PyPhi: A toolbox for integrated information theory

Integrated information theory provides a mathematical framework to fully characterize the cause-effect structure of a physical system. Here, we introduce PyPhi, a Python software package that implements this framework for causal analysis and unfolds the full cause-effect structure of discrete dynamical systems of binary elements. The software allows users to easily study these structures, serves as an up-to-date reference implementation of the formalisms of integrated information theory, and has been applied in research on complexity, emergence, and certain biological questions. We first provide an overview of the main algorithm and demonstrate PyPhi's functionality in the course of analyzing an example system, and then describe details of the algorithm's design and implementation. PyPhi can be installed with Python's package manager via the command 'pip install pyphi' on Linux and macOS systems equipped with Python 3.4 or higher. PyPhi is open-source and licensed under the GPLv3; the source code is hosted on GitHub at https://github.com/wmayner/pyphi . Comprehensive and continually-updated documentation is available at https://pyphi.readthedocs.io/ . The pyphi-users mailing list can be joined at https://groups.google.com/forum/#!forum/pyphi-users . A web-based graphical interface to the software is available at http://integratedinformationtheory.org/calculate.html .Comment: 22 pages, 4 figures, 6 pages of appendices. Supporting information "S1 Calculating Phi" can be found in the ancillary file

Correspondence between theoretical objects and PyPhi objects.

<p>Correspondence between theoretical objects and PyPhi objects.</p

A network of nodes and its TPM.

<p>Each node has its own TPM—in this case, the truth-table of a deterministic logic gate. Yellow signifies the “ON” state; white signifies “OFF”. The system’s TPM (right) is composed of the TPMs of its nodes (left), here shown in state-by-node form (see § Representation of the TPM and probability distributions). Note that in PyPhi’s TPM representation, the first node’s state varies the fastest, according to the little-endian convention (see § 2-dimensional state-by-node form).</p

System Integrated Information

Integrated information theory (IIT) starts from consciousness itself and identifies a set of properties (axioms) that are true of every conceivable experience. The axioms are translated into a set of postulates about the substrate of consciousness (called a complex), which are then used to formulate a mathematical framework for assessing both the quality and quantity of experience. The explanatory identity proposed by IIT is that an experience is identical to the cause–effect structure unfolded from a maximally irreducible substrate (a Φ-structure). In this work we introduce a definition for the integrated information of a system (φs) that is based on the existence, intrinsicality, information, and integration postulates of IIT. We explore how notions of determinism, degeneracy, and fault lines in the connectivity impact system-integrated information. We then demonstrate how the proposed measure identifies complexes as systems, the φs of which is greater than the φs of any overlapping candidate systems

Output.

<p><b>(A)</b> The SystemIrreducibilityAnalysis object is the main output of the software. It represents the results of the analysis of the system in question. It has several attributes (grey boxes): ‘ces’ is a CauseEffectStructure object containing all of the system’s Concepts; ‘cut’ is a Cut object that represents the minimum-information partition (MIP) of the system (the partition of the system that makes the least difference to its CES); ‘partitioned_ces’ is the CauseEffectStructure of Concepts specified by the system after applying the MIP; and ‘phi’ is the Φ value, which measures the difference between the unpartitioned and partitioned CES. <b>(B)</b> A Concept represents the maximally-irreducible cause (MIC) and maximally-irreducible effect (MIE) of a mechanism in a state. The ‘mechanism’ attribute contains the indices of the mechanism elements. The ‘cause’ and ‘effect’ attributes contain MaximallyIrreducibleCause and MaximallyIrreducibleEffect objects that describe the mechanism’s MIC and MIE, respectively; each of these contains a purview, repertoire, MIP, partitioned repertoire, and <i>φ</i> value. The ‘phi’ attribute contains the concept’s <i>φ</i> value, which is the minimum of the <i>φ</i> values of the MIC and MIE.</p

Algorithm schematic at the system level.

<p>PyPhi functions are named in boxes, with arguments in grey. Arrows point from callee to caller. Functions are organized by the postulate they correspond to (left). denotes the power set.</p

Algorithm schematic at the mechanism level.

<p>PyPhi functions are named in boxes, with arguments in grey. Arrows point from callee to caller. Functions are organized by the postulate they correspond to (left). ⊗ denotes the tensor product; denotes the power set.</p