The Size of Power Automata K. Sutner Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 sutner@cs.cmu.edu, www.cs.cmu.edu/~sutner Abstract We describe a class of simple transitive semiautomata that exhibit full expo- nential blow-up during deterministic simulation. For arbitrary semiautomata we show that it is PSPACE-complete to decide whether the size of the accessible part of their power automata exceeds a given bound. 1 Motivation Consider the following semiautomaton A = 〈[n], Σ,δ〉 where [n]= {1,...,n},Σ= {a, b, c} and the transition function is given by δ a a cyclic shift on [n], δ b the transposition that interchanges 1 and 2, δ c sends 1 and 2 to 2, identity elsewhere. It is well-known that A has a transition semigroup of maximal size n n , see [13]. In other words, every function f :[n] → [n] is already of the form δ w for some word w. Note that δ a ,δ b can be replaced by any other pair of generators for the symmetric group on n points, and δ c can be replaced by any function whose range has cardinality n − 1. It was shown by Salomaa that, for a three-letter alphabet Σ, those are the only choices that produce a maximal transition semigroup, see [15, 16]. If we think of the transition function as operating on sets of states, it follows that for all A, B ⊆ [n] such that |A|≥|B|≥ 1, there is a word w such that δ w (A)= B. Now suppose we reverse all transitions in A. In rev(A), for any B ⊆ [n], there is a word w such that δ w ({1})= B. Indeed, if we augment the semiautomaton rev(A) 1