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### From Higher-Order to First-Order Rewriting [chapter]

Eduardo Bonelli, Delia Kesner, Alejandro Ríos
2001 Lecture Notes in Computer Science
We show how higher-order rewriting may be encoded into rst-order rewriting modulo an equational theory E.  ...  We obtain a characterization of the class of higher-order rewriting systems which can be encoded by rst-order rewriting modulo an empty theory (that is, E = ;).  ...  Simulating Higher-Order rewriting via First-Order rewriting In order to simulate higher-order rewriting in a rst-order framework we have to deal with the conversion of valid valuations into assignments  ...

### Twenty Years Later [chapter]

Jean-Pierre Jouannaud
2005 Lecture Notes in Computer Science
coming from term rewriting.  ...  To quote a celebrated sentence that I have read in many papers: Equations are ubiquitous in computer science. This is the first reason : we all like to use equations for modeling problems.  ...  Acknowledgments: to all those I met during these 20 years and helped me understand that their problems were more important than mine.  ...

### Higher-order port-graph rewriting

Maribel Fernández, Sébastien Maulat
2012 Electronic Proceedings in Theoretical Computer Science
We illustrate the expressive power of higher-order port-graphs with examples taken from proof-net reduction systems.  ...  To facilitate the specification of proof normalisation procedures via graph rewriting, in this paper we add higher-order features to the original port-graph syntax, along with a generalised notion of graph  ...  The "hat" notation (as in " hat") is used to distinguish the higher-order functions and entities from the first-order ones.  ...

### Weak orthogonality implies confluence: The higher-order case [chapter]

Vincent Oostrom, Femke Raamsdonk
1994 Lecture Notes in Computer Science
In ~his pape~ we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known 'confluence by orthogonality' results.  ...  This corresponds to what is done in (first-order) term rewriting systems.  ...  Introduction This paper deals with higher-order rewriting. There seems to be no generally accepted definition of 'higher-order'.  ...

### Higher-order proof by consistency [chapter]

Henrik Linnestad, Christian Prehofer, Olav Lysne
1996 Lecture Notes in Computer Science
PBC may be seen as well-founded induction over an ordering which contains the rewrite relation, and in this paper we extend this method to the higher-order rewrite relation due to Nipkow.  ...  We investigate an integration of the first-order method of proof by consistency (PBC), also known as term rewriting induction, into theorem proving in higher-order specifications.  ...  In his treatment of higher-order narrowing, Loría-Sáenz [15] restricts candidate equations to quasi-first-order and rewrite rules to simple rewrite rules.  ...

### Harnessing First Order Termination Provers Using Higher Order Dependency Pairs [chapter]

Carsten Fuhs, Cynthia Kop
2011 Lecture Notes in Computer Science
Many functional programs and higher order term rewrite systems contain, besides higher order rules, also a significant first order part.  ...  We discuss how an automatic termination prover can split a rewrite system into a first order and a higher order part.  ...  While the first order dependency pairs classically still need to be regarded together with all rules from the underlying system (some of which are higher order), it may be possible to remove these higher  ...

### A transformation system combining partial evaluation with term rewriting [chapter]

Françoise Bellegarde
1994 Lecture Notes in Computer Science
Partial evaluation is used to eliminate higher-order functions. First-order rewriting is used to process the transformation.  ...  This paper presents a new approach to optimizing functional programs based on combining partial evaluation and rewriting. Programs are composed of higher-order primitives.  ...  The variable-only criteria is di erent from the restrictions given to higher-order functions that can be de ned using higher-order macros as advocated in 16]: the functional arguments of these higher-order  ...

