On structural similarities of finite automata and Turing machine enumerability classes

There are different ways of measuring the complexity of functions that map words to words. Well-known measures are time and space complexity. Enumerability is another possible measure. It is used in recursion theory, where it plays a key rôle in bounded query theory, but also in resource-bounded complexity theory, especially in connection with nonuniform computations. This dissertation transfers enumerability to automata theory. It is shown that enumerability behaves similarly in recursion theory and in automata theory, but differently in complexity theory. The enumerability of a function f is the smallestm such that there exists an m-enumerator for f . An m-enumerator is a machine that produces, for every input word w, a set of up to m possibilities for f(w). By varying the parameter m and the class of allowed enumerators, different enumerability classes can be defined. In recursion theory, one allows arbitrary Turing machines as enumerators; in automata theory, only finite automata. A deep structural result that holds both for finite automata and for Turing machine enumerability is the following cross product theorem: if f × g is (n + m)-enumerable, then either f is n-enumerable or g is m-enumerable. In contrast, this theorem does not hold for polynomial-time enumerability. Enumerability can be used to quantify the difficulty of a language A by asking how difficult it is to enumerate its n-fold characteristic function χA and cardinality function # n A. A language is (m,n)-verbose if χ n A is m-enumerable. The inclusion structures of Turing machine and of finite automata verboseness classes are identical: all (m,n)-Turing-verbose languages are (h, k)-Turing-verbose iff all (m,n)-finite-automata-verbose languages are (h, k)-finite-automata-verbose. The structure of polynomialtime verboseness classes is different. The enumerability of # A has been studied in detail in recursion theory. Kummer’s cardinality theorem states that if # A is n-enumerable by a Turing machine, then A must be recursive. Evidence is gathered that this theorem also holds for finite automata: it is shown that the nonspeedup theorem, the cardinality theorem for two words, and the restricted cardinality theorem all hold for finite automata. The cardinality theorem does not hold for polynomial-time computations. The central proofs rely on two proof techniques that promise to be applicable in other situations as well: generic proofs and branch diagonalisation. Generic proofs use elementary definitions, a concept from logic, to define enumerators in terms of other enumerators. They can be instantiated for all computational models that are closed under elementary definitions. Examples of such models are finite automata, but also Presburger arithmetic and ordinal number arithmetic. The second technique is a new diagonal-