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Second-Order Matching via Explicit Substitutions [chapter]

Flávio L. C. de Moura, Fairouz Kamareddine, Mauricio Ayala-Rincón
2005 Lecture Notes in Computer Science  
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.  ...  The past few years have established the benefit of using explicit substitutions for theorem proving and higher-order unification.  ...  Second-order Matching via Explicit Substitutions The language of the λσ-calculus is a non-trivial extension of the language of the λ-calculus, and hence, the decidability of second-order matching arises  ... 
doi:10.1007/978-3-540-32275-7_29 fatcat:bfzqutroerhbtcdgys4mro6yei

Functional Programming With Higher-order Abstract Syntax and Explicit Substitutions

Brigitte Pientka
2007 Electronical Notes in Theoretical Computer Science  
This paper sketches a foundation for programming with higher-order abstract syntax and explicit substitutions based on contextual modal type theory [9] .  ...  In this paper, we extend these ideas even further by adding first-class contexts and substitutions so that a program can pass and access code with free variables and an explicit environment, and link them  ...  By structural induction on the second derivation.2 Δ box(Ψ. M) . = p −→ e Branch p matches box(Ψ.  ... 
doi:10.1016/j.entcs.2006.10.037 fatcat:dwyffkssurg7pg6fxkx56vkyh4

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  ...  Niehren, Parallelism Constraints • P. de Groote, Higher-order linear matching is NP-complete • E. Bonelli, D. Kesner, and A.  ... 
doi:10.11309/jssst.18.306 fatcat:6eaopcvv3ndk7aanv7u76laz7y

A type-theoretic foundation for programming with higher-order abstract syntax and first-class substitutions

Brigitte Pientka
2008 SIGPLAN notices  
We present a novel type-theoretic foundation based on contextual modal types which allows us to recursively analyze open terms via higher-order pattern matching.  ...  substitution and renaming.  ...  Linear higher-order pattern matching Data-level terms represent our data which we analyze via pattern matching in computations.  ... 
doi:10.1145/1328897.1328483 fatcat:f3al7mdaazhcnbv7kte5k2qsou

Programming with proofs and explicit contexts

Brigitte Pientka, Jana Dunfield
2008 Proceedings of the 10th international ACM SIGPLAN symposium on Principles and practice of declarative programming - PPDP '08  
In addition, we present an operational semantics for this language based on higher-order pattern matching for dependently typed objects. Based on this development, we prove progress and preservation.  ...  To support reasoning about open data, our foundation makes contexts explicit. This provides us with a concise characterization of open data, which is crucial to elegantly describe proofs.  ...  higher-order pattern matching [13, 17] and are explicit in the source program.  ... 
doi:10.1145/1389449.1389469 dblp:conf/ppdp/PientkaD08 fatcat:oz4q4kbldvcyzjdstkdrsv5bfa

A type-theoretic foundation for programming with higher-order abstract syntax and first-class substitutions

Brigitte Pientka
2008 Proceedings of the 35th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '08  
We present a novel type-theoretic foundation based on contextual modal types which allows us to recursively analyze open terms via higher-order pattern matching.  ...  substitution and renaming.  ...  Linear higher-order pattern matching Data-level terms represent our data which we analyze via pattern matching in computations.  ... 
doi:10.1145/1328438.1328483 dblp:conf/popl/Pientka08 fatcat:pq6lxbndpzda5jtmj6sfil42ke

A Weak Calculus with Explicit Operators for Pattern Matching and Substitution [chapter]

Julien Forest
2002 Lecture Notes in Computer Science  
In this paper we propose a Weak Lambda Calculus called λPw having explicit operators for Pattern Matching and Substitution.  ...  This formalism is able to specify functions defined by cases via pattern matching constructors as done by most modern functional programming languages such as OCAML.  ...  The calculus presented in this paper, called λP w , is a calculus with explicit pattern matching and explicit substitutions.  ... 
doi:10.1007/3-540-45610-4_13 fatcat:sl5cnzmscbdwhdeb3d77qgukqa

Unidirectional Derivation Semantics for Synchronous Tree-Adjoining Grammars [chapter]

