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In the following sections we show, by a simple direct construction, that computations done by Turing machines can be duplicated by a very simple symbol manipulation process. The process is described by a simple form of Post Canonical system with some very strong restrictions. First, the system is monogenic; each formula (string of symbols) of the system can be affected by one and only one production (rule of inference) to yield a unique result. Accordingly, if we begin with a single axiom (initial string) the system generates a simply ordered sequence of formulas, and this operation of a monogenic system brings to mind the idea of a machine. The Post canonical system is further restricted to be of the "Tag" variety, described briefly below. It was shown in [1] that Tag systems are equivalent to Turing machines. The proof in [1] is very complicated and uses lemmas concerned with a variety of two-tape non-writing Turing machines. Our proof here avoids these otherwise interesting machines and strengthens the main result, obtaining the theorem with a best possible "deletion number" P ?? Also, the representation of the Turing machine in the present system has a lower degree of exponentiation, which may be of significance in applications. These systems seem to be of value in establishing unsolvability of combinatorial problems. |
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