Solomon Marcus Contributions to Theoretical Computer Science and Applications

Abstract

Solomon Marcus (1925–2016) was one of the founders of the Romanian theoretical computer science. His pioneering contributions to automata and formal language theories, mathematical linguistics and natural computing have been widely recognised internationally. In this paper we briefly present his publications in theoretical computer science and related areas, which consist in almost ninety papers. Finally we present a selection of ten Marcus books in these areas.

1. Introduction

In 2005, on the occasion of the 80th birthday anniversary of Professor Solomon Marcus, the editors of the present volume, both his disciples, together with his friend Professor G. Rozenberg, from Leiden, The Netherlands, have edited a special issue of Fundamenta Informaticae (vol. 64), with the title Contagious Creativity. This syntagma describes accurately the activity and the character of Marcus, a Renaissance-like personality, with remarkable contributions to several research areas (mathematical analysis, mathematical linguistics, theoretical computer science, semiotics, applications of all these in various areas, history and philosophy of science, education), with many disciples in Romania and abroad and with a wide recognition all around the world. Marcus projected his mathematical thinking in all domains in which he worked. Here is an example from semiotics, in the words of the Finnish musicologist and semiologist E. Tarasti (President of the International Association for Semiotic Studies (2004–2014), see N.-S. Drăgan, In Memoriam Solomon Marcus, “Hide and seek” with Solomon Marcus and Umberto Eco, Book of Abstracts, First Edition of the International Conference Semiosis in Communication, 1–3, Bucharest, 2016.):

No other semiotician is so accurate and challenging in his reasoning about fundamental issues of our discipline.

In what follows we only briefly describe his contributions to theoretical computer science and related areas, especially to automata and formal language theories, natural computing (DNA and membrane computing), applications of grammars in various domains, recursive function theory and provability in mathematics, as well as a selection of his many books in these areas. Some re-printed in S. Marcus, Words and Languages Everywhere, Polimetrica, Milano, 2007, but almost all collected in the two-volume book G. Păun (ed.), Solomon Marcus, Selected Papers—Computer Science, Spandugino Publ. House, Bucharest, 2018, abbreviated SPCS. Our choices have been guided by SPCS.

Marcus’ pioneering book Gramatici şi automate finite (Grammars and Finite Automata), published in 1964 in Romanian is one of the first monographs in the world on this subject. This book, written in a rigorous mathematical language at a time when the domain was in infancy, covers automata and language theories, closely linking finite automata and Chomsky regular grammars. The book ends with a chapter on the relations between natural languages and regular grammars, a theme which motivated Marcus’ interest and his many publications in mathematical linguistics. Unfortunately, the book, written in Romanian, was not translated into any other language; hence, it remained almost unknown internationally. This is not the case with many of his subsequent books, specifically those in mathematical linguistics, some of which will be listed in this paper. These books have been translated in several languages (French, English, German, Russian, Italian, Czech, Spanish, Greek and other languages) and then published by Academic Press, Dunod, Nauka and other well-known international publishers. Without exception, they had a very high international audience and impact.

His first paper in formal language theory was published in 1963 and it is illustrative for his permanent interest in building bridges between apparently disjoint research areas; in this case, finite automata, regular grammars, arithmetical progressions. Symmetrically, his last paper, published 50 years later, returns to bio-informatics, a domain which he somehow prognosticated (too early) in the beginning of the 70’s.

2. A Working Classification

It is difficult to classify the theoretical computer science papers of Marcus because of their inter/multi-disciplinarity. In SPCS, the papers have been classified into four large categories: Formal language theory, applications of formal language theory, bio-informatics, and recursive function theory. We will use this classification here too.

In the first class there are papers dealing with finite state grammars and automata, contextual grammars, the history of formal language theory, combinatorics on words and on infinite sequences (periodicity and quasi-periodicity, unavoidable patterns, density of words of a given length), mathematical analysis notions adapted to formal language theory, and so on.

The last category deserves a closer study, which we only suggest here: To systematically extend notions/ideas from mathematical analysis to formal language theory in general and to combinatorics on words in particular (a symmetric study is worth carrying out for applications of formal languages to other mathematical areas, e.g., number theory by classifying various classes of numbers in Chomsky’s hierarchy, characterising them with grammars, etc.). This was a direction of research programmatically explored by Marcus. The title of his 1999 paper is explicit and significant in this respect: From real analysis to discrete mathematics and back, followed by details: Symmetry, convexity, almost periodicity, and strange attractors. In the beginning of this paper he wrote:

Despite its importance, the relation between continuous and discrete mathematics is a rather neglected topic. (...) Working in real analysis in the fifties and in the sixties and then in discrete mathematics (the mathematical theory of languages), I became interested to look for the discrete analog of some facts belonging to continuous mathematics.

