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### Decidability of higher-order matching

2009 Logical Methods in Computer Science
We show that the higher-order matching problem is decidable using a game-theoretic argument.  ...  Acknowledgement I am indebted to Luke Ong for many extremely helpful discussions about matching.  ...  Although higher-order unification is undecidable (even if free variables are only second-order [6] ), higher-order matching was conjectured to be decidable by Huet [7] .  ...

### Introduction to Decidability of Higher-Order Matching [chapter]

Colin Stirling
2010 Lecture Notes in Computer Science
Matching is an instance of two undecidable problems, higher-order unification and higher-order β-matching.  ...  Decidability In this section we prove decidability of higher order matching; the proof essentially depends on the game theoretic characterisation of interpolation.  ...  In the play of Figure 7 on the tree of The second property appeals to sequences of positions that "correspond" when they pass through the same nodes of the interpolation tree. Definition 31.  ...

### Higher-Order Beta Matching with Solutions in Long Beta-Eta Normal Form

Kristian Støvring
2006 BRICS Report Series
<br /> <br />We show that higher-order matching up to beta-eta equivalence is decidable if and only if a restricted form of higher-order matching up to beta equivalence is decidable: the restriction is  ...  Loader has recently shown that higher-order matching up to beta equivalence is undecidable, but decidability of higher-order matching up to beta-eta equivalence is a long-standing open problem.  ...  Stirling has recently proposed a game-theoretic argument that higher-order matching is decidable [23] .  ...

### Complexity of the Higher Order Matching [chapter]

ToMasz Wierzbicki
1999 Lecture Notes in Computer Science
We use the standard encoding of Boolean values in simply typed lambda calculus to develop a method of translating SAT problems for various logics into higher order matching.  ...  the term "pattern" has multiple meanings in the theory of higher order unification, e.g., the one introduced by Dale Miller [15] , we would rather avoid it when speaking about unification of an arbitrary  ...  Moreover, the decidability of higher order matching with so called delta functions is equivalent to the decidability of lambda definability.  ...

### Linear Higher-Order Matching Is NP-Complete [chapter]

Philippe de Groote
2000 Lecture Notes in Computer Science
We consider the problem of higher-order matching restricted to the set of linear λ-terms (i.e., λ-terms where each abstraction λx. M is such that there is exactly one free occurrence of x in M ).  ...  We prove that this problem is decidable by showing that it belongs to NP. Then we prove that this problem is in fact NP-complete. Finally, we discuss some heuristics for a practical algorithm.  ...  Rusinowitch for interesting discussions about some of the material in this paper.  ...

### Twenty Years Later [chapter]

Jean-Pierre Jouannaud
2005 Lecture Notes in Computer Science
Twenty years later is the title of a novel by the french writer Alexandre Dumas.  ...  What has happened is that many research areas, such as programming languages, constraint solving, first-order provers, proof assistants, security theory, and verification have all been fertilized by ideas  ...  modulo based on pattern matching modulo, normalized rewriting, normal rewriting, higher-order rewriting based on plain pattern matching, higher-order rewriting based on higher-order pattern matching,  ...

### Improving the Efficiency of Higher Order Unification

Jared L. Darlington
1977 International Joint Conference on Artificial Intelligence
The sources of inefficiency in currently existing higher order unification algorithms are investigated.  ...  Huet 1976, Paterson and Wegman 1976 , and work referred to by them) has, with few exceptions, not been matched by a corresponding effort to improve the efficiency of higher order unification.  ...

### Second-Order Matching via Explicit Substitutions [chapter]

Flávio L. C. de Moura, Fairouz Kamareddine, Mauricio Ayala-Rincón
2005 Lecture Notes in Computer Science
The past few years have established the benefit of using explicit substitutions for theorem proving and higher-order unification.  ...  In this paper, we will make use of explicit substitutions to facilitate matching: we develop a second-order matching algorithm via the λσ-style of explicit substitutions.  ...  Third and fourth order matching are decidable [Dow94, Pad00] , but for higher orders, it remains unknown (for almost thirty years) whether this problem is decidable [Hue76] .  ...

### Decidable higher-order unification problems [chapter]

Christian Prehofer
1994 Lecture Notes in Computer Science
Second-order uni cation is undecidable in general. Miller showed that uni cation of so-called higher-order patterns is decidable and unitary.  ...  W e s h o w that the uni cation of a linear higher-order pattern s with an arbitrary second-order term that shares no variables with s is decidable and nitary.  ...  Furthermore we o p e n t h e w ay for nding decidable second-order matching problems w.r.t. higher-order equational theories.  ...

### Page 4109 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews
It is therefore promising to treat sorts in higher order theorem proving as well, in order to achieve a similar gain.  ...  In our system the partial ordering on the base sorts induces a partial ordering on the higher types by covariance in the rangesort and the natural inclusion of base function sorts.  ...

### A restricted form of higher-order rewriting applied to an HDL semantics [chapter]

Richard J. Boulton
1995 Lecture Notes in Computer Science
An algorithm for a restricted form of higher-order matching is described. The intended usage is for rewrite rules that use functionvalued variables in place of some unknown term structure.  ...  The feature of the language's semantics that necessitates the higher-order rewriting is described.  ...  Acknowledgements The author thanks David Wolfram and the anonymous referees for their constructive comments on an earlier version of this paper.  ...

### Higher-order associative commutative pattern matching for component retrieval

David Hemer
2004 Electronical Notes in Theoretical Computer Science
In this paper we describe a higher-order associative commutative pattern matching algorithm.  ...  In order to support adaptation of library components we assume the library components may contain higher-order parameters (representing types, functions and relations) -components are adapted by instantiating  ...  Acknowledgements This work was funded by Australian Research Council Discovery Grant DP0208046, Compilation of Specifications.  ...

### Linear interpolation for the higher-order matching problem [chapter]

Aleksy Schubert
1997 Lecture Notes in Computer Science
We present here a particular case of the higher order matching problem | the linear interpolation problem.  ...  We prove recursive equivalence of the higher order matching problem and the linear interpolation problem.  ...  Tiuryn for encouragement to deal with the higher-order matching problem and for discussions on the topic we had. I also thank prof. P.  ...

### A Fast Fractal Image Compression Algorithm Based on Average-Variance Function

C. ZHOU
2006 IEICE transactions on information and systems
In order to improve the efficiency and speed of match seeking in fractal compression, this paper presents an Average-Variance function which can make the optimal choice more efficiently.  ...  It can offer much higher optimal choice efficiency, higher reconstructive quality and rapid speed.  ...  It's not difficult to find the accurate ratio that match block D with small value of mcov(R) − mcov(D) can offer better match is higher than original functions.  ...

### Page 2767 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews
“In Chapter 6 we describe the problems that are involved in the extension of the formal language of Higher Order Logic with pattern matching and how these problems can be solved.  ...  First we introduce Higher Order Logic, a typed version of A-calculus, and illustrate how mathematics can be formalized in this language.  ...
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