Compact Normal Form for Regular Languages as Xor Automata

Abstract

The only presently known normal form for a regular language ${\mathcal{L}}\in{\mathcal{R}\mathrm{eg}}$ is its Minimal Deterministic Automaton ${\mathrm{MDA}}({\mathcal{L}})$. We show that a regular language is also characterized by a finite dimension $\dim({\mathcal{L}})$, which is always smaller than the number $|{\mathrm{MDA}}({\mathcal{L}})|$ of states, and often exponentially so. The dimension is also the minimal number of states of all Nondeterministic Xor Automaton (NXA) which accept the language. NXAs combine the advantages of deterministic automata (normal form, negation, minimization, equivalence of states, accessibility) and of nondeterministic ones (compactness, mirror language). We present an algorithmic construction of the Minimal Non Deterministic Xor Automaton ${\mathrm{MXA}}(\mathcal{L})$, in cubic time from any NXA for ${\mathcal{L}}\in{\mathcal{R}\mathrm{eg}}$. The MXA provides another normal form: ${\mathcal{L}}=\mathcal{L}^{\prime}\Leftrightarrow{\mathrm{MXA}}({\mathcal{L}})={\mathrm{MXA}}(\mathcal{L}^{\prime})$. Our algorithm establishes a missing connection between Brzozowski’s mirror-based minimization method for deterministic automata, and algorithms based on state-equivalence.