Abstract not found

Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically negative. Our results will be formulated in the language of membrane computing (P systems). Some mathematical results presented here are interesting in themselves. In contrast with most speed-up methods which are based on non-determinism, our results rest upon some universality results proved for deterministic P systems. These results will be used for building "accelerated P systems". In contrast with the case of Turing machines, acceleration is a part of the hardware (not a quality of the environment) and it is realised either by decreasing the size of "reactors" or by speeding-up the communication channels.

The aim of this paper is to provide a probabilistic, but non-quantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random N-bit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k> 0, we can effectively compute a time bound T such that the probability that an N-bit program will eventually halt given that it has not halted by T is smaller than 2 −k. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that “long ” runtimes are effectively rare. More formally, the set of times at which an N-bit program can stop after the time 2 N+ constant has effectively zero density.

In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of (very) simple programs. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied.

Abstract. Cristian Calude et al. in [5] have recently introduced the idea of measuring the degree of difficulty of a mathematical problem (even those still given as conjectures) by the length of a program to verify or refute the statement. The method to evaluate and compare problems, in some objective way, will be discussed in this note. For the practitioner, wishing to apply this method using a standard universal register machine language, we provide (for the first time) some “small ” core subroutines or library for dealing with array data structures. These can be used to ease the development of full programs to check mathematical problems that require more than a predetermined finite number of variables. 1