John Cocke
Marvin Minsky
Abstract
By a simple direct construction it is shown 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.
This 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 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 nonwriting Turing machines. The proof here avoids these otherwise interesting machines and strengthens the main result; obtaining the theorem with a best possible deletion number P = 2.
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.
- 1 MINsKY, M. lecursive unsolvability of Post's probleln of Tag and other topics in theory of Turig moehirms. Ann. Math. 7g, 3 (Nov. 1961), 437-455.Google Scholar
- 2 For further results along these lines, see: WANG, HAO. Tag systems and Lag systems. To apper.Google Scholar
- 3 MINgKY, M, Size and sgrtwture of universal Turing machines using Tag systems: a 4-symbol 7-share machine. In Proc. Symposium a Recursive FuncZion Theory, Am. Math. See., Providence, R, I., 1962.Google Scholar
- 4 SHHEDSON, J. C., AD STUaIs, H. E. Coputatiopnaly of recursie fmctions. J. ACM, 10 (Apr. 1963) 217-255. Google Scholar
- 5 WANG, HAo. A variant of Turing's theory of computing machines. J. ACM 4 (Apr. 1957), 63. Google Scholar
Index Terms
- Universality of Tag Systems with P = 2
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