Solomon Marcus

Contextual grammars

Let V be a finite non-void set ; V lary. Every finite sequence of elements in ia called a vocnbuV is said to be a s t r i n g o n V. G i v e n a s t r i n g x = a l a 2 . . . a n , t h e n u m b e r n i s c a l l e d t h e l e n g t h o f x . The s t r i n g o f l e n g t h z e r o i s c a l l e d t h e n t r i n g a n d i s d e n o t e d b y r~J . Any s e t o f s t r i n g s o n V i s c a l l e d a l a n g u a g e o n V. The s e t o f a l l s t r i n g s on V ( t h e n u l l s t r i n g i n c l u s i v e l y ) i s c a l l e d t h e u n i v e r s a l l a n g u a g e o n V. By a nwe denote the string a...a, where a is iterated n times. Any ordered pair (u,v~ of strings on V_ is said to be a contex~ on V. The string x is admitted by the context <u,v> With respect to the language L if u~ G L. Let .~ be a finite set of strings on the vocabulary V~ and let@be a finite se@ of contexts on V. The triple (v,~, ~)) (1)

Contextual grammars and natural languages

The systematic investigation of natural languages by means of algebraic, combinatorial and set-theoretic models begun in the 1950s concomitantly in Europe and the U.S.A. An important year in this respect seems to be 1957, when Chomsky published his pioneering book [6] concerning the new generative approach to syntactic structures and some Russian mathematicians proposed a conceptual framework for the study of general morphological categories [10], of the category of grammatical case (A. N. Kolmogorov; see [74]), and of the category of part of speech [75], giving the start in the development of analytical mathematical models of languages.

Symmetry in the simplest case: The real line

Abstract Various types and degrees of symmetry in the set of real numbers are investigated. It is shown that symmetry and its polar opposite, antisymmetry, are submitted to similar restrictions.

A topological characterization of random sequences

The set of random sequences is large in the sense of measure, but small in the sense of category. This is the case when we regard the set of infinite sequences over a finite alphabet as a subset of the usual Cantor space. In this note we will show that the above result depends on the topology chosen. To this end we will use a relativization of the Cantor topology, the Uδ-topology introduced by Staiger [RAIRO Inform. Theor 21 (1987) 147-173]. This topology is also metric, but the distance between two sequences does not depend on their longest common prefix (Cantor metric), but on the number of their common prefixes in a given language U. The resulting space is complete, but not always compact. We will show how to derive a computable set U from a universal Martin Lof test such that the set of non random sequences is nowhere dense in the Uδ-topology. As a byproduct we obtain a topological characterization of the set of random sequences. We also show that the Law of Large Numbers, which fails with respect to the usual topology, is true for the Uδ-topology.

Bridging P Systems and Genomics: A Preliminary Approach

Bringing genomics within the framework of P systems could give to the former the possibility to take profit of the computational capacities of the latter. Moreover, suggestions coming from genomics could enrich the study of P systems with new biological and computational ideas. In what follows, a first attempt is made in this respect.

The Paradox of the Heap of Grains in Respect to Roughness, Fuzziness and Negligibility

In a first step, roughness and fuzziness fail to account for the type of grad-uality (vagueness) involved in the concept of a heap, as it is conceived in the famous Eubulides’ paradox. One can partially bridge this gap by means of tolerance rough sets. Even in this case, a non-concordance persists between the empirical finiteness and the theoretical infinity of a heap. Another way to approach this problem could be via negligibility (be it cardinal, measure-theoretic or topological)

Marcus contextual languages consisting of primitive words

In this paper we prove that the language of all primitive (strongly primitive) words over a nontrivial alphabet can be generated by certain types of Marcus contextual grammars.

Mathematical Proofs at a Crossroad

For more than 2000 years, from Pythagoras and Euclid to Hilbert and Bourbaki, mathematical proofs were essentially based on axiomatic-deductive reasoning. In the last decades, the increasing length and complexity of many mathematical proofs led to the expansion of some empirical, experimental, psychological and social aspects, yesterday only marginal, but now changing radically the very essence of proof. In this paper, we try to organize this evolution, to distinguish its different steps and aspects, and to evaluate its advantages and shortcomings. Axiomatic-deductive proofs are not a posteriori work, a luxury we can marginalize nor are computer-assisted proofs bad mathematics. There is hope for integration!

Contextual Grammars with Distributed Catenation and Shuffle

We introduce a new type of contextual grammars. Instead of considering the catenation operation we use the distributed catenation operation. The contexts are distributed catenated with words from the language, defining in this way new words from the language. We investigate several properties of the languages generated by distributed catenated contextual grammars. Finally, we also present the relations to contextual grammars with shuffled contexts and some new results of such grammars.

Fifty-Two Oppositions Between Scientific and Poetic Communication

Scientific and poetic communication are two forms of a more general type of human communication, belonging to the family of languages of discovery. So, there are many common features between them. This is just the reason why it is interesting to investigate the differences, the oppositions between these two types of communication. Trubetzkoy has pointed out that oppositions between relatively similar things are more interesting than oppositions between completely different things [27]. So, in English it is more interesting to study the opposition between the phonemes s and z, which are similar except for one feature (nonvoiced-voiced) then the opposition between s and n, which differ with respect to several features (strident-nonstrident, nonnasal-nasal, continuant- noncontinuant, nonvoiced-voiced).