The notion of EOL forms was introduced by Maurer, Salomaa and Wood recwtly. The study of complete... more The notion of EOL forms was introduced by Maurer, Salomaa and Wood recwtly. The study of complete EOL form is of particular interest since it suggests to attempt a systematic classification of all possible short normal forms for EOL systems. In this paper WI: invest;g: _ under which conditions an EOL form with a single nonterminal S and a single terminal a with the length of the iig?nt side of each production one or two is complete. We can summarize the main results in this paper as follows. (1) We present a complete analysis of the case when both produc:ions a + S and S -+ S occur. (2) We: show that each complete form must contain at least one of a + S, S + S. (3) We are not able to give a complete analysis if exactly one of a + S, S + S cccurs but we show that in both cases complete forms with o! ly four productions exist.
Parikh matrices recently introduced have turned out to be a powerful tool in the arithmetizing of... more Parikh matrices recently introduced have turned out to be a powerful tool in the arithmetizing of the theory of words. In particular, many inequalities be- tween (scattered) subword occurrences have been obtained as consequences of the properties of the matrices. This paper continues the investigation of Parikh matrices and subword occurrences. In particular, we study certain inequali- ties, as
We investigate factorizations of regular languages in terms of prime languages. A language is sai... more We investigate factorizations of regular languages in terms of prime languages. A language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps. We give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition. We show that there exist non-regular unary languages that do not have prime decompositions. We also consider infinite factorizations of unary languages.
We turn once more to the study of EOL forms by focussing on synchronized EOL forms. We show that ... more We turn once more to the study of EOL forms by focussing on synchronized EOL forms. We show that there are complete propagating synchro-EOL forms in contrast to the situation for EOL forms. Turning to the syntax analysis of synchro-EOL form families, we are lead into the study of ambiguity in EOL systems. We show that there are languages which are n-CF-ambiguous, for arbitrarily large values of n 2 1, but which are EOL-unambiguous. Finally we characterize when synchro-{S, a}-forms are complete and also characterize when short synchro-{S, A, a}-forms are complete, where A is a terminal-like nonterminal. These results are also in contrast with EOL forms, where even short {S, c})-forms have only a restricted characterization with respect to completeness.
Maurer, H.A., A. Salomaa and D. Wood, Bounded delay L codes (Note), Theoretical Computer Science ... more Maurer, H.A., A. Salomaa and D. Wood, Bounded delay L codes (Note), Theoretical Computer Science 84 (1991) 265-279. Three natural notions of bounded delay for L codes are investigated. All interrelations, many of them rather surprising, between ordinary codes of bounded delay and L codes of bounded delay of various types are established.
WatsonCrick complementarity has been used as a basis for massive parallelism in DNA computing. A... more WatsonCrick complementarity has been used as a basis for massive parallelism in DNA computing. Also its use in an operational sense has turned out to be very promising in the study of WatsonCrick D0L systems. The latter generalize D0L systems in a way not ...
The inclusion problem for (nonerasing) pattern languages was raised by Angluin [1] in 1980. It ha... more The inclusion problem for (nonerasing) pattern languages was raised by Angluin [1] in 1980. It has been open ever since. In this paper, we settle this open problem and show that inclusion is undecidable for (both erasing and nonerasing) pattern languages. In addition, ...
DNA computing, or, more generally, molecular computing, is an exciting fast developing interdisci... more DNA computing, or, more generally, molecular computing, is an exciting fast developing interdisciplinary area. Research in this area concerns theory, experiments, and applications of DNA computing. In this paper, we demonstrate the theoretical developments by discussing a number of selected topics. We also give an introduction to the basic structure of DNA and the basic DNA processing tools.
When words are characterized in terms of numerical quantities, awkward considerations due to the ... more When words are characterized in terms of numerical quantities, awkward considerations due to the noncommutativity of words are avoided. The numerical quantity investigated in this paper is |w| u , the number of occurrences of a word u as a (scattered) subword of a word w. Parikh matrices recently introduced have these quantities as their entries. According to the main result in this paper, no entry in a Parikh matrix, no matter how high the dimension, can be computed in terms of the other entries. Consequences concerning various inference problems between numbers |w| u themselves, as well as of the word w from these numbers, are obtained.
... There are obvious limits to miniaturization with current computer tech-nologies. ... A typica... more ... There are obvious limits to miniaturization with current computer tech-nologies. ... A typical example would be the cryptanalysis of a ciphertext: all possible keys can be tried ... for computing because, as we will see latter, complementarity brings the universal twin-shuffle language ...
We investigate languages consisting of words following one of the given finitely many patterns. T... more We investigate languages consisting of words following one of the given finitely many patterns. The issues concerning such multi-pattern languages are relevant in inductive inference, theory of learning and term rewriting. We obtain results about decidability, characterization, hierarchies and special classes of multi-pattern languages. Some open problems are also presented.
