The Expressive Power of Transitive Closue and 2-way Multihead Automata

It is known, that over 1-dimensional strings, the expressive power of 2-way multihead (non) deterministic automata and (non) deterministic Transitive Closure formulas is (non) deterministic log space [Ib73, Im88]. However, the subset of formulas needed to simulate exactly k heads is unknown. It is also unknown if the automata and formulas have the same expressive power over more general structures such as multidimensional grids. We define a reduction from k-head automata to formulas of arity k, which works also for grids. The method used is a generalization of [K156], and the formulas obtained are a generalization of regular expressions to multihead automata and to grid languages. As simple applications, we use the reduction to show that the power of formulas of arity 1 over strings define (classical) regular languages, to give a simpler equivalent of the L=NL open problem, and to establish the equivalence of the automata and formulas over grids.