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# 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 {class}`~pyphi.models.ces.CauseEffectStructure`, returned as
`analysis.ces`:

```{code-cell} python
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))
```

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:

```{code-cell} python
ces.to_pandas()
```

This is the complete answer to the two questions from the [overview](overview.md):
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) | {class}`~pyphi.substrate.Substrate`, e.g. {func}`~pyphi.examples.iit4_2023_fig1a_substrate` |
| $\mathcal{T}_U$ | transition probability matrix (Eq. 1) | `substrate.tpm`, {attr}`substrate.factored_tpm <pyphi.substrate.Substrate.factored_tpm>` |
| $S$ | candidate system in a state | {class}`~pyphi.system.System`; analyzed via {func}`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 {doc}`intrinsic-information`) | `analysis.phi` (`analysis.sia.phi`) |
| $\varphi_s^{\ast}$ | maximal system integrated information (the complex's) | the $\varphi_s$ of a complex from {meth}`~pyphi.substrate.Substrate.complexes` |
| complex | maximal substrate | {meth}`~pyphi.substrate.Substrate.complexes`; the analyzed candidate is `analysis.system` |
| mechanism | subset specifying a distinction | `distinction.mechanism` |
| purview | subset a mechanism constrains | {attr}`distinction.cause_purview <pyphi.models.distinction.Distinction.cause_purview>`, {attr}`distinction.effect_purview <pyphi.models.distinction.Distinction.effect_purview>` |
| distinction | mechanism with its cause and effect state | {class}`~pyphi.models.distinction.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 | {class}`~pyphi.models.ces.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.