230 research outputs found

    Numbers and Languages

    Get PDF
    The thesis presents results obtained during the authors PhD-studies. First systems of language equations of a simple form consisting of just two equations are proved to be computationally universal. These are systems over unary alphabet, that are seen as systems of equations over natural numbers. The systems contain only an equation X+A=B and an equation X+X+C=X+X+D, where A, B, C and D are eventually periodic constants. It is proved that for every recursive set S there exists natural numbers p and d, and eventually periodic sets A, B, C and D such that a number n is in S if and only if np+d is in the unique solution of the abovementioned system of two equations, so all recursive sets can be represented in an encoded form. It is also proved that all recursive sets cannot be represented as they are, so the encoding is really needed. Furthermore, it is proved that the family of languages generated by Boolean grammars is closed under injective gsm-mappings and inverse gsm-mappings. The arguments apply also for the families of unambiguous Boolean languages, conjunctive languages and unambiguous languages. Finally, characterizations for morphisims preserving subfamilies of context-free languages are presented. It is shown that the families of deterministic and LL context-free languages are closed under codes if and only if they are of bounded deciphering delay. These families are also closed under non-codes, if they map every letter into a submonoid generated by a single word. The family of unambiguous context-free languages is closed under all codes and under the same non-codes as the families of deterministic and LL context-free languages.Siirretty Doriast

    Reduced languages as omega-generators

    Get PDF
    International audienceWe consider the following decision problem: “Is a rational omega-language generated by a code ?” Since 1994, the codes admit a characterization in terms of infinite words. We derive from this result the definition of a new class of languages, the reduced languages. A code is a reduced language but the converse does not hold. The idea is to “reduce" easy-to-obtain minimal omega-generators in order to obtain codes as omega-generators

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

    Full text link
    Let AA be a finite or countable alphabet and let θ\theta be literal (anti)morphism onto AA^* (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under θ\theta (θ\theta-invariant for short).We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin θ\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 θ\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 θ\theta-invariant code may be embedded into a complete one.Comment: To appear in Acts of WORDS 201

    Codes and infinite words

    Get PDF

    Acta Cybernetica : Volume 11. Number 4.

    Get PDF

    Embedding a θ\theta-invariant code into a complete one

    Full text link
    Let A be a finite or countable alphabet and let θ\theta be a literal (anti-)automorphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under θ\theta (θ\theta-invariant for short) that is, languages L such that θ\theta (L) is a subset of L.We establish an extension of the famous defect theorem. With regards to the so-called notion of completeness, we provide a series of examples of finite complete θ\theta-invariant codes. Moreover, we establish a formula which allows to embed any non-complete θ\theta-invariant code into a complete one. As a consequence, in the family of the so-called thin θ\theta--invariant codes, maximality and completeness are two equivalent notions.Comment: arXiv admin note: text overlap with arXiv:1705.0556

    Completing circular codes in regular submonoids

    Get PDF
    AbstractLet M be an arbitrary submonoid of the free monoid A∗, and let X⊆M be a variable length code (for short a code). X is weakly M-complete iff any word in M is a factor of some word in X∗ [J. Néraud, C. Selmi, Free monoid theory: Maximality and completeness in arbitrary submonoids, Internat. J. Algebra Comput. 13 (5) (2003) 507–516]. Given a regular submonoid M, and given an arbitrary code X⊆M, we are interested in the existence of a weakly M-complete code Xˆ that contains X. Actually, in [J. Néraud, Completing a code in a regular submonoid, in: Acts of MCU’2004, Lect. Notes Comput. Sci. 3354 (2005) 281–291; J. Néraud, Completing a code in a submonoid of finite rank, Fund. Inform. 74 (2006) 549–562], by presenting a general formula, we have established that, in any case, such a code Xˆ exists. In the present paper, we prove that any regular circular code X⊆M may be embedded into a weakly M-complete one iff the minimal automaton with behavior M has a synchronizing word. As a consequence of our result an extension of the famous theorem of Schützenberger is stated for regular circular codes in the framework of regular submonoids. We study also the behaviour of the subclass of uniformly synchronous codes in connection with these questions

    Completion of codes with finite bi-decoding delays

    Get PDF
    AbstractLet A∗ be a free monoid generated by a set A and let X⊆A∗ be a code with property P. The embedding of X into a complete code Y⊆A∗ with the same property P is called the completion of X. The method of completion of rational bifix codes and codes with finite decoding delays have been investigated by a number of authors. In this paper, we provide a general method of construction for completing the codes with finite bi-decoding delays. As a consequence, the completion method of rational bifix codes and codes with finite decoding delays is extended and applied to codes with finite bi-decoding delays
    corecore