfully to a lot of theories: equational axioms are removed from the theory but mixed with the unification process, leading to equational unification [Plo72] Thus, higher order unification is merely equational unification for fij. But it is not ....

....for this particular theory. This algorithm can be understood as a kind of optimized narrowing. At last we show that in contrast with the reduction of higher order unification to equational unification in a combinatory language [Dou93] the unification algorithm for oe can simulate the algorithm of [Hue75], i.e. every step of this algorithm can be simulated by a sequence of steps of our algorithm. We show also that this is not the more efficient way to use the unification algorithm of oe as this simulation includes a lot of steps that functionally code and decode scoping constraints, which can in ....

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G. Huet. A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1(1):27--57, 1975.

....c. Consequently, in many Clam methods, the input pattern is a term H G, which matches any input sequent. By contrast, the Clam proof planner has a pattern language which is based on Prolog. Patterns may therefore contain higher order meta variables. A higher order unification algorithm (Huet, 1975) is used to match patterns with input sequents. This allows considerably more powerful patterns, and consequently methods in Clam are more declarative than in Clam, and the conditions c play a lesser role in determining the applicability of methods. In both Clam and Clam , the conditions c are ....

Huet, G. (1975). A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1:27--57.

....additional input. The work on general purpose unification algorithms derives from work on term rewriting (see x4) since the axioms must usually be represented as a confluent rewrite rule set. One of the major achievements of built in unification was Huet s higher order unification algorithm, Huet, 1975; Jouannaud and Kirchner, 1991 ] which builds in the ff, fi and optionally the j rules of calculus. This algorithm makes automated higher order theorem proving possible. Higher order unification is a badly behaved problem: there can be infinitely many unifiers and the problem is undecidable. ....

G. Huet. A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1:27--57, 1975.

....Here we exclude the identity and projection functions as subterms. This is essential to guarantee that there exists least generalization in the application ordering. The intuition behind this is that when we match two higher order terms, in general there are imitation rule and projection rule [11]. Here only imitation rule is used. For our purpose, it is projection rule that brings about the unpleasant results and additional complexities in higher order generalizations. Definition 8 ( S ) 8 Higher order generalization Given two terms t and s. t is more general than s by subterms ....

G.P.Huet, A unification algorithm for typed lambda calculus, Theoretical Computer Science, 1 (1975), 27-57. Higher order generalization 23

....systems. A complete method for search of proof trees based on resolution and unification was formulated by Robinson [30] for the first order logic, and by Huet [15] for the higher order logic. In type systems, higher order unification (HOU) algorithms are known for the simply typed calculus [16] and for the Pi calculus of dependent types [12, 28] For the cube type systems, Dowek [8, 9] re formulates the unification procedure and generalizes it as a method of term enumeration. Recently, Cornes [6] proposes an extension of Dowek s method to the Calculus of Constructions with Inductive ....

G. Huet. A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1(1):27--57, 1975.

....[109] or conditional equational [48] laws such as associativity, commutativity, and idempotence. Several unification algorithms (e.g. 114] 7] or see [109] for such term algebras have been presented. Kapur and Narendran [55] showed that most of these unification problems are NP hard. Huet [47,46] investigated third and higher order unification and proved that it is recursively undecidable. Goldfarb [30] showed that second order unification is also undecidable. Unification has permeated the field of resolution based and even non resolution based theorem proving [5] With the ....

G. Huet. A unification algorithm for typed lambda-calculus. Theoretical Computer Science, 1(1):27--57, 1975.

....and bind functional variables. For doing so the binder is used in higher order logic and functional programming. Unfortunately, even unification constraints become undecidable for the so called lambda terms. However, they are semi decidable, and it turns out that Huet s semi decision procedure [12] is used in most proof development systems. An important decidable subcase of terms called pattern is used in Miller s language PROLOG, a powerful extension of ordinary PROLOG. Function symbols have a fixed number of arguments. It is sometimes convenient to represent a term as a record, by using ....

G. Huet. A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1(1):27--57, 1975.

....Here we exclude the identity and projection functions as subterms. This is essential to guarantee there exists least generalization in the application ordering. The intuitive behind this is that when we match two higher order terms, in general there are imitation rule and projection rule [6]. Here only imitation rule is used. We regard it is projection rule that brings about the unpleasant results and the complexities in higher order generalizations. Definition 5 ( S ) Given two terms t and s. t is more general than s by subterms (denoted as t S s) if there exists a sequence of r ....

