Topological Entropy of Formal Languages

In this thesis we will introduce topological automata and the topological entropy of a topological automaton, which is the topological entropy of the dynamical system contained in the automaton. We will use these notions to define a measure of complexity for formal languages. We assign to every language the topological entropy of the unique minimal topological automaton accepting it. We contribute several new results. We use a preexisting characterization of the topological entropy of a formal language in terms of Myhill-Nerode congruence classes to compute the topological entropy of several new example languages. We determine the entropy of the Dyck languages and the deterministic palindrom language. Also we will further develop an idea from Schneider and Borchmann [5] to solve the previously open question of whether the entropy function is surjective. Furthermore, we show that all languages accepted by deterministic real-time multi-counter automata have zero entropy and all languages accepted by deterministic real-time multi-push-down automata have finite entropy, bounded in terms of the sizes of the stack alphabets of the automaton. In particular this proves that all deterministic real-time context-free languages have finite entropy. We also give an example of a deterministic context-free language with infinite entropy, proving that not all context-free languages have finite entropy. We show that there are encodings of SAT, 3COLORING, and CLIQUE such that these languages have infinite entropy.