### Higher-Order Rewriting: Framework, Confluence and Termination [chapter]

Jean-Pierre Jouannaud
2005 Lecture Notes in Computer Science
Acknowledgments: This paper would not exist without the pionneering work of Klop on higher-order rewriting and the work I have done myself with my coauthors Femke van Raamsdonk and Albert Rubio, my colleague  ...  Higher-order rewriting is a useful generalization of first-order rewriting: by rewriting higher-order functional expressions, one can process abstract syntax as done for example in program verification  ...  Theorem Higher-order rewriting. Replacing joinability by higher-order joinability does not allow us to get a similar result for higher-order rewriting [24] .  ...

### Intersection type assignment systems with higher-order algebraic rewriting

Franco Barbanera, Maribel Fernández
1996 Theoretical Computer Science
A powerful type assignment system, the one based on intersection types, is extended with first-and higherorder algebraic rewriting, and relevant properties, like strong normalization and completeness,  ...  In the present paper this subject is addressed from the point of view of fype assignment.  ...  The first author is also grateful to Tiziana Pemarella and Sandro Costantini for their steadfast encouragement. Appendix: Combining many-sorted rewrite systems with I-"  ...

### Formalizing Higher-Order Termination in Coq [article]

Deivid Vale, Niels van der Weide
2021 arXiv   pre-print
We describe a formalization of higher-order rewriting theory and formally prove that an AFS is strongly normalizing if it can be interpreted in a well-founded domain.  ...  To do so, we use Coq, which is a proof assistant based on dependent type theory.  ...  Note that Contejean et al. deal with the first order case, while CoLoR also includes some methods applicable to higher-order rewriting.  ...

### Lightweight Higher-Order Rewriting in Haskell [chapter]

Emil Axelsson, Andrea Vezzosi
2016 Lecture Notes in Computer Science
Matching is defined as an instance of Miller's higher-order pattern unification and has the same complexity as first-order matching.  ...  We present a generic Haskell library for expressing rewrite rules with a safe treatment of variables and binders.  ...  Extending the Library to Higher-Order Rewriting We will now show how to extend the first-order library from Section 2 to higherorder rewriting.  ...

### Church-Rosser Properties of Normal Rewriting

Jean-Pierre Jouannaud, Jianqi Li, Marc Herbstritt
2012 Annual Conference for Computer Science Logic
New concrete Church-Rosser results are obtained, in particular for higher-order rewriting at higher types.  ...  This is achieved by studying abstract normal rewriting in a way that allows to incorporate positions at the abstract level.  ...  Acknowledgements: to Delia Kesner, Femke van Raamsdonk and Albert Rubio for discussing actively this topic in the past.  ...

### Logic Program Synthesis in a Higher-Order Setting [chapter]

David Lacey, Julian Richardson, Alan Smail
2000 Lecture Notes in Computer Science
We describe a system for the synthesis of logic programs from specifications based on higher-order logical descriptions of appropriate refinement operations.  ...  The generality of the approach is such that its extension to allow synthesis of higher-order logic programs was straightforward. Some illustrative examples are given.  ...  different from synthesising first-order programs in the following way: -The specification of the programs involves quantification over higher-order objects.  ...

### Logical Semantics for the Rewriting Calculus

Aaron Stump, Carsten Schürmann
2005 Electronical Notes in Theoretical Computer Science
The latter is demonstrated by adding support to the Rewriting Calculus for what we call higher-form rewriting, where rules rewrite other rules.  ...  The Rewriting Calculus has been proposed as a language for defining term rewriting strategies. Rules are explicitly represented as terms, and are applied explicitly to other terms to transform them.  ...  members of the PROTHEO team for many excellent discussions on the Rewriting Calculus.  ...

### A Higher-Order Iterative Path Ordering [chapter]

Cynthia Kop, Femke van Raamsdonk
2008 Lecture Notes in Computer Science
The higher-order recursive path ordering (HORPO) defined by Jouannaud and Rubio provides a method to prove termination of higher-order rewriting.  ...  We present an iterative version of HORPO by means of an auxiliary term rewriting system, following an approach originally due to Bergstra and Klop.  ...  We are very grateful to the referees of an earlier version and to the referees of LPAR 2008 for their remarks.  ...
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