Formal Languages in Dynamical Systems

We treat here the interrelation between formal languages and those dynamical systems that can be described by cellular automata (CA). There is a well-known injective map which identifies any CA-invariant subshift with a central formal language. However, in the special case of a symbolic dynamics, i.e. where the CA is just the shift map, one gets a stronger result: the identification map can be extended to a functor between the categories of symbolic dynamics and formal languages. This functor additionally maps topological conjugacies between subshifts to empty-string-limited generalized sequential machines between languages. If the periodic points form a dense set, a case which arises in a commonly used notion of chaotic dynamics, then an even more natural map to assign a formal language to a subshift is offered. This map extends to a functor, too. The Chomsky hierarchy measuring the complexity of formal languages can be transferred via either of these functors from formal languages to symbolic dynamics and proves to be a conjugacy invariant there. In this way it acquires a dynamical meaning. After reviewing some results of the complexity of CA-invariant subshifts, special attention is given to a new kind of invariant subshift: the trapped set, which originates from the theory of chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe

On the Properties of Language Classes Defined by Bounded Reaction Automata

Reaction automata are a formal model that has been introduced to investigate the computing powers of interactive behaviors of biochemical reactions([14]). Reaction automata are language acceptors with multiset rewriting mechanism whose basic frameworks are based on reaction systems introduced in [4]. In this paper we continue the investigation of reaction automata with a focus on the formal language theoretic properties of subclasses of reaction automata, called linearbounded reaction automata (LRAs) and exponentially-bounded reaction automata (ERAs). Besides LRAs, we newly introduce an extended model (denoted by lambda-LRAs) by allowing lambda-moves in the accepting process of reaction, and investigate the closure properties of language classes accepted by both LRAs and lambda-LRAs. Further, we establish new relationships of language classes accepted by LRAs and by ERAs with the Chomsky hierarchy. The main results include the following : (i) the class of languages accepted by lambda-LRAs forms an AFL with additional closure properties, (ii) any recursively enumerable language can be expressed as a homomorphic image of a language accepted by an LRA, (iii) the class of languages accepted by ERAs coincides with the class of context-sensitive languages.Comment: 23 pages with 3 figure

Parametrized Stochastic Grammars for RNA Secondary Structure Prediction

We propose a two-level stochastic context-free grammar (SCFG) architecture for parametrized stochastic modeling of a family of RNA sequences, including their secondary structure. A stochastic model of this type can be used for maximum a posteriori estimation of the secondary structure of any new sequence in the family. The proposed SCFG architecture models RNA subsequences comprising paired bases as stochastically weighted Dyck-language words, i.e., as weighted balanced-parenthesis expressions. The length of each run of unpaired bases, forming a loop or a bulge, is taken to have a phase-type distribution: that of the hitting time in a finite-state Markov chain. Without loss of generality, each such Markov chain can be taken to have a bounded complexity. The scheme yields an overall family SCFG with a manageable number of parameters.Comment: 5 pages, submitted to the 2007 Information Theory and Applications Workshop (ITA 2007

Calibrating Generative Models: The Probabilistic Chomsky-Schützenberger Hierarchy

A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning

Splicing Systems from Past to Future: Old and New Challenges

A splicing system is a formal model of a recombinant behaviour of sets of double stranded DNA molecules when acted on by restriction enzymes and ligase. In this survey we will concentrate on a specific behaviour of a type of splicing systems, introduced by P\u{a}un and subsequently developed by many researchers in both linear and circular case of splicing definition. In particular, we will present recent results on this topic and how they stimulate new challenging investigations.Comment: Appeared in: Discrete Mathematics and Computer Science. Papers in Memoriam Alexandru Mateescu (1952-2005). The Publishing House of the Romanian Academy, 2014. arXiv admin note: text overlap with arXiv:1112.4897 by other author

Lingvistiset kieliopit, joilla on on hyvin matala kompleksisuus

We have presented an overview of the FSIG approach and related FSIG gram- mars to issues of very low complexity and parsing strategy. We ended up with serious optimism according to which most FSIG grammars could be decom- posed in a reasonable way and then processed efficiently.Peer reviewe

Computation in Finitary Stochastic and Quantum Processes

We introduce stochastic and quantum finite-state transducers as computation-theoretic models of classical stochastic and quantum finitary processes. Formal process languages, representing the distribution over a process's behaviors, are recognized and generated by suitable specializations. We characterize and compare deterministic and nondeterministic versions, summarizing their relative computational power in a hierarchy of finitary process languages. Quantum finite-state transducers and generators are a first step toward a computation-theoretic analysis of individual, repeatedly measured quantum dynamical systems. They are explored via several physical systems, including an iterated beam splitter, an atom in a magnetic field, and atoms in an ion trap--a special case of which implements the Deutsch quantum algorithm. We show that these systems' behaviors, and so their information processing capacity, depends sensitively on the measurement protocol.Comment: 25 pages, 16 figures, 1 table; http://cse.ucdavis.edu/~cmg; numerous corrections and update