# Algebraic theory of minimal nondeterministic finite automata with applications

#### Abstract

Since the 1950s, the theory of deterministic and nondeterministic finite automata (DFAs and NFAs, respectively) has been a cornerstone of theoretical computer science. In this dissertation, our main object of study is * minimal NFAs.* In contrast with minimal DFAs, minimal NFAs are computationally challenging: first, there can be more than one minimal NFA recognizing a given language; second, the problem of converting an NFA to a minimal equivalent NFA is NP-hard, even for NFAs over a unary alphabet. Our study is based on the development of two main theories, *inductive bases* and * partials,* which in combination form the foundation for an incremental algorithm, ibas, to find minimal NFAs. ^ An *inductive basis* is a collection of languages with the property that it can generate (through union) each of the left quotients of its elements. We prove a fundamental *characterization theorem* which says that a language can be recognized by an *n*-state NFA if and only if it can be generated by an *n*-element inductive basis. A *partial* is an incompletely-specified language. We say that an NFA recognizes a partial if its language *extends* the partial, meaning that the NFA’s behavior is unconstrained on unspecified strings; it follows that a minimal NFA for a partial is also minimal for its language. We therefore direct our attention to minimal NFAs recognizing a given partial. Combining inductive bases and partials, we generalize our characterization theorem, showing that a partial can be recognized by an *n*-state NFA if and only if it can be generated by an *n*-element partial inductive basis. ^ We apply our theory to develop and implement ibas, an incremental algorithm that finds minimal partial inductive bases generating a given partial. In the case of unary languages, ibas can often find minimal NFAs of up to 10 states in about an hour of computing time; with brute-force search this would require many trillions of years. ^

#### Subject Area

Computer Science

#### Recommended Citation

Cazalis, Daniel, "Algebraic theory of minimal nondeterministic finite automata with applications" (2007). *ProQuest ETD Collection for FIU*. AAI3299197.

http://digitalcommons.fiu.edu/dissertations/AAI3299197