Closing the Gap Between Runtime Complexity and Polytime Computability

Abstract

In earlier work, we have shown
that for confluent TRSs, innermost polynomial runtime complexity
induces polytime computability of the functions defined.  
In this paper, we generalise this result to full rewriting, for that
we exploit graph rewriting. We give a new proof of the adequacy of 
graph rewriting for full rewriting that allows for
a precise control of the resources copied. In sum we
completely describe an implementation of rewriting
on a Turing machine (TM for short). We show that the runtime complexity of
the TRS and the runtime complexity of the TM is polynomially
related.
Our result strengthens the evidence that the complexity of
a rewrite system is truthfully represented through the length
of derivations. Moreover our result allows the classification of
nondeterministic polytime-computation based on runtime 
complexity analysis of rewrite systems.