Computational universes

Suspicions that the world might be some sort of a machine or algorithm existing ``in the mind'' of some symbolic number cruncher have lingered from antiquity. Although popular at times, the most radical forms of this idea never reached mainstream. Modern developments in physics and computer science have lent support to the thesis, but empirical evidence is needed before it can begin to replace our contemporary world view.Comment: Several corrections of typos and smaller revisions, final versio

First Steps Towards a Geometry of Computation

We introduce a geometrical setting which seems promising for the study of computation in multiset rewriting systems, but could also be applied to register machines and other models of computation. This approach will be applied here to membrane systems (also known as P systems) without dynamical membrane creation. We discuss the role of maximum parallelism and further simplify our model by considering only one membrane and sequential application of rules, thereby arriving at asynchronous multiset rewriting systems (AMR systems). Considering only one membrane is no restriction, as each static membrane system has an equivalent AMR system. It is further shown that AMR systems without a priority relation on the rules are equivalent to Petri Nets. For these systems we introduce the notion of asymptotically exact computation, which allows for stochastic appearance checking in a priori bounded (for some complexity measure) computations. The geometrical analogy in the lattice Nd0 ; d 2 N, is developed, in which a computation corresponds to a trajectory of a random walk on the directed graph induced by the possible rule applications. Eventually this leads to symbolic dynamics on the partition generated by shifted positive cones C+ p , p 2 Nd0 , which are associated with the rewriting rules, and their intersections. Complexity measures are introduced and we consider non-halting, loop-free computations and the conditions imposed on the rewriting rules. Eventually, two models of information processing, control by demand and control by availability are discussed and we end with a discussion of possible future developments

O-Minimal Invariants for Linear Loops

The termination analysis of linear loops plays a key rôle in several areas of computer science, including program verification and abstract interpretation. Such deceptively simple questions also relate to a number of deep open problems, such as the decidability of the Skolem and Positivity Problems for linear recurrence sequences, or equivalently reachability questions for discrete-time linear dynamical systems. In this paper, we introduce the class of o-minimal invariants, which is broader than any previously considered, and study the decidability of the existence and algorithmic synthesis of such invariants as certificates of non-termination for linear loops equipped with a large class of halting conditions. We establish two main decidability results, one of them conditional on Schanuel’s conjecture

Probabilistic cellular automata, invariant measures, and perfect sampling

A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The cells are updated synchronously and independently, according to a distribution depending on a finite neighborhood. We investigate the ergodicity of this Markov chain. A classical cellular automaton is a particular case of PCA. For a 1-dimensional cellular automaton, we prove that ergodicity is equivalent to nilpotency, and is therefore undecidable. We then propose an efficient perfect sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm does not assume any monotonicity property of the local rule. It is based on a bounding process which is shown to be also a PCA. Last, we focus on the PCA Majority, whose asymptotic behavior is unknown, and perform numerical experiments using the perfect sampling procedure

μ\mu-Limit Sets of Cellular Automata from a Computational Complexity Perspective

This paper concerns $\mu$-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial $\mu$-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, $\mu$-limit sets can have a $\Sigma\_3^0$-hard language, second, they can contain only $\alpha$-complex configurations, third, any non-trivial property concerning them is at least $\Pi\_3^0$-hard. We prove complexity upper bounds, study restrictions of these questions to particular classes of CA, and different types of (non-)convergence of the measure of a word during the evolution.Comment: 41 page