Analyzing Innermost Runtime Complexity of Term Rewriting by Dependency Pairs

Abstract

We present a modular framework to analyze the innermost runtime complexity of term rewrite systems automatically. Our method is based on the dependency pair framework for termination analysis. In contrast to previous work, we developed a direct adaptation of successful termination techniques from the dependency pair framework in order to use them for complexity analysis. By extensive experimental results, we demonstrate the power of our method compared to existing techniques.

Analyzing Runtime and Size Complexity of Integer Programs

“…To this end, we plan to combine the framework presented in this paper with earlier work on complexity analysis of term rewriting systems [Noschinski et al 2013], and use terms to represent inductive heap data structures (such as lists or trees).…”

“…Complexity analysis has also become a topic of intense study for term rewrite systems (TRSs), see, e.g., [Avanzini and Moser 2013;Noschinski et al 2013]. To deduce bounds on the runtime complexity of TRSs, most techniques adapt suitable techniques for termination proving (e.g., polynomial interpretations [Lankford 1979;Fuhs et al 2007], which are very similar to our PRFs).…”

“…To prove bounds on a program's runtime complexity, it is often crucial to also derive (possibly non-linear) bounds on the size of program variables, which may be modified repeatedly in loops. Our approach to find such bounds builds upon the well-known observation that polynomial ranking functions for termination proofs also provide a runtime complexity bound [Alias et al 2010;Albert et al 2011a;Albert et al 2012;Avanzini and Moser 2013;Noschinski et al 2013]. However, this only holds for proofs using a single polynomial ranking function.…”

Analyzing Program Termination and Complexity Automatically with AProVE

“…2.1 yield rewrite systems where it suffices to consider innermost rewriting in the backend. (Polynomial) upper bounds on the runtime complexity are inferred by an adaption of dependency pairs for complexity analysis [51]. To solve the resulting search problems, AProVE re-uses the techniques from termination analysis to generate suitable well-founded orders.…”

“…A graphical overview of our approach is shown below. Technical details on the techniques for transforming programs to (int-) TRSs and for analyzing rewrite systems can be found in, e.g., [10,11,12,14,17,25,26,27,28,29,30,31,32,33,34,36,37,39,40,41,43,44,51,52,57,58]. Since the current paper is a system description, we focus on the implementation of these techniques in AProVE, which we have made available as a plug-in for the popular Eclipse software development environment [23].…”