State Complexity of Unambiguous Operations on Deterministic Finite Automata

The paper determines the number of states in a deterministic finite automaton (DFA) necessary to represent “unambiguous” variants of the union, concatenation, and Kleene star operations on formal languages. For the disjoint union of languages represented by an m-state and an n-state DFA, the state complexity is \(mn-1\); for the unambiguous concatenation, it is known to be \(m2^{n-1} - 2^{n-2}\) (Daley et al. “Orthogonal concatenation: Language equations and state complexity”, J. UCS, 2010), and this paper shows that this number of states is necessary already over a binary alphabet; for the unambiguous star, the state complexity function is determined to be \(\frac{3}{8}2^n+1\). In the case of a unary alphabet, disjoint union requires up to \(\frac{1}{2}mn\) states, unambiguous concatenation has state complexity \(m+n-2\), and unambiguous star requires \(n-2\) states in the worst case.

[1]  Jeffrey Shallit,et al.  Unary Language Operations, State Complexity and Jacobsthal's Function , 2002, Int. J. Found. Comput. Sci..

[2]  Galina Jirásková,et al.  Complexity in Prefix-Free Regular Languages , 2010, DCFS.

[3]  Alexander Okhotin,et al.  State Complexity of Operations on Two-Way Deterministic Finite Automata over a Unary Alphabet , 2011, DCFS.

[4]  Michael Domaratzki,et al.  Orthogonal Concatenation: Language Equations and State Complexity , 2010, J. Univers. Comput. Sci..

[5]  Galina Jirásková,et al.  State Complexity of Intersection and Union of Suffix-Free Languages and Descriptional Complexity , 2009, NCMA.

[6]  Derick Wood,et al.  Operational State Complexity of Prefix-Free Regular Languages , 2009, Automata, Formal Languages, and Related Topics.

[7]  Galina Jirásková,et al.  Basic Operations on Binary Suffix-Free Languages , 2011, MEMICS.

[8]  Sheng Yu,et al.  The State Complexities of Some Basic Operations on Regular Languages , 1994, Theor. Comput. Sci..

[9]  Alexander Okhotin,et al.  State complexity of operations on two-way finite automata over a unary alphabet , 2012, Theor. Comput. Sci..

[10]  Alexander Okhotin.  Unambiguous finite automata over a unary alphabet , 2012, Inf. Comput..

[11]  Yo-Sub Han,et al.  State complexity of basic operations on suffix-free regular languages , 2009, Theor. Comput. Sci..

[12]  Yo-Sub Han,et al.  Nondeterministic State Complexity for Suffix-Free Regular Languages , 2010, DCFS.

[13]  Derick Wood,et al.  Nondeterministic State Complexity of Basic Operations for Prefix-Free Regular Languages , 2009, Fundam. Informaticae.

[14]  Jozef Jirásek,et al.  State complexity of concatenation and complementation , 2005, Int. J. Found. Comput. Sci..

[15]  Janusz A. Brzozowski,et al.  Complexity of suffix-free regular languages , 2015, J. Comput. Syst. Sci..

[16]  Alexander Okhotin,et al.  State Complexity of Union and Intersection for Two-way Nondeterministic Finite Automata , 2011, Fundam. Informaticae.

[17]  Alexander Okhotin,et al.  On the state complexity of operations on two-way finite automata , 2017, Inf. Comput..

[18]  Martin Kutrib,et al.  Nondeterministic Descriptional Complexity Of Regular Languages , 2003, Int. J. Found. Comput. Sci..

[19]  Jozef Jirásek,et al.  Operations on Unambiguous Finite Automata , 2016, DLT.

[20]  Jeffrey Shallit,et al.  State complexity of unique rational operations , 2009, Theor. Comput. Sci..

[21]  Alexander Okhotin,et al.  Formal Languages over GF(2) , 2018, LATA.