The non-uniform version of Turing machines with an extra advice input tape that depends on the length of the input but not the input itself is a well-studied model in complexity theory. We investigate the same notion of non-uniformity in weaker models, namely one-way finite automata. In particular, we are interested in the power of two-sided bounded-error randomization, and how it compares to determinism and non-determinism. We show that for unlimited advice, randomization is strictly stronger than determinism, and strictly weaker than non-determinism. However, when the advice is restricted to polynomial length, the landscape changes: the expressive power of determinism and randomization does not change, but the power of non-determinism is reduced to the extent that it becomes incomparable with randomization.

The investigation of the computational power of randomized computations is one of the central tasks of current complexity and algorithm theory. In this paper for the first time a "strong" separation between the power of determinism, Las Vegas randomization, and nondeterminism for a computing model is proved. The computing models considered here are finite automata with two-dimensional input tapes (i.e., finite automata recognizing picture languages).

We introduce new techniques for deriving lower bounds on the message complexity in asynchronous distributed computation. These techniques combine the choice of specific patterns of communication delays and crossing sequence arguments with consideration of the speed of propagation of messages, together with careful counting of messages in different parts of the network. They enable us to prove the following results, settling two open problems: An Ω(n log* n) lower bound for the number of messages sent by an asynchronous algorithm for computing any nonconstant function on a bidirectional ring of n anonymous processors. An Ω(n log n) lower bound for the average number of messages sent by any maximum finding algorithm on a ring of n processors, in case n is known.

We prove the following four results on communication complexity:

It is known that k tapes are no better than two tapes for nondeterministic machines. We show here that two tapes are better than one. In fact, we show that two pushdown stores are better than one tape. Also, k tapes are no better than two for nondeterministic reversal-bounded machines; and we show that even two reversal-bounded pushdown stores are better than one reversal-bounded tape. We also show that for one-tape nondeterministic machines, unrestricted operation is better than reversal-bounded operation.

We prove the following lower bounds for on line computation.

It is known that k tapes are no better than two tapes for nondeterministic machines. We show here that two tapes are better than one. In fact, we show that two pushdown stores are better than one tape. Also, k tapes are no better than two for nondeterministic reversal-bounded machines. We show here that two tapes are better than one for such machines. In fact, we show that two reversal-bounded pushdown stores are better than one reversal-bounded tape. We also show that for one-tape nondeterministic machines, unrestricted machines are better than reversal-bounded machines.

We define a language L and show that its time and space complexities T and S must satisfy T2S ≥ cn3 even allowing machines with multiple (non random) access to the input.

We define a language L and show that is cannot be recognized by any two way deterministic counter machine. It is done by fooling any given such machine; i.e. showing that if it accepts L' ⊇ L, then L'-L ≠ φ. For this purpose, an argument stronger than the well known crossing sequence argument needs to be introduced. Since L is accepted by a two-way deterministic pushdown automaton, we consequently show that one pushdown stack is more powerful than one counter for deterministic two-way machines.