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RPO, Second-order Contexts, and Lambda-calculus

Pietro Di Gianantonio, Furio Honsell, Marina Lenisa, Roberto Amadio
2009 Logical Methods in Computer Science  
To overcome this problem, we introduce the general notion of second-order context category.  ...  Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies.  ...  The Lambda Calculus First, we recall the λ-calculus syntax together with lazy and cbv reduction strategies and observational equivalences.  ... 
doi:10.2168/lmcs-5(3:6)2009 fatcat:h5ez25kfvfffvmlfhdlgobv3hq

Page 1159 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews  
Confluence results are obtained for lambda calculus with beta, eta and omega rules, lambda cal- culi with restricted expansion rules and for (first- and higher-order) term rewriting systems for which all  ...  Recursive path orderings (RPO) and their extensions are the most widely used in theorem proving systems such as RRL, REVE, and LP.  ... 

RPO, Second-Order Contexts, and λ-Calculus [chapter]

Pietro Di Gianantonio, Furio Honsell, Marina Lenisa
2008 Lecture Notes in Computer Science  
To overcome this problem, we introduce the general notion of second-order context category.  ...  We apply Leifer-Milner RPO approach to the λ-calculus, endowed with lazy and call by value reduction strategies.  ...  In Section 5, we introduce the notion of second-order context category, and we apply the RPO approach to CL viewed as a second-order rewriting system, thus obtaining characterizations of lazy and cbv observational  ... 
doi:10.1007/978-3-540-78499-9_24 fatcat:pf7no7ubovesxag35ylphb3p5y

Strong Normalisation for a Gentzen-like Cut-Elimination Procedure [chapter]

Christian Urban
2001 Lecture Notes in Computer Science  
Inspired by work of Bloo and Geuvers concerning the λx-calculus, we shall show that a simple trick allows us to preserve this property in our cut-elimination procedure.  ...  We shall establish this property using the recursive path ordering by Dershowitz.  ...  This is similar to the situation with the lambda-calculus and λx: strong normalisation for the explicit substitution calculus does not follow directly from strong normalisation of the lambda-calculus.  ... 
doi:10.1007/3-540-45413-6_32 fatcat:nwatdcl5mbdojecblmazk7ekyu

Higher-Order Termination: From Kruskal to Computability [chapter]

Frédéric Blanqui, Jean-Pierre Jouannaud, Albert Rubio
2006 Lecture Notes in Computer Science  
Termination is a major question in both logic and computer science. In logic, termination is at the heart of proof theory where it is usually called strong normalization (of cut elimination).  ...  Later generalized by Girard under the name of reducibility or computability candidates, it showed very effective in proving the strong normalization property of typed lambda-calculi...  ...  In Section 3, we recall the computability argument for this variation of the simply typed lambda calculus. Using a computability argument again, we show in Section 4 that RPO is well-founded.  ... 
doi:10.1007/11916277_1 fatcat:7zpraikmwvdknnvywiblbuluwa

Pattern matching as cut elimination

Serenella Cerrito, Delia Kesner
2004 Theoretical Computer Science  
We present a typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen  ...  Our calculus is inspired by the Curry-Howard Isomorphism, in the sense that types, both for patterns and terms, correspond to propositions, terms correspond to proofs, and term reduction corresponds to  ...  We also wish to thank Roy Dyckho and Jean Goubault-Larrecq, for their  ... 
doi:10.1016/j.tcs.2004.03.032 fatcat:bsuozy3ar5ajbb7de3chn64xz4

A Lambda-Free Higher-Order Recursive Path Order [chapter]

Jasmin Christian Blanchette, Uwe Waldmann, Daniel Wand
2017 Lecture Notes in Computer Science  
It appears promising as the basis of a higher-order superposition calculus.  ...  We generalize the recursive path order (RPO) to higher-order terms without λ-abstraction.  ...  We are grateful to Stephan Merz, Tobias Nipkow, and Christoph Weidenbach for making this research possible; to Heiko Becker and Dmitriy Traytel for proving some theorems on the lexicographic and multiset  ... 
doi:10.1007/978-3-662-54458-7_27 fatcat:tyswf6bl3japhgppwbknvg546u

Term-graph Rewriting in Tom Using Relative Positions

Emilie Balland, Paul Brauner
2008 Electronical Notes in Theoretical Computer Science  
We note n, p the elements of Z * and ω r , ω r , . . . the elements of Rpos.  ...  Definition 2.1 (Relative position) The set Rpos of relative positions is the monoid (Z * , .) with neutral element Λ where Z * = Z \ {0}.  ...  Finally, we will illustrate the use of this extension by an implementation of lambda-calculus normalization.  ... 
doi:10.1016/j.entcs.2008.03.030 fatcat:jkkw4inc4bcivkwxuqtamqeuri

Modularity of termination using dependency pairs [chapter]

