We introduce asynchronous variants of the PC systems of pushdown automata of Csuhaj-Varju et. al. These are obtained by using a response symbol in addition to the usual query symbols. Our main result states that centralized asynchronous PC systems of pushdown automata of degree n that work in returning mode have exactly the same expressive power as n-head pushdown automata. This holds in the nondeterministic as well as in the deterministic case.

Asynchronous PC Systems of Pushdown Automata

Several types of systems of parallel communicating restarting automata are introduced and studied. The main result shows that, for all types of restarting automata X, centralized systems of restarting automata of type X have the same computational power as non-centralized systems of restarting automata of the same type and the same number of components. This result is proved by a direct simulation. In addition, it is shown that for one-way restarting automata without auxiliary symbols, systems of degree at least two are more powerful than the component automata themselves. Finally, a lower and an upper bound are given for the expressive power of systems of parallel communicating restarting automata.

Systems of Parallel Communicating Restarting Automata.

The linear speedup theorem states, informally, that constants do not matter: It is essentially always possible to find a program solving any decision problem a factor of 2 faster. This result is a classical theorem in computing, but also one of the most debated. The main ingredient of the typical proof of the linear speedup theorem is tape compression, where a fast machine is constructed with tape alphabet or number of tapes far greater than that of the original machine. In this paper, we prove that limiting Turing machines to a fixed alphabet and a fixed number of tapes rules out linear speedup. Specifically, we describe a language that can be recognized in linear time (e. g., 1:51n), and provide a proof, based on Kolmogorov complexity, that the computation cannot be sped up (e. g., below 1:49n). Without the tape and alphabet limitation, the linear speedup theorem does hold and yields machines of time complexity of the form (1+e)n for arbitrarily small e > 0. Earlier results negating linear speedup in alternative models of computation have often been based on the existence of very efficient universal machines. In the vernacular of programming language theory: These models have very efficient self-interpreters. As the second contribution of this paper, we define a class, PICSTI, of computation models that exactly captures this property, and we disprove the Linear Speedup Theorem for every model in this class, thus generalizing all similar, model-specific proofs.

/pdf/computational-models-with-no-linear-speedup-29c9bydo0n.pdf

Computational Models with No Linear Speedup

In (Csuhaj-Varju et. al. 2000) Parallel Communicating Systems of Pushdown Automata (PCPA) were introduced and shown to be able to simulate nondeterministic one-way multi-head pushdown automata in returning mode, even if communication is restricted to be one-way having a single target component. A simulation of such centralized PCPA by one-way multi-head pushdown automata (Balan 2009) turned out to be incomplete (Otto 2012). Subsequently it was shown that centralized returning PCPA are universal even if the number of components is two (Petersen 2013) and thus are separated from one-way multi-head pushdown automata. Another line of research modified the definition of PCPA such that communication is asynchronous (Otto 2013). While the simulation of one-way multi-head pushdown automata is still possible, now a converse construction shows this model in returning mode to be equivalent to the one-way multi-head pushdown automaton in a very precise sense. It was left open, whether non-centralized returning PCPA of degree two are universal. In the first part of the paper we show this to be the case.

A Note on Pushdown Automata Systems

In (Csuhaj-Varju et. al. 2000) Parallel Communicating Systems of Pushdown Automata (PCPA) were introduced and shown to be able to simulate nondeterministic one-way multi-head pushdown automata in returning mode, even if communication is restricted to be one-way having a single target component. A simulation of such centralized PCPA by one-way multi-head pushdown automata (Balan 2009) turned out to be incomplete (Otto 2012). Subsequently it was shown that centralized returning PCPA are universal even if the number of components is two (Petersen 2013) and thus are separated from one-way multi-head pushdown automata. Another line of research modified the definition of PCPA such that communication is asynchronous (Otto 2013). While the simulation of one-way multi-head pushdown automata is still possible, now a converse construction shows this model in returning mode to be equivalent to the one-way multi-head pushdown automaton in a very precise sense. It was left open, whether non-centralized returning PCPA of degree two are universal. In the first part of the paper we show this to be the case. Then we turn our attention to Uniform Distributed Pushdown Automata Systems (UDPAS). These systems of automata work in turn on a single tape. We show that UPDAS accepting with empty stack do not form a hierarchy depending on the number of components and that the membership problem is complete in NP, answering two open problems from (Arroyo et. al. 2012).

In (Csuhaj-Varju et. al. 2000) parallel communicating systems of pushdown automata (PCPA) were introduced and in their centralized variants shown to be able to simulate nondeterministic one-way multi-head pushdown automata. A claimed converse simulation for returning mode (Balan 2009) turned out to be incomplete (Otto 2012) and a language was suggested for separating these PCPA of degree two (number of pushdown automata) from nondeterministic one-way two-head pushdown automata. We show that the suggested language can be accepted by the latter computational model. We present a different decidable example over a single letter alphabet indeed ruling out the possibility of a simulation between the models. The open question about the power of centralized PCPA working in returning mode is then settled by showing them to be universal. Since the construction is possible using systems of degree two, this also improves the previous bound three for accepting all recursively enumerable languages with non-centralized systems. A similar technique can be applied to centralized PCPA working in non-returning mode improving the previous bound on the number of components to two for accepting all recursively enumerable languages. Finally PCPAs are restricted in such a way that a simulation by multi-head automata is possible.

The Power of Centralized PC Systems of Pushdown Automata

Parallel communicating systems of pushdown automata (PCPA) were introduced in (Csuhaj-Varj{\'u} et. al. 2000) and in their centralized variants shown to be able to simulate nondeterministic one-way multi-head pushdown automata. A claimed converse simulation for returning mode (Balan 2009) turned out to be incomplete (Otto 2012) and a language was suggested for separating these PCPA of degree two (number of pushdown automata) from nondeterministic one-way two-head pushdown automata. We show that the suggested language can be accepted by the latter computational model. We present a different example over a single letter alphabet indeed ruling out the possibility of a simulation between the models. The open question about the power of centralized PCPA working in returning mode is then settled by showing them to be universal. Since the construction is possible using systems of degree two, this also improves the previous bound three for generating all recursively enumerable languages. Finally PCPAs are restricted in such a way that a simulation by multi-head automata is possible.

We show that deterministic finite automata equipped with k two-way heads are equivalent to deterministic machines with a single two-way input head and k−1 linearly bounded counters if the accepted language is strictly bounded, i.e., a subset of $a_1^*a_2^*\cdots a_m^*$ for a fixed sequence of symbols a1, a2,…, am. Then we investigate linear speed-up for counter machines. Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines. For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters.

Holger Petersen

Papers

The Power of Centralized PC Systems of Pushdown Automata

The Power of Centralized PC Systems of Pushdown Automata

Bounded counter languages

A Note on Pushdown Automata Systems

A Note on Pushdown Automata Systems