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Higher Order Unification via Explicit Substitutions

Gilles Dowek, Thérèse Hardin, Claude Kirchner
2000 Information and Computation  
Higher order uni cation is equational uni cation for -conversion. But it is not rst order equational uni cation, as substitution has to avoid capture.  ...  Such a separation and delay are provided by a calculus of explicit substitutions.  ...  In this way we reduce higher order uni cation to rst order equational uni cation in .  ... 
doi:10.1006/inco.1999.2837 fatcat:smdrvm24ybfozgesugh2y67qlm

Higher-order equational unification via explicit substitutions [chapter]

Claude Kirchner, Christophe Ringeissen
1997 Lecture Notes in Computer Science  
Kirchner for performing uni cation of simply typed -terms in a rst-order setting via the -calculus of explicit substitutions. Additional rules are used to deal with the interaction between E and .  ...  a calculus of explicit substitutions.  ...  Kirchner DHK95] have shown how to reduce the higher-order uni cation problem into a rst-order uni cation problem modulo a rst-order theory of explicit substitutions.  ... 
doi:10.1007/bfb0027003 fatcat:bsgnksilt5ed3fqruscrxvjrbm

Higher-order unification via explicit substitutions

G. Dowek, T. Hardin, C. Kirchner
Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science  
In this paper higher-order uni cation is reduced to rstorder equational uni cation in a suitable theory: the -calculus of explicit substitutions.  ...  Higher-order uni cation is equational uni cation for -conversion. But it is not rst-order equational uni cation, as substitution has to avoid capture.  ...  At last, this work suggests that higher-order logic itself should be expressed with explicit substitutions. Then, higher-order resolution would be equational resolution in this theory.  ... 
doi:10.1109/lics.1995.523271 dblp:conf/lics/DowekHK95 fatcat:x6kws5xsdzc7pfrttnlcnt77zy

Page 32 of Mathematical Reviews Vol. , Issue 2001A [page]

2001 Mathematical Reviews  
order unification via explicit substitutions.  ...  Summary: “Higher-order unification is equational unification for fn-conversion. But it is not first-order equational unification, as substitution has to avoid capture.  ... 

Report on 11th International Conference on Rewriting Techniques and Applications (RTA2000)
第11回 書き換え技法と応用国際会議 (RTA2000) 参加報告

Yoshihito Toyama
2001 Konpyuta sofutowea  
Unification via lambda-s e style explicit substitution • Stephane Vaillant, Expressing Set Theory in First-Order Predicate Logic • Mark Utting and Steve Reeves, Implementing Z C Substitutions in Ergo  ...  Rios, A de Bruijn notation for higher-order rewriting De Bruijn • C. Munoz and N. Bjorner, Absolute Explicit Unification • F.  ... 
doi:10.11309/jssst.18.306 fatcat:6eaopcvv3ndk7aanv7u76laz7y

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.  ...  Using the λσ-style of explicit substitutions has the well known advantage of reducing higher-order unification problems into equivalent first-order equational unification problems, and in this way, the  ... 
doi:10.1007/978-3-540-32275-7_29 fatcat:bfzqutroerhbtcdgys4mro6yei

The Metalanguage λprolog and Its Implementation [chapter]

Gopalan Nadathur
2001 Lecture Notes in Computer Science  
unification • Provides an explicit representation for unification problems that supports sharing • Has efficient mechanisms for realizing branching in unification Higher-Order Pattern Unification Decidability  ...  Realizing Higher-Order Unification Multiple most general unifiers may exist.  ... 
doi:10.1007/3-540-44716-4_1 fatcat:iq3a6msg3bc53f7qgtmb4dpsui

Optimizing Higher-Order Pattern Unification [chapter]

Brigitte Pientka, Frank Pfenning
2003 Lecture Notes in Computer Science  
The presented modal framework explains a number of features of the current implementation of higher-order unification in Twelf and provides insight into several optimizations.  ...  This allows us to justify optimizations to pattern unification such as linearization, which eliminates many unnecessary occurs-checks.  ...  Higher-order unification is implemented via Huet's algorithm [5] and special mechanisms are molded into the WAM instruction set to support branching and carrying unification problems.  ... 
doi:10.1007/978-3-540-45085-6_40 fatcat:t3c2cnvthrd7zfggww3kxrfsle

