Positive Versions of Polynomial Time

We show that restricting a number of characterizations of the complexity class P to be positive (in natural ways) results in the same class of (monotone) problems which we denote by posP. By a well-known result of Razborov, posP is a proper subclass of the class of monotone problems in P. We exhibit complete problems for posP via weak logical reductions, as we do for other logically defined classes of problems. Our work is a continuation of research undertaken by Grigni and Sipser, and subsequently Stewart; indeed, we introduce the notion of a positive deterministic Turing machine and consequently solve a problem posed by Grigni and Sipser.