Randomness, Structure and Causality - Agenda

Abstracts

Effective Complexity of Stationary Process Realizations

Ay, Nihat (nay@mis.mpg.de)
SFI & Max Planck Institute

Learning Out of Equilibrium
Bell, Tony (tony@salk.edu)
UC Berkeley

The Transmission of Sense Information

Bergstrom, Carl (cbergst@u.washington.edu)
SFI & University of Washington

Optimizing Information Flow in Small Genetic Networks

Bialek, William (wbialek@Princeton.EDU)
Princeton University

To a Mathematical Theory of Evolution and Biological Creativity

Chaitin, Gregory (gjchaitin@gmail.com)
IBM Watson Research Center

We present an information-theoretic analysis of Darwin’s theory of evolution, modeled as a hill-climbing algorithm on a fitness landscape. Our space of possible organisms consists of computer programs, which are subjected to random mutations. We study the random walk of increasing fitness made by a single mutating organism. In two different models we are able to show that evolution will occur and to characterize the rate of evolutionary progress, i.e., the rate of biological creativity.

Framing Complexity

Crutchfield, James (chaos@cse.ucdavis.edu)
SFI & UC Davis

Is there a theory of complex systems? And who should care, anyway?

The Vocabulary of Grammar-Based Codes and the Logical Consistency of Texts

Debowski, Lukasz (ldebowsk@ipipan.waw.pl)

We will present a new explanation for the distribution of words in natural language which is grounded in information theory and inspired by recent research in excess entropy. Namely, we will demonstrate a theorem with the following informal statement: If a text of length ${\displaystyle n}$ describes ${\displaystyle n^{\beta }}$ independent facts in a repetitive way then the text contains at least ${\displaystyle n^{\beta }/\log n}$ different words.  In the formal statement, two modeling postulates are adopted. Firstly, the words are understood as nonterminal symbols of the shortest grammar-based encoding of the text. Secondly, the text is assumed to be emitted by a finite-energy strongly nonergodic source whereas the facts are binary IID variables predictable in a shift-invariant way. Besides the theorem, we will exhibit a few stochastic processes to which this and similar statements can be related.

Prediction, Retrodiction, and the Amount of Information Stored in the Present

Ellison, Christopher (cellison@cse.ucdavis.edu)
Complexity Sciences Center, UC Davis

We introduce an ambidextrous view of stochastic dynamical systems, comparing their forward-time and reverse-time representations and then integrating them into a single time-symmetric representation. The perspective is useful theoretically, computationally, and conceptually. Mathematically, we prove that the excess entropy--a familiar measure of organization in complex systems--is the mutual information not only between the past and future, but also between the predictive and retrodictive causal states. Practically, we exploit the connection between prediction and retrodiction to directly calculate the excess entropy. Conceptually, these lead one to discover new system invariants for stochastic dynamical systems: crypticity (information accessibility) and causal irreversibility. Ultimately, we introduce a time-symmetric representation that unifies all these quantities, compressing the two directional representations into one. The resulting compression offers a new conception of the amount of information stored in the present.

'Complexity Measures and Frustration

Feldman, David (dave@hornacek.coa.edu)
College of the Atlantic

Complexity, Parallel Computation and Statistical Physics

Machta, Jon (machta@physics.umass.edu)
SFI & University of Massachusetts

In this talk I will present some new results applying complexity measures to frustrated systems, and I will also comment on some frustrations I have about past and current work in complexity measures. I will conclude with a number of open questions and ideas for future research.

I will begin with a quick review of the excess entropy/predictive information and argue that it is a well understood and broadly applicable measure of complexity that allows for a comparison of information processing abilities among very different systems. The vehicle for this comparison is the complexity-entropy diagram, a scatter-plot of the entropy and excess entropy as model parameters are varied. This allows for a direct comparison in terms of the configurations' intrinsic information processing properties. To illustrate this point, I will show complexity-entropy diagrams for: 1D and 2D Ising models, 1D Cellular Automata, the logistic map, an ensemble of Markov chains, and an ensemble of epsilon-machines.

I will then present some new work in which a local form of the 2D excess entropy is calculated for a frustrated spin system. This allows one to see how information and memory are shared unevenly across the lattice as the system enters a glassy state. These results show that localised information theoretic complexity measures can be usefully applied to heterogeneous lattice systems. I will argue that local complexity measures for higher-dimensional and heterogeneous systems is a particularly fruitful area for future research.

Finally, I will conclude by remarking upon some of the areas of complexity-measure research that have been sources of frustration. These include the persistent notions of a universal "complexity at the edge of chaos," and the relative lack of applications of complexity measures to empirical data and/or multidimensional systems. These remarks are designed to provoke dialog and discussion about interesting and fun areas for future research.

Crypticity and Information Accessibility

Mahoney, John (jmahoney3@ucmerced.edu)
UC Merced

We give a systematic expansion of the crypticity--a recently introduced measure of the inaccessibility of a stationary process's internal state information. This leads to a hierarchy of k-cryptic processes and allows us to identify finite-state processes that have infinite crypticity--the internal state information is present across arbitrarily long, observed sequences. The crypticity expansion is exact in both the finite- and infinite-order cases. It turns out that k-crypticity is complementary to the Markovian finite-order property that describes state information in processes. One application of these results is an efficient expansion of the excess entropy--the mutual information between a process's infinite past and infinite future--that is finite and exact for finite-order cryptic processes.

Automatic Identification of Information-Processing Structures in Cellular Automata

Mitchell, Melanie (mm@cs.pdx.edu)
SFI & Portland State University

Cellular automata have been widely used as idealized models of natural spatially-extended dynamical systems.  An open question is how to best understand such systems in terms of their information-processing capabilities.   In this talk we address this question by describing several approaches to automatically identifying the structures underlying information processing in cellular automata. In particular, we review the computational mechanics methods of Crutchfield et al.,  the local sensitivity and local statistical complexity filters proposed by Shalizi et al., and the information theoretic filters proposed by Lizier et al.   We illustrate these methods by applying them to several one- and two-dimensional cellular automata that have been designed to perform the so-called density (or majority) classification task.

Statistical Mechanics of Interactive Learning

Still, Suzanne (sstill@hawaii.edu)
University of Hawaii at Manoa

Measuring the Complexity of Psychological States

Tononi, Guilio (gtononi@wisc.edu)
University of Michigan

Ergodic Parameters and Dynamical Complexity

Vilela-Mendes, Rui (vilela@cii.fc.ul.pt)
University of Lisbon

Using a cocycle formulation, old and new ergodic parameters beyond the Lyapunov exponent are rigorously characterized. Dynamical Renyi entropies and fluctuations of the local expansion rate are related by a generalization of the Pesin formula. How the ergodic parameters may be used to characterize the complexity of dynamical systems is illustrated by some examples: Clustering and synchronization, self-organized criticality and the topological structure of networks.