D0L Systems Research Articles

Networks of Watson-Crick D0L systems with communication by substrings

Watson-Crick D0L systems (WD0L systems) are augmented variants of D0L systems defined over a DNA-like alphabet, where each letter has a complementary letter and this relation is symmetric. WD0L systems operate under a control that is inspired by the well-known phenomenon of Watson-Crick complementarity of the double helix of DNA. Depending on a trigger, the standard D0L rewriting step is applied either to the string or to its complementary string. In this paper, we examine extended networks of standard Watson-Crick D0L systems (ENSWD0L systems) with a variant of incomplete communication. An NWD0L system is a finite set of WD0L systems defined over a common DNA-like alphabet and operating in a synchronized manner. After rewriting their own strings in the WD0L manner, they communicate copies of certain generated strings (the so-called good strings) to the other nodes. In some previous papers, it was shown that ENSWD0L systems are computationally complete, and their computational power does not change if the communicated string is a non-empty prefix (non-empty suffix) of the generated string. We strengthen the previous results, namely we show that ENSWD0L systems are computationally complete even if the communicated string is an arbitrary substring of the generated string.

Watson-crick (D)0 L systems: a survey

AbstractWatson-Crick L systems are string generating formal models obtained by augmenting Lindenmayer systems with the Watson-Crick morphism inspired by the Watson-Crick complementarity principle. The Watson-Crick morphism is controlled by a trigger—a boolean condition eventually met by the generated word. Despite their simplicity, Watson-Crick (D)0 L systems have interesting mathematical properties, a strong generative power, and they can also characterize some open decision problems. Furthermore, networks of Watson-Crick D0L systems are capable of trading space for time, and thus of characterizing linear time solutions to intractable problems. We provide a survey of results in this field since its origin in 1997 to the present, and we conclude with a series of open research problems and ideas.

D0L-System Inference from a Single Sequence with a Genetic Algorithm

In this paper, we proposed a new method for image-based grammatical inference of deterministic, context-free L-systems (D0L systems) from a single sequence. This approach is characterized by first parsing an input image into a sequence of symbols and then, using a genetic algorithm, attempting to infer a grammar that can generate this sequence. This technique has been tested using our test suite and compared to similar algorithms, showing promising results, including solving the problem for systems with more rules than in existing approaches. The tests show that it performs better than similar heuristic methods and can handle the same cases as arithmetic algorithms.

Discrete Watson–Crick dynamical systems

We generalize previous ideas by Mihalache and Salomaa to define discrete Watson–Crick dynamical systems. We study decision problems for these systems and apply our results to Watson–Crick D0L systems.

A new bound for the D0L language equivalence problem

We study the language equivalence problem for smooth and loop-free D0L systems. We show that the number of initial terms in the associated D0L sequences we have to consider to decide language equivalence depends only on the cardinality of the underlying alphabet.

The sequence equivalence problem for primitive D0L systems

The D0L sequence equivalence problem consists of deciding, given two morphisms g:X⁎→X⁎, h:X⁎→X⁎ and a word w∈X⁎, whether or not gi(w)=hi(w) for all i⩾0. We show that in case of primitive morphisms, to decide the D0L sequence equivalence problem, it suffices to consider the terms of the sequences with i<7n3nlogn, where n is the cardinality of X. A smaller bound is obtained for primitive looping morphisms.

Marked D0L systems and the [formula omitted]-conjecture

We show that to test the equivalence of two D0L sequences over an n-letter alphabet generated by marked morphisms it suffices to compare the first 2n+1 initial terms of the sequences. Under an additional condition it is enough to consider the 2n initial terms.

COMPARING SUBWORD OCCURRENCES IN BINARY D0L SEQUENCES

We investigate binary words and ω-words, mainly those generated by D0L systems. We compare the numbers of occurrences of ab and ba as (scattered) subwords by introducing a 'difference function'. Of special interest are the 'balanced' or 'fair' words for which the function assumes the value 0. If the three first words in a D0L sequence are fair, so are all words in the sequence. Two kinds of matrices constitute a useful technical tool in the study. An explicit formula, leading to various consequences, is obtained for the difference function.

THE D0L ω-EQUIVALENCE PROBLEM

Culik II and Harju have shown that the ω-equivalence problem is decidable for D0L systems. We give a new proof of this result.