We introduce and investigate new methods to define parallel composition of words and languages as... more We introduce and investigate new methods to define parallel composition of words and languages as well as of w-words and o-languages. The operation of parallel composition leads to new shuffle-like operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to define parallel composition of processes. It is also applicable to parallel computation. The operations are introduced using a uniform method based on the notion of trajectory. As a consequence, we obtain a very intuitive geometrical interpretation of the parallel composition operation. These operations lead in a natural way to a large class of semirings. The approach is amazingly flexible, diverse concepts from the theory of concurrency can be introduced and studied in this framework. For instance, we provide examples of applications to fairness property and to parallelization of non-context-free languages in terms of context-free and even regular languages. This paper concentrates on syntactic constraints. Semantic constraints will be dealt with in a forthcoming contribution.
The notion of an I, code is introduced as a generalization of the customary notion of a code. Att... more The notion of an I, code is introduced as a generalization of the customary notion of a code. Attention is focused on unary L codes. where the target alphabet consists of one letter only. Unary L codes can be identified with such number system5 where the representation is unique. Surprisingly many unary morphisms are L codes. Furthermore, even for non-L-codes there is only one 'exceptional length' We also give a complete classification for the I)inary case.
We study a Lindenmayer-like parallel rewriting system to model the growth of filaments (arrays of... more We study a Lindenmayer-like parallel rewriting system to model the growth of filaments (arrays of cells) in which developmental errors may occur. In essence this model is the fuzzy analogue of the derivation-controlled iteration grammar. Under minor assumptions on the family of control languages and on the family of fuzzy languages in the underlying iteration grammar, we show that (i) regular control does not provide additional generating power to the model, (ii) the number of fuzzy substitutions in the underlying iteration grammar can be reduced to two, and (iii) the resulting family of fuzzy languages possesses strong closure properties, viz. it is a full hyper-AFFL, i.e., a hyper-algebraically closed full Abstract Family of Fuzzy Languages.
Page 1. DNA Complementarity and Paradigms of Computing Arto Salomaa Turku Centre for Computer Sci... more Page 1. DNA Complementarity and Paradigms of Computing Arto Salomaa Turku Centre for Computer Science Lemminkäisenkatu 14, 20520 Turku, Finland [email protected] Abstract. Watson-Crick complementarity is one ...
Representations of nite languages as a product (catenation) of languages are investigated, where ... more Representations of nite languages as a product (catenation) of languages are investigated, where the factor languages are \prime", that is, cannot be decomposed further in a nontrivial manner. In general, such prime decompositions are not unique -even the number of factors can vary exponentially. The paper investigates the uniqueness of prime decompositions, as well as the commuting of the factors. Interconnections to languages more general than nite are pointed out. In the case of regular languages, the notion of a decomposition set turns out to be a powerful tool.
The notion of EOL forms was introduced by Maurer, Salomaa and Wood recwtly. The study of complete... more The notion of EOL forms was introduced by Maurer, Salomaa and Wood recwtly. The study of complete EOL form is of particular interest since it suggests to attempt a systematic classification of all possible short normal forms for EOL systems. In this paper WI: invest;g: _ under which conditions an EOL form with a single nonterminal S and a single terminal a with the length of the iig?nt side of each production one or two is complete. We can summarize the main results in this paper as follows. (1) We present a complete analysis of the case when both produc:ions a + S and S -+ S occur. (2) We: show that each complete form must contain at least one of a + S, S + S. (3) We are not able to give a complete analysis if exactly one of a + S, S + S cccurs but we show that in both cases complete forms with o! ly four productions exist.
Parikh matrices recently introduced have turned out to be a powerful tool in the arithmetizing of... more Parikh matrices recently introduced have turned out to be a powerful tool in the arithmetizing of the theory of words. In particular, many inequalities be- tween (scattered) subword occurrences have been obtained as consequences of the properties of the matrices. This paper continues the investigation of Parikh matrices and subword occurrences. In particular, we study certain inequali- ties, as
We investigate factorizations of regular languages in terms of prime languages. A language is sai... more We investigate factorizations of regular languages in terms of prime languages. A language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps. We give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition. We show that there exist non-regular unary languages that do not have prime decompositions. We also consider infinite factorizations of unary languages.
We turn once more to the study of EOL forms by focussing on synchronized EOL forms. We show that ... more We turn once more to the study of EOL forms by focussing on synchronized EOL forms. We show that there are complete propagating synchro-EOL forms in contrast to the situation for EOL forms. Turning to the syntax analysis of synchro-EOL form families, we are lead into the study of ambiguity in EOL systems. We show that there are languages which are n-CF-ambiguous, for arbitrarily large values of n 2 1, but which are EOL-unambiguous. Finally we characterize when synchro-{S, a}-forms are complete and also characterize when short synchro-{S, A, a}-forms are complete, where A is a terminal-like nonterminal. These results are also in contrast with EOL forms, where even short {S, c})-forms have only a restricted characterization with respect to completeness.
Maurer, H.A., A. Salomaa and D. Wood, Bounded delay L codes (Note), Theoretical Computer Science ... more Maurer, H.A., A. Salomaa and D. Wood, Bounded delay L codes (Note), Theoretical Computer Science 84 (1991) 265-279. Three natural notions of bounded delay for L codes are investigated. All interrelations, many of them rather surprising, between ordinary codes of bounded delay and L codes of bounded delay of various types are established.
