Partial Derivatives of Regular Expressions and Finite Automaton Constructions

We introduce a notion of partial derivative of a regular expression and apply it to finite automaton constructions. The notion is a generalization of the known notion of word derivative due to Brzozowski: partial derivatives are related to non-deterministic finite automata (NFA’s) in the same natural way as derivatives are related to deterministic ones (DFA’s). We give a constructive definition of partial derivatives and prove several facts, in particular: (1) any derivative of a regular expression r can be represented by a finite set of partial derivatives of r; (2) the set of all partial derivatives of r is finite and its cardinality is less than or equal to one plus the number of occurrences of letters from d appearing in r; (3) any partial derivative of r is either a regular unit, or a subterm of r, or a concatenation of several such subterms. These theoretical results lead us to a new algorithm for turning regular expressions into relatively small NFA’s and allow us to provide certain improvements to Brzozowski’s algorithm for constructing DFA’s. We also report on a prototype implementation of OUT NFA construction and present several examples.