scholarly journals COMPLEXITY OF ATOMS OF REGULAR LANGUAGES

2013 ◽  
Vol 24 (07) ◽  
pp. 1009-1027 ◽  
Author(s):  
JANUSZ BRZOZOWSKI ◽  
HELLIS TAMM
Keyword(s):  
Upper Bound ◽  
Regular Language ◽  
Regular Languages ◽  
State Complexity ◽  
Empty Intersection ◽  
Language L ◽  
The Right

The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs of a given language, i.e., those NFAs that have no equivalent states. We prove that, for any language L with quotient complexity n, the quotient complexity of any atom of L with r complemented quotients has an upper bound of 2n − 1 if r = 0 or r = n; for 1 ≤ r ≤ n − 1 the bound is[Formula: see text] For each n ≥ 2, we exhibit a language with 2n atoms which meet these bounds.

STATE COMPLEXITY OF ADDITIVE WEIGHTED FINITE AUTOMATA

2007 ◽  
Vol 18 (06) ◽  
pp. 1407-1416 ◽  
Author(s):  
KAI SALOMAA ◽  
PAUL SCHOFIELD
Keyword(s):  
Upper Bound ◽  
Finite Automaton ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Weighted Finite Automata

It is known that the neighborhood of a regular language with respect to an additive distance is regular. We introduce an additive weighted finite automaton model that provides a conceptually simple way to reprove this result. We consider the state complexity of converting additive weighted finite automata to deterministic finite automata. As our main result we establish a tight upper bound for the state complexity of the conversion.

State Complexity of Insertion

2016 ◽  
Vol 27 (07) ◽  
pp. 863-878 ◽  
Author(s):  
Yo-Sub Han ◽  
Sang-Ki Ko ◽  
Timothy Ng ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Regular Language ◽  
Finite Automata ◽  
The State ◽  
Tight Bound ◽  
State Complexity ◽  
Nondeterministic State

It is well known that the resulting language obtained by inserting a regular language to a regular language is regular. We study the nondeterministic and deterministic state complexity of the insertion operation. Given two incomplete DFAs of sizes m and n, we give an upper bound (m+2)·2mn−m−1·3m and find a lower bound for an asymp-totically tight bound. We also present the tight nondeterministic state complexity by a fooling set technique. The deterministic state complexity of insertion is 2Θ(mn) and the nondeterministic state complexity of insertion is precisely mn+2m, where m and n are the size of input finite automata. We also consider the state complexity of insertion in the case where the inserted language is bifix-free or non-returning.

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 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.

State Complexity of k-Union and k-Intersection for Prefix-Free Regular Languages

2015 ◽  
Vol 26 (02) ◽  
pp. 211-227 ◽  
Author(s):  
Hae-Sung Eom ◽  
Yo-Sub Han ◽  
Kai Salomaa
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Finite Automata ◽  
Constant Factor ◽  
The State ◽  
Regular Languages ◽  
Deterministic Finite Automata ◽  
State Complexity ◽  
Binary Alphabet ◽  
Multiple Intersections

We investigate the state complexity of multiple unions and of multiple intersections for prefix-free regular languages. Prefix-free deterministic finite automata have their own unique structural properties that are crucial for obtaining state complexity upper bounds that are improved from those for general regular languages. We present a tight lower bound construction for k-union using an alphabet of size k + 1 and for k-intersection using a binary alphabet. We prove that the state complexity upper bound for k-union cannot be reached by languages over an alphabet with less than k symbols. We also give a lower bound construction for k-union using a binary alphabet that is within a constant factor of the upper bound.

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

2020 ◽  
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.

NON-UNIQUENESS AND RADIUS OF CYCLIC UNARY NFAs

2005 ◽  
Vol 16 (05) ◽  
pp. 883-896 ◽  
Author(s):  
MICHAEL DOMARATZKI ◽  
KEITH ELLUL ◽  
JEFFREY SHALLIT ◽  
MING-WEI WANG
Keyword(s):  
Diophantine Equation ◽  
Regular Language ◽  
Regular Languages ◽  
Language L ◽  
Minimal Radius ◽  
Unary Languages

In this paper we study some properties of cyclic unary regular languages. We find a connection between the uniqueness of the minimal NFA for certain cyclic unary regular languages and a Diophantine equation studied by Sylvester. We also obtain some results on the radius of unary languages. We show that the nondeterministic radius of a cyclic unary regular language L is not necessarily obtained by any of the minimal NFAs for L. We give a class of examples which demonstrates that the nondeterministic radius of a regular language cannot necessarily even be approximated by the minimal radius of its minimal NFAs.