Among the most fruitful ideas of this kind we mention several variants of the Darboux property for languages, the basic one being the following: If we have three families of languages, ${\mathcal{L}}_1\subset {\mathcal{L}}_2\subset {\mathcal{L}}_3$, conceivably belonging to a larger hierarchy of families of languages, possibly infinite, and two languages $L_1\in {\mathcal{L}}_1$, $L_3\in {\mathcal{L}}_3\backslash {\mathcal{L}}_2$, can we find a language $L_2\in {\mathcal{L}}_2$ such that $L_1\subset L_2\subset L_3$? Various definitions of symmetry, attractors, periodicity, convexity, etc., have been extended to strings. In all cases, Marcus used to define a series of subtle variants, of the type left-, right-, almost-, pseudo-, weak-, strong-, etc. Marcus had an unbounded creativity to pose open problems, and these papers never missed them; quite a few papers solved such problems, some of them with Marcus as a coauthor.

Actually, formulating open problems and suggesting research directions is one of the specific features of “Marcus’ style”. Many of the questions formulated by Marcus were addressed by his disciples, collaborators, by researchers in mathematics and computer science from Romania and other countries. Some problems were, partially or totally, solved—many of them are still waiting for solutions.

3. A Constant Interest for Bio-Informatics

We mentioned before that in the 1970s Marcus published “too early” a paper dealing with applications of mathematical linguistics and formal language theory in biology, specifically in the genomics area. The year was 1974 and the title of the paper is Linguistic structures and generative devices in molecular genetics.

Bio-informatics can be understood in two senses, as an attempt to use computer science in biology, providing notions, tools, techniques to the biologist and, mainly in the last decades, in the opposite direction, to utilise ideas inspired from biology in developing algorithms in computer science, and in hardware too, as is the case in DNA computing—DNA molecules do computations. In his paper, Marcus considered both directions. In the first direction of research he synthesised previous approaches and results; in the second one he proposed new research vistas for using mathematical (linguistic) tools in addressing questions in the genetic area, to model the DNA and its biochemistry. Speculations about using DNA molecules as a support for computations were published only later (by M. Conrad, R. Feynman, C. H. Bennet), while the first computing model based on an operation specific to DNA recombination was introduced only in 1987 by T. Head (another friend of Marcus). However, it is worth emphasising the attention paid by Marcus, in this first paper and also in many others, to a 1965 proposal formulated by the Polish mathematician Z. Pawlak (famous for introducing in early 1990s, the rough sets), to generate proteins starting from amino acids; the method used a specific representation of amino acids and certain picture grammars. (This is the reason Marcus considered Z. Pawlak a precursor of picture grammars, a type of generative mechanisms developed later.)

Over the years, Marcus was constantly interested in the (mathematical) linguistic approach to cellular biology, to applications in genomics and life sciences. For instance, after the apparition of DNA computing in 1994, and especially after the initiation of membrane computing in 1998, he had contributed to these areas with a series of papers and participated to several international meetings dedicated to these subjects, in Romania and abroad. As expected, the inter-disciplinary approach, typical to Marcus, is always present in his contributions—here are two illustrative titles of papers in membrane computing, Membranes versus DNA and Bridging P systems and genomics, presented at the first meetings devoted to membrane computing (Curtea de Argeş, Romania, 2001, 2002). Actually, in 2002, he proposed a slogan which became folklore in this research area:

Life = DNA software + membrane hardware.

As expected, in this area too he proposed several research directions, some of them truly “non-standard” (“too” inter-disciplinary) at the first sight. We only cite two examples of ideas not yet explored: To consider membranes with a topology different from the usual one (vesicle-like membranes), where the separation between inside and outside is crisp (for example, to study membranes similar to Klein’s bottle), and, respectively, to use multisets, the sets with a multiplicity associated with their elements (the usual data structure in membrane computing) described by Pawlak rough sets.

4. Marcus Contextual Grammars

In a paper simply called “Contextual grammars” (published in 1969 in Revue Roumaine de Mathématiques Pures et Appliquées) Marcus has introduced the grammars which are now called Marcus contextual grammars, a branch of formal language theory. In fact, the paper was presented one year before in an international linguistics conference held in Stockholm, Sweden.

