Abstract A reaction system consists of a background set of entities and a set of reactions. Reactions are specified by three sets of entities: reactants, inhibitors, and products. A reaction is enabled by a state (a subset of entities), if all its reactants are present in that state and none of its inhibitors. The result of a set of reactions on a given state is a new state that consists of the products of the reactions that were enabled at the original state. In this paper, we further investigate enabling equivalence. This relation equates two sets of reactions for which the states that enable all their reactions simultaneously, are the same and, moreover, their results on those states are the same. From the point of view of enabling equivalence, sets of reactions act as if they were a single (combined) reaction. We show how combined reactions characterize enabling equivalence classes. Furthermore, enabling equivalence induces a partial order in the form of a cover relation on its equivalence classes. The resulting partially ordered set turns out to be a lattice and we demonstrate how this lattice relates to the enabling cover relation introduced earlier for single reactions.

Reaction systems are a computational model originally introduced to formalize the interactions between biochemical reactions that are the basis of the functioning of the living cell. Subsets of reactions of a reaction system define result functions which leads to a concept of functional equivalence. This equivalence in turn induces a functional cover relation on the reactions of a reaction system which captures redundancies in the system. In this paper, the functional cover relation is transferred to functional equivalence classes of sets of reactions. We introduce so-called atoms as building blocks of reactions. Atoms provide a characterization of functional equivalence classes of sets of reactions and are used to prove that the functional cover relation on functional equivalence classes of sets of reactions is a lattice.

For a family of sets we consider elements that belong to the same sets within the family as companions. The global dynamics of a reactions system (as introduced by Ehrenfeucht and Rozenberg) can be represented by a directed graph, called a transition graph, which is uniquely determined by a one-out subgraph, called the 0-context graph. We consider the companion classes of the outsets of a transition graph and introduce a directed multigraph, called an essential motion, whose vertices are such companion classes. We show that all one-out graphs obtained from an essential motion represent 0-context graphs of reactions systems with isomorphic transition graphs. All such 0-context graphs are obtained from one another by swapping the outgoing edges of companion vertices.

We investigate regular languages in the context of the forbidding-enforcing systems introduced by Ehrenfeucht and Rozenberg in the variant where one fe-system defines a single language. In general, these systems may have infinite sets of rules, allowing one to define arbitrary languages. On the oth er hand when restricted to finite sets, one obtains a strict subclass of the regular languages, between the strictly locally testable and locally testable languages. We further investigate classes of enforcing systems that characterize the regular languages. These systems have infinite sets of enforcers, but can be defined using regular languages (finite state automata).

DNA expressions constitute a formal notation for DNA molecules that may contain nicks and gaps. Different DNA expressions may denote the same DNA molecule. We describe an algorithm to rewrite a given DNA expression into a DNA expression of minimal length denoting the same molecule.

DNA expressions constitute a formal notation for DNA molecules that may contain nicks and gaps. Different DNA expressions may denote the same DNA molecule. We define a (minimal) normal form for the language of DNA expressions, and describe an algorithm to rewrite a given DNA expression into the normal form.

We show that the symmetric-difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition, we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal “null space” while the third has a larger dimension.

We consider the Schur complement operation for symmetric matrices over GF(2), which we identify with graphs through the adjacency matrix representation. It is known that Schur complementation for such a matrix (i.e., for a graph) can be decomposed in

In a biological nervous system, astrocytes play an important role in the functioning and interaction of neurons, and astrocytes have excitatory and inhibitory influence on synapses. In this work, with this biological inspiration, a class of computation devices that consist of neurons and astrocytes is introduced, called spiking neural P systems with astrocytes (SNPA systems). The computation power of SNPA systems is investigated. It is proved that SNPA systems with simple neurons (all neurons have the same rule, one per neuron, of a very simple form) are Turing universal in both generative and accepting modes. If a bound is given on the number of spikes present in any neuron along a computation, then the computation power of SNPA systems is diminished. In this case, a characterization of semilinear sets of numbers is obtained.

