A state of a dynamic computational structure distributed in an environment: a model and its corollarie

In this work a collective of interacting stateless automata in a discrete geometric environment is considered as an integral automata-like computational dynamic object. For such distributed on the environment object different approaches to definition of the measure of state transition are possible. We propose an approach for defining what a state is.В работе коллектив взаимодействующих в дискретной среде автоматов рассматривается как цельный автоматоподобный вычислительный динамический объект. Для таких распределённых в среде объектов предлагается метод определения их функциональной эквивалентности, инвариантной относительно динамики.У роботі колектив взаємодіючих автоматів з одним станом у дискретному середовищі розглядається як цільний автоматоподібний обчислювальний динамічний об'єкт. Для таких розподілених по середовищу об'єктів пропонується метод визначення їхньої функціональної еквівалентності, інваріантної щодо динаміки

Reachability problems for PAMs

Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.Comment: 16 page

Deciding Reachability for Piecewise Constant Derivative Systems on Orientable Manifolds

© 2019 Springer-Verlag. This is a post-peer-review, pre-copyedit version of a paper published in Reachability Problems: 13th International Conference, RP 2019, Brussels, Belgium, September 11–13, 2019, Proceedings. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-030-30806-3_14A hybrid automaton is a finite state machine combined with some k real-valued continuous variables, where k determines the number of the automaton dimensions. This formalism is widely used for modelling safety-critical systems, and verification tasks for such systems can often be expressed as the reachability problem for hybrid automata. Asarin, Mysore, Pnueli and Schneider defined classes of hybrid automata lying on the boundary between decidability and undecidability in their seminal paper ‘Low dimensional hybrid systems - decidable, undecidable, don’t know’ [9]. They proved that certain decidable classes become undecidable when given a little additional computational power, and showed that the reachability question remains unsolved for some 2-dimensional systems. Piecewise Constant Derivative Systems on 2-dimensional manifolds (or PCD2m) constitute a class of hybrid automata for which decidability of the reachability problem is unknown. In this paper we show that the reachability problem becomes decidable for PCD2m if we slightly limit their dynamics, and thus we partially answer the open question of Asarin, Mysore, Pnueli and Schneider posed in [9]

Reachability problems in low-dimensional iterative maps

Abstract. In this paper we analyse the dynamics of one-dimensional piecewise maps (PAMs). We show that one-dimensional PAMs are equivalent to pseudo-billiard or so called “strange billiard ” systems. We also show that the more general class of rational functions leads to undecidability of reachability problem for one-dimensional piecewise maps with a finite number of intervals

Reachability Problems for Hierarchical Piecewise Constant Derivative Systems

In this paper, we investigate the computability and complexity of reachability problems for two-dimensional hierarchical piecewise constant derivative (HPCD) systems. The main interest in HPCDs stems from the fact that their reachability problem is on the border between decidability and undecidability, since it is equivalent to that of reachability for one-dimensional piecewise affine maps (PAMs) which is a long standing open problem. Understanding the most expressive hybrid system models that retain decidability for reachability has generated a great deal of interest over the past few years. In this paper, we show a restriction of HPCDs (called RHPCDs) which leads to the reachability problem becoming decidable. We then study which additional powers we must add to the RHPCD model to render it 1D PAM-equivalent. Finally, we show NP-hardness of reachability for nondeterministic RHPCDs

From Post Systems to the Reachability Problems for Matrix Semigroups and Multicounter Automata

Abstract. The main result of this paper is the reduction of PCP(n) to the vector reachability problem for a matrix semigroup generated by n 4 \Theta 4 integral matrices. It follows that the vector reachability problem is undecidable for a semigroup generated by 7 integral matrices of dimension 4. The question whether the vector reachability problem is decidable for n=2 and n=3 remains open. Also we show that proposed technique can be applied to Post's tag-systems. As a result we define new classes of counter automata that lie on the border between decidability and undecidability. 1 Introduction In this paper we show the connection between decision problems for Post systems and the reachability problems for matrix semigroups and counter automata. We start from the vector reachability problem for a matrix semigroup, which is a generalisation of the orbit problem [12]. The vector reachability problem is formulated as follows: "Let S be a given finitely generated semigroup of n \Theta n matrices from