The paper has ten pages, but currently there probably exist more than 400 papers on contextual grammars, about two dozen of PhD and Master Theses, as well as two monographs, one published by the Publishing House of the Romanian Academy, Bucharest, 1982 in Romanian), and one by Kluwer Publishing, The Netherlands, in 1997 (Marcus Contextual Grammars), both of them authored by Gh. Păun. In the second volume of the massive Handbook of Formal Languages, Springer-Verlag, 1997 (three volumes), edited by G. Rozenberg and A. Salomaa, there are two chapters dedicated to this topic, one by Marcus, “Contextual grammars and natural languages”, which discusses motivations and developments in this area, and another more technical one, “Contextual grammars and formal languages”, by A. Ehrenfeucht, Gh. Păun, and G. Rozenberg.

The idea has the origins in algebraic linguistics: For a natural language L (over an alphabet V), with every word w over V one associates a set of contexts $⟨u,v⟩$ over V which accept w with respect to L (that is, $uwv\in L$). Can we use this process of selecting words by contexts, in order to describe a language? One can also conversely state it. The answer was initially given in the form of simple contextual grammars, triples of the form $G=(V,A,C)$, where V is an alphabet, A is a finite language over V (its elements are called axioms), and C is a finite set of contexts over V. Such a grammar generates a language $L\left(G\right)$ which contains (1) all axioms in A and (2) all strings obtained from axioms by adjoining contexts to them. More formally, $L\left(G\right)$ contains all strings of the form $u_n\cdots u_{1}x{v}_1\cdots v_n$, where $x\in A$ and $⟨u_i,v_i⟩\in C$ for all $1\le i\le n$, with $n\ge 0$; for $n=0$ the string is an axiom from A.

This simple model does not have a powerful generative capacity. Moreover, it does not take into consideration the string-contexts selectivity mentioned above. However, at the end of the paper, Marcus also proposes the contextual grammars with choice, $G=(V,A,C,\phi )$, where $\phi :V^{*}\to 2^C$ is the selection mapping (of contexts by the strings). This time, a string is in $L\left(G\right)$ if it is of the form $u_n\cdots u_{1}x{v}_1\cdots v_n$ as above with $x\in A$, $⟨u_1,v_1⟩\in \phi \left(x\right)$, and $⟨u_i,v_i⟩\in \phi (u_{i-1}\cdots u_{1}x{v}_1\cdots u_{i-1})$ for all $i=2,\cdots ,n$.

A great research program started from there, following the usual questions of formal language theory: Variants (extensions and restrictions), characterisations, generative power, comparisons of the obtained families among them and with the known families of languages, especially with those in the Chomsky hierarchy, closure and decidability properties, parsing complexity, equivalent automata, etc.

An important detail, which makes Marcus contextual grammars so attractive is the fact that they are not using, like the Chomsky grammars, nonterminal symbols, categorial auxiliary symbols: They are intrinsic grammars as each derived string belongs to the generated language.

Still, there was an embarrassing restriction in the initial model, the possibility to adjoin contexts only in the ends of the current string. A real breakthrough was proposed at the end of the 1970s, when the Vietnamese Nguyen Xuan My came to Romania to start a PhD with Marcus. In a joint paper Nguyen-Păun, the inner contextual grammars have been introduced: The contexts can be added in any place inside the current string, under the control of the selection mapping. (Formally, an inner contextual grammar is a usual contextual grammar with choice, $G=(V,A,C,\phi )$, with $\phi :V^{*}\to 2^C$, with the language $L\left(G\right)$ defined as the smallest language $L\subseteq V^{*}$ such that (i) $A\subseteq L$ and (ii) if $x_1{x}_2{x}_3\in L$ and $⟨u,v⟩\in \phi \left(x_2\right)$, then $x_{1}u{x}_{2}v{x}_3\in L$.) In this way, the generative capacity has significantly increased, the flexibility (hence the adequacy) of the model has been accordingly augmented.

Another important advance in this area was made at the beginning of the 1990s, when G. Rozenberg, A. Salomaa, A. Ehrenfeucht became interested in contextual grammars. Details can be found in Kluwer’s monograph mentioned before and in two chapters in the Handbook of Formal Languages.

Progress was rather rapid. Certain classes of contextual grammars have been proved to be relevant for modelling typical constructions in natural languages (duplication, multiple agreements, crossed agreements) and classes of contextual grammars which are mildly context sensitive in the sense requested by linguists (A. K. Joshi and others) have been introduced. They are parsable in polynomial time and contain strings whose lengths do not make large jumps—sometimes one asks only that the language be semilinear.