Matthias Büchse, Andreas Maletti, Heiko Vogler
2012 Lecture Notes in Computer Science  
substitution.  ...  We prove that the unidirectional semantics coincides with the bimorphism-based semantics with the help of a strong correspondence to linear and nondeleting extended top-down tree transducers with explicit  ...  Then we match the input tree fragment to the input tree of the rule by matching variables governed by states with input rank 0 using first-order substitution, and second-order substitution otherwise.  ... 
doi:10.1007/978-3-642-31653-1_33 fatcat:fshn6xeexvhtlkaghhe34xykj4

Case Analysis of Higher-Order Data

Jana Dunfield, Brigitte Pientka
2009 Electronical Notes in Theoretical Computer Science  
We discuss coverage checking for data that is dependently typed and is defined using higher-order abstract syntax.  ...  More generally, our work is a foundation for proofs by case analysis in systems that reason about higher-order abstract syntax.  ...  We use single capital letters U, V, W for contextual variables, which are instantiated via higher-order pattern matching. We examine the second function, cntV, first.  ... 
doi:10.1016/j.entcs.2008.12.117 fatcat:locdbf3wazeylozsaerrmfgvyu

F-ing modules

Andreas Rossberg, Claudio V. Russo, Derek Dreyer
2010 Proceedings of the 5th ACM SIGPLAN workshop on Types in language design and implementation - TLDI '10  
We begin by defining a subset of the language supporting a Standard ML-like language with second-class modules and generative functors.  ...  To demonstrate the scalability of our "F-ing" semantics, we use it to define a representative, higher-order ML-style module language, encompassing all the major features of existing ML module dialects  ...  the following equivalences need to hold for a suitable choice of τ: β 1 = λα.τ α (via t) λα.β 1 α = τ (via u) Substituting the solution for τ, given by the second equation, into the first reveals that  ... 
doi:10.1145/1708016.1708028 dblp:conf/tldi/RossbergRD10 fatcat:xuxx4vbj2bc4tdzkayhinucoge

Currying Second-Order Unification Problems [chapter]

Jordi Levy, Mateu Villaret
2002 Lecture Notes in Computer Science  
We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one binary function symbol.  ...  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

F-ing modules

ANDREAS ROSSBERG, CLAUDIO RUSSO, DEREK DREYER
2014 Journal of functional programming  
We begin by defining a subset of the language supporting a Standard ML-like language with second-class modules and generative functors.  ...  To demonstrate the scalability of our "F-ing" semantics, we use it to define a representative, higher-order ML-style module language, encompassing all the major features of existing ML module dialects  ...  the following equivalences need to hold for a suitable choice of τ: β 1 = λα.τ α (via t) λα.β 1 α = τ (via u) Substituting the solution for τ, given by the second equation, into the first reveals that  ... 
doi:10.1017/s0956796814000264 fatcat:sli6rtl525dxdicsank7camdom

The evolution of traveling waves in a KPP reaction–diffusion model with cut‐off reaction rate. II. Evolution of traveling waves

Alex D. O. Tisbury, David J. Needham, Alexandra Tzella
2020 Studies in applied mathematics (Cambridge)  
On matching expansion (131) in region (as → 0 − ) with expansion (81b) in region (as → −∞), via Van Dyke's matching principle, 19 we readily obtain that = ln −∞ . (132) Thus, via (131) and (132) ,  ...  An explicit expression of * ( ) is in general not known.  ... 
doi:10.1111/sapm.12352 fatcat:hgehowv3hfbbdpeydtoox237am

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  
One is via a general modal operator 2 Ψ , and the other is via a new quantifier Π 2 u::(Ψ A 1 ). A 2 . We will briefly discuss these choices.  ...  The definitions of explicit and modal substitutions readily extend to frames, as do the corresponding substitution principles.  ... 
doi:10.1145/976571.976582 dblp:conf/icfp/NanevskiPP03 fatcat:j5avipv5angovdghjekeuetaoe

Higher-order matching for program transformation

Oege de Moor, Ganesh Sittampalam
2001 Theoretical Computer Science  
Our algorithm ÿnds more matches than the standard second order matching algorithm of Huet and Lang, but it has an equally simple speciÿcation, and it is better suited to the transformation of programs  ...  A substitution that satisÿes the above equation is said to be a match. Later on, we shall reÿne the notion of a match. Unlike ordinary ÿrst-order matching, there is no canonical choice for .  ...  Furthermore, in Section 5 it is shown that match sets include all second-order matches.  ... 
doi:10.1016/s0304-3975(00)00402-3 fatcat:4gmxtrfwwbddrc7b66anfgaguu
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