A complete worked example#
Download this page as a Jupyter notebook
This page follows the worked example of the IIT 4.0 paper (Albantakis et al.
2023) from start to finish, reproducing its published numbers: Figure 1 —
is a set of units a complex, and how irreducible is it (\(\varphi_s\))? Figure 2 — what distinctions compose its
cause-effect structure? Figure 4 — how do those distinctions bind into
relations? All three use the same small network:
pyphi.examples.iit4_2023_fig1a_substrate().
The paper’s \(\varphi_s\) values were computed before the system’s intrinsic information became part of the definition of \(\varphi_s\), so this page pins the version of IIT the paper used (see Reproduce results from earlier versions of IIT).
import pyphi
from pyphi.conf import presets
pyphi.config.progress_bars = False
# Reproduce the paper's published values under the version of IIT it used.
pyphi.config.iit = presets.iit4_2023["iit"]
The substrate: three logistic units#
Fig 1A defines three units \(A\), \(B\), \(C\). Each unit’s probability of turning ON is a logistic function (slope \(k = 4\)) of its weighted inputs, with the inputs read as \(\pm 1\) (paper Eq. 60). \(A\) and \(B\) excite each other strongly (\(\pm 0.7\)), \(C\) inhibits \(B\) (\(-0.8\)), and each unit weakly affects itself.
substrate = pyphi.examples.iit4_2023_fig1a_substrate()
substrate
| Units | A, B, C |
|---|---|
| State space | binary |
| A | B | C | |
|---|---|---|---|
| A | 1 | 1 | 1 |
| B | 1 | 1 | · |
| C | · | 1 | 1 |
| state | A | B | C |
|---|---|---|---|
| (0,0,0) | 0.119203 | 0.768525 | 0.167982 |
| (1,0,0) | 0.026597 | 0.998887 | 0.5 |
| (0,1,0) | 0.973403 | 0.401312 | 0.167982 |
| (1,1,0) | 0.880797 | 0.994514 | 0.5 |
| (0,0,1) | 0.119203 | 0.0054863 | 0.5 |
| (1,0,1) | 0.026597 | 0.598688 | 0.832018 |
| (0,1,1) | 0.973403 | 0.00111254 | 0.5 |
| (1,1,1) | 0.880797 | 0.231475 | 0.832018 |
The paper analyzes the state written “aBC”: \(A\) off, \(B\) and \(C\) on. (Lowercase marks an OFF unit.)
state = (0, 1, 1)
Figure 1: integration and exclusion#
Fig 1E asks which candidate systems exist. We compute \(\varphi_s\) for the three candidates the paper compares — the single unit \(\{A\}\), the pair \(\{A, B\}\) (“aB”), and the whole substrate:
for subset in [(0,), (0, 1), (0, 1, 2)]:
analysis = pyphi.analyze(substrate, state, subset=subset)
print(subset, round(analysis.phi, 4))
(0,) 0.0396
(0, 1) 0.1719
(0, 1, 2) 0.1339
These reproduce the paper’s Fig 1E values: \(0.04\), \(0.17\), and \(0.13\). The pair aB beats both its subset and its superset — and in fact every candidate that overlaps it — so aB is a complex. The exhaustive competition over all candidates returns aB and one other, non-overlapping complex:
for complex_ in substrate.complexes(state):
print(complex_.node_indices, round(float(complex_.phi), 4))
(2,) 0.2122
(0, 1) 0.1719
The single unit \(\{C\}\) has the globally maximal \(\varphi_s\) here; since it does not overlap aB, both exist. The paper presents aB as a complex: maximal among the candidates that share its units.
Where does aB’s \(\varphi_s\) come from? Fig 1D splits it by temporal direction. Integration is measured separately over the system’s causes and its effects, and the system is only as integrated as its weaker direction:
aB = pyphi.analyze(substrate, state, subset=(0, 1))
print("φ_c =", round(float(aB.sia.cause.phi), 4))
print("φ_e =", round(float(aB.sia.effect.phi), 4))
print("φ_s =", round(aB.phi, 4))
φ_c = 0.2446
φ_e = 0.1719
φ_s = 0.1719
\(\varphi_c = 0.24\) and \(\varphi_e = 0.17\), the paper’s published split; the
effect side is the weaker one, so \(\varphi_s = \varphi_e\). The partition
responsible — the minimum partition — is on the analysis as
aB.sia.partition.
