Universality of Tag Systems with P = 2

  • Authors:

  • John Cocke;Marvin Minsky

  • Affiliations:

  • International Business Machines Corp., Watson Research Center, Yorktown Heights, N.Y.;Massachusetts Institute of Technology, Computation Center and Department of Electrical Engineering, Cambridge, Mass

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1964

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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.