System Description: Twelf — A Meta-Logical Framework for Deductive Systems [chapter]

Frank Pfenning, Carsten Schürmann
1999 Lecture Notes in Computer Science  
Source and binary distributions are accessible via the Twelf home page  ...  The theorem prover generates explicit representations of the proofs it finds as higher-order judgments in LF.  ...  In contrast to earlier implementations, terms are represented in spine notation [PS98] with explicit substitutions, where we take advantage of normal forms in the λσ-calculus.  ... 
doi:10.1007/3-540-48660-7_14 fatcat:ysgd47xcbzbybccjwjaecilgke

Page 4623 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
Cvetkovié (Belgrade) 2002g:03022 03B40 68N18 Ayala-Rincon, Mauricio (BR-BRSL,; Brasilia); Kamareddine, Fairouz (4-HWAT2-CEL; Edinburgh) Unification via the As,-style of explicit substitutions.  ...  Summary: “A unification method based on the /s,-style of explicit substitution is proposed.  ... 

Currying Second-Order Unification Problems [chapter]

Jordi Levy, Mateu Villaret
2002 Lecture Notes in Computer Science  
We also discuss about the difficulties of applying the same ideas to third or higher order unification.  ...  This is roughly what it is done in explicit unification.  ...  We acknowledge Roberto Nieuwenhuis for suggesting us the use of currying in second-order unification problems. We also thank all the anonymous referees for their helpful comments.  ... 
doi:10.1007/3-540-45610-4_23 fatcat:ccuvx5vf3nfdfovij6jhsgupoe

Functional Pearl: Dependent type inference via free higher-order unification [article]

Nikolai Kudasov
2022 arXiv   pre-print
The cornerstone of dependent type inference is higher-order unification. Even though many algorithms exist, state-of-the-art is implementing it directly for a particular proof assistant.  ...  These constructions allow us to handle scopes in a language-agnostic way and provide generic higher-order unification algorithms, which serve as a foundation for dependent type inference.  ...  Second, we should make higher-order unification more efficient by optimizing the representation of free scoped monads and taking into account the types of unified terms.  ... 
arXiv:2204.05653v1 fatcat:xznkynhgnbgjjolho6o4eqqfpq

Page 2781 of Mathematical Reviews Vol. , Issue 99d [page]

1991 Mathematical Reviews  
{For the entire collection see MR 99b:68008. } 99d:68218 68T15 03B35 03B40 68N17 Kirchner, Claude; Ringeissen, Christophe Higher-order equational unification via explicit substitutions.  ...  Kirchner for perform- ing unification of simply typed A-terms in a first-order setting via the Ac-calculus of explicit substitutions.  ... 

A modal foundation for meta-variables

Aleksandar Nanevski, Brigitte Pientka, Frank Pfenning
2003 Proceedings of the 2003 workshop on Mechanized reasoning about languages with variable binding - MERLIN '03  
C(x, y(x)), but higher-order unification can also be described directly with respect to a mixed prefix.  ...  In related work we have shown [10, 8] that the presented framework provides insight into several important optimizations and implementation techniques such as higher-order pattern unification and higher-order  ... 
doi:10.1145/976571.976582 dblp:conf/icfp/NanevskiPP03 fatcat:j5avipv5angovdghjekeuetaoe

Abstract Syntax for Variable Binders: An Overview [chapter]

Dale Miller
2000 Lecture Notes in Computer Science  
, substitution, etc., are not addressed naturally by the structure of first-order terms (labeled trees).  ...  In particular, we shall focus on a representation of syntax called higher-order abstract syntax and on a more primitive version of that representation called λ-tree syntax.  ...  Both systems implemented versions of unification for simply typed λ-terms modulo α, β, η conversion (often called higher-order unification).  ... 
doi:10.1007/3-540-44957-4_16 fatcat:z56auyjhu5dxnhmx5lcvg3xpay
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