Thomas Arts, Jürgen Giesl
1998 Lecture Notes in Computer Science  
In this paper we present a re nement of this framework in order to prove termination in a modular way.  ...  The framework of dependency pairs allows automated termination and innermost termination proofs for many TRSs where such proofs were not possible before.  ...  Renaming in the Lambda Calculus (Simpli ed Variant) The following TRS is a shortened and simpli ed variant of a system for renaming in the lambda calculus.  ... 
doi:10.1007/bfb0052373 fatcat:gdfsn2a4orfibdkxzhp2g45dpm

Variable binding, symmetric monoidal closed theories, and bigraphs [article]

Richard Garner, Tom Hirschowitz
2009 arXiv   pre-print
In our setting, one first specifies an SMC theory T, which may express binding operations, in a way reminiscent from higher-order abstract syntax.  ...  We apply our approach to Jensen and Milner's (abstract binding) bigraphs, which are linear w.r.t. processes. This leads to an alternative category of bigraphs, which we compare to the original.  ...  Formulae of higher ranks yield a form of higher-order contexts.  ... 
arXiv:0905.4200v1 fatcat:us4ht2xabjc4hjksoskr3ut6ne

Formalizing the metatheory of logical calculi and automatic provers in Isabelle/HOL (invited talk)

Jasmin Christian Blanchette
2019 Proceedings of the 8th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2019  
In this paper, I describe and reflect on three verification subprojects to which I contributed: a first-order resolution prover, an imperative SAT solver, and generalized term orders for λ-free higher-order  ...  IsaFoL (Isabelle Formalization of Logic) is an undertaking that aims at developing formal theories about logics, proof systems, and automatic provers, using Isabelle/HOL.  ...  The following fragment corresponds to my first attempt at defining a lexicographic path order (LPO), a special case of RPO, on ground terms: datatype ′ c tm = F ′ c ( ′ c tm list) context fixes ≺ c ::  ... 
doi:10.1145/3293880.3294087 dblp:conf/cpp/Blanchette19 fatcat:n7delklh2nejvh4zspbcljamwy

On the Construction of Sorted Reactive Systems [chapter]

Lars Birkedal, Søren Debois, Thomas Hildebrandt
2008 Lecture Notes in Computer Science  
This construction is an improvement over previous attempts in that it produces smaller and much more natural sortings, as witnessed by our recovery of local bigraphs as a sorting.  ...  We then lift both this result and previous work on sortings for reactive systems [15, 6] to the present setting of bigraphical reactive systems.  ...  For the application of local bigraphs in [10] , encoding of the lambda calculus as a bigraphical reactive system, this extra flexibility is not exploited: For any term or reaction rule containing binders  ... 
doi:10.1007/978-3-540-85361-9_20 fatcat:fjmdmxigsbbrfdujqy7eiydjpi

A Transfinite Knuth–Bendix Order for Lambda-Free Higher-Order Terms [chapter]

Heiko Becker, Jasmin Christian Blanchette, Uwe Waldmann, Daniel Wand
2017 Lecture Notes in Computer Science  
Transfinite weights and argument coefficients can also be supported. The order appears promising as the basis of a higher-order superposition calculus.  ...  The order has many useful properties, including transitivity, the subterm property, compatibility with contexts (monotonicity), stability under substitution, and well-foundedness.  ...  Popescu and Christian Sternagel for advice with extending a partial well-founded order to a total one in the mechanized proof of Lemma 3; to Andrei Voronkov for the enlightening discussion about KBO at  ... 
doi:10.1007/978-3-319-63046-5_27 fatcat:t3y2q3czrrawzixu5ffoiyhbei

Explicit substitutions with de bruijn's levels [chapter]

Pierre Lescanne, Jocelyne Rouyer-Degli
1995 Lecture Notes in Computer Science  
Termination (or strong normalisation) of χ 1 implies this of χ and is proved by the recursive path ordering > rpo derived from the precedence C > ren > Λ > App > x.  ...  Our calculus raises interesting open issues on implementation and on higher order unification.  ... 
doi:10.1007/3-540-59200-8_65 fatcat:zvasi5oxyzddjig6l7vly3rcl4

Deriving structural labelled transitions for mobile ambients

Julian Rathke, Pawel Sobociński
2010 Information and Computation  
The key feature of this lts is that it is the fruit of ongoing work on developing a systematic procedure for deriving ltss in the structural style from the underlying reduction semantics and observability  ...  We present a new labelled transition system (lts) for the ambient calculus on which ordinary bisimilarity coincides with contextual equivalence.  ...  In the remainder of this section we describe how to analyse the skeletons in order to obtain the process-view rules and how this combines with the context-view.  ... 
doi:10.1016/j.ic.2010.06.001 fatcat:7alm4vl7hfg3pbmgzhz6cwo6f4
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