Dennis Pixton

Regularity of splicing languages

Abstract Motivated by the recombinant behavior of DNA, Tom Head introduced a scheme for the evolution of formal languages called splicing . We give a simpler proof of the fundamental fact that the closure of a regular language under iterated splicing using a finite number of splicing rules is again regular. We then extend this result in two directions, by incorporating circular strings and by using infinite, but regular, sets of splicing rules.

Language theory and molecular genetics: generative mechanisms suggested by DNA recombination

The stimulus for the development of the theory presented in this chapter is the string behaviors exhibited by the group of molecules often referred to collectively as the informational macromolecules. These include the molecules that play central roles in molecular biology and genetics: DNA, RNA, and the polypeptides. The discussion of the motivation for the generative systems is focused here on the recombinant behaviors of double stranded DNA molecules made possible by the presence of specific sets of enzymes. The function of this introduction is to provide richness to the reading of this chapter. It indicates the potential for productive interaction between the systems discussed and molecular biology, biotechnology, and DNA computing. However, the theory developed in this chapter can stand alone. It does not require a concern for its origins in molecular phenomena. Accordingly, only the most central points concerning the molecular connection are given here. An appendix to this chapter is included for those who wish to consider the molecular connection and possible applications in the biosciences. Here we present only enough details to motivate each term in the definition of the concept of a splicing rule that is given in the next section. The splicing rule concept is the foundation for the present chapter.

The Ehrhart Polynomial of the Birkhoff Polytope

Abstract The nth Birkhoff polytope is the set of all doubly stochastic n × n matrices, that is, those matrices with nonnegative real coefficients in which every row and column sums to one. A wide open problem concerns the volumes of these polytopes, which have been known for n

Linear and circular splicing systems

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Splicing in abstract families of languages

8. We present a new, complex-analytic way to compute the Ehrhart polynomial of the Birkhoff polytope, that is, the function counting the integer points in the dilated polytope. One reason to be interested in this counting function is that the leading term of the Ehrhart polynomial is—up to a trivial factor—the volume of the polytope. We implemented our methods in the form of a computer program, which yielded the Ehrhart polynomial (and hence the volume) of the ninth Birkhoff polytope, as well as the volume of the tenth Birkhoff polytope.

Recognizing splicing languages: Syntactic monoids and simultaneous pumping

Considers closure properties of classes of languages under the operation of iterated splicing. The main result is that full abstract families of languages are closed under splicing using a regular set of splicing rules. The author has the same result for families of circular strings, with two extra assumptions: the languages in the abstract family must be closed under cyclic permutations and the splicing scheme must be reflective. In both cases the hypotheses are satisfied by the families of regular languages and of context-free languages.<<ETX>>

Splicing to the Limit

We show that the iterated splicing operation determined by a regular H scheme (with some necessary restrictions) preserves membership in any full abstract family of languages. This involves translation of an H scheme into two alternative forms. The first form, which is closely related to the underlying biochemical operations, uses cutting and pasting rather than splicing. The second form uses matrices of languages, and in this formulation the splicing operation is translated into standard formal language operations (concatenation and quotient). Moreover, in the matrix formulation the splicing language itself may be expressed in terms of standard formal language operations, and this provides an algorithm for calculating the splicing language. As an application we use the cutting and pasting approach to extend the closure result to circular strings.

Semi-simple splicing systems

We use syntactic monoid methods, together with an enhanced pumping lemma, to investigate the structure of splicing languages. We obtain an algorithm for deciding whether a regular language is a reflexive splicing language, but the general question remains open.

Splicing Systems: Regularity and Below

We consider the result of a wet splicing procedure after the reaction has run to its completion, or limit, and we try to describe the molecules that will be present at this final stage. In language theoretic terms the splicing procedure is modeled as an H system, and the molecules that we want to consider correspond to a subset of the splicing language which we call the limit language. We give a number of examples, including one based on differential equations, and we propose a definition for the limit language. With this definition we prove that a language is regular if and only if it is the limit language of a reflexive and symmetric splicing system.

Planar homoclinic points

Generalizing a notion introduced by Păun and his coworkers, we introduce semi-simple splicing systems, in which all splicing rules have the form (a, 1; b,1) where a and b are single symbols. We find a simple graph representation of these systems, and from this representation we show that semi-simple splicing languages are reflexive splicing languages, that they contain constants, and that they are, in fact, strictly locally testable.