Abstract
We consider a problem of hyper-minimisation of an automaton [2,3]: given a DFA M we want to compute a smallest automaton N such that the language L(M) ΔL(N) is finite, where Δ denotes the symmetric difference. We improve the previously known \(\mathcal O (|\Sigma|n^2)\) solution by giving an expected \(\mathcal O (|\delta|\log n)\) time algorithm for this problem, where |δ| is the size of the (potentially partial) transition function. We also give a slightly slower deterministic \(\mathcal O(|\delta|\log^2 n)\) version of the algorithm.
Then we introduce a similar problem of k-minimisation: for an automaton M and number k we want to find a smallest automaton N such that L(M) ΔL(N) ⊆ Σ< k, i.e. the languages they recognize differ only on words of length less than k. We characterise such minimal automata and give algorithm with a similar complexity for this problem.
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Gawrychowski, P., Jeż, A. (2009). Hyper-minimisation Made Efficient. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_31
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DOI: https://doi.org/10.1007/978-3-642-03816-7_31
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