In a biological system, if a long enough time interval is given, an enabled chemical reaction will finish its reaction in the given time interval. With this motivation, it is natural to impose a bound on the time interval when an enabled spiking rule in a spiking neural P system (SN P system, for short) remains unused. In this work, a new working mode of SN P systems is defined, which is called limited asynchronous mode. In an SN P system working in limited asynchronous mode, if a rule is enabled at some step, this rule is not obligatorily used. From this step on, if the unused rule may be used later, it should be used in the given time interval. If further spikes make the rule non-applicable, then the computation continues in the new circumstances. The computation result of a computation in an SN P system working in limited asynchronous mode is defined as the total number of spikes sent into the environment by the system. It is proved that limited asynchronous SN P systems with standard spiking rules are universal. If the number of spikes present in each neuron of a limited asynchronous SN P system with standard spiking rules is bounded during a computation, then the power of a limited asynchronous SN P system with standard spiking rules falls drastically, and we get a characterization of semilinear sets of numbers.

A variant of spiking neural P systems with positive or negative weights on synapses is introduced, where the rules of a neuron fire when the potential of that neuron equals a given value. The involved values-weights, firing thresholds, potential consumed by each rule-can be real (computable) numbers, rational numbers, integers, and natural numbers. The power of the obtained systems is investigated. For instance, it is proved that integers (very restricted: 1, -1 for weights, 1 and 2 for firing thresholds, and as parameters in the rules) suffice for computing all Turing computable sets of numbers in both the generative and the accepting modes. When only natural numbers are used, a characterization of the family of semilinear sets of numbers is obtained. It is shown that spiking neural P systems with weights can efficiently solve computationally hard problems in a nondeterministic way. Some open problems and suggestions for further research are formulated.

Given a genome, i.e., a long string over a fixed finite alphabet, the problem is to find short (dis)similar substrings. This computationally intensive task has many biological applications. We first describe an algorithm to detect substrings that have edit distances to a fixed substring at most equal to a given. We then propose an algorithm that finds the set of all substrings that have edit distances larger than to all others. Several applications are given, where attention is paid to practical biological issues such as hairpins and GC percentage. An experiment shows the potential of the methods.

The pebble tree automaton and the pebble tree transducer are enhanced by additionally allowing an unbounded number of "invisible" pebbles (as opposed to the usual ("visible" ones). The resulting pebble tree automata recognize the regular tree languages (i.e., can validate all generalized DTD's) and hence can find all matches of MSO definable n-ary patterns. Moreover, when viewed as a navigational device, they lead to an XPath-like formalism that has a path expression for every MSO definable binary pattern. The resulting pebbletree transducers can apply arbitrary MSO definable tests to (the observable part of) their configurations, they (still) have a decidable typechecking problem, and they can model the recursion mechanism of XSLT. The time complexity ofthe typechecking problem for conjunctive queries that use MSO definable binary patterns can often be reduced through the use of invisible pebbles.

String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k=1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.

We extend a classic result of Büchi, Elgot, and Trakhtenbrot: MSO definable string transductions i.e., string-to-string functions that are definable by an interpretation using monadic second-order (MSO) logic, are exactly those realized by deterministic two-way finite-state transducers, i.e., finite-state automata with a two-way input tape and a one-way output tape. Consequently, the equivalence of two mso definable string transductions is decidable. In the nondeterministic case however, MSO definable string tranductions, i.e., binary relations on strings that are mso definable by an interpretation with parameters, are incomparable to those realized by nondeterministic two-way finite-state transducers. This is a motivation to look for another machine model, and we show that both classes of MSO definable string transductions are characterized in terms of Hennie machines, i.e., two-way finite-state transducers that are allowed to rewrite their input tape, but may visit each position of their input only a bounded number of times.

A text can be defined as a word w together with a (second) linear order on its domain {1,..., |w|}. This second order may be used to define a hierarchical, tree-like, structure representing the text. The family of context-free sets of texts is investigated, i.e., sets of texts defined by context-free text grammars. In particular, those sets of texts are studied in the framework of universal algebra. This allows to compare the classical notions of equational and recognizable families in an algebra with context-free sets in the “algebra of texts”. Within this algebra the notion of equational sets coincides with the context-free sets. A grammatical characterization of the family of recognizable sets is given as a subfamily of the context-free sets of texts.