In this way, the impressive bibliography we mentioned above has been accumulated—and this bibliography is still growing.

5. Applications of Formal Language Theory

In this class we have included the papers devoted to applications of grammars and automata. This was a really central and continuous interest of Marcus, also passed onto his students and collaborators. The domains of applicability are very diverse: Natural and programming languages, the semiotics of folklore fairy tales, the modelling of economic processes, diplomatic negotiations, the medical diagnosis, the semiotics of theatre, action theory, learning theory, chemistry, genetics.

These applications should be placed in a more general context under the slogan linguistics as a pilot-science, a catchphrase coined by C. Levi-Strauss: Adopted, extended and transformed by Marcus it became a real research program for his Romanian school of mathematical linguistics and formal language theory.

The grounding assumption, also explored by M. Nowakowska in her book Languages of Action, Languages of Motivations, Mouton, The Hague, 1973, was that many processes/activities can be described as sequences of elementary actions (“semantic marks”), sequences which are governed by precise restrictions which can be described by syntactic rules. Thus, languages describing actions and grammars describing languages of actions came into stage. Combined with the Chomskian hypothesis that the linguistic competence is innate and influences all other competences of the human brain, Levi-Strauss’s slogan became Marcus’ formal linguistics as a pilot-science. Indeed, a large variety of processes, from fairy tales description to economic processes proved to be described, at convenient levels of abstraction, by grammars of the types initially developed in linguistics.

6. Recursive Function Theory and Provability

The last category of papers we mention deals with recursive functions and provability in mathematics; it contains fewer papers, but some of these papers have a special significance, as they clarify an important paternity in the history of computability. Specifically, they proved that the first example of a recursive function which is not primitive recursive was constructed by G. Sudan in 1927, simultaneously with and independently of W. Ackermann, who was credited before with this achievement (1928). The problem was examined by Marcus in collaboration with C. Calude and I. Ţevy, following a suggestion coming from G. C. Moisil.

It is important to mention that Marcus was constantly concerned with adequately valuing the history of the Romanian mathematics: Pointing out the priorities in this area was already one of the main goals of his well-known book Din gândirea matematică românească (From the Romanian Mathematical Thinking), Scientific and Encyclopaedic Publishing House, Bucharest, 1975.

This group also includes a few papers on provability in mathematics, at different levels of formalisation and with various tools, including proof-assistants.

7. Papers

7.1. Formal Language Theory

• S. Marcus, Automates finis, progressions arithmétiques et grammaires à un nombre fini d’etats. Comptes rendus de l’Academie des Sciences Paris, 256, 17 (1963), 3571–3574.

• S. Marcus, Sur un modéle de H. B. Curry pour le langage mathématique. Comptes rendus de l’Academie des Sciences Paris, 258, 7 (1964), 1954–1956.

• S. Marcus, Sur les grammaires à un nombre fini d’états. Cahiers de Linguistique Théorique et Appliquée, 2 (1965), 146–164.

• S. Marcus, Analytique et génératif dans la linguistique algébrique. In To Honor Roman Jakobson II, Mouton, The Hague, 1967, 1252–1261.

• S. Marcus, Contextual grammars. Revue Roumaine de Mathématiques Pures et Appliquée, 14, 10 (1969), 1525–1534; also, Preprint nr. 48, Intern. Conf. Comput. Ling., Stockholm, 1968.

• S. Marcus, Deux types nouveaux de grammaires génératives. Cahiers de Linguistique Théorique et Appliquées, 6 (1969), 67–74.

• S. Marcus, Darboux property and formal languages. Revue Roumaine de Mathématiques Pures et Appliquées, 22, 10 (1977), 1449–1451.

• S. Marcus, Problems. Bulletin of the European Association for Theoretical Computer Science, 27 (1985), 245.

• S. Marcus, Formal languages before Axel Thue? Bulletin of the European Association for Theoretical Computer Science, 34 (1988), 62.

• S. Marcus, Din istoria limbajelor formale. Al doilea Colocviu Naţional de Limbaje, Logică, Lingvistică Matematică, Braşov, iunie 1888, 1–9.

• S. Marcus, Gh. Păun, Langford strings, formal languages and contextual ambiguity, Intern. J. Computer Math., 26, 3 + 4 (1989), 179–191.

