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We introduce a new measure of descriptional complexity on finite automata, called the number of active states. Roughly speaking, the number of active states of an automaton A on input w counts the number of different states visited during the most economic computation of the automaton A for the word w. This concept generalizes to finite automata and regular languages in a straightforward way. We show that the number of active states of both finite automata and regular languages is computable, even with respect to nondeterministic finite automata. We further compare the number of active states to related measures for regular languages. In particular, we show incomparability to the radius of regular languages and that the difference between the number of active states and the total number of states needed in finite automata for a regular language can be of exponential order.

A synchronization problem in cellular automata has been known as the Firing Squad Synchronization Problem (FSSP), where the FSSP gives a finite-state protocol for synchronizing a large scale of cellular automata. A quest for smaller state FSSP solutions has been an interesting problem for a long time. It has been shown by Balzer (Inf Control 10:22–42, 1967), Sanders (in: Jesshope, Jossifov, Wilhelmi (eds) Proceedings of the VI international workshop on parallel processing by cellular automata and arrays, Akademie, Berlin, 1994) and Berthiaume et al. (Theoret Comput Sci 320:213–228, 2004) that there exists no 4-state FSSP solution in arrays and rings. The number four is the state lower bound in the class of FSSP protocols. Umeo et al. (Parallel Process Lett 19(2):299–313, 2009), by introducing a concept of full versus partial FSSP solutions, provided a list of the smallest 4-state symmetric powers-of-2 FSSP protocols that can synchronize any one-dimensional (1D) ring cellular automata of length (n=2^{k}) for any positive integer (k ge 1). Afterwards, Ng (in: Partial solutions for the firing squad synchronization problem on rings, ProQuest Publications, Ann Arbor, MI, 2011) also added a list of asymmetric FSSP partial solutions, thus completing the 4-state powers-of-2 FSSP partial solutions. A question whether there are any 4-state partial solutions for ring lengths other than powers-of-2 has remained open. In this paper, we answer the question by providing a new class of the smallest symmetric and asymmetric 4-state FSSP protocols that can synchronize any 1D ring of length (n=2^{k}-1) for any positive integer (k ge 2). We show that the class includes a rich variety of FSSP protocols that consists of 39 symmetric and 132 asymmetric solutions, ranging from minimum to linear synchronization time. In addition, we make an investigation into several interesting properties of those partial solutions, such as swapping general states, transposed protocols, a duality property between them, and an inclusive property of powers-of-2 solutions.

We consider the notion of a constant length queue automaton—i.e., a traditional queue automaton with a built-in constant limit on the length of its queue—as a formalism for representing regular languages. We show that the descriptional power of constant length queue automata greatly outperforms that of traditional finite state automata, of constant height pushdown automata, and of straight line programs for regular expressions, by providing optimal exponential and double-exponential size gaps. Moreover, we prove that constant height pushdown automata can be simulated by constant length queue automata paying only by a linear size increase, and that removing nondeterminism in constant length queue automata requires an optimal exponential size blow-up, against the optimal double-exponential cost for determinizing constant height pushdown automata. Finally, we investigate the size cost of implementing Boolean language operations on deterministic and nondeterministic constant length queue automata.

In this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.

We present an automated system repair framework for cyber-physical systems. The proposed framework consists of three main steps: (1) system simulation and fault detection to generate a labeled dataset, (2) identification of the repairable temporal properties leading to the faulty behavior and (3) repairing the system to avoid the occurrence of the cause identified in the second step. We express the cause as a past time signal temporal logic (ptSTL) formula and present an efficient monotonicity-based method to synthesize a ptSTL formula from a labeled dataset. Then, in the third step, we modify the faulty system by removing all behaviors that satisfy the ptSTL formula representing the cause of the fault. We apply the framework to two rich modeling formalisms: discrete-time dynamical systems and timed automata. For both of them, we define repairable formulae, the corresponding repair procedures, and illustrate them over case studies.

We investigate finite automata whose state graphs are undirected. This means that for any transition from state p to q consuming some letter a from the input there exists a symmetric transition from state q to p consuming a letter a as well. So, the corresponding language families are subregular, and in particular in the deterministic case, subreversible. In detail, we study the operational descriptional complexity of deterministic and nondeterministic undirected finite automata. To this end, the different types of automata on alphabets with few letters are characterized. Then, the operational state complexity of the Boolean operations as well as the operations concatenation and iteration is investigated, where tight upper and lower bounds are derived for unary as well as arbitrary alphabets under the condition that the corresponding language classes are closed under the operation considered.