Undecidability in binary tag systems and the Post correspondence problem for four pairs of words

TL;DR

This paper proves that binary tag systems with only two symbols are universal, demonstrates their polynomial-time simulation of Turing machines, and applies this to improve undecidability results for the Post correspondence problem and matrix mortality problem.

Contribution

It shows that binary tag systems are universal with only two symbols and reduces the Post correspondence problem's pairs for undecidability from 7 to 4.

Findings

Binary 2-symbol tag systems are universal.

Post correspondence problem undecidable for 4 pairs of words.

Matrix mortality problem undecidable for smaller matrix sets.

Abstract

Since Cocke and Minsky proved 2-tag systems universal, they have been extensively used to prove the universality of numerous computational models. Unfortunately, all known algorithms give universal 2-tag systems that have a large number of symbols. In this work, tag systems with only 2 symbols (the minimum possible) are proved universal via an intricate construction showing that they simulate cyclic tag systems. Our simulation algorithm has a polynomial time overhead, and thus shows that binary tag systems simulate Turing machines in polynomial time. We immediately find applications of our result. We reduce the halting problem for binary tag systems to the Post correspondence problem for 4 pairs of words. This improves on 7 pairs, the previous bound for undecidability in this problem. Following our result, only the case for 3 pairs of words remains open, as the problem is known to be decidable for 2 pairs. As a further application, we find that the matrix mortality problem is undecidable for sets with five $3\times 3$ matrices and for sets with two $15\times 15$ matrices. The previous bounds for the undecidability in this problem was seven $3\times 3$ matrices and two $ 21\times 21$ matrices.