Prefix-primitive annihilators of languages under some operations

This paper studies some properties of prefix-primitive annihilators of languages under the catenation, shuffle product and bi-catenation operations. We prove that for every finite language L under the catenation operation, the left prefix-primitive annihilator of L is not equal to the right prefix-primitive annihilator of L, the left prefix-primitive annihilator of languages is not regular for any finite language, and the left prefix-primitive annihilator of any thin languages is not empty. Moreover, we also characterize the prefix-primitive annihilators of non-empty language under the shuffle product and bi-catenation operations over the alphabet with two letters.     

d-Words and d-languages

Let X be a finite alphabet containing more than one letter. A d-primitive word u overX is a non-overlapping word in the sense that no proper prefix of u is a suffix of u. D(1) is the set of all d-primitive words over X and D is the set of all positive powers of all words in D (1). Every language in D will be called a d-language. In this paper, we study some algebraic properties of d-primitive words and d-languages relative to formal language theory and codes. We show that there are infinitely many cyclic-square-free words over alphabet with three letters. A characterization of three elements codes in D (1) is obtained and we prove that every regular component in D (1) is either a prefix code or a suffix code. Received: 22 September 1997 / 7 January 1998     

Classifications of Dense Languages

Let X be a finite alphabet containing more than one letter. A dense language over X is a language containing a disjunctive language. A language L is an n-dense language if for any distinct n words there exist two words such that In this paper we classify dense languages into strict n-dense languages and study some of their algebraic properties. We show that for each n ≥ 0, the n-dense language exists. For an n-dense language L, n ≠ 1, the language L ∩ Q is a dense language, where Q is the set of all primitive words over X. Moreover, for a given n ≥ 1, the language L is such that , then for some m, n ≤ m ≤ 2n + 1. Characterizations on 0-dense languages and n-dense languages are obtained. It is true that for any dense language L, there exist such that for some. We show that everyn-dense language, n ≥ 0, can be split into disjoint union of infinitely many n-dense languages.     

Palindrome words and reverse closed languages

A word which is equal to its mirror image is called a palindrome word. Any language consisting of palindrome words is called a palindrome language. In this paper we investigate properties of palindrome words and languages. We show that there is no dense regular language consisting of palindrome words. A language contains all the mirror images of its elements is called a reverse closed language. Clearly, every palindrome language is reverse closed. We show that whether a given regular or context-free language is reverse closed is decidable. We study certain properties concerning reverse closed finite maximal prefix codes in this paper. Properties of languages that commute with reverse closed languages are investigated too.     

A strongly primitive word of arbitrary length and its application †

It is known that both the class of all (n,k)-languages and the class of all time-variant languages over a finite alphabet contain the class of all regular languages. In this paper we show that in general neither one of these two classes of languages contains the other by constructing an infinite sequence of strongly primitive words over an alphabet with four letters.     

The expressive power of the shuffle product

There is an increasing interest in the shuffle product on formal languages, mainly because it is a standard tool for modeling process algebras. It still remains a mysterious operation on regular languages.Antonio Restivo proposed as a challenge to characterize the smallest class of languages containing the singletons and closed under Boolean operations, product and shuffle. This problem is still widely open, but we present some partial results on it. We also study some other smaller classes, including the smallest class containing the languages composed of a single word of length 2 which is closed under Boolean operations and shuffle by a letter (resp. shuffle by a letter and by the star of a letter). The proof techniques have both an algebraic and a combinatorial flavor.     

On the structure of bispecial Sturmian words

A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that palindromic bispecial Sturmian words are precisely the maximal internal factors of primitive Christoffel words. We extend this result by showing that bispecial Sturmian words are precisely the maximal internal factors of all Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the language of Sturmian words.     

The syntactic monoid of hairpin-free languages

The study of hairpin-free words has been initiated in the context of DNA computing. DNA strands that, theoretically speaking, are finite strings over the alphabet {A, G, C, T} are used in DNA computing to encode information. Due to the fact that A is complementary to T and G to C, DNA single strands that are complementary can bind to each other or to themselves in either intended or unintended ways. One of the structures that is usually undesirable for biocomputation, since it makes the affected DNA string unavailable for future interactions, is the hairpin: if some subsequences of a DNA single string are complementary to each other, the string will bind to itself forming a hairpin-like structure. This paper continues the theoretical study of hairpin-free languages. We study algebraic properties of hairpin-free words and hairpins. We also give a complete characterization of the syntactic monoid of the language consisting of all hairpin-free words over a given alphabet and illustrate it with an example using the DNA alphabet.     

