D. J. Dougherty, Higher-order unification via combinators, Theoretical Computer Science, 114 (1993), pp. 273--298.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....such as narrowing. In fact we will design a more efficient algorithm, for this particular theory. This algorithm can be understood as a kind of optimised narrowing. At last we show that in contrast with the reduction of higher order unification to equational unification in a combinatory language [16], the unification algorithm for oe can simulate Huet s algorithm, since every step of this algorithm can be simulated by a sequence of steps of our algorithm. We also show that this is not the most efficient way to use the unification algorithm of oe as this simulation includes a lot of steps that ....

....and these approaches is that we consider a substitution calculus, entirely devoted to the evaluation of calculus, as part of the framework and thus we design unification for this recent equational theory. Another reduction of higher order unification in a first order framework has been given by [16] using combinatory logic and an extension of narrowing for handling extensionality. But this reduction inherits the defaults of the translation of calculus on combinatory logic: coming back to calculus is rather intricate. We hope that the new framework we propose, that allowed us to ....

D. J. Dougherty, Higher-order unification via combinators, Theoretical Computer Science, 114 (1993), pp. 273--298.

....such as narrowing. In fact we will design a more efficient algorithm, 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 ....

....with these approaches is that we consider a substitution calculus, entirely devoted to the evaluation of calculus, as part of the framework and thus we design unification for this recent equational theory. Another reduction of higher order unification in a first order framework has been given by [Dou93] using combinatory logic and an extension of narrowing for handling extensionality. But 42 this reduction inherits the defaults of the translation of calculus on combinatory logic : coming back to calculus is rather intricate. We hope that the new framework we propose, that allowed us to ....

D. J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science, 114:273--298, 1993.

....with the MAG system [10] there seem to be a handful of techniques to deal with failed matches (for instance raising a rule by introducing explicit abstractions) so we feel that the disadvantage is not too serious. There is a wealth of related work on higher order matching and unification [5, 7, 11, 12, 13, 16, 18, 24, 25], to name just a few. One important concept identified in some of these works (in particular [16, 18] is that of a restricted notion of higher order pattern. To wit, a restricted pattern is a normal term where every occurrence of a free function variable is applied to a list of distinct local ....

D. J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science, 114:273--298, 1993.

....instead resembling the rules for the combinators S and K. A calculus following the sequent calculus more closely will be the study of further work. This also leads naturally to the study of linear systems. Explicit substitution leads to a cleaner treatment of unification [8] and Dougherty [7] has used combinators as the basis for higher order unification. We believe that our calculus could be used in a similar way. Combined with an extension to dependent types, this would lead to a more formal foundation for proof development in type theory based proof assistants. Extending our ....

D. J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science, 114(2):273--298, 1993.

....with the MAG system [9] there seem to be a handful of techniques to deal with failed matches (for instance raising a rule by introducing explicit abstractions) so we feel that the disadvantage is not too serious. There is a wealth of related work on higher order matching and unification [5, 7, 11 13, 16, 18, 24, 25], to name just a few. One important concept identified in some of these works (in particular [16, 18] is that of a restricted notion of higher order pattern. To wit, a restricted pattern is a normal term where every occurrence of a free function variable is applied to a list of distinct local ....

D. J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science, 114:273--298, 1993.

.... Finally, in our opinion, another important reason for undertaking the jump to IFLP is that once the properties and behaviour of different inverted narrowing techniques are established, the step to higher order induction may be easier to bridge based on the deductive higher order counterparts [9, 27]. The work is organized as follows. In Section 2, we recall the main concepts of ILP and we formalize the narrowing semantics we focus on. Section 3 presents the general IFLP framework and gives the overall strategy for searching the program space. The search is guided by measures of program ....

....are feasible. In this way, the first steps towards Higher Order Induction are being taken by Bowers et al. 6] An intended higher order inverse narrowing first requires the choice of a proper higher order narrowing from some higher order unification methods which have been presented to date [9, 27]. Although the greater expressible power of higher order logic can make hypotheses shorter and more consilient, function invention is the problem that highlights the necessity of higher order representation languages for induction. Of course, we lose the conveniences of first order languages, ....

Daniel J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science, 114(2):273--298, 21 1993.

....to combine higher order features with first order equational reasoning (including [Breazu Tannen, 1988; Dougherty, 1991] and others) These proposals deal with the combination of lambda calculus and a first order equational theory. Recently, Dougherty and Johann ( Dougherty and Johann, 1992; Dougherty, 1993]) proposed a method for higher order reasoning by transforming lambda calculus terms to combinatory logic, that is, they use a combination of combinatory logic with an equational theory as the formulation of higher order reasoning. In [Dougherty and Johann, 1992] they also provide a complete set ....

D. J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science, Volume 114, pages 273--298, 1993.

....with the MAG system [8] there seem to be a handful of techniques to deal with failed matches (for instance raising an equation by introducing explicit abstractions) so we feel that the disadvantage is not too serious. There is a wealth of related work on higher order matching and unification [5, 9, 10, 11, 15, 16, 22], to name just a few. One important concept identified in some of these works (in particular [15, 16] is that of a restricted notion of higher order pattern. To wit, a restricted pattern is a tight term where every occurrence of a free function variable is applied to a list of distinct bound ....

D. J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science, 114:273--298, 1993.

....formal similarity between SKInT and the fans, croissants and brackets of (Gonthier et al. 1992b) SKInT has identities I m but lacks abstraction and application nodes and plugs. Sequent Combinators 39 Combinators and explicit substitution leads to a cleaner treatment of higher order unification (Dougherty, 1993; Dowek et al. 1995) We believe that our calculi can be profitably used in a similar way. Extending our approach to dependent types could also allow the incorporation of functional programming techniques (Peyton Jones, 1986) in proof checkers and lead to simpler implementations. 8. Conclusions ....

Dougherty, D. J. (1993). Higher-order unification via combinators. Theoretical Computer Science, 114(2):273--298.

....to an order sorted higher order calculus with arbitrary term declarations and functional base sorts. Although for automated deduction purposes higher order logic is typically presented in terms of the lambda calculus, recent successes in developing combinatory logic based unification algorithms ([Dou93], DJ92] Vit92] suggest that this alternate, algebraic formulation of higher order logic can provide a computational framework for the mechanization of pure higher order logic and its more expressive extensions. Transporting to a combinatory logic setting the features of the lambda calculi ....

D. J. Dougherty. Higher-order Unification via Combinators. Theoretical Computer Science 114, pp. 273 -- 298, 1993.

....a difficulty: even when E is empty, the equality generated by the basic CL equations is not that induced by the translation from LC to CL. Furthermore, no convergent term rewriting system is known for this induced equality. This difficulty is solved here generalizing the techniques of [Dou93] by defining a certain notion of reduction on systems of CL terms. When E has a convergent presentation, this reduction captures the induced equality described above and supports the syntactic unification strategy of narrowing (and so yields a solution to the unification problem in its usual ....

....is a type. Our dual substitutions behave in most ways just as ordinary substitutions. A substitution is idempotent iff both 0 and 1 are idempotent, for example, and if two (LC or CL ) terms are syntactically unifiable they possess a most general unifier, or mgu. Details are worked out in [Dou93]. An instance of the higher order E unification problem is a two element multiset hM; N i of LC terms; a solution is a substitution such that M = fijE N . Suppose W is a set of type variables and indeterminates. Whenever = is a notion of equality on terms the notation = 0 [W] means ....

[Article contains additional citation context not shown here]

D. J. Dougherty. Higher-order unification via combinators. Theoretical Computer Science B 114, pp. 273--298, 1993.

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