Alternating Automatic Register Machines

This paper introduces and studies a new model of computation called an Alternating Automatic Register Machine (AARM). An AARM possesses the basic features of a conventional register machine and an alternating Turing machine, but can carry out computations using bounded automatic relations in a single step. One finding is that an AARM can recognise some NP-complete problems, including SAT (using a particular coding), in $\log^* n + O(1)$ steps. On the other hand, if all problems in P can be solved by an AARM in $O(\log^*n)$ rounds, then $\text{P} \subset \text{PSPACE}$. Furthermore, we study an even more computationally powerful machine, called a Polynomial-Size Padded Alternating Automatic Register Machine (PAARM), which allows the input to be padded with a polynomial-size string. It is shown that the polynomial hierarchy can be characterised as the languages that are recognised by a PAARM in $\log^*n + O(1)$ steps. These results illustrate the power of alternation when combined with computations involving automatic relations, and uncover a finer gradation between known complexity classes.