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Higher Order Unification via Explicit Substitutions
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]
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
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
2001
Konpyuta sofutowea
第11回 書き換え技法と応用国際会議 (RTA2000) 参加報告
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]
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]
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]
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]
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]
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]
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
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]
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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