T. Nipkow. Higher Order Critical Pairs. In Sixth Annual Symposium on Logic in Computer Science, pages 342--349. IEEE, July 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the interest in higher order systems such as higher order logics and higherorder deduction systems [And86,Pau94,GLM97,And01,Pfe01] higher order (functional) programming languages [BMS80,Tur85,Pau91,Bar90,Bir98] higherorder logic programming languages [Mil91,HKMN95] higher order rewriting [Nip91,Klo92,DJ90] and higher order uni cation [Hue75,Dow01] It is well known that second order uni cation hence higher order uni cation is undecidable ( Gol81,Far91,LV00a] Higher order uni cation procedures were already described by in [Hue75,JP76] The undecidability results triggered research into ....

Tobias Nipkow. Higher-order critical pairs. In Proc. 6th IEEE Symp. LICS, pages 342-349, 1991.

....other, internally bound variables. Substitution of general values is not part of the equality theory in the # term syntax approach: it must be coded as 4 a separate judgment via logic. This weaker approach notion of equality gives rise to L # unification (also, higher order pattern unification) [Mil91a,Nip91], which is decidable and unary. The relationship between the # tree approach where abstractions are applied to only internally bound variable and HOAS where abstractions can be applied to general terms is rather similar to the distinctions made in the # calculus between # I , which only allows ....

Tobias Nipkow. Higher-order critical pairs. In G. Kahn, editor, LICS91, pages 342--349. IEEE, July 1991.

.... for functional programming and some connections can be drawn also from our work to the system F of [3] and to dynamic types proposed in [1] Concerning the operationalisation of deduction, we have a purely first order mechanism and avoid higher order rewriting or unification as developed in [28] as well as combinatory reduction systems [16] In contrast, in the theorem proving domain, HOL [8] and Isabelle [31] are based on type theory but provide means to reason in set theory. Relation to object oriented languages: Through their use of sets, both for types and for higher order ....

T. Nipkow. Higher-order critical pairs. In Proc. 6th IEEE symposium on Logic in Computer Science (LICS), pages 342--349. IEEE Computer Society Press, Los Alamitos, July 1991.

....can actually be seen as a particular case of the latter, resulting in a uniform formalism with a strong rewriting flavor. In the sequel, we assume the reader familiar with the notions of calculus and term rewriting, as presented in [4] for the simply typed calculus, 16] for term rewriting and [31,39,49] for the several variants of higher order rewriting existing in the literature. We first introduce the term language before to move on with the definition of higher order rewrite rules and of the new formulation of the General Schema. 2.1 The language In this subsection, we introduce ....

....if u is an instance of l by some substitution . Matching here is syntactic, that is, u is ff convertible to the instance of l. In contrast, the more sophisticated notions of higher order rewriting defined by Klop (Combinatory Reduction Systems [30,31] Nipkow (Higher order Rewrite Systems [39,34]) and van Raamsdonk and van Oostrom (Higher Order Rewriting Systems [49,50] generalizing both) are based on higher order pattern matching, that is, u must be fijff convertible to the instance of l. Definition 6 (Rewrite rules and rewriting) A rewrite rule is a pair l r of terms such that: 1) ....

[Article contains additional citation context not shown here]

T. Nipkow. Higher-order critical pairs. In Proc. of the 6th Symp. on Logic in Computer Science, IEEE Computer Society, 1991.

....term is an application of the symbol s to n arguments and replaces it by its argument list. ISABST is the corresponding instruction for abstractions. Matching fails if the current term does not have the required form. NEXT is used to select the next argument. In Higher Order Rewrite Systems [13, 6], an ISABST instruction would not have to check anything, as the type system already guarantees that the current term is an abstraction. In a certain sense, ISABST checks for the presence of the symbol introduced by the translation from CRSs into HRSs, see [6, 15] exec (CHECK vs) let fun ....

Tobias Nipkow. Higher order critical pairs. LICS'91, pages 342--349.

....Okada [10] introduced the Algebraic Functional Language allowing to define higher order functional constants by rewrite rules which follows a general schema. Also here modularity results are given, in particular with respect to termination aspects. Another approach is given in the work of Nipkow [16] where it is presented a rewriting relation modulo fi and j conversion. The decidability of validating an equation between higher order terms E mail: Olav.Lysne ifi.uio.no. On leave from Department of informatics, University of Oslo, P.O. box 1080 Blindern, 0316 Oslo, Norway. Supported by the ....

....on the set of fi normal forms: s R t , 9(l r) 2 R; p 2 Pos(s) oe : sj p = ff loe t = ff s[roe] p # A sequence of rewrite steps is written t 1 R t 2 R t 3 : and consists only of terms in fi normal forms. It has been shown that local confluence of this rewrite relation is decidable [16], and that given convergence this relation may be used to decide ff[fi [j[R equivalence. 3 RPO for higher order rewriting As a first approach to a termination ordering for higher order terms we present a way to interpret fi normal higher order terms as first order ones. These first order terms ....

[Article contains additional citation context not shown here]

T. Nipkow. Higher order critical pairs. In Proceedings 6th IEEE Symposium on Logic in Computer Science, pages 342--349, 1991.

....role in the operational semantics. Besides uni cation other problems such as generalization or complement also arise frequently. In this paper we are concerned with the problem of pattern complement in a setting where patterns may contain binding operators, so called higher order patterns [Mil91, Nip91]. Higher order patterns have found applications in logic programming [Mil91, Pfe91a] logical frameworks [DPS97] term rewriting [Nip93] and functional logic programming [HP96] Higher order patterns inherit many pleasant properties from the rst order case. In particular, most general uni ers ....

