An Implicit Characterization of the Polynomial-Time Decidable Sets by Cons-Free Rewriting

We define the class of constrained cons-free rewriting systems and show that this class characterizes P , the set of languages decidable in polynomial time on a deterministic Turing machine. The main novelty of the characterization is that it allows very liberal properties of term rewriting, in particular non-deterministic evaluation: no reduction strategy is enforced, and systems are allowed to be non-confluent. We present a class of constructor term rewriting systems that characterizes the complexity class P—the set of languages decidable in polynomial time on a deterministic Turing machine. The class is an analogue of similar classes in functional programming that use cons-freeness–the inability of a program to construct new compound data during its evaluation–to characterize a range of complexity classes, including L and P [1, 2], and for higher-order programs PSPACE and hierarchies of exponential space and time classes [3]. The primary novelty is that while previous work has crucially utilized the deterministic evaluation (in particular, call-by-value) and typing disciplines usually found in functional programming languages, we allow for the full rewriting relation to be used, and we allow non-orthogonal systems. The ability to use non-orthogonal and non-confluent systems means that we do not have access to standard results on orthogonality such as normalization or finite developments of sets of redexes, and we cannot appeal to results connecting deterministic Turing machines to confluent rewriting [4], or to functional programming without overlapping function declarations [1, 3, 5]. These are the main reasons that our proofs are substantially more difficult than similar work by Bonfante showing that introducing non-determinism to a cons-free functional language characterizes P [2].