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An ordinal calculus for proving termination in term rewriting [chapter]

E. A. Cichon, H. Touzet
1996 Lecture Notes in Computer Science  
In this article, we are concerned with the proofs of termination of rewrite systems by the method of interpretations.  ...  As the corresponding ordinal terms are primitive recursively closed, we get a concise and intuitive proof for the closure of the class of primitive recursive functions under these schemes.  ...  Introduction Proving termination of a rewrite system means establishing the well-foundedness of the induced rewrite ordering.  ... 
doi:10.1007/3-540-61064-2_40 fatcat:eh2kog27abcedlt6uftgoesxha

Problems in rewriting III [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1995 Lecture Notes in Computer Science  
Many thanks to Stefan Blom for substantial help producing this list.  ...  An interesting system for doing arithmetic by rewriting was presented in [24] . Unfortunately, its termination has not been proved. Termination of a related system is proved in [103] .  ...  Termination of string-rewriting systems is known to be undecidable [49] . Termination of a single term-rewriting rule was proved undecidable in [32; 64] .  ... 
doi:10.1007/3-540-59200-8_82 fatcat:in5aspimbrbvngnrhzcm7tfwei

Inductive-data-type systems

Frédéric Blanqui, Jean-Pierre Jouannaud, Mitsuhiro Okada
2002 Theoretical Computer Science  
In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed lambda-calculus  ...  This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive  ...  In the sequel, we assume the reader familiar with the notions of λ-calculus and term rewriting, as presented in [4] for the simply-typed λ-calculus, [16] for term rewriting and [31, 39, 49] for the  ... 
doi:10.1016/s0304-3975(00)00347-9 fatcat:qfnumnedlzga7hccjpriygadbu

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

Frédéric Blanqui, Jean-Pierre Jouannaud, Albert Rubio
2006 Lecture Notes in Computer Science  
In computer science, termination has always been an important issue for showing programs correct.  ...  In the sixties, Tait invented a new method for showing cut elimination of natural deduction, based on a predicate over the set of terms, such that the membership of an expression to the predicate implied  ...  The above rule-schemas define a rewrite system which is known to be terminating, a result proved in Section 3.  ... 
doi:10.1007/11916277_1 fatcat:7zpraikmwvdknnvywiblbuluwa

Page 3883 of Mathematical Reviews Vol. , Issue 97F [page]

1997 Mathematical Reviews  
{For the entire collection see MR 97f:68005. } 97f:68086 68Q42 03D55 Cichon, E. A.; Touzet, H. An ordinal calculus for proving termination in term rewriting.  ...  We then prove that for typeable rewrite systems satisfying a general scheme for re- cursive definitions, every typeable term has an approximant of the same type.  ... 

More problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1993 Lecture Notes in Computer Science  
How can termination orderings for term rewriting be adapted to cover those cases in which graph rewriting is terminating although term rewriting is not? Problem 7'5 (D. Plump).  ...  An interesting system for doing arithmetic by rewriting was presented in [Cohen and Watson, 1991] . Unfortunately, its termination has not been proved. Problem 66 (F. Baader, K. Schulz [1992] ).  ... 
doi:10.1007/3-540-56868-9_39 fatcat:cibthhywyrgmvcsv73tryp3v2e

Open problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1991 Lecture Notes in Computer Science  
-J. de Vries for helping prepare this list, and Ron Book, for going out of his way t o m a k e it possible.  ...  Can strong normalization termination of the typed lambda calculus be proved by a reasonably straightforward mapping from typed terms to a well-founded ordering?  ...  egx; y Trans nite 0 -induction is required for a proof of termination. Must any termination ordering have the Howard ordinal as its order type, as conjectured in 18 ? Problem 24.  ... 
doi:10.1007/3-540-53904-2_120 fatcat:bsnbx2vghrc2tmv6xcywwijbjq

Termination of algebraic type systems: The syntactic approach [chapter]

Gilles Baxthe, Femke van Raamsdonk
1997 Lecture Notes in Computer Science  
In 14], they prove that the combination of the polymorphic -calculus {system F{ with the curried version of a terminating rst-order term rewriting system is terminating.  ...  The proof consists of extending a termination model of the term rewriting system to a termination model for simply typed -calculus.  ... 
doi:10.1007/bfb0027010 fatcat:wrtsahg6zbbkphaw6juzgojawm

