ON SUBWORD SYMMETRY OF WORDS

2008 ◽  
Vol 19 (01) ◽  
pp. 243-250 ◽  
Author(s):  
ANTON ČERNÝ
Keyword(s):  
Initial Segment ◽  
Language L ◽  
Finite Language

We call a word L-symmetric with respect to a finite language L if it contains the same number of scattered subwords u as of uR for every word u from L. We show that increasing the size of the language L may lead to an unlimited refinement of the language of L-symmetric words. Further we prove that if a long enough initial segment of a D0L-sequence consists entirely of L-symmetric words, then all words in the sequence are L-symmetric.

Separating the Words of a Language by Counting Factors

2021 ◽  
Vol 180 (4) ◽  
pp. 375-393
Author(s):  
Aleksi Saarela
Keyword(s):  
Regular Languages ◽  
Infinite Words ◽  
Language L ◽  
Finite Language

For a given language L, we study the languages X such that for all distinct words u, v ∈ L, there exists a word x ∈ X that appears a different number of times as a factor in u and in v. In particular, we are interested in the following question: For which languages L does there exist a finite language X satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.

A TIME AND SPACE EFFICIENT ALGORITHM FOR MINIMIZING COVER AUTOMATA FOR FINITE LANGUAGES

2003 ◽  
Vol 14 (06) ◽  
pp. 1071-1086 ◽  
Author(s):  
HEIKO KÖRNER
Keyword(s):  
Data Structure ◽  
Programming Language ◽  
Finite Automaton ◽  
Efficient Algorithm ◽  
Previous Method ◽  
Deterministic Finite Automaton ◽  
Time And Space ◽  
Language L ◽  
Finite Language ◽  
The Common

A deterministic finite automaton (DFA) [Formula: see text] is called a cover automaton (DFCA) for a finite language L over some alphabet Σ if [Formula: see text], with l being the length of some longest word in L. Thus a word w ∈ Σ* is in L if and only if |w| ≤ l and [Formula: see text]. The DFCA [Formula: see text] is minimal if no DFCA for L has fewer states. In this paper, we present an algorithm which converts an n–state DFA for some finite language L into a corresponding minimal DFCA, using only O(n log n) time and O(n) space. The best previously known algorithm requires O(n2) time and space. Furthermore, the new algorithm can also be used to minimize any DFCA, where the best previous method takes O(n4) time and space. Since the required data structure is rather complex, an implementation in the common programming language C/C++ is also provided.

scholarly journals Separating the Classes of Recursively Enumerable Languages Based on Machine Size

2015 ◽  
Vol 26 (06) ◽  
pp. 677-695
Author(s):  
Jan van Leeuwen ◽  
Jiří Wiedermann
Keyword(s):  
Nineteen Sixties ◽  
Turing Machines ◽  
Recursively Enumerable ◽  
Language L ◽  
Finite Language ◽  
Recursively Enumerable Languages

In the late nineteen sixties it was observed that the r.e. languages form an infinite proper hierarchy [Formula: see text] based on the size of the Turing machines that accept them. We examine the fundamental position of the finite languages and their complements in the hierarchy. We show that for every finite language L one has that L, [Formula: see text] for some [Formula: see text] where m is the length of the longest word in L, c is the cardinality of L, and [Formula: see text]. If [Formula: see text], then [Formula: see text] for some [Formula: see text]. We also prove that for every n, there is a finite language Ln with [Formula: see text] such that [Formula: see text] but Ln, [Formula: see text] for some [Formula: see text]. Several further results are shown that how the hierarchy can be separated by increasing chains of finite languages. The proofs make use of several auxiliary results for Turing machines with advice.

COUNTING SUBWORDS USING A TRIE AUTOMATON

2011 ◽  
Vol 22 (06) ◽  
pp. 1457-1469 ◽  
Author(s):  
HAMED M. K. ALAZEMI ◽  
ANTON ČERNÝ
Keyword(s):  
Matrix Representation ◽  
Input Word ◽  
Tree Representation ◽  
Nondeterministic Automaton ◽  
Prefix Tree ◽  
Language L ◽  
The Matrix ◽  
Finite Language ◽  
Original Concept

We use the concept of trie (prefix tree) representation of a prefix-closed finite language L to design a simple nondeterministic automaton. Each computation of this trie automaton corresponds to a subword occurrence of a word from L in the input word. The matrix representation of the trie automaton leads to a fairly general extension of the original concept of the Parikh matrix from [7].

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