Infinite Networks, Halting and Local Algorithms

The immediate past has witnessed an increased amount of interest in local algorithms, i.e., constant time distributed algorithms. In a recent survey of the topic (Suomela, ACM Computing Surveys, 2013), it is argued that local algorithms provide a natural framework that could be used in order to theoretically control infinite networks in finite time. We study a comprehensive collection of distributed computing models and prove that if infinite networks are included in the class of structures investigated, then every universally halting distributed algorithm is in fact a local algorithm. To contrast this result, we show that if only finite networks are allowed, then even very weak distributed computing models can define nonlocal algorithms that halt everywhere. The investigations in this article continue the studies in the intersection of logic and distributed computing initiated in (Hella et al., PODC 2012) and (Kuusisto, CSL 2013).Comment: In Proceedings GandALF 2014, arXiv:1408.556

Experimental Study of the Shortest Reset Word of Random Automata

In this paper we describe an approach to finding the shortest reset word of a finite synchronizing automaton by using a SAT solver. We use this approach to perform an experimental study of the length of the shortest reset word of a finite synchronizing automaton. The largest automata we considered had 100 states. The results of the experiments allow us to formulate a hypothesis that the length of the shortest reset word of a random finite automaton with $n$ states and 2 input letters with high probability is sublinear with respect to $n$ and can be estimated as $1.95 n^{0.55}.

Tree transducers, L systems, and two-way machines

A relationship between parallel rewriting systems and two-way machines is investigated. Restrictions on the “copying power” of these devices endow them with rich structuring and give insight into the issues of determinism, parallelism, and copying. Among the parallel rewriting systems considered are the top-down tree transducer; the generalized syntax-directed translation scheme and the ETOL system, and among the two-way machines are the tree-walking automaton, the two-way finite-state transducer, and (generalizations of) the one-way checking stack automaton. The. relationship of these devices to macro grammars is also considered. An effort is made .to provide a systematic survey of a number of existing results

On Factor Universality in Symbolic Spaces

The study of factoring relations between subshifts or cellular automata is central in symbolic dynamics. Besides, a notion of intrinsic universality for cellular automata based on an operation of rescaling is receiving more and more attention in the literature. In this paper, we propose to study the factoring relation up to rescalings, and ask for the existence of universal objects for that simulation relation. In classical simulations of a system S by a system T, the simulation takes place on a specific subset of configurations of T depending on S (this is the case for intrinsic universality). Our setting, however, asks for every configurations of T to have a meaningful interpretation in S. Despite this strong requirement, we show that there exists a cellular automaton able to simulate any other in a large class containing arbitrarily complex ones. We also consider the case of subshifts and, using arguments from recursion theory, we give negative results about the existence of universal objects in some classes

Synchronizing Data Words for Register Automata

Register automata (RAs) are finite automata extended with a finite set of registers to store and compare data from an infinite domain. We study the concept of synchronizing data words in RAs: does there exist a data word that sends all states of the RA to a single state? For deterministic RAs with k registers (k-DRAs), we prove that inputting data words with 2k+1 distinct data from the infinite data domain is sufficient to synchronize. We show that the synchronization problem for DRAs is in general PSPACE-complete, and it is NLOGSPACE-complete for 1-DRAs. For nondeterministic RAs (NRAs), we show that Ackermann(n) distinct data (where n is the size of the RA) might be necessary to synchronize. The synchronization problem for NRAs is in general undecidable, however, we establish Ackermann-completeness of the problem for 1-NRAs. Another main result is the NEXPTIME-completeness of the length-bounded synchronization problem for NRAs, where a bound on the length of the synchronizing data word, written in binary, is given. A variant of this last construction allows to prove that the length-bounded universality problem for NRAs is co-NEXPTIME-complete