• L. Kari, S. Marcus, Gh. Păun, A. Salomaa, In the prehistory of formal languages, Gauss languages. Bulletin EATCS, 46 (1992), 124–139.

• S. Marcus, Fivefold symmetry: A generative approach. In Caiet de Semiotică. Univ. Timişoara, 9 (1992), 1–23.

• S. Marcus, Thirty-six years ago. The beginning of the formal language theory. In Salodays in Theoretical Computer Science, May 1992 (A. Atanasiu, C.S. Calude, eds.), Univ. Hyperion, Bucharest, 1993.

• S. Marcus, Symbols in a multidimensional space. In SEMIOTICS 1990 (K. Haworth, J. Deely, T. Prewitt, eds.) with SYMBOLICITY (J. Bernard, J. Deely, V. Voigt, G. Withalm, eds.), The Semiotic Soc. of America, 1993, 115–126.

• J. Dassow, S. Marcus, Gh. Păun, Iterated reading of numbers and “black-holes". Periodica Mathematica Hungarica, 27, 2 (1993), 137–152.

• J. Dassow, S. Marcus, Gh. Păun, Iterative reading of numbers; Parikh mappings, parallel rewriting, infinite sequences. Preprint of. Tech. Univ. Otto von Guericke Univ., Magdeburg, July 1993, 18 pp.

• J. Dassow, S. Marcus, Gh. Păun, Iterative reading of numbers: The ordered case. In Developments in Language Theory. At the Crossroad of Mathematics, Computer Science and Biology (G. Rozenberg, A. Salomaa, eds.), World Sci. Publ., Singapore, 1994, 157–168.

• S. Marcus, Gh. Păun, On symmetry in languages. Intern. J. Computer Math., 52, 1/2 (1994), 1–15.

• S. Marcus, Gh. Păun, Infinite words and their associated formal languages. In Salodays in Auckland (C. Calude, M.J.J. Lennon, H. Maurer, eds.), Auckland Univ. Press, 1994, 95–99.

• S. Marcus, Al. Mateescu, Gh. Păun, A. Salomaa, On symmetry in strings, sequences and languages. Intern. J. Computer Math., 54, 1/2 (1994), 1–13.

• S. Marcus, Gh. Păun, Infinite (almost periodic) words, formal languages, and dynamical systems. Bulletin EATCS, 54 (1994), 224–231.

• M. Kudlek, S. Marcus, A. Mateescu, Contextual grammars with distributed catenation and shuffle. Found. of Computation Theory, FCT, LNCS 1279 (B.S. Chlebus, L. Czeja, eds.), Springer, Berlin, 1997, 269–280.

• J. Dassow, S. Marcus, Gh. Păun, Convex and anti-convex languages. Intern. J. Computer Math., 69, 1–2 (1998), 1–16.

• S. Marcus, C. Martin-Vide, Gh. Păun, On the power of internal contextual grammars with maximal use of selectors. Conf. Automata and Formal Languages, Salgotarjan, 1996, Publicationes Mathematicae, Debrecen, 54 (1999), 933–947.

• S. Marcus, On the length of words. In Jewels are Forever. Contributions on Theoretical Computer Science in Honor of Arto Salomaa (J. Karhumaki, H. Maurer, Gh. Păun, G. Rozenberg, eds.), Springer, Berlin, 1999, 194–203.

• S. Marcus, From real analysis to discrete mathematics and back: symmetry, convexity, almost periodicity and strange attractors. Real Analysis Exchange, 25, 1 (1999-2000), 125–128.

• S. Marcus, Pseudo-slender languages and their infinite hierarchy. Ninth Intern. Conf. Automata and Formal Languages, Vasscecseny, Hungary, August 1999, 1-2.

• S. Marcus, C. Martin-Vide, V. Mitrana, Gh. Păun, A new–old class of linguistically motivated regulated grammars. Computational Linguistics in the Netherlands, 2000, Rodopi, New York, 2001, 111–125.

• S. Marcus, Bridging two hierarchies of infinite words. Journal of Universal Computer Sci., 8, 2 (2002), 292–296.

• S. Marcus, Quasiperiodic infinite words. Bulletin EATCS, 82 (2004), 170–174.

• L. Ilie, I. Petre, S. Marcus, Periodic and Sturmian languages. Information Processing Letters, 98, 6 (2006), 242–246.

• S. Marcus, Mild context-sensitivity, after twenty years. Fundamenta Informaticae, 73, 1/2 (2006), 203–204.

