On Union-Free and Deterministic Union-Free Languages

Jirásková, Galina; Nagy, Benedek

doi:10.1007/978-3-642-33475-7_13

Cited by 5 publications

(4 citation statements)

References 8 publications

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“…As new subregular classes of languages, we may consider the finite unions of deterministic union-free languages [8].…”

Section: N-wordsmentioning

confidence: 99%

“…Later on, in [4], their description by equations were examined, and it was shown that this class cannot be axiomatized by a finite set of equations. Automata theoretical characterisation was given in [11] allowing to define the deterministic counterpart of the class, the family of deterministic union-free languages [3,7,8].…”

Section: Introductionmentioning

confidence: 99%

“…The union-complexity of the regular languages is defined subsequently based on minimal decompositions [1,10,12]. However, there are regular languages that cannot be obtained as a finite union of deterministic union-free languages [8]. On the other hand allowing infinite unions one is able to describe every recursively enumerable language.…”

Section: Introductionmentioning

confidence: 99%

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Union-Freeness, Deterministic Union-Freeness and Union-Complexity

Nagy

2019

Descriptional Complexity of Formal Systems

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Union-free expressions are regular expressions without using the union operation. Consequently, union-free languages are described by regular expressions using only concatenation and Kleene star. The language class is also characterised by a special class of finite automata: 1CFPAs have exactly one cycle-free accepting path from each of their states. Obviously such an automaton has exactly one accepting state. The deterministic counterpart of such class of automata defines the deterministic union-free languages. A regular expression is in union (disjunctive) normal form if it is a finite union of union-free expressions. By manipulating regular expressions, each of them has equivalent expression in union normal form. By the minimum number of union-free expressions needed to describe a regular language, its union-complexity is defined. For any natural number n there are languages such that their union complexity is n. However, there is not known any simple algorithm to determine the union-complexity of any language. Regarding the deterministic union-free languages, there are regular languages such that they cannot be written as a union of finitely many deterministic union-free languages.

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“…As new subregular classes of languages, we may consider the finite unions of deterministic union-free languages [8].…”

Section: N-wordsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Union-Freeness, Deterministic Union-Freeness and Union-Complexity

Nagy

2019

Descriptional Complexity of Formal Systems

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“…It was shown in [15] that the class of languages defined by DFAs with the one-cycle-free-path property is a proper subclass of that defined by one-cycle-free-path NFAs; the former class is called the class of deterministic union-free languages. In 2012 Jirásková and Nagy [16] proved that the class of finite unions of deterministic union-free languages is a proper subclass of the class of regular languages. They also showed that every deterministic union-free language is accepted by a special kind of a one-cycle-free-path DFA called a balloon DFA.…”

Section: Introductionmentioning

confidence: 99%

Most Complex Deterministic Union-Free Regular Languages

Brzozowski

Davies

2018

Descriptional Complexity of Formal Systems

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A regular language L is union-free if it can be represented by a regular expression without the union operation. A union-free language is deterministic if it can be accepted by a deterministic one-cycle-freepath finite automaton; this is an automaton which has one final state and exactly one cycle-free path from any state to the final state. Jirásková and Masopust proved that the state complexities of the basic operations reversal, star, product, and boolean operations in deterministic unionfree languages are exactly the same as those in the class of all regular languages. To prove that the bounds are met they used five types of automata, involving eight types of transformations of the set of states of the automata. We show that for each n 3 there exists one ternary witness of state complexity n that meets the bound for reversal and product. Moreover, the restrictions of this witness to binary alphabets meet the bounds for star and boolean operations. We also show that the tight upper bounds on the state complexity of binary operations that take arguments over different alphabets are the same as those for arbitrary regular languages. Furthermore, we prove that the maximal syntactic semigroup of a union-free language has n n elements, as in the case of regular languages, and that the maximal state complexities of atoms of union-free languages are the same as those for regular languages. Finally, we prove that there exists a most complex union-free language that meets the bounds for all these complexity measures. Altogether this proves that the complexity measures above cannot distinguish union-free languages from regular languages.

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Union-Complexities of Kleene Plus Operation

Nagy

2022

Descriptional Complexity of Formal Systems

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On Union-Free and Deterministic Union-Free Languages

Cited by 5 publications

References 8 publications

Union-Freeness, Deterministic Union-Freeness and Union-Complexity

Union-Freeness, Deterministic Union-Freeness and Union-Complexity

Most Complex Deterministic Union-Free Regular Languages

Union-Complexities of Kleene Plus Operation

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