Rational series and their languages
Rational series and their languages
SIAM Journal on Computing
On the number of distinct languages accepted by finite automata with n states
Journal of Automata, Languages and Combinatorics - Third international workshop on descriptional complexity of automata, grammars and related structures
Minimization of unary symmetric difference NFAs
SAICSIT '04 Proceedings of the 2004 annual research conference of the South African institute of computer scientists and information technologists on IT research in developing countries
On binary ⊕-NFAs and succinct descriptions of regular languages
Theoretical Computer Science - Implementation and application of automata
On the State Minimization of Nondeterministic Finite Automata
IEEE Transactions on Computers
Finite automata and their decision problems
IBM Journal of Research and Development
Decomposition of constraint automata
FACS'10 Proceedings of the 7th international conference on Formal Aspects of Component Software
Minimal DFA for symmetric difference NFA
DCFS'12 Proceedings of the 14th international conference on Descriptional Complexity of Formal Systems
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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.