Operational State Complexity and Decidability of Jumping Finite Automata

We consider jumping finite automata and their operational state complexity and decidability status. Roughly speaking, a jumping automaton is a finite automaton with a non-continuous input. This device has nice relations to semilinear sets and thus to Parikh images of regular sets, which will be exhaustively used in our proofs. In particular, we prove upper bounds on the intersection and complementation. The latter result on the complementation upper bound answers an open problem from [G. J. Lavado, G. Pighizzini, S. Seki: Operational State Complexity of Parikh Equivalence, 2014]. Moreover, we correct an erroneous result on the inverse homomorphism closure. Finally, we also consider the decidability status of standard problems as regularity, disjointness, universality, inclusion, etc. for jumping finite automata.

Spiking Neural P Systems with a Generalized Use of Rules

Spiking neural P systems (SN P systems) are a class of distributed parallel computing devices inspired by spiking neurons, where the spiking rules are usually used in a sequential way (an applicable rule is applied one time at a step) or an exhaustive way (an applicable rule is applied as many times as possible at a step). In this letter, we consider a generalized way of using spiking rules by “combining” the sequential way and the exhaustive way: if a rule is used at some step, then at that step, it can be applied any possible number of times, nondeterministically chosen. The computational power of SN P systems with a generalized use of rules is investigated. Specifically, we prove that SN P systems with a generalized use of rules consisting of one neuron can characterize finite sets of numbers. If the systems consist of two neurons, then the computational power of such systems can be greatly improved, but not beyond generating semilinear sets of numbers. SN P systems with a generalized use of rules consisting of three neurons are proved to generate at least a non-semilinear set of numbers. In the case of allowing enough neurons, SN P systems with a generalized use of rules are computationally complete. These results show that the number of neurons is crucial for SN P systems with a generalized use of rules to achieve a desired computational power.

Spiking Neural P Systems with Thresholds

Spiking neural P systems with weights are a new class of distributed and parallel computing models inspired by spiking neurons. In such models, a neuron fires when its potential equals a given value (called a threshold). In this work, spiking neural P systems with thresholds (SNPT systems) are introduced, where a neuron fires not only when its potential equals the threshold but also when its potential is higher than the threshold. Two types of SNPT systems are investigated. In the first one, we consider that the firing of a neuron consumes part of the potential (the amount of potential consumed depends on the rule to be applied). In the second one, once a neuron fires, its potential vanishes (i.e., it is reset to zero). The computation power of the two types of SNPT systems is investigated. We prove that the systems of the former type can compute all Turing computable sets of numbers and the systems of the latter type characterize the family of semilinear sets of numbers. The results show that the firing mechanism of neurons has a crucial influence on the computation power of the SNPT systems, which also answers an open problem formulated in Wang, Hoogeboom, Pan, Păun, and Pérez-Jiménez ( 2010 ).