Glossary#

Terms as PyPhi and the IIT 4.0 papers use them. Each entry points to the page that treats it. Where the IIT wiki uses a different word for the same thing, the entry says so.

substrate#: A set of units and the probability of each unit’s next state given the current state of all of them: a causal model. The wiki calls the full set of units under analysis the network and uses substrate for any candidate subset of it. Substrate; Substrate and system; Build a substrate.
system#: A candidate subset of a substrate’s units in a definite state, analyzed from its own perspective; the rest of the substrate supplies its background conditions. System; Substrate and system.
background conditions#
background#: The units outside a system. They contribute no cause–effect power of their own. They are causally marginalized conditional on the current state: held at their current state for effects, with their past states weighted by their probability given the current state for causes. Substrate and system.
transition probability matrix (TPM)#: The substrate’s probabilities of next states given current states, in state-by-node form: one row per current state (first unit changing fastest), one column per unit. Build a substrate; Transition probability matrix conventions.
little-endian#: PyPhi’s state order: the first unit is the least significant bit, so the state (1, 0, 0) is row 1 of the matrix and (0, 0, 1) is row 4. Transition probability matrix conventions.
unit#: A constituent of a substrate at the grain it is observed and manipulated in. PyPhi units may have any finite number of states; the wiki’s units are binary. Transition probability matrix conventions.
repertoire#: A probability distribution over the states of a purview, as constrained by a mechanism or a system in its current state: the cause repertoire over past states, the effect repertoire over future states. Also the set of those states itself, as in a system’s repertoire of alternatives. Distinctions and relations.
mechanism#: A subset of the system’s units, in their current state, considered for the cause–effect state it specifies. The wiki calls this a candidate mechanism and reserves mechanism for one that specifies a distinction. Distinctions and relations.
purview#: The subset of units over which a mechanism’s cause or effect is assessed; for a relation, the units of its congruent overlap. In result cards a purview is written in the state it is specified in: uppercase ON, lowercase OFF. Read a result.
order#: The number of units in a mechanism or purview: a first-order mechanism is a single unit. Distinctions and relations.
cause–effect state#
specified state#: The cause state and effect state that maximize intrinsic information: what a mechanism selects over a purview, or a system selects over itself. Only distinctions whose specified states are congruent with the system’s cause–effect state belong to its Φ-structure. System integrated information; Distinctions and relations.
congruent#: Two specified states agree on every unit they share. A distinction must be congruent with the system’s cause–effect state, and a relation exists only where purviews overlap congruently. Distinctions and relations.
intrinsic information#: For a mechanism or a system: how selectively and informatively it picks out its specified state. The system’s intrinsic information ii(s) is the smaller of its intrinsic specification and intrinsic differentiation over both directions. The intrinsic-information requirement.
intrinsic specification#: The selectivity times informativeness of the specified state (Mayner et al. 2026, Eqs. 7 and 9); Albantakis et al. (2023) call the same quantity intrinsic information. The intrinsic-information requirement.
intrinsic differentiation#: The surprisal of the specified state: how much of a repertoire of alternatives the system provides itself. Zero for a deterministic transition. The intrinsic-information requirement.
intrinsic-information requirement#: φₛ = min{φ_c, φ_e, ii(s)}: the system’s integrated information cannot exceed its intrinsic information (Mayner et al. 2026, Eq. 23). The intrinsic-information requirement.
partition#: A set of cuts that severs the connections between parts of a system, between a mechanism and its purview, or among the distinctions of a relation, to test irreducibility. System integrated information.
minimum partition (MIP)#: The partition that makes the least difference, judged on normalized φ: the system’s weakest link. Earlier papers call it the minimum information partition. Reported on every system irreducibility analysis. System integrated information.
normalized φ#: A partition’s φ divided by the maximum number of connections it could sever. The minimum partition is chosen on this value; the reported φ is the unnormalized value at the winning partition. System integrated information.
partition scheme#: The family of partitions a search ranges over: one setting for system partitions and one for mechanism partitions, fixed by the theory. Computational complexity.
integrated information (φ)#: How much a partition changes what is specified. φ_d for a distinction, φ_r for a relation, φₛ for a system. What IIT computes.
system integrated information (φₛ)#: The irreducibility of a system’s specified cause–effect state over its minimum partition; whether the system exists as one whole. analysis.phi. System integrated information.
system irreducibility analysis (SIA)#: The record of one system’s integration test: its specified cause–effect state, intrinsic information, minimum partition, φ_c, φ_e, and φₛ. analysis.sia; Read a result.
structure integrated information (Φ)#: The sum of φ over every distinction and relation of a Φ-structure; how much structure the system specifies. analysis.big_phi; not defined under IIT 3.0, whose Φ is the system-level value. The Φ-structure.
distinction#: A mechanism together with the cause and effect states it irreducibly specifies over its maximally irreducible purviews, with its φ_d. IIT 3.0’s name is concept. Distinctions and relations.
relation#: A congruent overlap among the purviews of one or more distinctions (the same units specified in the same states), with its φ_r. A single distinction whose cause and effect purviews overlap congruently has a self-relation. Distinctions and relations; Query relational structure.
relation face#: The congruent overlap of a subset of a relation’s purviews; a component of the relation. Query relational structure.
degree#: Of a relation, the number of distinctions it binds; a self-relation has degree 1. Of a relation face, the number of purviews that overlap. Query relational structure.
Φ-structure#
cause–effect structure (CES)#: The distinctions a complex specifies and the relations among them. IIT 3.0’s name is conceptual structure. CauseEffectStructure; The Φ-structure.
unfolding#: Computing the Φ-structure a complex specifies: its distinctions and the relations among them. The Φ-structure.
complex#: A set of units whose φₛ is maximal among all sets overlapping it; the exclusion postulate’s answer to which units exist as one whole. complexes(); Recursive exclusion: how complexes divide a substrate.
condensation#: The recursive exclusion cascade that splits a substrate into disjoint complexes: take the system with maximal φₛ, set its units aside, and repeat on the rest. The wiki calls the first the major complex and the others minor complexes. pyphi.condensation; Recursive exclusion: how complexes divide a substrate.
maximally irreducible cause and effect (MICE)#: For a mechanism: the cause purview and effect purview with the highest φ, the search a distinction’s computation performs. Distinctions and relations.
selection margin#: How far the winning partition (or state) was from its nearest competitor; zero means an effective tie. Control tie-breaking.
tie#: Two candidates (states, purviews, partitions, or systems) whose values agree at the configured precision. Ties are resolved by the papers’ cascades and reported with a zero margin. Control tie-breaking.
formalism#: The set of rules that turns a system into results. PyPhi computes IIT; it also implements the earlier IIT 3.0 and IIT 4.0 as published in 2023, for reproducing published results, and actual causation, which is a separate analysis. Reproduce results from earlier versions of IIT.
preset#: The complete configuration of an earlier version of IIT: iit3 or iit4_2023 in pyphi.conf.presets, applied with config.override(**preset); analyze(..., formalism="IIT_4_0_2023") selects one for a single run. Reproduce results from earlier versions of IIT.
actual causation#: The analysis of what caused what in one observed transition, measured in α (bits); a separate formalism from φ and Φ. Actual causation.
grain#: The scale at which units are defined: the constituent grain (which smaller units make up a unit) and the update grain (how many of their updates one step of the unit spans). The exclusion postulate picks the grain that maximizes φₛ. Macro units and grains.
macro unit#: A unit defined over several micro units, by coarse-graining at update grain 1 or by blackboxing over a window of updates, admitted when it is maximally irreducible within. Macro units and grains.