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State Complexity of GF(2)-Concatenation and GF(2)-Inverse on Unary Languages

Abstract : The paper investigates the state complexity of two operations on regular languages, known as GF(2)-concatenation and GF(2)-inverse (Bakinova et al., “Formal languages over GF(2)” , LATA 2018), in the case of a one-symbol alphabet. The GF(2)-concatenation is a variant of the classical concatenation obtained by replacing Boolean logic in its definition with the GF(2) field; it is proved that GF(2)-concatenation of two unary languages recognized by an m-state and an n-state DFA is recognized by a DFA with 2mn states, and this number of states is necessary in the worst case, as long as m and n are relatively prime. This operation is known to have an inverse, and the state complexity of the GF(2)-inverse operation over a unary alphabet is proved to be exactly $$2^{n-1}+1$$.
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Submitted on : Friday, November 29, 2019 - 4:35:34 PM
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Alexander Okhotin, Elizaveta Sazhneva. State Complexity of GF(2)-Concatenation and GF(2)-Inverse on Unary Languages. 21th International Conference on Descriptional Complexity of Formal Systems (DCFS), Jul 2019, Košice, Slovakia. pp.248-259, ⟨10.1007/978-3-030-23247-4_19⟩. ⟨hal-02387287⟩

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