• T. Monteil, S. Marcus, Quasiperiodic words: multi-scale case and dynamical properties. https://arxiv.org/abs/math/0603354, March 2006.

• P. Dömösi, M. Ito, S. Marcus, Marcus contextual languages consisting of primitive words. Discrete Mathematics, 308 (2008), 4877–4881.

7.2. Applications of Formal Languages

• S. Marcus, Linguistique générative, modèles analytiques et linguistique générale. Revue Roumaine de Linguistique, 14, 4 (1969), 313–326.

• S. Marcus, Linguistics for programming languages. Revue Roumaine de Linguistique. Cahiers de Linguistique Théorique et Appliquée, 16, 1 (1970), 29–39.

• S. Fotino, S. Marcus, Gramatica basmului (I). Revista de Etnografie şi Folclor, 18, 4 (1973), 255–277.

• S. Fotino, S. Marcus, Gramatica basmului (II). Revista de Etnografie şi Folclor, 18, 5 (1973), 349–363.

• E. Celan, S. Marcus, Le diagnostique comme langage (I). Cahiers de Linguistique Théorique et Appliquée, 10, 2 (1973), 163–173.

• S. Marcus, Linguistics as a pilot science. In Current Trends in Linguistics (Th. Sebeok, ed.), Mouton, The Hague, 1974, 2871–2887, şi în Studii şi cercetări lingvistice, 20, 3 (1969), 235–245.

• S. Marcus, Applications de la théorie des langages formels en économie et organisation, Cahiers de Linguistique Théorique et Appliquée, 13, 2 (1976), 583–594. Also published in Annales de la Faculté des sciences de l’Université Nationale de Zaïre, Kinshasa, vol. 3, 1977, nr. 1, p. 125–147

• S. Marcus, Languages, grammars and negotiations. Some suggestions. In Mathematical Approaches to International Relations (M. Bunge, J. Galtung, M. Maliţa, eds.), vol. 2, Romanian Acad. of Social and Political Sci., Bucharest, 1977, 378–385.

• S. Marcus, A new generative approach to fairy-tales. Ethnologica, annexe á la publication Recherches sur l’histoire comparative des constitutions et du droit, Bucharest, 1978, 14–17.

• C. Calude, S. Marcus, Gh. Păun, The universal grammar as a hypothetical brain, Rev. Roumaine Ling. 25, 5 (1979), 479–489.

• S. Marcus, Lingvistica şi logica. Studii şi cercetări lingvistice, 30, 3 (1979), 247–249.

• Al. Balaban, M. Barasch, S. Marcus, Computer programs for the recognition of acyclic regular isoprenoid structures. MATCH - Mathematical Chemistry, 5 (1979), 239–261.

• S. Marcus, Learning, as a generative process. Revue Roumaine de Linguistique, 24 (1979), Cahiers de Linguistique Théorique et Appliquée, 16, 2 (1979), 117–130.

• S. Marcus, Semiotics of theatre: A mathematical linguistic approach, Revue roumaine de linguistique, 25, 3 (1980), 161–189.

• Al. Balaban, M. Barasch, S. Marcus, Picture grammars in Chemistry. Generation of acyclic isoprenoid structures. MATCH - Mathematical Chemistry, 8 (1980), 193–213.

• Al. Balaban, M. Barasch, S. Marcus, Computer program for the recognition of standard isoprenoid structures. MATCH - Mathematical Chemistry, 8 (1980), 215–268.

• Al. Balaban, M. Barasch, S. Marcus, Codification of acyclic isoprenoid structures using context-free grammars and pushdown automata. MATCH - Mathematical Chemistry, 12 (1981), 25–64.

• S. Marcus, La lecture générative. Degrés, 28 (1981), 61–66.

• S. Marcus, Limbaje naturale şi limbaje artificiale. Lucrările primului Colocviu Naţional de Limbaje, Logică, Lingvistică Matematică, Braşov, iunie 1886, 1–18.

• S. Marcus, Interplay of innate and acquired in some mathematical models of language learning. Revue Roumaine de Linguistique, 34, 2 (1989), 101–116.

• S. Marcus, Semiotics and formal artificial languages. In Encyclopaedia of Computer Science and Technology (A. Kent, J.G. Williams, eds.), vol. 29, Marcel Dekker Inc., New York, 1994, 393–405.

• S. Marcus, C. Martin-Vide, Gh. Păun, Contextual grammars versus natural languages. Speech and Computers Conf., SPECOM 96, St. Petersburg, 1996, 28–33.

