Generalizations of Code Languages with Marginal Errors

2021 ◽  
pp. 1-21
Author(s):  
Sang-Ki Ko ◽  
Yo-Sub Han ◽  
Kai Salomaa
Keyword(s):  
Time Complexity ◽  
Regular Language ◽  
Efficient Algorithms ◽  
Context Free ◽  
Free Language

The [Formula: see text]-prefix-free, [Formula: see text]-suffix-free and [Formula: see text]-infix-free languages generalize the prefix-free, suffix-free and infix-free languages by allowing marginal errors. For example, a string [Formula: see text] in a [Formula: see text]-prefix-free language [Formula: see text] can be a prefix of up to [Formula: see text] different strings in [Formula: see text]. We also define finitely prefix-free languages in which a string [Formula: see text] can be a prefix of finitely many strings. We present efficient algorithms that determine whether or not a given regular language is [Formula: see text]-prefix-free, [Formula: see text]-suffix-free or [Formula: see text]-infix-free, and analyze the time complexity of the algorithms. We establish undecidability results for deciding these properties for (linear) context-free languages.

scholarly journals Growth in Baumslag–Solitar groups I: subgroups and rationality

2011 ◽  
Vol 14 ◽  
pp. 34-71 ◽  
Author(s):  
Eric M. Freden ◽  
Teresa Knudson ◽  
Jennifer Schofield
Keyword(s):  
Turing Machine ◽  
Time Complexity ◽  
Machine Construction ◽  
Convolution Product ◽  
Growth Series ◽  
Context Sensitive ◽  
Context Free Language ◽  
Time And Space Complexity ◽  
Context Free ◽  
Free Language

AbstractThe computation of growth series for the higher Baumslag–Solitar groups is an open problem first posed by de la Harpe and Grigorchuk. We study the growth of the horocyclic subgroup as the key to the overall growth of these Baumslag–Solitar groups BS(p,q), where 1<p<q. In fact, the overall growth series can be represented as a modified convolution product with one of the factors being based on the series for the horocyclic subgroup. We exhibit two distinct algorithms that compute the growth of the horocyclic subgroup and discuss the time and space complexity of these algorithms. We show that when p divides q, the horocyclic subgroup has a geodesic combing whose words form a context-free (in fact, one-counter) language. A theorem of Chomsky–Schützenberger allows us to compute the growth series for this subgroup, which is rational. When p does not divide q, we show that no geodesic combing for the horocyclic subgroup forms a context-free language, although there is a context-sensitive geodesic combing. We exhibit a specific linearly bounded Turing machine that accepts this language (with quadratic time complexity) in the case of BS(2,3) and outline the Turing machine construction in the general case.

THE EDIT-DISTANCE BETWEEN A REGULAR LANGUAGE AND A CONTEXT-FREE LANGUAGE

2013 ◽  
Vol 24 (07) ◽  
pp. 1067-1082 ◽  
Author(s):  
YO-SUB HAN ◽  
SANG-KI KO ◽  
KAI SALOMAA
Keyword(s):  
Regular Language ◽  
Edit Distance ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Optimal Alignment ◽  
Worst Case ◽  
Context Free Language ◽  
Context Free ◽  
Free Language

The edit-distance between two strings is the smallest number of operations required to transform one string into the other. The distance between languages L1and L2is the smallest edit-distance between string wi∈ Li, i = 1, 2. We consider the problem of computing the edit-distance of a given regular language and a given context-free language. First, we present an algorithm that finds for the languages an optimal alignment, that is, a sequence of edit operations that transforms a string in one language to a string in the other. The length of the optimal alignment, in the worst case, is exponential in the size of the given grammar and finite automaton. Then, we investigate the problem of computing only the edit-distance of the languages without explicitly producing an optimal alignment. We design a polynomial time algorithm that calculates the edit-distance based on unary homomorphisms.

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.

The word problem of ℤ n is a multiple context-free language

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Meng-Che Ho
Keyword(s):  
Word Problem ◽  
Regular Language ◽  
Formal Languages ◽  
Strong Connection ◽  
Context Free Language ◽  
Multiple Context ◽  
Context Free ◽  
Free Language

Abstract The word problem of a group {G=\langle\Sigma\rangle} can be defined as the set of formal words in {\Sigma^{*}} that represent the identity in G. When viewed as formal languages, this gives a strong connection between classes of groups and classes of formal languages. For example, Anīsīmov showed that a group is finite if and only if its word problem is a regular language, and Muller and Schupp showed that a group is virtually-free if and only if its word problem is a context-free language. Recently, Salvati showed that the word problem of {\mathbb{Z}^{2}} is a multiple context-free language, giving the first example of a natural word problem that is multiple context-free, but not context-free. We generalize Salvati’s result to show that the word problem of {\mathbb{Z}^{n}} is a multiple context-free language for any n.

scholarly journals The Insertion Encoding of Permutations

10.37236/1944 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Michael H. Albert ◽  
Steve Linton ◽  
Nik Ruškuc
Keyword(s):  
Polynomial Time ◽  
Generating Functions ◽  
Regular Language ◽  
Necessary Conditions ◽  
Polynomial Time Algorithms ◽  
Context Free Language ◽  
Context Free ◽  
Free Language

We introduce the insertion encoding, an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Applications of the insertion encoding to the evaluation of generating functions for classes of permutations, construction of polynomial time algorithms for enumerating such classes, and the illustration of bijective equivalence between classes are demonstrated.

WHEN CHURCH-ROSSER BECOMES CONTEXT FREE

2007 ◽  
Vol 18 (06) ◽  
pp. 1293-1302 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER
Keyword(s):  
Linear Time ◽  
Membership Problem ◽  
Closure Properties ◽  
Context Free Language ◽  
Arithmetic Hierarchy ◽  
Context Free ◽  
Free Language

We investigate the intersection of Church-Rosser languages and (strongly) context-free languages. The intersection is still a proper superset of the deterministic context-free languages as well as of their reversals, while its membership problem is solvable in linear time. For the problem whether a given Church-Rosser or context-free language belongs to the intersection we show completeness for the second level of the arithmetic hierarchy. The equivalence of Church-Rosser and context-free languages is Π1-complete. It is proved that all considered intersections are pairwise incomparable. Finally, closure properties under several operations are investigated.

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