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Journal of Logic and Computation 2005 15(6):901-947; doi:10.1093/logcom/exi050
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Vol. 15 No. 6, © The Author, 2005. Published by Oxford University Press. All rights reserved.

Original Articles

Relating Higher-order and First-order Rewriting

Eduardo Bonelli1, Delia Kesner2 and Alejandro Rios3

1 CONICET and LIFIA, Facultad de Informática, Universidad Nacional de La Plata, Argentina. Email: eduardo{at}sol.info.unlp.edu.ar, 2 PPS, CNRS and Université Paris 7, France. Email: kesner{at}pps.jussieu.fr, 3 Departamento de Computación, Universidad de Buenos Aires, Argentina. Email: rios{at}dc.uba.ar

We define a formal encoding from higher-order rewriting into first-order rewriting modulo an equational theory E. In particular, we obtain a characterization of the class of higher-order rewriting systems which can be encoded by first-order rewriting modulo an empty equational theory (that is, E = {emptyset}). This class includes of course the {lambda}-calculus. Our technique does not rely on the use of a particular substitution calculus but on an axiomatic framework of explicit substitutions capturing the notion of substitution in an abstract way. The axiomatic framework specifies the properties to be verified by a substitution calculus used in the translation. Thus, our encoding can be viewed as a parametric translation from higher-order rewriting into first-order rewriting, in which the substitution calculus is the parameter of the translation.

Keywords: Higher-order rewriting, first-order rewriting, explicit substitutions

Received May 2002.


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