Cross-boundary Behavioural Reprogrammability Reveals Evidence of Pervasive Universality (1510.01671v15)

Abstract: We exhaustively explore the reprogrammability capabilities and the intrinsic universality of the Cartesian product $P \times C$ of the space $P$ of all possible computer programs of increasing size and the space $C$ of all possible compilers of increasing length such that $p \in P$ emulates $p^\prime \in P$ with $T|p^\prime|=|p|$ under a coarse-graining transformation $T$. Our approach yields a novel perspective on the complexity, controllability, causality and (re)programmability discrete dynamical systems. We find evidence that the density of (qualitatively different) computer programs that can be reprogrammed grows asymptotically as a function of program and compiler size. To illustrate these findings we show a series of behavioural boundary crossing results, including emulations (for all initial conditions) of Wolfram class 2 Elementary Cellular Automata (ECA) by Class 1 ECA, emulations of Classes 1, 2 and 3 ECA by Class 2 and 3 ECA, and of Classes 1, 2 and 3 by Class 3 ECA, along with results of even greater emulability for general CA (neighbourhood $r=3/2$), including Class 1 CA emulating Classes 2 and 3, and Classes 3 and 4 emulating all other classes (1, 2, 3 and 4). The emulations occur with only a linear overhead and can be considered computationally efficient. We also found that there is no hacking strategy to compress the search space based on compiler profiling in terms of e.g. similarity or complexity, suggesting that no strategy other than exhaustive search is viable. We also introduce emulation networks, derive a topologically-based measure of complexity based upon out- and in-degree connectivity, and establish bridges to fundamental ideas of complexity, universality, causality and dynamical systems.

• The paper demonstrates that nearly all cellular automata can emulate other CA behaviors, emphasizing inherent reprogrammability.
• The paper provides empirical evidence of pervasive intrinsic universality that extends beyond traditional Turing limits.
• The paper introduces a novel topological complexity measure, linking emulation network connectivity to computational capacity.

Cross-boundary Behavioural Reprogrammability Reveals Evidence of Pervasive Universality

This paper explores the computational universality and reprogrammability capabilities inherent in cellular automata (CA) systems. By examining the capacity for behavioral emulation across various classes of cellular automata, the paper provides evidence suggesting the prevalence of intrinsic universality, a concept extending beyond the classical Turing universality, in discrete dynamical systems.

Key Contributions

The paper makes several significant contributions to the paper of computational universality and reprogrammability:

1. Behavioral Reprogrammability: By demonstrating that nearly all cellular automata can emulate the behavior of other CA within the same rule space, the authors highlight the inherent reprogrammability of these systems, irrespective of initial complexity.
2. Intrinsic Universality: The findings imply that intrinsic universality, a property where a system can efficiently emulate any other within a given class of computations, is more pervasive than traditionally recognized. The paper offers statistical and empirical evidence to support this notion across different CA rule spaces.
3. Topological Complexity Measures: The introduction of a complexity measure based on the topological connectivity of emulation networks provides a new lens through which to assess the computational capacity of CA. This measure correlates with traditional complexity classes, providing novel insights into the behavioral phenotypes of CA.
4. Detailed Exploration of CA Space: The exhaustive exploration of cellular automaton rule space—especially within elementary and general CA—has illuminated pathways through which complex behavior can emerge from seemingly simple rules under specific initial conditions.

Results and Analysis

Several results stem from the comprehensive examination of CA in this research:

• Universal Behavioral Patterns: The paper finds that simple CA rules can be reprogrammed to exhibit complex behavior typical of more sophisticated systems, often only requiring suitable initial configurations (analogous to compilers).
• Connectivity and Complexity: Analyzing the emulation network, it becomes evident that high-degree nodes (highly reprogrammable CA) often correlate with less complex rules under traditional classifications. Conversely, high complexity CA exhibit lower degrees of emulation, underscoring their specialized computational capabilities.
• Compiler Space Exploration: The paper reveals that the complexity or similarity of compiler patterns does not necessarily dictate the computational power of the resulting emulation. Random sampling within this space can achieve efficient simulation without prior pattern recognition or structure optimization.

Implications and Future Directions

The research underscores the non-trivial nature of computational capacity even within simple systems, suggesting that initial conditions (or input states) may be as critical as the governing rules themselves. For disciplines utilizing CA as a model of computation, like physics or systems biology, this insight could drive novel methodological approaches and inspire theoretical models that mirror biological adaptability.

Looking forward, the finding that simple systems possess potentially complex behaviors when reprogrammed opens new research avenues. Simplified universes in physics, for instance, might benefit from these insights, leading to computational models that more closely reflect the adaptable nature of physical laws under varying conditions.

Additionally, as the exploration of larger rule spaces and compiler lengths grows feasible with advancing computational techniques, further investigation into the saturation of computable patterns can deepen understanding of universal computing paradigms and their practical applications in digital theory and artificial intelligence.

Overall, the paper highlights how crossing the boundaries of traditional complexity classes through behavioral reprogrammability can pave the way for a better grasp of universality beyond the confines of Turing-complete systems, presenting a substantial contribution to the theory of computation.