Unavoidable Sets of Words of Uniform Length

A set of words X over a finite alphabet A is said to be unavoidable if all but finitely many words in A* have a factor in X. We examine the problem of calculating the cardinality of minimal unavoidable sets of words of uniform length; we correct an error in [8], state a conjecture offering a formula for the minimum size of these so called n-good sets for all values of n, and show that the conjecture is correct in an infinite number of cases.     

Bounded hairpin completion

Hairpin completion is a formal operation inspired from biochemistry. Here we consider a restricted variant of hairpin completion called bounded hairpin completion. Applied to a word encoding a single stranded molecule x such that either a suffix or a prefix of x is complementary to a subword of x, hairpin completion produces a new word z, which is a prolongation of x to the right or to the left by annealing.Although this operation is a purely mathematical one and the biological reality is just a source of inspiration, it seems rather unrealistic to impose no restriction on the length of the prefix or suffix added by the hairpin completion. The restriction considered here concerns the length of all prefixes and suffixes that are added to the current word by hairpin completion. They cannot be longer than a given constant. Closure properties of some classes of formal languages under the non-iterated and iterated bounded hairpin completion are investigated. We consider the bounded hairpin completion distance between two words and generalize this distance to languages and discuss algorithms for computing them. Finally also the inverse operation, namely bounded hairpin reduction, as well as the set of all primitive bounded hairpin roots of a regular language are considered.     

Superposition Based on Watson–Crick-Like Complementarity

In this paper we propose a new formal operation on words and languages, called superposition. By this operation, based on a Watson–Crick-like complementarity, we can generate a set of words, starting from a pair of words, in which the contribution of a word to the result need not be one subword only, as happens in classical bio-operations of DNA computing. Specifically, starting from two single stranded molecules x and y such that a suffix of x is complementary to a prefix of y, a prefix of x is complementary to a suffix of y, or x is complementary to a subword of y, a new word z, which is a prolongation of x to the right, to the left, or to both, respectively, is obtained by annealing. If y is complementary to a subword of x, then the result is x. This operation is considered here as an abstract operation on formal languages. We relate it to other operations in formal language theory and we settle the closure properties under this operation of classes in the Chomsky hierarchy. We obtain a useful result by showing that unrestricted iteration of the superposition operation, where the "parents" in a subsequent iteration can be any words produced during any preceding iteration step, is equivalent to restricted iteration, where at each step one parent must be a word from the initial language. This result is used for establishing the closure properties of classes in the Chomsky hierarchy under iterated superposition. Actually, since the results are formulated in terms of AFL theory, they are applicable to more classes of languages. Then we discuss "adult" languages, languages consisting of words that cannot be extended by further superposition, and show that this notion might bring us to the border of recursive languages. Finally, we consider some operations involved in classical DNA algorithms, such as Adleman's, which might be expressed through iterated superposition.     

On the structure of systematic prefix codes

Systematic prefix codes play an important role in coding theory, we relate them with the problem of the partition of a free (sub-) monoid into two free sub-monoids. We show too that among the dual codes of a systematic prefix code A there exists one and only one which appears in the automaton recognizing A ^*. The characterization of this automaton and some corollaries stated here will allow us to show in further note that systematic prefix codes are involved in the structure of any regular prefix code. Work done under CNR contract No. R-l7-02-417-0-A.     

Shuffle languages are in P

In this paper we show that shuffle languages are contained in one-way-NSPACE(log n) thus in P. We consider the class of shuffle languages which emerges from the class of finite languages through regular operations (union, concatenation, Kleene star) and shuffle operations (shuffle and shuffle closure). For every shuffle expression E we construct a shuffle automaton which accepts the language generated by E and we show that the automaton can be simulated by a one-way nondeterministic Turing machine in logarithmic space.     

Disjunctive languages related to p-primitive words

It is known that the set of all primitive words and the set of all $d$ -primitive words are disjunctive languages. In this paper we prove that the set of all $p$ -primitive words is disjunctive. We also prove that the set of all primitive but not $p$ -primitive words, the set of all balanced but not $p$ -primitive words, and the set of all $d$ -primitive but not $p$ -primitive words are disjunctive languages.     

