On the limits of the communication complexity technique for proving lower bounds on the size of minimal NFA’s

AbstractIn contrast to the minimization of deterministic finite automata (DFA’s), the task of constructing a minimal nondeterministic finite automaton (NFA) for a given NFA is PSPACE-complete. Moreover, there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal NFA’s.Since one is unable to efficiently estimate the size of a minimal NFA in an efficient way, one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal NFA for a given regular language. Here we consider the robust and most successful lower bound proof technique that is based on communication complexity. In this paper it is proved that even a strong generalization of this method fails for some concrete regular languages.“To fail” is considered here in a very strong sense. There is an exponential gap between the size of a minimal NFA and the achievable lower bound for a specific sequence of regular languages.The generalization of the concept of communication protocols is also strong here. It is shown that cutting the input word into 2O(n1/4) pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata. It seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head

Problems Related to Shortest Strings in Formal Languages

In formal language theory, studying shortest strings in languages, and variations thereof, can be useful since these strings can serve as small witnesses for properties of the languages, and can also provide bounds for other problems involving languages. For example, the length of the shortest string accepted by a regular language provides a lower bound on the state complexity of the language. In Chapter 1, we introduce some relevant concepts and notation used in automata and language theory, and we show some basic results concerning the connection between the length of the shortest string and the nondeterministic state complexity of a regular language. Chapter 2 examines the effect of the intersection operation on the length of the shortest string in regular languages. A tight worst-case bound is given for the length of the shortest string in the intersection of two regular languages, and loose bounds are given for two variations on the problem. Chapter 3 discusses languages that are defined over a free group instead of a free monoid. We study the length of the shortest string in a regular language that becomes the empty string in the free group, and a variety of bounds are given for different cases. Chapter 4 mentions open problems and some interesting observations that were made while studying two of the problems: finding good bounds on the length of the shortest squarefree string accepted by a deterministic finite automaton, and finding an efficient way to check if a finite set of finite words generates the free monoid. Some of the results in this thesis have appeared in work that the author has participated in \cite{AngPigRamSha,AngShallit}