• S. Marcus, Contextual grammars and natural languages. In Handbook of Formal Languages (G. Rozenberg, A. Salomaa, eds.), vol. II, Springer, Berlin, 1997, 215–235.

• S. Marcus, C. Martin-Vide, Gh. Păun, Contextual grammars as generative models of natural languages. Fourth Meeting on Mathematics of Language, MOL 4, Philadelphia, 1995, Computational Linguistics, 24, 2 (1998), 245–274.

• S. Marcus, Linguistic and semiotic preliminaries of contextual languages. Math. and Comput. Analysis of Natural Languages. Proc. Second Intern. Conf. on Math. Linguistics (C. Martin-Vide, ed.), Tarragona, 1996, John Benjamins, Amsterdam, 1998, 47–57.

• S. Marcus, Contextual grammars, learning processes and the Kolmogorov-Chaitin metaphor. Math. Found. Computer Sci. Workshop on Mathematical Linguistics, Brno, August 1998, Bericht 213, Univ. Hamburg, Juli 1998, 1–12.

• S. Marcus, Reading numbers as a metaphor of the universe. In BRIDGES - Math. Connections in Art, Music and Science, Southwestern College, Winfield, Kansas, 1999 (R. Sarhangi, ed.), Gilliland Printing, Maryland, 1999, 302.

• S. Marcus, History as text: Xenopol’s series between structuralism and generative formal grammars. Romanian J. Information Sci. and Technology, 5, 1-2 (2002), 5–8.

• S. Marcus, Formal languages: Foundations, prehistory, sources, and applications. In Formal Languages and Applications (C. Martín-Vide, V. Mitrana, Gh. Păun, eds.), Springer, Berlin, 2004, 11–53.

7.3. Bio-Informatics

• S. Marcus, Linguistic structures and generative devices in molecular genetics. Cahiers de Linguistique Théorique et Appliquée, 11, 1 (1974), 77–104.

• C. Calude, S. Marcus, Gh. Păun, The universal grammar as a hypothetical brain. Revue Roumaine de Linguistique, 24, 5 (1979), 479–489.

• S. Marcus, Hidden grammars. In Developments in Language Theory (G. Rozenberg, A. Salomaa, eds.), World Sci. Publ., Singapore, 1994, 453–460.

• S. Marcus, Language, at the crossroad of computation and biology. In Computing with Biomolecules. Theory and Experiments (Gh. Păun, ed.), Springer, Singapore, 1998, 1–34.

• S. Marcus, Bags and beyond them. Pre-proc. Workshop on Multiset Processing, Curtea de Argeş, Romania, 21-25 August 2000, Report CDMTCS-140, C.S. Calude, M.J. Dinneen, Gh. Păun, eds., 191–192.

• S. Marcus, Tolerance multisets. In Multiset Processing. Mathematical, Computer Science and Molecular Computing Points of View (C.S. Calude, Gh. Păun, G. Rozenberg, A. Salomaa, eds.), LNCS 2235, Springer, Berlin, 2001, 217–223.

• S. Marcus, Membranes versus DNA. Pre-proc. Workshop on Membrane Computing (WMC-CdA 2001) (C. Martin-Vide, Gh. Păun, eds.), Rovira i Virgili Univ., Tarragona, Spain, 2001, 193–198, and Fundamenta Informaticae, 49, 1-3 (2002), 223–227.

• S. Marcus, Bridging P systems and genomics. In Membrane Computing. International Workshop, WMC-CdeA 2002, Curtea de Argeş, Romania, August 2002, Revised Papers (Gh. Păun, G. Rozenberg, A. Salomaa, C. Zandron, eds.), LNCS 2597, Springer, Berlin, 2003, 371–376.

• S. Marcus, Symmetry phenomena in infinite words, with biological, philosophical and aesthetic relevance. Symmetry: Culture and Science, 14/15 (2003-2004), 477–487.

• S. Marcus, The duality of patterning in molecular genetics. In Aspects of Molecular Computing. Essays Dedicated to Tom Head on the Occasion of His 70th Birthday (N. Jonoska, Gh. Păun, eds.), LNCS 2950, Springer, Berlin, 2004, 318–321.

• S. Marcus, Z. Pawlak, a precursor of DNA computing and of picture grammars. Fundamenta Informaticae, 75, 1/4 (2007), 331–334.