Decision problems concerning thinness and slenderness of formal languages

A language L is called thin if for almost all n there is at most one word of length n in L. A language L is called slender if there is a positive integer k such that for any n there are at most k words of length n in L. The notions of Parikh thin and Parikh slender languages are defined similarly by counting the words with the same Parikh vectors instead of the words of the same length. In this paper we discuss decision problems concerning these four properties. It is shown that all four properties are decidable for bounded semilinear languages but undecidable for DT0L languages. As a consequence all these problems are decidable for context-free languages. Received 9 June 1997 / 3 December 1997     

Expanded subalphabets in the theories of languages and semigroups†

A new type of code, called an expanded subalphabet, is introduced. It is shown that the following four conditions on a subset L of the free monoid A ^? over a finite or infinite alphabet A are equivalent: (1) L is the submonoid generated by an expanded subalphabet; (2) L is a retract; (3) L is the fixed language of an endomorphism; (4) L is the stationary language of an endomorphism. Expanded subalphabets are used as a tool for the investigation of fixed languages (=retracts). Special results for the case of a finite alphabet are given and the relationship with the theory of L-systems is indicated.     

Context-free and stationary languagest†

A stationary word of a OL system G=(X,P,w) is a word that generates itself after a positive number of direct derivations. The stationary set of G is the set of stationary words of G and the stationary language of G is the set of the stationary words generated by w. The stationary set of G is a submonoid of X ^* containing the adult set of G.

A language not containing the empty word is context-free if and only if it is the stationary language of a propagating system. Every TOL language is the image under a coding of the stationary language of a TOL system.     

The equational theory of regular words

Courcelle introduced the study of regular words, i.e., words isomorphic to frontiers of regular trees. Heilbrunner showed that a nonempty word is regular iff it can be generated from the singletons by the operations of concatenation, omega power, omega-op power, and the infinite family of shuffle operations. We prove that the algebra of nonempty regular words on the set A, equipped with these operations, is freely generated by A in a variety which is axiomatizable by an infinite collection of some natural equations. We also show that this variety has no finite equational basis and that its equational theory is decidable in polynomial time.     

Some properties of the disjunctive languages contained in <Emphasis Type="Italic">Q</Emphasis>

The set of all primitive words Q over an alphabet X was first defined and studied by Shyr and Thierrin (Proceedings of the 1977 Inter. FCT-Conference, Poznan, Poland, Lecture Notes in Computer Science 56. pp. 171–176 (1977)). It showed that for the case |X| ≥ 2, the set along with \({Q^{(i)} = \{f^i\,|\,f \in Q\}, i\geq 2}\) are all disjunctive. Since then these disjunctive sets are often be quoted. Following Shyr and Thierrin showed that the half sets \({Q_{ev} = \{f \in Q\,|\,|f| = {\rm even}\}}\) and Q _od = Q \ Q _ev of Q are disjunctive, Chien proved that each of the set \({Q_{p,r}= \{u\in Q\,|\,|u|\equiv r\,(mod\,p) \},\,0\leq r < p}\) is disjunctive, where p is a prime number. In this paper, we generalize this property to that all the languages \({Q_{n,r}= \{u\in Q\,|\,|u|\equiv r\,(mod\,n) \},\, 0\leq r < n}\) are disjunctive languages, where n is any positive integer. We proved that for any n ≥ 1, k ≥ 2, (Q _n,0) ^k are all regular languages. Some algebraic properties related to the family of languages {Q _n,r | n ≥ 2, 0 ≤ r < n } are investigated.     

Succinct representation of regular languages by boolean automata

Boolean automata are a generalization of finite automata in the sense that the ‘next state’, i.e. the result of the transition function given a state and a letter, is not just a single state (deterministic automata) or a union of states (nondeterministic automata) but a boolean function of states. Boolean automata accept precisely regular languages; furthermore they correspond in a natural way to certain language equations as well as to sequential networks. We investigate the succinctness of representing regular languages by boolean automata. In particular, we show that for every deterministic automaton A with m states there exists a boolean automaton with [log₂m] states which accepts the reverse of the language accepted by A (m≥1). We also show that for every n≥1 there exists a boolean automation with n states such that the smallest deterministic automaton accepting the same language has 2⁽²ⁿ⁾ states; moreover this holds for an alphabet with only two letters.