Figure 2: the distinctions#
With the complex fixed, the composition postulate unfolds what exists within it. Every subset of aB’s units — every mechanism — is tested for an irreducible cause and effect. The irreducible ones are the complex’s distinctions, collected in the cause-effect structure:
ces = aB.ces
for d in ces.distinctions:
print(
f"{d.mechanism_label:>3} φ_d = {float(d.phi):.4f} "
f"cause {d.cause_purview} effect {d.effect_purview}"
)
a φ_d = 0.3327 cause (1,) effect (1,)
B φ_d = 0.3236 cause (0,) effect (0, 1)
aB φ_d = 0.0714 cause (1,) effect (0, 1)
Three distinctions, matching Fig 2: the first-order mechanisms \(a\) (\(\varphi_d = 0.33\)) and \(B\) (\(0.32\)), and the second-order mechanism \(aB\) (\(0.07\)). Each specifies the purviews shown — the units its cause and effect power is about. A single distinction prints its full detail, including the repertoires (the probability distributions it specifies over its purviews):
ces.distinctions[1]
| Mechanism | B |
|---|---|
| φd | 0.323594 |
| Purview | a |
|---|---|
| φ | 0.323594 |
| Specified state | (1,) |
| Purview | aB |
|---|---|
| φ | 0.861095 |
| Specified state | (1, 0) |
The whole set converts to a table with to_pandas, handy for sorting,
filtering, or exporting:
ces.to_pandas()
| phi | mechanism_state | cause_purview | cause_state | effect_purview | effect_state | |
|---|---|---|---|---|---|---|
| mechanism | ||||||
| (A,) | 0.332728 | (0,) | (B,) | (0,) | (B,) | (0,) |
| (B,) | 0.323594 | (1,) | (A,) | (1,) | (A, B) | (1, 0) |
| (A, B) | 0.071440 | (0, 1) | (B,) | (0,) | (A, B) | (1, 0) |
Figure 4: the relations#
Distinctions whose purviews overlap congruently — same units, same specified
state — bind together into relations. Fig 4 works out the relation
between the distinctions \(a\) and \(aB\), which overlap over unit \(b\). By
default PyPhi computes the relation set analytically — a closed-form summary
that answers aggregate queries without building relation objects. To examine
individual relations, enumerate them explicitly with materialize():
relations = ces.relations.materialize()
relation = next(
r for r in relations
if {tuple(m) for m in r.mechanisms} == {(0,), (0, 1)}
)
print("φ_r =", round(float(relation.phi), 4))
print("faces:", relation.num_faces)
φ_r = 0.0357
faces: 9
\(\varphi_r = 0.036\) with all \(9\) faces, the paper’s Fig 4 relation (quoted there as \(0.035\), from the rounded \(\varphi_d(aB) = 0.07\) divided over the two-unit purview union). The structure has seven relations in all — including self-relations, where a single distinction’s own cause and effect purviews overlap:
for r in relations:
mechs = [tuple(m) for m in r.mechanisms]
print(f"φ_r = {float(r.phi):.4f} mechanisms {mechs}")
φ_r = 0.0357 mechanisms [(0, 1), (0,)]
φ_r = 0.0357 mechanisms [(0, 1)]
φ_r = 0.1618 mechanisms [(0,), (1,)]
φ_r = 0.3327 mechanisms [(0,)]
φ_r = 0.0714 mechanisms [(0, 1), (1,)]
φ_r = 0.0357 mechanisms [(0, 1), (1,), (0,)]
φ_r = 0.1618 mechanisms [(1,)]
The Φ-structure, summed#
Distinctions and relations together are the complex’s \(\Phi\)-structure, and their summed \(\varphi\) is the structure integrated information \(\Phi\):
print("Σ φ_d =", round(float(ces.sum_phi_distinctions), 4))
print("Σ φ_r =", round(float(ces.sum_phi_relations), 4))
print("Φ =", round(float(ces.big_phi), 4))
Σ φ_d = 0.7278
Σ φ_r = 0.8349
Φ = 1.5627
Note that \(\Phi\) (the structure’s total, \(1.56\) here) is a different quantity
from \(\varphi_s\) (the system’s irreducibility over its minimum partition,
\(0.17\) here). Both are reported on the analysis: aB.phi is \(\varphi_s\);
aB.ces.big_phi is \(\Phi\).
Summary#
For the paper’s Fig 1A network in state aBC:
\(\varphi_s = 0.04 / 0.17 / 0.13\) for \(\{A\}\) / aB / aBC (Fig 1E), with \(\varphi_c = 0.24\), \(\varphi_e = 0.17\) for aB (Fig 1D);
aB as a complex, alongside the non-overlapping complex \(\{C\}\);
aB’s three distinctions with \(\varphi_d = 0.33, 0.32, 0.07\) and their purviews (Fig 2);
the relation \(r(\{a, aB\})\) with \(\varphi_r = 0.035\) and 9 faces (Fig 4);
the summed structure, \(\Phi = 1.56\).
Where to go next#
Theory — the same pipeline, quantity by quantity, with the full map from paper symbols to code.
The intrinsic-information requirement — how the system’s intrinsic information enters \(\varphi_s\), and why a deterministic network has \(\varphi_s = 0\).
The IIT 4.0 demo notebook — the paper’s own supplementary notebook, going deeper into the algorithm.