On the Cascaded Decomposition of Automata its Complexity and its Application to Logic Draft

The primary decomposition theorem due to Krohn and Rhodes KR which has been considered as one of the fundamental results in the theory of automata and semigroups states that every automaton is homomorphic to a cascaded de composition wreath product of simpler automata of two kinds reset automata and permutation automata If the automaton is non counting and correspond ingly its transformation semigroup is group free then it can be decomposed using only reset components There exist various proofs and partial proofs for the primary decomposition theorem e g HS Ze a Ze b Gi MT La We Ei None of them give explicit bounds on the size of the decomposition In this paper we give tight exponential bounds on the size of the decomposition as a function of the size of the original automaton For the upper bound we give an exponential algorithm by modifying the implicit construction appearing in Ei Our algorithm is con structive enough to allow implementation We apply the algorithm to give an exponential upper bound on transforming star free regular resp regular sets expressed by counter free automata resp automata into past resp future temporal logic formulae These upper bounds improve upon previously known non elementary translations MNP LPZ Zu Some of the results in this paper has been presented in O Maler A Pnueli Tight Bounds on the Complexity of Cascaded Decomposition of Automata Proc st Annual Symposium on Foundations of Computer Science St Louis Missouri IEEE Press To quote the last paragraph in Ginzburg s book Gi Finally notice that the above theory does not indicate how many particular basic building blocks are needed to construct a cascade product covering of a given semiautomaton Our decomposition construction is proved to be optimal by showing the exis tence of a family of automata such that the size of their minimal permutation free decomposition is exponential in the size of the automaton by size we refer to the total number of de ned transitions