Foreword

Corrado Böhm, Irène Guessarian
1999 Theoretical Computer Science  
Pinto provides a confluent and weakly normalizing rewriting system for terms representing cut-free sequent calculus derivations.  ...  Termination of Permutative Conversions in Intuitionistic Gentzen Calculi by H. Schwichtenberg proves the result stated in the title by referring to a I-calculus with explicit substitution.  ... 
doi:10.1016/s0304-3975(98)00133-9 fatcat:ea62itnvanbttcep6sq5qojv5y

Termination and Confluence of Higher-Order Rewrite Systems [chapter]

Frédéric Blanqui
2000 Lecture Notes in Computer Science  
In Section 4, we adapt the General Schema to this new calculus and prove in Section 5 that the rewrite systems that follow this schema are strongly normalizing (every reduction sequence is nite).  ...  It appears that most of the known approaches can be obtained by using an appropriate substitution calculus.  ...  Jouannaud for their suggestions and remarks on previous versions of this paper. I also thank the anonymous referees for their useful comments.  ... 
doi:10.1007/10721975_4 fatcat:yk77zf3ctjhune3mphq7hwyaxm

Zenon Modulo: When Achilles Outruns the Tortoise Using Deduction Modulo [chapter]

David Delahaye, Damien Doligez, Frédéric Gilbert, Pierre Halmagrand, Olivier Hermant
2013 Lecture Notes in Computer Science  
The theory of deduction modulo is an extension of predicate calculus, which allows us to rewrite terms as well as propositions, and which is well suited for proof search in axiomatic theories, as it turns  ...  Finally, we describe an additional backend for Zenon that outputs proof certificates for Dedukti, which is a proof checker based on the λΠ-calculus modulo. ⋆ This work is supported by the BWare project  ...  Jouvelot for their detailed comments on this paper, to G. Burel for helpful discussions, and to the Deducteam Inria research team for the many interactions.  ... 
doi:10.1007/978-3-642-45221-5_20 fatcat:oovkbtnk7vaztfia4h7cjpdreq

An on-line problem database [chapter]

Nachum Dershowitz, Ralf Treinen
1998 Lecture Notes in Computer Science  
For the current proceedings, the main subjects were Term rewriting systems Symbolic and algebraic computation Uni cation and matching Completion techniques String and graph rewriting Conditional and typed  ...  rewriting Rewriting-based theorem proving Parallel rewriting and deduction Constrained rewriting and deduction Constraint s o l v i n g Higher-order rewriting Lambda calculi Functional and logic programming  ...  Termination of a single term-rewriting rule was proved undecidable in 16,38 .  ... 
doi:10.1007/bfb0052380 fatcat:x57lyotlxrh5bdan42az6tileq

A Type-Based Termination Criterion for Dependently-Typed Higher-Order Rewrite Systems [chapter]

Frédéric Blanqui
2004 Lecture Notes in Computer Science  
We extend these works to general rewriting and dependent types, hence providing a powerful termination criterion for the combination of rewriting and β-reduction in the Calculus of Constructions.  ...  Several authors devised type-based termination criteria for ML-like languages allowing non-structural recursive calls.  ...  I thank Ralph Matthes for having invited me in Münich in February 2002, Andreas Abel for his useful comments on a previous version of this work (his technical report [2] and the discussions we had in  ... 
doi:10.1007/978-3-540-25979-4_2 fatcat:fptdfpb4rfctzm6t2tcr5thl4y

Highlights in infinitary rewriting and lambda calculus

Jörg Endrullis, Dimitri Hendriks, Jan Willem Klop
2012 Theoretical Computer Science  
We present some highlights from the emerging theory of infinitary rewriting, both for first-order term rewriting systems and λ-calculus.  ...  We present a recent observation concerning the continuity of infinitary rewriting. In the second section we present an excursion to the infinitary λ-calculus.  ...  For finite rewrite sequences σ , the argument for termination of this procedure is precisely as in [36] .  ... 
doi:10.1016/j.tcs.2012.08.018 fatcat:ulhkxwoambaj3pjimztxxlz6jm

Page 7467 of Mathematical Reviews Vol. , Issue 94m [page]

1994 Mathematical Reviews  
Also, the decidability of the existence of an equivalent ground term rewriting system and some other results are proved.”  ...  This paper considers the order type of termination orderings as they relate to the complexity of term rewriting systems: specifically, they consider order types which are ordinals.  ... 
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