An [formula omitted]-bound for the ultimate equivalence problem of certain D0L systems over an n-letter alphabet

The ultimate equivalence problem for D0L systems consists of deciding, given two morphisms g : X * ⟶ X * , h : X * ⟶ X * and a word w ∈ X * , whether or not g i ( w ) = h i ( w ) for all but finitely many i ⩾ 0 . We show that for a large class of D0L systems, to decide the ultimate equivalence problem, it suffices to check whether or not g i ( w ) = h i ( w ) holds for suitably chosen card ( X ) 2 consecutive values of i.

The Structure of Infinite Solutions of Marked and Binary Post Correspondence Problems

In the infinite Post Correspondence Problem an instance (h,g) consists of two morphisms h and g, and the problem is to determine whether or not there exists an infinite word ω such that h(ω) = g(ω). This problem is undecidable in general, but it is known to be decidable for binary and marked instances. A morphism is binary if the domain alphabet is of size 2, and marked if each image of a letter begins with a different letter. We prove that the solutions of a marked instance form a set Eω ⋃ E* (P ⋃ F), where P is a finite set of ultimately periodic words, E is a finite set of solutions of the PCP, and F is a finite set of morphic images of fixed points of D0L systems. We also establish the structure of infinite solutions of the binary PCP.

Bounds for the D0L language equivalence problem

We have recently proved that there is a bound for the sequence equivalence problem of polynomially bounded D0L systems depending only on the cardinality of the underlying alphabet. In this paper we deduce a similar bound for the language equivalence problem of polynomially bounded D0L systems. More generally, we prove that if a given class of D0L systems (satisfying certain natural conditions) has a uniform bound for sequence equivalence then it also has a uniform bound for language equivalence.

Watson–Crick D0L systems: generative power and undecidable problems

The properties of Watson–Crick D0L system, a language–theoretical formalism inspired by natural DNA processing, are studied. The model incorporates the iterated D0L-like morphism and the DNA complementarity principle represented by a letter-to-letter morphism. These two morphisms are connected by a natural condition called the trigger.We show first that this very simple model has rather unexpected power; it can closely and simply simulate any Minsky register machine. As a consequence, any recursively enumerable language can be obtained as a projection of the language of some standard Watson–Crick D0L system. Finally, we show that the graph reachability problem, equivalence problems and some other problems of standard Watson–Crick D0L systems are undecidable.

A bound for theω-equivalence problem of polynomial D0L systems

We give a bound for the ω -equivalence problem of polynomially bounded D0L systems which depends only on the size of the underlying alphabet.

Decidability results for Watson–Crick D0L systems with nonregular triggers

We show that sequence equivalence, language equivalence, growth equivalence and road equivalence are decidable for standard Watson–Crick D0L systems having bounded balance.

Watson–Crick D0L systems: the power of one transition

We investigate the class of functions computable by uni-transitional Watson–Crick D0L systems: only one complementarity transition is possible during each derivation. The class is characterized in terms of a certain min-operation applied to Z-rational functions. We also exhibit functions outside the class, and show that the basic decision problems are equivalent or harder than a celebrated open problem. For instance, the latter alternative applies to the growth-bound problem for functions in the class.

The Equivalence Problem of Polynomially Bounded D0L Systems— a Bound Depending Only on the Size of the Alphabet

Abstract. We give a bound for the sequence equivalence problem of polynomially bounded D0L systems which depends only on the size of the underlying alphabet.

REMARKS CONCERNING THE D0L ω-EQUIVALENCE PROBLEM

Culik II and Harju have shown that the ω-equivalence problem is decidable fot D0L systems. We give a partially new proof of this result by using elementary morphisms.

Universal computation with Watson-Crick D0L systems

Watson-Crick D0L systems, introduced in 1997 by Mihalache and Salomaa, arise from two major principles: the Lindenmayer rewriting and the Watson-Crick complementarity principle. Complementarity can be viewed as a purely language-theoretic operation. Majority of a certain type of symbols in a string (purines vs. pyrimidines) triggers a transition to the complementary string. The paper deals with an expressive power of deterministic interactionless Watson-Crick Lindenmayer systems. A rather surprising result is obtained: these systems, consisting of iterated morphism and a basic DNA operation, are able to compute any Turing computable function.

The Equivalence Problem for DF0L Languages and Power Series

We show that equivalence is decidable for D0L systems with finite axiom sets. We discuss also DF0L power series and solve their equivalence problem over computable fields.