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Relating Higher-order and First-order Rewriting

We define a formal encoding from higher-order rewriting into first-order rewriting modulo an equational theory. 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, = o). This class includes of course the λ-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.