Tobias Nipkow. Higher-order critical pairs. In G. Kahn, editor, Sixth Annual IEEE Symposium on Logic in Computer Science, pages 342-349, Amsterdam, The Netherlands, July 1991.

....Combinatory Rewrite Systems (CRSs) 15, 16] to generalize both first order term rewriting and rewrite systems with bound variables like Church s calculus. In 1991, after Miller s decidability result of the pattern unification problem [20] Nipkow introduced Higher order Rewrite Systems (HRSs) [23] (called Pattern Rewrite Systems (PRSs) in [18] to investigate the metatheory of logic programming languages and theorem provers like Prolog [21] or Isabelle [25] In particular, he extended to the higher order case the decidability result of Knuth and Bendix about local confluence of ....

T. Nipkow. Higher-order critical pairs. In Proc. of LICS'91, IEEE Computer Society.

.... the interest in higher order systems such as higher order logics and higherorder deduction systems [And86,Pau94,GLM97,And01,Pfe01] higher order (functional) programming languages [BMS80,Tur85,Pau91,Bar90,Bir98] higherorder logic programming languages [Mil91,HKMN95] higher order rewriting [Nip91,Klo92,DJ90] and higher order uni cation [Hue75,Dow01] It is well known that second order uni cation hence higher order uni cation is undecidable ( Gol81,Far91,LV00a] Higher order uni cation procedures were already described by in [Hue75,JP76] The undecidability results triggered research into ....

Tobias Nipkow. Higher-order critical pairs. In Proc. 6th IEEE Symp. LICS, pages 342-349, 1991.

....a proof search algorithm for first order logic only, one is naturally led into this fragment of higher order logic, where the algorithm works as well. In doing this we rely heavily on Miller s [1] who has introduced this type of restriction to higher order terms (called patterns by Nipkow [2]) noted its relevance for extensions of logic programming, and showed that the unification problem for patterns is solvable and admits most general unifiers. The present paper was motivated by the desire to use Miller s approach as a basis for an implementation of a simple proof search engine for ....

....for an implementation of a simple proof search engine for (first and higher order) minimal logic. This goal prompted us into several simplifications, optimizations and extensions, in particular the following. ffl Instead of arbitrarily mixed prefixes we only use those of the form 898. Nipkow in [2] already had presented a version of Miller s pattern unification algorithm for such prefixes, and Miller in [1, Section 9.2] notes that in such a situation any two unifiers can be transformed into each other by a variable renaming substitution. Here we restrict ourselves to 898 prefixes ....

Tobias Nipkow. Higher-order critical pairs. In R. Vemuri, editor, Proceedings of the Sixth Annual IEEE Symposium on Logic in Computer Science, pages 342--349, Los Alamitos, 1991. IEEE Computer Society Press. 12

No context found.

T. Nipkow. Higher Order Critical Pairs. In Sixth Annual Symposium on Logic in Computer Science, pages 342--349. IEEE, July 1991.

No context found.

Tobias Nipkow. Higher-order critical pairs. In Sixth Annual Symposium on Logic in Computer Science. IEEE, July 1991. To appear.

No context found.

Tobias Nipkow. Higher-order critical pairs. In G. Kahn, editor, Sixth Annual IEEE Symposium on Logic in Computer Science, pages 342--349, Amsterdam, The Netherlands, July 1991.

No context found.

T. Nipkow. Higher-order critical pairs. In Proc. of the 6th IEEE Symp. on Logic in Computer Science, 1991. Extended version in [89].

No context found.

T. Nipkow. Higher-order critical pairs. In Proc. of the 6th IEEE Symp. on Logic in Computer Science, 1991.

No context found.

Tobias Nipkow. Higher-order critical pairs. In Kahn [Kah91], pages 342-349.

No context found.

Tobias Nipkow. Higher order critical pairs. In Proc. IEEE Symp. on Logic in Comp. Science, Amsterdam, 1991.

No context found.

Tobias Nipkow. Higher-order critical pairs. In Proc. 6th IEEE Symp. Logic in Computer Science, 1991.

No context found.

T. Nipkow. Higher-order critical pairs. In Proceedings of the sixth annual IEEE Symposium on Logic in Computer Science, pages 342--349. IEEE Computer Society Press, 1991.

No context found.

Tobias Nipkow. Higher-order critical pairs. In Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, pages 342--349, Amsterdam, The Netherlands, 15--18 July 1991. IEEE Computer Society Press.

No context found.

Nipkow, Tobias. (1991). Higher-order critical pairs. In Proceedings of the 6th IEEE Symposium on Logic in Computer Science, pages 342--349. IEEE Computer Society Press.

No context found.

Tobias Nipkow. Higher-order critical pairs. In G. Kahn, editor, LICS91, pages 342--349. IEEE, July 1991.

No context found.

Tobias Nipkow. Higher-order critical pairs. In Proc. 6th IEEE Symp. Logic in Computer Science, pages 342--349, 1991.

No context found.

T. Nipkow. Higher order critical pairs. In Proc. IEEE Symp. on Logic in Comp. Science, Amsterdam, 1991.

No context found.

T. Nipkow. Higher-order critical pairs. In Sixth Annual Symposium on Logic in Computer Science. IEEE, 1991.

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