On Constructor Rewrite Systems and the Lambda-Calculus (Long Version)

TL;DR

This paper demonstrates a polynomially equivalent simulation between orthogonal constructor term rewrite systems and lambda-calculus with weak call-by-value reduction, establishing their computational complexity relationship with Turing machines.

Contribution

It proves mutual simulation with linear overhead between these systems, linking their computational complexity to Turing machine performance.

Findings

Weak call-by-value beta-reduction can be simulated by orthogonal constructor TRS in the same steps.

Each TRS reduction can be simulated by a constant number of beta-reduction steps.

Both systems are polynomially related to Turing machine complexity.

Abstract

We prove that orthogonal constructor term rewrite systems and lambda-calculus with weak (i.e., no reduction is allowed under the scope of a lambda-abstraction) call-by-value reduction can simulate each other with a linear overhead. In particular, weak call-by-value beta-reduction can be simulated by an orthogonal constructor term rewrite system in the same number of reduction steps. Conversely, each reduction in a term rewrite system can be simulated by a constant number of beta-reduction steps. This is relevant to implicit computational complexity, because the number of beta steps to normal form is polynomially related to the actual cost (that is, as performed on a Turing machine) of normalization, under weak call-by-value reduction. Orthogonal constructor term rewrite systems and lambda-calculus are thus both polynomially related to Turing machines, taking as notion of cost their natural parameters.