G.P.Huet, A unification algorithm for typed lambda calculus, Theoretical Computer Science, 1 (1975), 27-57.

....the source. Their analysis of such structures consists of: a) determining the parallel structure of source and target; b) determining which are parallel elements in source and target (e.g. Dan and George are parallel elements in the example) c) using Huet s higher order unification algorithm [8] for finding a property P such that P (s 1 ; s n ) S, where s 1 through s n are the interpretations of the parallel elements of the source, and s is the interpretation of the source itself. Only solutions which do not contain a primary occurrence of the parallel elements are considered ....

G. Huet. A unification algorithm for typed lambda-calculus. Theoretical Computer Science, 1:27--57, 1975.

....by two independent lines of research. The first was denotational semantics as carried out by Scott and others, which laid a foundation for understanding and reasoning about programs, including the equivalence of program schemata. The second was the development of higher order unification by Huet [14], originally developed for resolution proofs in higher order logic, but shortly thereafter applied to program transformation. For example, Huet and Lang in [15] combine these two strands: they use Scott style domain theory (e.g. LCF) to reason about the equivalence of program schemata formulated ....

G'erard Huet. A unification algorithm for typed lambda-calculus. Theoretical Computer Science, pages 27--57, 1975.

....something like (x:A:x) but x is a bound variable in (x:A:tu) and then it is not possible to do this substitution. We recall that the substitution operation in calculus takes care of renaming bound variables. One solution to the above problem, taken from the Higher Order Unification tradition ( Hue75] Ell89] is the functional handle of scope. With this technique, the information that a variable can indeed occur in the substitution needs to be functionally handled by the variable tu. Thus we do not consider the term (x:A:tu) but the term (x:A: tu x) where tu is of type A A. Now, to ....

G. Huet. A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1(1):27--57, 1975.

....ffl Semantic unification (see for example [53] which corresponds to solving equations in a given equational theory. ffl Sorted unification, either many sorted or order sorted [111, 112, 99, 80, 104, 53] where type constraints are added to variables in equations. ffl Higher order unification [49, 82], which corresponds to solving equations between expressions. ffl Disunification [22] which corresponds to solving not only equalities but also negated equalities. ffl Solution of equalities and inequalities in a theory, as for example the solution of numerical constraints built into the ....

G. Huet, A unification algorithm for typed lambda calculus, Theoretical Computer Science 1(1), 1973, pages 27--57.

....that respects bound variables and to ensure that all constraints that are generated have the simple rigid rigid or flex rigid form that can be solved eagerly. These simplifications enable us to implement a more efficient reconstruction algorithm. Because higher order unification is undecidable [6] and expensive in general, Miller [10] proposes syntactic restrictions so that to ensure that the only unification problems that occur can be solved by a simple extension of the first order unification, as in our case. This approach is the approach taken in the language L . Unfortunately, these ....

Huet, G. A unification algorithm for typed lambda calculus. Theoretical Computer Science 1, 1 (1973), 27--57.

....Matching and Substitution Matching routines are at the heart of several tactics such as the rewriting tactics (see Chapter 4) and the chaining tactics (see Section 3. 6) Nuprl V4.1 s matching routine is based on a second order restriction [HL78] of Huet s higher order unification algorithm [Hue75] This second order routine handles patterns with bound variables, in contrast to Nuprl V3 s first order routine which did not. The advantage of using this second order algorithm, rather than the full higher order algorithm, is that it is much more controlled; unique most general substitions (or ....

Gerard P. Huet. A unification algorithm for the typed lambdacalculus. Theoretical Computer Science, 1(1):27--58, 1975.

....variable y i . The definition of the j long form of a variable is by 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 ) ....

G. Huet. A unification algorithm for typed lambda calculus. Theoretical Computer Science, 1, 1 (1975) pp. 27--57.

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G. Huet, A unification algorithm for typed lambda calculus, Theoretical Computer Science 1(1) (1973) 27--57.

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. A unification algorithm for typed lambda-calculus. Theoretical Computer Science,

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Huet, G. 1975. "A Unification Algorithm for Typed Lambda Calculus". Theoretical Computer Science 1:27--57.

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