Unambiguous Auxiliary Pushdown Automata and Semi-unbounded Fan-in Circuits

Notions of unambiguity for uniform circuits and AuxPDAs are studied and related to each other. In particular, a coincidence for counting and unambiguous versions of AuxPDAs and semi-unbounded fan-in circuits is shown. Moreover, an improved simulation of LOGUCFL (the class of languages logspace many-one reducible to unambiguous context-free languages) by unambiguous circuits and AuxPDAs is developed. Next, an inductive counting technique on semi-unbounded fan-in circuits is presented and employed for several applications, especially an alternative proof for the closure under complementation of LOGCFL. A cost-free simulation of polynomially ambiguity bounded AuxPDAs by unambiguous ones is given. A rst nontrivial upper bound for a circuit class deened by Lange and its closure under complementation are indicated. Finally, a normal form for AuxPDAs is investigated. Inter alia it is shown that for unambiguous AuxPDAs operating in polynomial time and logarithmic space a push-down height of O(log 2 n) suuces, thus paralleling results for deterministic and nondeterministic AuxPDAs. It is pointed out that without loss of generality the underlying machines of the most important AuxPDA classes work obliviously. 1. Introduction The major aim of computational complexity theory is to decide which problems are eeciently solvable. To tackle this question, in general three cases are distinguished (Parberry, 1987): (a) EEciency in the sequential case means polynomial time computations, (b) eeciency in the parallel case with unlimited parallelism leads to complexity classes within polylogarithmic space due to the parallel computation thesis (Goldschlager, 1982). (c) eeciency in the parallel case with a limited amount of hardware, i.e., a polynomial number of processors, yields the class of problems commonly called NC. In all three cases the nondeterminism vs. determinism problem plays a decisive role for the development of eecient algorithms. In this paper we will deal with classes within the NC-hierarchy. NC is a fairly robust class. and polynomially time bounded auxiliary push-down automata (AuxPDAs) (Cook, 1971). For the question of determinism vs. nondeterminism within NC AuxPDAs are the most suitable model. An AuxPDA is a space bounded Turing machine with an additional unbounded push-down store. Since AuxPDAs are special Turing machines we immediately have deterministic and nondeterministic versions. For the moment, we only consider AuxPDAs that are simultaneously logarithmically space bounded and polynomially time bounded. These machines show strong relations to context free languages (CFLs) or, to be more precise, to their closure under log-space many-one reductions: Sudborough (1978) characterized LOGCFL by nondeterministic AuxPDAs and LOGDCFL …