Grammatical codes of trees

AbstractThe problem of coding (chain free) trees by words where the length of the word coding a tree t equals the number of leaves of t is investigated. The notion of an insertive strict code is introduced and investigated—these are codes of a grammatical nature. It is shown that there are exactly 120 insertive strict codes. A characterization of these codes (and their various subclasses) is given in grammatical terms

On the subword complexity of square-free DOL languages

AbstractThe subword complexity of a language K is the function which to every positive integer n assigns the number of different subwords of length n occuring in words of K. A language K is square-free if no word in it contains a subword of the form xx where x is a nonempty word. The (best) upper and lower bounds on the subword complexity of infinite square-free DOL languages are established

On regularity of context-free languages

AbstractThis paper considers conditions under which a context-free language is regular and conditions which imposed on (productions of) a rewriting system generating a context-free language will guarantee that the generated language is regular. In particular: 1.(1) necessary and sufficient conditions on productions of a unitary grammar are given that guarantee the generated language to be regular (a unitary grammar is a semi-Thue system in which the left-hand of each production is the empty word), and2.(2) it is proved that commutativity of a linear language implies its regularity. To obtain the former result, we give a generalization of the Myhill–Nerode characterization of the regular languages in terms of well-quasi orders, along with a generalization of Higman's well-quasi order result concerning the subsequence embedding relation on Σ*. In obtaining the latter results, we introduce the class of periodic languages, and demonstrate how they can be used to characterize the commutative regular languages. Here we also utilize the theory of well-quasi orders

On the membership problem for regular DNLC grammars

AbstractThere are (at least) three motivations to study the class of regular directed node-label controlled graph grammars (regular DNLC grammars for shor): (1) it fits very well into the hierarchy of subclasses of DNLC grammars, (2) it generalizes naturally right-linear string grammars and (3) it provides a useful framework for the theory of concurrent systems based on the theory of traces.The complexity of (the membership problem for) the class of regular DNLC grammars is investigated

Invariance: a Theoretical Approach for Coding Sets of Words Modulo Literal (Anti)Morphisms

Let $A$ be a finite or countable alphabet and let $\theta$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short).We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin $\theta$-invariant codes, maximality and completeness are two equivalent notions. We prove that a similar property holds in the framework of some special families of $\theta$-invariant codes such as prefix (bifix) codes, codes with a finite deciphering delay, uniformly synchronized codes and circular codes. For a special class of involutive antimorphisms, we prove that any regular $\theta$-invariant code may be embedded into a complete one.Comment: To appear in Acts of WORDS 201

New Algorithms for Position Heaps

We present several results about position heaps, a relatively new alternative to suffix trees and suffix arrays. First, we show that, if we limit the maximum length of patterns to be sought, then we can also limit the height of the heap and reduce the worst-case cost of insertions and deletions. Second, we show how to build a position heap in linear time independent of the size of the alphabet. Third, we show how to augment a position heap such that it supports access to the corresponding suffix array, and vice versa. Fourth, we introduce a variant of a position heap that can be simulated efficiently by a compressed suffix array with a linear number of extra bits

On the hardness of switching to a small number of edges

Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other one by a sequence of switches. Jel\'inkov\'a et al. [DMTCS 13, no. 2, 2011] presented a proof that it is NP-complete to decide if the input graph can be switched to contain at most a given number of edges. There turns out to be a flaw in their proof. We present a correct proof. Furthermore, we prove that the problem remains NP-complete even when restricted to graphs whose density is bounded from above by an arbitrary fixed constant. This partially answers a question of Matou\v{s}ek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014].Comment: 19 pages, 7 figures. An extended abstract submitted to COCOON 201

The Fibers and Range of Reduction Graphs in Ciliates

The biological process of gene assembly has been modeled based on three types of string rewriting rules, called string pointer rules, defined on so-called legal strings. It has been shown that reduction graphs, graphs that are based on the notion of breakpoint graph in the theory of sorting by reversal, for legal strings provide valuable insights into the gene assembly process. We characterize which legal strings obtain the same reduction graph (up to isomorphism), and moreover we characterize which graphs are (isomorphic to) reduction graphs.Comment: 24 pages, 13 figure

On-line construction of position heaps

We propose a simple linear-time on-line algorithm for constructing a position heap for a string [Ehrenfeucht et al, 2011]. Our definition of position heap differs slightly from the one proposed in [Ehrenfeucht et al, 2011] in that it considers the suffixes ordered from left to right. Our construction is based on classic suffix pointers and resembles the Ukkonen's algorithm for suffix trees [Ukkonen, 1995]. Using suffix pointers, the position heap can be extended into the augmented position heap that allows for a linear-time string matching algorithm [Ehrenfeucht et al, 2011].Comment: to appear in Journal of Discrete Algorithm