Decremental Subset Construction

Abstract

Finite automata are exploited in several domains, including language processing, artificial intelligence, automatic control, and software engineering. Let ${\mathcal {N}}$ be a nondeterministic finite automaton (NFA) and ${\mathcal {D}}$ the deterministic finite automaton (DFA) equivalent to ${\mathcal {N}}$. Assume to decrement ${\mathcal {N}}$ by $\varDelta {\mathcal {N}}$, thus obtaining the decremented NFA ${\mathcal {N}}' = {\mathcal {N}}\setminus \varDelta {\mathcal {N}}$, where some states and/or transitions are missing. Consider determinizing ${\mathcal {N}}'$. Instead of determinizing ${\mathcal {N}}'$ from scratch using the classical Subset Construction algorithm, we propose Decremental Subset Construction, an algorithm which generates the DFA ${\mathcal {D}}'$ equivalent to ${\mathcal {N}}'$ by updating ${\mathcal {D}}$ based on ${\mathcal {N}}$ and $\varDelta {\mathcal {N}}$. This way, only the actions necessary to transform ${\mathcal {D}}$ into ${\mathcal {D}}'$ are applied. Although evidence from worst-case complexity analysis indicates that Decremental Subset Construction is not better than Subset Construction, in practice, when ${\mathcal {N}}$ is large and $\varDelta {{\mathcal {N}}}$ relatively small, Decremental Subset Construction may outperform Subset Construction significantly.