Counter Machines and Distributed Automata: A Story about Exchanging Space and Time

We prove the equivalence of two classes of counter machines and one class of distributed automata. Our counter machines operate on finite words, which they read from left to right while incrementing or decrementing a fixed number of counters. The two classes differ in the extra features they offer: one allows to copy counter values, whereas the other allows to compute copyless sums of counters. Our distributed automata, on the other hand, operate on directed path graphs that represent words. All nodes of a path synchronously execute the same finite-state machine, whose state diagram must be acyclic except for self-loops, and each node receives as input the state of its direct predecessor. These devices form a subclass of linear-time one-way cellular automata.Comment: 15 pages (+ 13 pages of appendices), 5 figures; To appear in the proceedings of AUTOMATA 2018

Reasoning about transfinite sequences

We introduce a family of temporal logics to specify the behavior of systems with Zeno behaviors. We extend linear-time temporal logic LTL to authorize models admitting Zeno sequences of actions and quantitative temporal operators indexed by ordinals replace the standard next-time and until future-time operators. Our aim is to control such systems by designing controllers that safely work on $\omega$-sequences but interact synchronously with the system in order to restrict their behaviors. We show that the satisfiability problem for the logics working on $\omega^k$-sequences is EXPSPACE-complete when the integers are represented in binary, and PSPACE-complete with a unary representation. To do so, we substantially extend standard results about LTL by introducing a new class of succinct ordinal automata that can encode the interaction between the different quantitative temporal operators.Comment: 38 page

Proceedings of the Eindhoven FASTAR Days 2004 : Eindhoven, The Netherlands, September 3-4, 2004

The Eindhoven FASTAR Days (EFD) 2004 were organized by the Software Construction group of the Department of Mathematics and Computer Science at the Technische Universiteit Eindhoven. On September 3rd and 4th 2004, over thirty participants|hailing from the Czech Republic, Finland, France, The Netherlands, Poland and South Africa|gathered at the Department to attend the EFD. The EFD were organized in connection with the research on finite automata by the FASTAR Research Group, which is centered in Eindhoven and at the University of Pretoria, South Africa. FASTAR (Finite Automata Systems|Theoretical and Applied Research) is an in- ternational research group that aims to lead in all areas related to finite state systems. The work in FASTAR includes both core and applied parts of this field. The EFD therefore focused on the field of finite automata, with an emphasis on practical aspects and applications. Eighteen presentations, mostly on subjects within this field, were given, by researchers as well as students from participating universities and industrial research facilities. This report contains the proceedings of the conference, in the form of papers for twelve of the presentations at the EFD. Most of them were initially reviewed and distributed as handouts during the EFD. After the EFD took place, the papers were revised for publication in these proceedings. We would like to thank the participants for their attendance and presentations, making the EFD 2004 as successful as they were. Based on this success, it is our intention to make the EFD into a recurring event. Eindhoven, December 2004 Loek Cleophas Bruce W. Watso

Decidability Problems for Self-induced Systems Generated by a Substitution

International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems