The Φ-structure#

The distinctions and relations of a complex, taken together, are its \(\Phi\)-structure — the cause–effect structure that, in IIT, corresponds to the quality of an experience. Its total irreducibility is the structure integrated information \(\Phi\) (“big phi”), the sum of the integrated information of every distinction and relation it contains (Albantakis et al., 2023):

\[ \Phi = \sum_{d} \varphi_d \;+\; \sum_{r} \varphi_r. \]

In PyPhi the \(\Phi\)-structure is a CauseEffectStructure, returned as analysis.ces:

import pyphi

pyphi.config.progress_bars = False

analysis = pyphi.analyze(pyphi.examples.iit4_2023_fig1a_substrate(), (0, 1, 1), subset=(0, 1))
ces = analysis.ces

(round(float(ces.sum_phi_distinctions), 3),
 round(float(ces.sum_phi_relations), 3),
 round(float(ces.big_phi), 3))

(0.728, 0.835, 1.563)

The three distinction \(\varphi_d\) values sum to \(0.728\), the seven relation \(\varphi_r\) values to \(0.835\), and their total is \(\Phi \approx 1.563\). The whole structure can be viewed as a table:

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)

This is the complete answer to the two questions from the overview: the system exists with system integrated information \(\varphi_s \approx 0.04\), and the experience it specifies has the \(\Phi\)-structure above, of quantity \(\Phi \approx 1.563\).

From paper to code#

Every named quantity of IIT 4.0 corresponds to a type or attribute in PyPhi. The table maps the symbols of Albantakis et al. (2023) and Mayner, Marshall & Tononi (2026) to the code that implements them, using the worked example (analysis = pyphi.analyze(substrate, (0, 1, 1), subset=(0, 1))).

Symbol	Quantity	In PyPhi
\(U\)	substrate (units and their interactions)	`Substrate`, e.g. `iit4_2023_fig1a_substrate()`
\(\mathcal{T}_U\)	transition probability matrix (Eq. 1)	`substrate.tpm`, `substrate.factored_tpm`
\(S\)	candidate system in a state	`System`; analyzed via `pyphi.analyze()`
\(\mathit{ii}\)	intrinsic information (maximal cause–effect state)	`analysis.sia.system_state` (per-direction `intrinsic_information`)
\(\varphi_c,\ \varphi_e\)	cause- and effect-side integrated information	`analysis.sia.cause.phi`, `analysis.sia.effect.phi`
MIP	minimum partition	`analysis.sia.partition`
\(\varphi_s\)	system integrated information, \(\min(\varphi_c, \varphi_e, \mathit{ii}(s))\) (see The intrinsic-information requirement)	`analysis.phi` (`analysis.sia.phi`)
\(\varphi_s^{\ast}\)	maximal system integrated information (the complex’s)	the \(\varphi_s\) of a complex from `complexes()`
complex	maximal substrate	`complexes()`; the analyzed candidate is `analysis.system`
mechanism	subset specifying a distinction	`distinction.mechanism`
purview	subset a mechanism constrains	`distinction.cause_purview`, `distinction.effect_purview`
distinction	mechanism with its cause and effect state	`Distinction`; `analysis.ces.distinctions`
\(\varphi_d\)	distinction integrated information	`distinction.phi`
relation	congruent overlap of distinctions	`Relation`; `analysis.ces.relations` (closed-form by default, enumerate with `.materialize()`/`.strongest()`)
\(\varphi_r\)	relation integrated information	`relation.phi`
\(\Phi\)-structure	the distinctions and relations together	`CauseEffectStructure`; `analysis.ces`
\(\Phi\)	structure integrated information	`analysis.ces.big_phi`

The worked example on the preceding pages walks this table from top to bottom.