Finding Upper Bounds for Nondeterministic State Complexity Is Hard

Finite automata are one of the oldest and most intensely investigated computational models. Historically, the theory of finite automata counts among the oldest topics in computer science, and is one of the basic building blocks of the theory of formal languages. Its origins can be tracked back to the 1940’s, when finite automata were used to model nerve nets [6]. Since its original discovery, finite automata have found applications both as a computing model and data structure in many application domains. Today, one of the most prominent applications of finite automata is text processing, e.g., in construction of compilers for programming languages. The power and usability of text processing tools based on regular languages like “sed” and “grep” for UNIX operating systems inspired the advent of the programming language “perl” with built-in capabilities for regular languages. This again prompted the decision-makers of today’s leading programming platforms to build in support for regular languages. But finite automata are also found widespread use in many other different areas such as circuit design, natural language processing, computational biology, parallel processing, image compression, to mention a few [1, 7, 8, 9]. As some of these applications deal with huge masses of data, the amount of space needed by finite automata is an important research topic. On the one hand, it is well known that while nondeterministic finite automata and deterministic finite automata are equal in expressive power, nondeterministic automata can be exponentially more succinct than deterministic ones. On the other hand, minimizing deterministic finite automata can be carried out efficiently, whereas the state minimization problem for nondeterministic finite state machines is PSPACE-complete, even if the regular language is specified as a deterministic finite automaton [5]. This prompted the authors of the aforementioned paper to ask whether there exist at least polynomial-time approximation algorithms with a reasonable performance guarantee. However, recent work [2] shows that this problem cannot be approximated within √ n polylog n for state minimization and n polylog n for transition minimization, provided some cryptographic assumption holds, namely that nonuniform logspace contains strong pseudo-random functions. As this is not a usual assumption under which inapproximability results are proved, the authors note the task of proving approximation hardness results under the weaker (and more familiar) assumption P 6= NP as an open problem. In this paper we solve this open problem. At first glance we show that minimizing nondeterministic finite automata, when given a deterministic finite state machine, is not approximable within n1− , for all > 0, for an input alphabet of size O(n), unless P = NP. The proof is based 2 H. Gruber and M. Holzer on a reduction from the biclique edge cover problem. This problem turns out to be central to our studies. Note, that recently this problem was also used in [3] to obtain lower bounds on the nondeterministic state complexity of finite automata. With a more sophisticated construction we can overcome the problem of the growing alphabet size. In this way, we prove that minimization of nondeterministic finite automata w.r.t. the number of states with binary input alphabet is not approximable within n 1 2 − , for all > 0, unless P = NP. When considering the number of transitions instead of the number of states, the non-approximation factor changes from n 1