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This paper is the second part of a series of two papers dealing with bulking: a way to define quasi-order on cellular automata by comparing space-time diagrams up to rescaling. In the present paper, we introduce three notions of simulation between cellular automata and study the quasi-order structures induced by these simulation relations on the whole set of cellular automata. Various aspects of these quasi-orders are considered (induced equivalence relations, maximum elements, induced orders, etc) providing several formal tools allowing to classify cellular automata.
Gauge-invariance is a fundamental concept in Physics---known to provide mathematical justification for the fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts directly in terms of Cellular Automata. More precisely, the notions of gauge-invariance and gauge-equivalence in Cellular Automata are formalized. A step-by-step gauging procedure to enforce this symmetry upon a given Cellular Automaton is developed, and three examples of gauge-invariant Cellular Automata are examined.
The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. We then provide explicit tools to prove or disprove the existence of such a simulation between two stochastic cellular automata, even though the intrinsic simulation relation is shown to be undecidable in dimension two and higher. The key result behind this is the caracterization of equality of stochastic global maps by the existence of a coupling between the random sources. We then prove that there is a universal non-deterministic cellular automaton, but no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality.
Gauge-invariance is a mathematical concept that has profound implications in Physics---as it provides the justification of the fundamental interactions. It was recently adapted to the Cellular Automaton (CA) framework, in a restricted case. In this paper, this treatment is generalized to non-abelian gauge-invariance, including the notions of gauge-equivalent theories and gauge-invariants of configurations
We introduce the notion of adaptive synchronisation for pushdown automata, in which there is an external observer who has no knowledge about the current state of the pushdown automaton, but can observe the contents of the stack. The observer would then like to decide if it is possible to bring the automaton from any state into some predetermined state by giving inputs to it in an emph{adaptive} manner, i.e., the next input letter to be given can depend on how the contents of the stack changed after the current input letter. We show that for non-deterministic pushdown automata, this problem is 2-EXPTIME-complete and for deterministic pushdown automata, we show EXPTIME-completeness. To prove the lower bounds, we first introduce (different variants of) subset-synchronisation and show that these problems are polynomial-time equivalent with the adaptive synchronisation problem. We then prove hardness results for the subset-synchronisation problems. For proving the upper bounds, we consider the problem of deciding if a given alternating pushdown system has an accepting run with at most $k$ leaves and we provide an $n^{O(k^2)}$ time algorithm for this problem.
We introduce homing vector automata, which are finite automata augmented by a vector that is multiplied at each step by a matrix determined by the current transition, and have to return the vector to its original setting in order to accept the input. The computational power and properties of deterministic, nondeterministic, blind, non-blind, real-time and one-w