We propose a new order-theoretic characterisation of the class of polytime computable functions. To this avail we define the small polynomial path order (sPOP * for short). This termination order entails a new syntactic method to analyse the innermost runtime complexity of term rewrite systems fully automatically: for any rewrite system compatible with sPOP… (More)
Bellantoni and Cook have given a function-algebra characterization of the polynomial-time computable functions via an unbounded recursion scheme which is called safe recursion. Inspired by their work, we characterize the exponential-time computable functions with the use of a safe variant of nested recursion.
In this paper we present a new path order for rewrite systems, the exponential path order EPO. Suppose a term rewrite system is compatible with EPO , then the runtime complexity of this rewrite system is bounded from above by an exponential function. Furthermore, the class of function computed by a rewrite system compatible with EPO equals the class of… (More)
The predicative lexicographic path order (PLPO for short), a syntactic restriction of the lex-icographic path order, is presented. As well as lexicographic path orders, several non-trivial primitive recursive equations, e.g., primitive recursion with parameter substitution, unnested multiple recursion, or simple nested recursion, can be oriented with PLPOs.… (More)
In this paper we present a new termination proof and complexity analysis of unfolding graph rewriting which is a specific kind of infinite graph rewriting expressing the general form of safe recursion. We introduce a termination order over sequences of terms together with an interpretation of term graphs into sequences of terms. Unfolding graph rewrite… (More)
It is known that (i) programs can be executed in polynomial space if they are compatible with lexicographic path orders (LPOs) and admit polynomial quasi-interpretations (PQIs), and (ii) LPO-termination proofs can be formalized in the Σ 0 2-induction fragment of Peano arithmetic. We show that LPO-termination proofs can be formalized in the second order… (More)
The general form of safe recursion (or ramified recurrence) can be expressed by an infinite graph rewrite system including unfolding graph rewrite rules introduced by Dal Lago, Martini and Zorzi, in which the size of every normal form by innermost rewriting is polynomially bounded. Every unfolding graph rewrite rule is precedence terminating in the sense of… (More)
In this paper we present a novel termination order the pred-icative lexicographic path order (PLPO for short), a syntactic restriction of the lexicographic path order. As well as lexicographic path orders, several non-trivial primitive recursive equations, e.g., primitive recur-sion with parameter substitution, unnested multiple recursion, or simple nested… (More)
Adopting former term rewriting characterisations of poly-time and exponential-time computable functions, we introduce a new reduction order, the Path Order for ETIME (POE * for short), that is sound and complete for ETIME computable functions. The proposed reduction order for ETIME makes contrasts to those related complexity classes clear.