Specific ergodicity: an informative indicator for invertible computational media

• Authors: Tommaso Toffoli;Lev B. Levitin
• Venue: Proceedings of the 2nd conference on Computing frontiers
• Year: 2005

Specific ergodicity asks, for an invertible cellular automaton, lattice gas, or similar indefinitely-extended computational medium, what fraction of the information needed to specify an individual state is still missing after one is told the computational trajectory to which that state belongs. While the well-known distinction between "ergodic" and "nonergodic" for a dynamical system is an all-or-nothing classification, specific ergodicity---with range in the [0,1] interval---provides a continuous parameter which may be interpreted as "degree of ergodicity." Moreover, while the property of a system's being ergodic can only refer to the system as a whole, specific ergodicity is an intensive quantity (ie it factors out the size of the system); thus, for a spatially-distributed, homogeneous computational system such as a cellular automaton or a lattice gas, in the limit of infinite system size this quantity reflects an intrinsic property of the material that makes up the computational medium, abstracting from its specific size or shapeWe provide the conceptual background, present theoretical and numerical results, and discuss the relevance of specific ergodicity to a number of concrete research questions.Values of specific ergodicity for a variety of systems of actual interest turn out to be well distributed over its entire range; in this sense, this quantity is an informative indicator. Indeed, besides representing a useful parameter in the classification of distributed computational media, specific ergodicity provides a "sense of direction" in issues such as protection from noise, design of self-organizing media, necessary conditions for the emergence and persistence of life, effectiveness of a parallel architecture as a "programmable medium," and a number of topics in nanotechnology