Succinct representation, leaf languages, and projection reductions

The concepts of succinct problem representation, and of NP leaf languages, were developed to characterize complexity classes above polynomial time. Here, we work out a descriptive complexity approach to succinctly represented problems, and prove a strictly stronger version of the Conversion Lemma from Balcazar et al (1992) which allows iterated application. Moreover, we prove that for every problem Π its succinct version sΠ is complete under projection reductions for the leaf language it defines. Projection reductions are a highly restrictive reducibility notion stemming from descriptive complexity theory. Our main tool is a characterization of polynomial time Turing machines in terms of circuits which are constructed uniformly by quantifier-free formulas. Finally, we show that an alternative succinct representation model allows to obtain completeness results for all syntactic complexity classes even under monotone projection reductions. Thus, we positively answer a question by Stewart (1991, 1994) for a large number of complexity classes