Reproduce results from earlier versions of IIT#

PyPhi computes IIT as formulated by Albantakis et al. (2023) and Mayner, Marshall & Tononi (2026). Much of the published literature was computed under earlier versions of the theory: IIT 4.0 as first published in 2023, and IIT 3.0 (Oizumi, Albantakis & Tononi, 2014). PyPhi implements both, so those results can be reproduced. This page describes how the versions differ, how to compute under an earlier one, and how to compare the numbers.

import pyphi

pyphi.config.progress_bars = False

How the versions differ#

The same substrate and state give a different system integrated information under each version, because each version defines that quantity differently. On the three-XOR network:

substrate = pyphi.examples.xor_substrate()

{version: round(float(pyphi.analyze(substrate, (0, 0, 0), formalism=version).phi), 4)
 for version in ("IIT_3_0", "IIT_4_0_2023", "IIT_4_0_2026")}

{'IIT_3_0': 1.875, 'IIT_4_0_2023': 1.5, 'IIT_4_0_2026': 0.0}

"IIT_4_0_2026" is what PyPhi computes when no version is given.

IIT 4.0 as published in 2023#

Albantakis et al. (2023) define system integrated information as the smaller of the cause-side and effect-side integration, \(\varphi_s = \min(\varphi_c, \varphi_e)\). IIT’s current definition also includes the system’s intrinsic information, \(\varphi_s = \min(\varphi_c, \varphi_e, \mathit{ii}(s))\) (Mayner, Marshall & Tononi, 2026; see The intrinsic-information requirement). Everything else is the same: the partition search, the specified-state search, the distinctions, the relations, and \(\Phi\).

The two therefore give different values only where \(\mathit{ii}(s)\) is the smallest of the three terms. That happens in deterministic systems, where \(\mathit{ii}(s) = 0\) but the 2023 value is positive, and in probabilistic systems whose intrinsic information is below their integration. The XOR network above is deterministic. Most of the classic worked examples in the literature are deterministic too, and their published nonzero \(\varphi_s\) values are 2023 values.

The two-unit system aB of the IIT 4.0 paper’s Fig. 1A network is a probabilistic example. Its cause-side and effect-side integration are \(\varphi_c \approx 0.245\) and \(\varphi_e \approx 0.172\), as in the paper’s Fig. 1D, so the 2023 value is \(\varphi_s = \varphi_e \approx 0.172\), the value printed in Fig. 1E. Its intrinsic information is smaller still, so PyPhi reports \(\varphi_s \approx 0.04\):

fig1a = pyphi.examples.iit4_2023_fig1a_substrate()
{version: round(float(pyphi.analyze(fig1a, (0, 1, 1), subset=(0, 1),
                                    formalism=version, compute="sia").phi), 4)
 for version in ("IIT_4_0_2023", "IIT_4_0_2026")}

{'IIT_4_0_2023': 0.1719, 'IIT_4_0_2026': 0.0405}

The quantity the 2023 paper calls “intrinsic information” is the one PyPhi calls intrinsic specification. A result computed under the 2023 version uses the paper’s name for it on its analysis card.

IIT 3.0#

IIT 3.0 has the same overall shape as IIT 4.0: mechanisms specify causes and effects, and a system’s irreducibility is measured over its minimum partition. It differs in what it builds and how it measures it:

  • A mechanism specifies a concept: its maximally irreducible cause and effect repertoires, with a small-\(\varphi\) value. The concepts of a system form its cause–effect structure (a constellation in concept space).

  • The system’s big-\(\Phi\) measures how irreducible the whole constellation is under the minimum partition, using the earth mover’s distance between constellations.

  • There are no relations. Relations between overlapping purviews were introduced in IIT 4.0, as was the intrinsic-difference measure that replaced the earth mover’s distance.

  • Units outside the system (its background) are held at their current state for causes as well as effects. This is PyPhi 1.x’s convention, so results computed with PyPhi 1.x reproduce. Oizumi et al. (2014) held the background at its actual past state for causes (IIT 4.0, S2 Text). IIT 4.0 instead causally marginalizes the background’s past conditional on its current state (Albantakis et al., 2023, Eqs. 3–4). The difference matters only for systems smaller than the whole substrate.

analysis = pyphi.analyze(pyphi.examples.basic_substrate(), (1, 1, 0), formalism="IIT_3_0")
analysis.phi

0.1875

The result’s analysis.sia is an IIT3SystemIrreducibilityAnalysis, and its cause–effect structure is a set of concepts rather than the distinctions and relations of a \(\Phi\)-structure. For the full IIT 3.0 formalism, see Oizumi, Albantakis & Tononi (2014).

Compute under an earlier version#

Each version is defined by a complete set of settings under pyphi.config.formalism.iit: the version itself, the distance measures, the partition schemes, the tie-resolution rules, and, for IIT 3.0, the background convention and precision. PyPhi packages each set as a preset. There are three ways to apply one, and they give the same results:

  • The formalism= argument of pyphi.analyze(), for a single call: pyphi.analyze(substrate, state, formalism="IIT_4_0_2023").

  • A preset in a pyphi.config.override block, for a block of code: with pyphi.config.override(**pyphi.iit4_2023): .... The presets are pyphi.iit4_2023 and pyphi.iit3, also available from pyphi.conf.presets.

  • Replacing the formalism layer for a whole session: pyphi.config.formalism = dataclasses.replace(pyphi.config.formalism, **pyphi.iit4_2023).

with pyphi.config.override(**pyphi.iit3):
    print("version:  ", pyphi.config.formalism.iit.version)
    print("precision:", pyphi.config.precision)

version:   IIT_3_0
precision: 6

Apply the whole preset. Setting formalism.iit.version on its own raises a ConfigurationError, because the other settings would stay at their current values and the result would match no published version. The same holds in a pyphi_config.yml file: to select a version there, write out every field the preset sets. The repository ships pyphi_config_3.0.yml as a complete example for IIT 3.0.

A result computed under an earlier version records it: analysis.formalism gives the version, and the analysis card shows it in a Formalism row.

Compare the numbers#

Compare \(\varphi_s\) values only when they were computed under the same version. Compare them with a tolerance: pyphi.numerics.eq(a, b) respects the configured precision, and == does not. When a number does not match a paper, first check which version the paper used, then check the substrate, the state, and the candidate system.

pyphi.sweep() can compute the same cells under several versions in one call through its formalisms= argument, which indexes the resulting table by version; see Sweep states and subsystems.

References#

  • Albantakis L, Barbosa L, Findlay G, Grasso M, et al. (2023). Integrated information theory (IIT) 4.0: Formulating the properties of phenomenal existence in physical terms. PLOS Computational Biology 19(10): e1011465.

  • Mayner WGP, Marshall W, Tononi G. (2026). Intrinsic cause–effect power: the tradeoff between differentiation and specification. Entropy 28(4): 410.

  • Oizumi M, Albantakis L, Tononi G. (2014). From the phenomenology to the mechanisms of consciousness: Integrated Information Theory 3.0. PLOS Computational Biology 10(5): e1003588.