This paper is concerned with the complexity analysis of constructor term rewrite systems and its ramification in implicit computational complexity. We introduce a path order with multiset status, the polynomial path order POP ∗ , that is applicable in two related, but distinct contexts. On the one hand POP ∗ induces polynomial innermost runtime complexity and hence may serve as a syntactic, and fully automatable, method to analyse the innermost runtime complexity of term rewrite systems. On the other hand POP ∗ provides an order-theoretic characterisation of the polytime computable functions: the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP ∗ .
Matrix interpretations can be used to bound the derivational complexity of term rewrite systems. In particular, triangular matrix interpretations over the natural numbers are known to induce polynomial upper bounds on the derivational complexity of (compatible) rewrite systems. Recently two different improvements were proposed, based on the theory of weighted automata and linear algebra. In this paper we strengthen and unify these improvements by using joint spectral radius theory. In this paper, we present a variant of the dependency pair method for analysing runtime complexities of term rewrite systems automatically. This method is easy to implement, but significantly extends the analytic power of existing direct methods. Our findings extend the class of TRSs whose linear or quadratic runtime complexity can be detected automatically. We provide ample numerical data for assessing the viability of the method. In this paper we introduce a modular framework which allows to infer (feasible) upper bounds on the (derivational) complexity of term rewrite systems by combining different criteria. All current investigations to analyze the derivational complexity are based on a single termination proof, possibly preceded by transformations. We prove that the modular framework is strictly more powerful than the conventional setting. Furthermore, the results have been implemented and experiments show significant gains in power. 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.  In this paper we study context dependent interpretations, a semantic termination method extending interpretations over the natural numbers, introduced by Hofbauer. We present two subclasses of context dependent interpretations and establish tight upper bounds on the induced derivational complexities. In particular we delineate a class of interpretations that induces quadratic derivational complexity. Furthermore, we present an algorithm for mechanically proving termination of rewrite systems with context dependent interpretations. This algorithm has been implemented and we present ample numerical data for the assessment of the viability of the method.