On elementary cellular automata asymptotic (a)synchronism sensitivity and complexity

TL;DR

It is shown that update schedule changes can lead to significant computational complexity jumps (from constant to superpolynomial ones) in terms of their temporal asymptotes in terms of their temporal asymptotes.

Abstract

Among the fundamental questions in computer science is that of the impact of synchronism/asynchronism on computations, which has been addressed in various fields of the discipline: in programming, in networking, in concurrence theory, in artificial learning, etc. In this paper, we tackle this question from a standpoint which mixes discrete dynamical system theory and computational complexity, by highlighting that the chosen way of making local computations can have a drastic influence on the performed global computation itself. To do so, we study how distinct update schedules may fundamentally change the asymptotic behaviors of finite dynamical systems, by analyzing in particular their limit cycle maximal period. For the message itself to be general and impacting enough, we choose to focus on a ``simple'' computational model which prevents underlying systems from having too many intrinsic degrees of freedom, namely elementary cellular automata. More precisely, for elementary cellular automata rules which are neither too simple nor too complex (the problem should be meaningless for both), we show that update schedule changes can lead to significant computational complexity jumps (from constant to superpolynomial ones) in terms of their temporal asymptotes.

Figures (5)

• Figure 1: Illustration of the execution over time of local transition functions of any BAN $f$ of size $4$ according to (top left) $\mu_\textsc{bs} = (\{0\}, \{2,3\}, \{1\})$, (top right center) $\mu_\textsc{bp} = \{(1), (2,0,3)\}$, and (bottom) $\mu_\textsc{lc} = ((1,3,2,2), (0,2,1,0))$. The $\checkmark$ symbols indicate the moments at which the automata update their states; the vertical dashed lines separate periodical time steps from each other.
• Figure 2: Order of inclusion of the defined families of periodic update modes, where per stands for "periodic".
• Figure 3: Space-time diagrams (time going downward) representing the $3$ first (periodical) steps of the evolution of configuration $x = (0,1,1,0,0,1,0,1)$ of dynamical systems (left) $(156, \mu_\textsc{bs})$, and (right) $(178, \mu_\textsc{bp})$, where $\mu_\textsc{bs} = (\{1,3,4\}, \{0,2,6\}, \{5,7\})$, and $\mu_\textsc{bp} = \{(1,3,4), (0,2,6), (5), (7)\}$, The configurations obtained at each step are depicted by lines with cells at state $1$ in black. Lines with cells at state $1$ in light gray represent the configurations obtained at substeps. Remark that $x$ belongs to a limit cycle of length $3$ (resp. $2$) in $(156, \mu_\textsc{bs})$ (resp. $(178, \mu_\textsc{bp})$).
• Figure 4: Space-time diagrams (time going downward) of configuration $0000001100000001$ following rule $156$ depending on: (a) the parallel update mode $\mu_\textsc{par} = (\llbracket 16\rrbracket)$, (b) the bipartite update mode $\mu_\textsc{bip} = (\{i \in \llbracket 16\rrbracket \mid i \equiv 0 \mod 2\}, \{i \in \llbracket 16\rrbracket \mid i \equiv 1 \mod 2\})$, (c) the block-sequential update mode $\mu_\textsc{bs} = (\{10,15\},\{0,1,5,7,8,12\}, \{4,6,9,11,14\},\{3,13\},\{2\})$, (d) the block-parallel update mode $\mu_\textsc{bp} = \{(0,1),(2,3,4),(5),(6,8,7),(11,10,9),(14,13,12),(15)\}$, (e) the local clocks update mode $\mu_\textsc{lc} = (P = (2,2,2,2,4,4,4,4,3,3,3,3,1,4,1,1), \Delta = (1,1,1,0,3,3,3,2,1,1,1,0,0,3,0,0))$.
• Figure 5: Space-time diagrams (time going downward) of configuration $0000011010111110$ following rule $178$ depending on: (a) the parallel update mode $\mu_\textsc{par} = (\llbracket 16\rrbracket)$, (b) the bipartite update mode $\mu_\textsc{bip} = (\{i \in \llbracket 16\rrbracket \mid i \equiv 0 \mod 2\}, \{i \in \llbracket 16\rrbracket \mid i \equiv 1 \mod 2\})$, (c) the block-sequential update mode $\mu_\textsc{bs} = (\{3,9,15\}, \{2,4,8,10,14\}, \{11,5,7,11,13\},\{0,6,12\})$, (d) the block-parallel update mode $\mu_\textsc{bp} =\{(0), (1), (2,3), (4,5), (6), (7), (8), (9), (10, 11), (12, 13), (14, 15)\}$, (e) the local clocks update mode $\mu_\textsc{lc} = (P = (1,1,2,2,2,2,1,1,1,4,4,4,4,4,4,4), \Delta = (0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0))$.

Theorems & Definitions (24)

• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• proof