Iterated Uniform Finite-State Transducers on Unary Languages

We consider the model of an iterated uniform finite-state transducer, which executes the same length-preserving transduction in iterative sweeps. The first sweep takes place on the input string, while any subsequent sweep works on the output of the previous one. We focus on unary languages. We show that any unary regular language can be accepted by a deterministic iterated uniform finite-state transducer with at most max{2⋅ϱ,p}+1max{2⋅ϱ,p}+1 states, where ϱϱ and p are the greatest primes in the factorization of the, respectively, pre-periodic and periodic part of the language. Such a state cost cannot be improved by using nondeterminism, and it turns out to be exponentially lower in the worst case than the state costs of equivalent classical models of finite-state automata. Next, we give a characterization of classes of unary languages accepted by non-constant sweep-bounded iterated uniform finite-state transducers in terms of time bounded one-way cellular automata. This characterization enables both to exhibit interesting families of unary non-regular languages accepted by iterated uniform finite-state transducers, and to prove the undecidability of several questions related to iterated uniform finite-state transducers accepting unary languages with an amount of sweeps that is at least logarithmic.