Generalized Computational Systems

Abstract

The definition of a computational system that I proposed in chapter 1 (definition 3) employs the concept of Turing computability. In this chapter, however, I will show that this concept is not absolute, but instead depends on the relational structure of the support on which Turing machines operate. Ordinary Turing machines operate on a linear tape divided into a countably infinite number of adjacent squares. But one can also think of Turing machines that operate on different supports. For example, we can let a Turing machine work on an infinite checkerboard or, more generally, on some n-dimensional infinite array. I call an arbitrary support on which a Turing machine can operate a pattern field. Depending on the pattern field F we choose, we in fact obtain different concepts of computability. At the end of this chapter (section 6), I will thus propose a new definition of a computational system (a computational system on pattern field F) that takes into account the relativity of the concept of Turing computability. If F is a doubly infinite tape, however, computational systems on F reduce to computational systems. Turing (1965) presented his machines as an idealization of a human being that transforms symbols by means of a specified set of rules. Turing based his analysis on four hypotheses: 1. The capacity to recognize, transform, and memorize symbols and rules is finite. It thus follows that any transformation of a complex symbol must always be reduced to a series of simpler transformations. These operations on elementary symbols are of three types: recognizing a symbol, replacing a symbol, and shifting the attention to a symbol that is contiguous to the symbol which has been considered earlier. 2. The series of elementary operations that are in fact executed is determined by three factors: first, the subject’s mental state at a given time; second, the symbol which the subject considers at that time; third, a rule chosen from a finite number of alternatives.