POWERS OF REGULAR LANGUAGES

2011 ◽  
Vol 22 (02) ◽  
pp. 323-330 ◽  
Author(s):  
SZILÁRD ZSOLT FAZEKAS
Keyword(s):  
Regular Language ◽  
Regular Languages ◽  
Language L ◽  
Unary Languages ◽  
Primitive Roots ◽  
Exponent Sets ◽  
Kleene Closure

In this paper we prove that it is decidable whether the set pow (L), which we get by taking all the powers of all the words in some regular language L, is regular or not. The problem was originally posed by Calbrix and Nivat in 1995. Partial solutions have been given by Cachat for unary languages and by Horváth et al. for various kinds of exponent sets for the powers and regular languages which have primitive roots satisfying certain properties. We show that the regular languages which have a regular power are the ones which are 'almost' equal to their Kleene-closure.

The complexity of concatenation on deterministic and alternating finite automata

2018 ◽  
Vol 52 (2-3-4) ◽  
pp. 153-168
Author(s):  
Michal Hospodár ◽  
Galina Jirásková
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Finite Automata ◽  
The State ◽  
Regular Languages ◽  
Final States ◽  
State Complexity ◽  
Deterministic Automata

We study the state complexity of the concatenation operation on regular languages represented by deterministic and alternating finite automata. For deterministic automata, we show that the upper bound m2n − k2n−1 on the state complexity of concatenation can be met by ternary languages, the first of which is accepted by an m-state DFA with k final states, and the second one by an n-state DFA with ℓ final states for arbitrary integers m, n, k, ℓ with 1 ≤ k ≤ m − 1 and 1 ≤ ℓ ≤ n − 1. In the case of k ≤ m − 2, we are able to provide appropriate binary witnesses. In the case of k = m − 1 and ℓ ≥ 2, we provide a lower bound which is smaller than the upper bound just by one. We use our binary witnesses for concatenation on deterministic automata to describe binary languages meeting the upper bound 2m + n + 1 for the concatenation on alternating finite automata. This solves an open problem stated by Fellah et al. [Int. J. Comput. Math. 35 (1990) 117–132].

Recognizing Lexicographically Smallest Words and Computing Successors in Regular Languages

2021 ◽  
pp. 1-22
Author(s):  
Lukas Fleischer ◽  
Jeffrey Shallit
Keyword(s):  
Regular Language ◽  
Formal Language ◽  
Equivalence Classes ◽  
Regular Languages ◽  
State Complexity ◽  
Finite State ◽  
Finite State Transducer ◽  
Enumeration Problem ◽  
Necessary And Sufficient ◽  
Input Formula

For a formal language [Formula: see text], the problem of language enumeration asks to compute the length-lexicographically smallest word in [Formula: see text] larger than a given input [Formula: see text] (henceforth called the [Formula: see text]-successor of [Formula: see text]). We investigate this problem for regular languages from a computational complexity and state complexity perspective. We first show that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are (in general) necessary and sufficient for an unambiguous finite-state transducer to compute [Formula: see text]-successors. As a byproduct, we obtain that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are sufficient for a DFA to recognize the subset [Formula: see text] of [Formula: see text] composed of its lexicographically smallest words. We give a matching lower bound that holds even if [Formula: see text] is represented as an NFA. It has been known that [Formula: see text]-successors can be computed in polynomial time, even if the regular language is given as part of the input (assuming a suitable representation of the language, such as a DFA). In this paper, we refine this result in multiple directions. We show that if the regular language is given as part of the input and encoded as a DFA, the problem is in [Formula: see text]. If the regular language [Formula: see text] is fixed, we prove that the enumeration problem of the language is reducible to deciding membership to the Myhill-Nerode equivalence classes of [Formula: see text] under [Formula: see text]-uniform [Formula: see text] reductions. In particular, this implies that fixed star-free languages can be enumerated in [Formula: see text], arbitrary fixed regular languages can be enumerated in [Formula: see text] and that there exist regular languages for which the problem is [Formula: see text]-complete.

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