PSPACE-completeness of majority automata networks

Rectangular hybrid automata (RHA) are finite state machines with additional skewed clocks that are useful for modeling realtime systems. This paper is concerned with the uniform verification of safety properties of networks with arbitrarily many interacting RHAs. Each automaton is equipped with a finite collection of pointers to other automata that enables it to read their state. This paper presents a small model result for such networks that reduces the verification problem for a system with arbitrarily many processes to a system with finitely many processes. The result is applied to verify and discover counterexamples of inductive invariant properties for distributed protocols like Fischer's mutual exclusion algorithm and the Small Aircraft Transportation System (SATS). We have implemented a prototype tool called Passel relying on the satisfiability modulo theories (SMT) solver Z3 to check inductive invariants automatically.

Computing Research Repository, 2011

This article is set in the field of regulation networks modeled by discrete dynamical systems. It focuses on Boolean automata networks. In such networks, there are many ways to update the states of every element. When this is done deterministically, at each time step of a discretised time flow and according to a predefined order, we say that the network is updated according to a blocksequential update schedule (blocks of elements are updated sequentially while, within each block, the elements are updated synchronously). Many studies, for the sake of simplicity and with some biologically motivated reasons, have concentrated on networks updated with one particular block-sequential update schedule (more often the synchronous/parallel update schedule or the sequential update schedules). The aim of this paper is to give an argument formally proven and inspired by biological considerations in favour of the fact that the choice of a particular update schedule does not matter so much in terms of the possible and likely dynamical behaviours that networks may display.

We present a SAT-based technique to analyse properties of a network of finite automata. In particular, we show how to encode any finite transition system via Boolean formulae, and we present a reachability algorithm, which can be easily extended to work with more complicated sort of automata (for example, timed automata). We exemplify the use of the technique by means of a classic problem-“mutual exclusion".

Information and Computation, 2011

Finite automata are probably best known for being equivalent to right-linear context-free grammars and, thus, for capturing the lowest level of the Chomsky-hierarchy, the family of regular languages. Over the last half century, a vast literature documenting the importance of deterministic, nondeterministic, and alternating finite automata as an enormously valuable concept has been developed. In the present paper, we tour a fragment of this literature. Mostly, we discuss developments relevant to finite automata related problems like, for example, (i) simulation of and by several types of finite automata, (ii) standard automata problems such as fixed and general membership, emptiness, universality, equivalence, and related problems, and (iii) minimization and approximation. We thus come across descriptional and computational complexity issues of finite automata. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.

Lecture Notes in Computer Science, 2017

In this article we study the minimum number κ of additional automata that a Boolean automata network (BAN) associated with a given blocksequential update schedule needs in order to simulate a given BAN with a parallel update schedule. We introduce a graph that we call NECC graph built from the BAN and the update schedule. We show the relation between κ and the chromatic number of the NECC graph. Thanks to this NECC graph, we bound κ in the worst case between n/2 and 2n/3 + 2 (n being the size of the BAN simulated) and we conjecture that this number equals n/2. We support this conjecture with two results: the clique number of a NECC graph is always less than or equal to n/2 and, for the subclass of bijective BANs, κ is always less than or equal to n/2 + 1.

International Journal of Foundations of Computer Science

A deterministic finite automaton in which every non-empty set of states occurs as the image of the whole state set under the action of a suitable input word is called completely reachable. We characterize such automata in terms of graphs and trees.

2006

We study the computational complexity of counting the fixed point configurations (FPs), the predecessor configurations and the ancestor configurations in certain classes of network automata viewed as discrete dynamical systems. Some early results of this investigation are presented in [38, 39]. In particular, it is proven in [39] that both exact and approximate counting of FPs in the two closely related classes of Boolean network automata, called Sequential and Synchronous Dynamical Systems (SDSs and SyDSs, respectively), are computationally intractable problems when each node is required to update according to a monotone Boolean function. In the present paper, we further strengthen those results by showing that the intractability of exact enumeration of FPs of a monotone Boolean SDS or SyDS still holds even when (i) the monotone update rules are restricted to linear threshold functions, and (ii) the underlying graph is uniformly sparse. By uniform sparseness we mean that every node in the graph has its degree bounded by Ç´½µ for a small value of constant. In particular, we prove that exactly enumerating FPs in such SDSs and SyDSs remains #P-complete even when no node degree exceeds ¿. Among other consequences, we show that this result also implies intractability of determining the exact memory capacity of discrete Hopfield networks with uniformly sparse and nonnegative integer weight matrices.

Complex Systems, 1988

A bstract. We study cellular automata. with addi tive rules on finite undi rected graphs. The addition is carried ou t in some finite abelian mono id. We investigate the prob lem of deciding wheth er a given configuration has a predecessor. Depending on th e underlying monoid this prob lem is solvable in polynomial time or NP-complete. Furt hermo re, we st udy the global reversibility of cellular graph automata based on addition mo dulo two . We give a linear time algor ithm to decide reversibility of unicyclic graphs.

2008

Signal Transition Graphs (STGs) are a popular formalism for the specification of asynchronous circuits. A necessary condition for the implementability of an STG is the existence of a consistent and complete state encoding. For an important subclass of STGs, the marked graph STGs, we show that checking consistency is polynomial, but checking the existence of a complete state coding is co-NP-complete. In fact, co-NP-completeness already holds for acyclic and 1-bounded marked graph STGs and for live and 1-bounded marked graph STGs. We add some relevant results for free-choice, bounded, and general STGs.

Lecture Notes in Computer Science, 2004

The cellular automata discrete dynamical system is considered as the two-stage process: the majority rule for the change in the automata state and the rule for the change in topological relations between automata. The influence of changing topology to the cooperative phenomena, namely zero-temperature ferromagnetic phase transition, is observed.