WatsonCrick complementarity has been used as a basis for massive parallelism in DNA computing. A... more WatsonCrick complementarity has been used as a basis for massive parallelism in DNA computing. Also its use in an operational sense has turned out to be very promising in the study of WatsonCrick D0L systems. The latter generalize D0L systems in a way not ...
The inclusion problem for (nonerasing) pattern languages was raised by Angluin [1] in 1980. It ha... more The inclusion problem for (nonerasing) pattern languages was raised by Angluin [1] in 1980. It has been open ever since. In this paper, we settle this open problem and show that inclusion is undecidable for (both erasing and nonerasing) pattern languages. In addition, ...
DNA computing, or, more generally, molecular computing, is an exciting fast developing interdisci... more DNA computing, or, more generally, molecular computing, is an exciting fast developing interdisciplinary area. Research in this area concerns theory, experiments, and applications of DNA computing. In this paper, we demonstrate the theoretical developments by discussing a number of selected topics. We also give an introduction to the basic structure of DNA and the basic DNA processing tools.
When words are characterized in terms of numerical quantities, awkward considerations due to the ... more When words are characterized in terms of numerical quantities, awkward considerations due to the noncommutativity of words are avoided. The numerical quantity investigated in this paper is |w| u , the number of occurrences of a word u as a (scattered) subword of a word w. Parikh matrices recently introduced have these quantities as their entries. According to the main result in this paper, no entry in a Parikh matrix, no matter how high the dimension, can be computed in terms of the other entries. Consequences concerning various inference problems between numbers |w| u themselves, as well as of the word w from these numbers, are obtained.
... There are obvious limits to miniaturization with current computer tech-nologies. ... A typica... more ... There are obvious limits to miniaturization with current computer tech-nologies. ... A typical example would be the cryptanalysis of a ciphertext: all possible keys can be tried ... for computing because, as we will see latter, complementarity brings the universal twin-shuffle language ...
We investigate languages consisting of words following one of the given finitely many patterns. T... more We investigate languages consisting of words following one of the given finitely many patterns. The issues concerning such multi-pattern languages are relevant in inductive inference, theory of learning and term rewriting. We obtain results about decidability, characterization, hierarchies and special classes of multi-pattern languages. Some open problems are also presented.
We introduce and investigate new methods to define parallel composition of words and languages as... more We introduce and investigate new methods to define parallel composition of words and languages as well as of w-words and o-languages. The operation of parallel composition leads to new shuffle-like operations defined by syntactic constraints on the usual shuffle operation. The approach is applicable to concurrency, providing a method to define parallel composition of processes. It is also applicable to parallel computation. The operations are introduced using a uniform method based on the notion of trajectory. As a consequence, we obtain a very intuitive geometrical interpretation of the parallel composition operation. These operations lead in a natural way to a large class of semirings. The approach is amazingly flexible, diverse concepts from the theory of concurrency can be introduced and studied in this framework. For instance, we provide examples of applications to fairness property and to parallelization of non-context-free languages in terms of context-free and even regular languages. This paper concentrates on syntactic constraints. Semantic constraints will be dealt with in a forthcoming contribution.
The notion of an I, code is introduced as a generalization of the customary notion of a code. Att... more The notion of an I, code is introduced as a generalization of the customary notion of a code. Attention is focused on unary L codes. where the target alphabet consists of one letter only. Unary L codes can be identified with such number system5 where the representation is unique. Surprisingly many unary morphisms are L codes. Furthermore, even for non-L-codes there is only one 'exceptional length' We also give a complete classification for the I)inary case.
We study a Lindenmayer-like parallel rewriting system to model the growth of filaments (arrays of... more We study a Lindenmayer-like parallel rewriting system to model the growth of filaments (arrays of cells) in which developmental errors may occur. In essence this model is the fuzzy analogue of the derivation-controlled iteration grammar. Under minor assumptions on the family of control languages and on the family of fuzzy languages in the underlying iteration grammar, we show that (i) regular control does not provide additional generating power to the model, (ii) the number of fuzzy substitutions in the underlying iteration grammar can be reduced to two, and (iii) the resulting family of fuzzy languages possesses strong closure properties, viz. it is a full hyper-AFFL, i.e., a hyper-algebraically closed full Abstract Family of Fuzzy Languages.
Page 1. DNA Complementarity and Paradigms of Computing Arto Salomaa Turku Centre for Computer Sci... more Page 1. DNA Complementarity and Paradigms of Computing Arto Salomaa Turku Centre for Computer Science Lemminkäisenkatu 14, 20520 Turku, Finland [email protected] Abstract. Watson-Crick complementarity is one ...
Representations of nite languages as a product (catenation) of languages are investigated, where ... more Representations of nite languages as a product (catenation) of languages are investigated, where the factor languages are \prime", that is, cannot be decomposed further in a nontrivial manner. In general, such prime decompositions are not unique -even the number of factors can vary exponentially. The paper investigates the uniqueness of prime decompositions, as well as the commuting of the factors. Interconnections to languages more general than nite are pointed out. In the case of regular languages, the notion of a decomposition set turns out to be a powerful tool.
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