scholarly journals Operational complexity and right linear grammars

2021 ◽  
Vol 58 (4) ◽  
pp. 281-299
Author(s):  
Jürgen Dassow
Keyword(s):  
Regular Language ◽  
Regular Languages ◽  
Natural Numbers

AbstractFor a regular language L, let $${{\,\mathrm{Var}\,}}(L)$$ Var ( L ) be the minimal number of nonterminals necessary to generate L by right linear grammars. Moreover, for natural numbers $$k_1,k_2,\ldots ,k_n$$ k 1 , k 2 , … , k n and an n-ary regularity preserving operation f, let $$g_f^{{{\,\mathrm{Var}\,}}}(k_1,k_2,\ldots ,k_n)$$ g f Var ( k 1 , k 2 , … , k n ) be the set of all numbers k such that there are regular languages $$L_1,L_2,\ldots , L_n$$ L 1 , L 2 , … , L n such that $${{\,\mathrm{Var}\,}}(L_i)=k_i$$ Var ( L i ) = k i for $$1\le i\le n$$ 1 ≤ i ≤ n and $${{\,\mathrm{Var}\,}}(f(L_1,L_2,\ldots , L_n))=k$$ Var ( f ( L 1 , L 2 , … , L n ) ) = k . We completely determine the sets $$g_f^{{{\,\mathrm{Var}\,}}}$$ g f Var for the operations reversal, Kleene-closures $$+$$ + and $$*$$ ∗ , and union; and we give partial results for product and intersection.

Further Remarks on the Operational Nonterminal Complexity

2021 ◽  
pp. 1-15
Author(s):  
Jürgen Dassow
Keyword(s):  
Regular Language ◽  
Regular Languages ◽  
Partial Result ◽  
Natural Numbers ◽  
Left Quotient

For a regular language [Formula: see text], let [Formula: see text] be the minimal number of nonterminals necessary to generate [Formula: see text] by right linear grammars. Moreover, for natural numbers [Formula: see text] and an [Formula: see text]-ary regularity preserving operation [Formula: see text], let the range [Formula: see text] be the set of all numbers [Formula: see text] such that there are regular languages [Formula: see text] with [Formula: see text] for [Formula: see text] and [Formula: see text]. We show that, for the circular shift operation [Formula: see text], [Formula: see text] is infinite for all [Formula: see text], and we completely determine the set [Formula: see text]. Moreover, we give a precise range for the left right quotient and a partial result for the left quotient. Furthermore, we add some values to the range for the operation intersection which improves the result of [2].

CONJUNCTIVE GRAMMARS GENERATE NON-REGULAR UNARY LANGUAGES

2008 ◽  
Vol 19 (03) ◽  
pp. 597-615 ◽  
Author(s):  
ARTUR JEŻ
Keyword(s):  
Negative Answer ◽  
The Body ◽  
Regular Languages ◽  
Natural Numbers ◽  
Open Problems ◽  
Conjunctive Grammars ◽  
Unary Languages ◽  
Context Free ◽  
Conjunctive Grammar ◽  
Context Free Grammars

Conjunctive grammars, introduced by Okhotin, extend context-free grammars by an additional operation of intersection in the body of any production of the grammar. Several theorems and algorithms for context-free grammars generalize to the conjunctive case. Okhotin posed nine open problems concerning those grammars. One of them was a question, whether a conjunctive grammars over a unary alphabet generate only regular languages. We give a negative answer, contrary to the conjectured positive one, by constructing a conjunctive grammar for the language {a4n : n ∈ ℕ}. We also generalize this result: for every set of natural numbers L we show that {an : n ∈ L} is a conjunctive unary language, whenever the set of representations in base-k system of elements of L is regular, for arbitrary k.

Bounded Prefix-Suffix Duplication: Language Theoretic and Algorithmic Results

2015 ◽  
Vol 26 (07) ◽  
pp. 933-952 ◽  
Author(s):  
Marius Dumitran ◽  
Javier Gil ◽  
Florin Manea ◽  
Victor Mitrana
Keyword(s):  
Efficient Algorithm ◽  
Regular Language ◽  
Regular Languages ◽  
Sufficient Condition ◽  
Membership Problem

We consider a restricted variant of the prefix-suffix duplication operation, called bounded prefix-suffix duplication. It consists in the iterative duplication of a length-bounded prefix or suffix of a given word. We give a sufficient condition for the closure of a class of languages under bounded prefix-suffix duplication. Consequently, we get that the class of regular languages is closed under bounded prefix-suffix duplication; furthermore, we propose an algorithm which decides whether a regular language is a finite k-prefix-suffix duplication language. An efficient algorithm solving the membership problem for the k-prefix-suffix duplication of a language is also presented. Finally, we define the k-prefix-suffix duplication distance between two words, extend it to languages and show how it can be computed for regular languages.