• G. Ciobanu, S. Marcus, Gh. Păun, New strategies of using the rules of a P system in a maximal way. Power and complexity. Romanian J. Information Sci. and Technology, 12, 2 (2009), 157–173.

• S. Marcus, The biological cell in spectacle. In Membrane Computing. 10th Intern. Workshop, WMC 2009, Curtea de Argeş, Romania, August 2009, Revised Selected and Invited Papers (Gh. Păun, M.J. Pérez-Jiménez, A. Riscos-Núñez, G. Rozenberg, A. Salomaa, eds.), LNCS 5957, Springer, Berlin, 2010, 95–103.

• S. Istrail, S. Marcus, Alan Turing and John von Neumann—Their Brains and Their Computers. Membrane Computing, 13th Intern. Conf., Budapest, August 2012, Revised Selected Papers (E. Csuhaj-Varjú, M. Gheorghe, G. Rozenberg, A. Salomaa, G. Vaszil, eds.), LNCS 7762, Springer, Berlin, 2013, 26–35.

7.4. Recursive Function Theory and Provability

• C. Calude, S. Marcus, I. Ţevy, Sur les fonctions récursives qui ne sont pas récursives primitives, Revue Roumaine des Sciences Sociales, Série de Philosophie et Logique 19, 3 (1975), 185–188.

• C. Calude, S. Marcus, I. Ţevy, The first example of a recursive function which is not primitive recursive, Historia Mathematica 6 (1979), 380–384.

• C. Calude, S. Marcus, I. Ţevy, Recursive properties of Sudan function, Revue Roum. Math. Pures Appl., 25, 4 (1980), 503–507.

• C. Calude, S. Marcus, Sudan’s recursive but not primitive recursive function: A retrospective look, Analele Universităţii din Bucureşti, Matematică-Informatică, 38, 2 (1989), 25–30.

• C. S. Calude, S. Marcus, L. Staiger, A topological characterization of random sequences, Inform. Process. Lett. 88 (2003), 245–250.

• C. S. Calude, E. Calude, S. Marcus, Passages of proof, Bull. Eur. Assoc. Theor. Comput. Sci. 84 (2004), 167–188.

• C. S. Calude, E. Calude, S. Marcus, Proving and programming, in C. S. Calude (ed.), Randomness & Complexity, from Leibniz to Chaitin, World Scientific, Singapore, 2007, 310–321.

• C. S. Calude, S. Marcus, Mathematical proofs at a crossroad? in J. Karhumäki, H. Maurer, Gh. Păun, G. Rozenberg (eds.), Theory Is Forever, Lectures Notes in Comput. Sci. 3113, Springer-Verlag, Berlin, 2004, 15–28.

• S. Marcus, S. M. Watt, What is an equation? Proc. 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, IEEE (2012), 23–29.

• S. Marcus, Proofs and mistakes: Their syntactics, semantics, and pragmatics, Semiotica 188 (2012), 139–155.

8. Selected Books

• S. Marcus, Lingvistica matematică. Modele matematice în lingvistică (Mathematical Linguistics. Mathematical Models in Linguistics), Ed. Didactică şi Pedagogică, Bucureşti, 1963. In Romanian)

• S. Marcus, Gramatici şi automate finite (Grammars and Finite Automata), Ed. Academiei, Bucureşti, 1964. In Romanian)

• S. Marcus, Introduction mathematique a la linguistique structurale, Dunod, Paris, 1967. In French)

• S. Marcus, Algebraic Linguistics; Analytical Models, Academic Press, New York, 1967.

• S. Marcus, Introduzione alla linguistica matematica, Casa editrice Riccardo Patron, Bologna, 1970. (with E. Nicolau and S. Stati) In Italian)

• S. Marcus, Teoretiko-mnozestvennye modeli jazykov, Ed. Nauka, Moscova, 1970. In Russian)

• S. Marcus (coordinator), La sémiotique formelle du folklore. Approche linguistico-mathématique, Ed. Klincksieck, Paris—Ed. Academiei, Bucureşti, 1978. In French)

• S. Marcus, Mathematische Poetik, Athenaeum Verlag, Frankfurt/Main, 1973. In German)

• S. Marcus, Snmeia gia ta snmeia, Ed. Pneumatikos, Atena, 1981. In Greek)

• S. Marcus, Contextual Ambiguities in Natural and in Artificial Languages, Communication and Cognition, Ghent, Belgium, vol.1, 1981; vol.2, 1983. In German)

Author Contributions

Authors have contributed in equal parts. Both authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

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