Computations with Uncertain Time Constraints: Effects on Parallelism and Universality

Abstract

It is known that a parallel computer can solve problems that are impossible to be solved sequentially. That is, any general purpose sequential model of computation, such as the Turing machine or the random access machine (RAM), cannot simulate certain computations, for example solutions to real-time problems, that are carried out by a specific parallel computer. This paper extends the scope of such problems to the class of problems with uncertain time constraints. The first type of time constraints refers to uncertain time requirements on the input data, that is when and for how long are input data available. A second type of time constraints refers to uncertain deadlines for tasks. The main contribution of this paper is to exhibit computational problems in which it is very difficult to find out (read compute) what to do and when to do it. Furthermore, problems with uncertain time constraints, as described in this paper, prove once more, that it is impossible to define a universal computer, that is, a computer able to simulate all computable functions.

On Computable Numbers, Nonuniversality, and the Genuine Power of Parallelism

“…Notice that the parallel (that is, multiprocessor) computer succeeded in performing C 0 satisfactorily. This demonstrates an important and often overlooked feature of parallelism: far from being simply a faster alternative to sequential computing, it is essential for the success of certain inherently parallel computations [18,19,46,47,48,49,50,51]. The twoprocess problem is uncomputable by a sequential computer and computable by a parallel one.…”

“…The same is obviously true for any practical computer which is built once and for all; it too can only perform a finite (and fixed) number of operations per time unit. Given the nonuniversality proof, it follows that unless the unreasonable assumption is made that a computer is capable at the outset of an infinite number of operations per time unit, universality cannot be achieved by any computer [6,7,8,9,10,11,12,13,14,15,16,17,18,46,47,48,49,50,51].…”