INFIX-FREE REGULAR EXPRESSIONS AND LANGUAGES

2006 ◽  
Vol 17 (02) ◽  
pp. 379-393 ◽  
Author(s):  
YO-SUB HAN ◽  
YAJUN WANG ◽  
DERICK WOOD
Keyword(s):  
Polynomial Time ◽  
Regular Language ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Regular Languages ◽  
Finite State Automaton ◽  
Primality Test ◽  
Finite State ◽  
State Pair ◽  
Free Decomposition

We study infix-free regular languages. We observe the structural properties of finite-state automata for infix-free languages and develop a polynomial-time algorithm to determine infix-freeness of a regular language using state-pair graphs. We consider two cases: 1) A language is specified by a nondeterministic finite-state automaton and 2) a language is specified by a regular expression. Furthermore, we examine the prime infix-free decomposition of infix-free regular languages and design an algorithm for the infix-free primality test of an infix-free regular language. Moreover, we show that we can compute the prime infix-free decomposition in polynomial time. We also demonstrate that the prime infix-free decomposition is not unique.

scholarly journals ON PANSIOT WORDS AVOIDING 3-REPETITIONS

2012 ◽  
Vol 23 (08) ◽  
pp. 1583-1594
Author(s):  
IRINA A. GORBUNOVA ◽  
ARSENY M. SHUR
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Regular Language ◽  
Numerical Estimate ◽  
Regular Languages ◽  
Finite Alphabet

The recently confirmed Dejean's conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with k ≥ 5 letters, Pansiot words avoiding 3-repetitions form a regular language, which is a rather small superset of the set of all threshold words. Using cylindric and 2-dimensional words, we prove that, as k approaches infinity, the growth rates of complexity for these regular languages tend to the growth rate of complexity of some ternary 2-dimensional language. The numerical estimate of this growth rate is ≈1.2421.

Roots and Powers in Regular Languages: Recognizing Nonregular Properties by Finite Automata

2020 ◽  
Vol 175 (1-4) ◽  
pp. 173-185
Author(s):  
Fabian Frei ◽  
Juraj Hromkovič ◽  
Juhani Karhumäki
Keyword(s):  
Lower Bound ◽  
Regular Language ◽  
Finite Automata ◽  
Regular Languages ◽  
Fractional Powers ◽  
Concrete Construction ◽  
Square Roots ◽  
Language L ◽  
Interesting Behavior ◽  
Fractional Length

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L, the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

scholarly journals Decidability of regular language genus computation

2019 ◽  
Vol 29 (9) ◽  
pp. 1428-1443
Author(s):  
Guillaume Bonfante ◽  
Florian L. Deloup
Keyword(s):  
Regular Language ◽  
New Genus ◽  
Regular Languages ◽  
Minimal Size ◽  
Deterministic Automaton ◽  
Letter Alphabet ◽  
New Family ◽  
Generic Class ◽  
Deterministic Automata ◽  
The Difference

AbstractThis article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.

scholarly journals QUOTIENT COMPLEXITY OF STAR-FREE LANGUAGES

2012 ◽  
Vol 23 (06) ◽  
pp. 1261-1276 ◽  
Author(s):  
JANUSZ BRZOZOWSKI ◽  
BO LIU
Keyword(s):  
Lower Bound ◽  
Regular Language ◽  
Regular Languages ◽  
Symmetric Difference ◽  
Boolean Operations ◽  
State Complexity

The quotient complexity, also known as state complexity, of a regular language is the number of distinct left quotients of the language. The quotient complexity of an operation is the maximal quotient complexity of the language resulting from the operation, as a function of the quotient complexities of the operands. The class of star-free languages is the smallest class containing the finite languages and closed under boolean operations and concatenation. We prove that the tight bounds on the quotient complexities of union, intersection, difference, symmetric difference, concatenation and star for star-free languages are the same as those for regular languages, with some small exceptions, whereas 2n-1 is a lower bound for reversal.

scholarly journals On the Commutative Equivalence of Algebraic Formal Series and Languages

2021 ◽  
pp. 1-27
Author(s):  
Arturo Carpi ◽  
Flavio D’Alessandro
Keyword(s):  
Exponential Growth ◽  
Regular Language ◽  
Formal Series ◽  
Regular Languages ◽  
Context Free Language ◽  
Context Free ◽  
Free Language

The problem of the commutative equivalence of context-free and regular languages is studied. Conditions ensuring that a context-free language of exponential growth is commutatively equivalent with a regular language are investigated.

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