A10 — Regularity / INUS Conditions

This framework asks: is this component an Insufficient but Necessary part of an Unnecessary but Sufficient set for producing the behavior?

Mackie’s INUS conditions (1965) provide a formal account of causation under redundancy: a cause is typically not sufficient on its own (Insufficient), but is a required part (Necessary) of some condition-set that is sufficient (Sufficient), though that set may not be the only sufficient set (Unnecessary). In circuit terms: a head is an INUS condition if it is necessary within its particular sub-circuit but the model has alternative sub-circuits that can also produce the behavior. This directly formalizes the “backup circuit” phenomenon observed in many transformer models.

This framework bridges A01’s necessity testing with A08’s redundancy analysis by asking a structured question: for each component, identify all minimal sufficient sets it belongs to, and test its necessity within each set. The result is a map of the model’s causal redundancy structure at a finer grain than binary “necessary / not necessary.”

Theoretical grounding

Source	Year	Key contribution
Mackie, “Causes and Conditions”	1965	INUS conditions: causes as parts of minimal sufficient sets
Pearl, Causality	2000/2009	Structural formalization of regularity accounts
McGrath et al., arXiv 2307.15771	2023	Hydra effect as empirical evidence of INUS structure in transformers
Wang et al., arXiv 2211.00593	2022	Multiple sufficient paths in the IOI circuit

Core concept: minimal sufficient sets

A minimal sufficient set (MSS) for behavior ( B ) is a set of components ( S ) such that: (1) ( S ) is sufficient for ( B ) (the circuit produces the behavior when only ( S ) is active), and (2) no proper subset of ( S ) is sufficient. Component ( c ) is an INUS condition if:

[ c \in S_i \text{ (Necessary within } S_i\text{)} \quad \text{and} \quad \exists S_j \neq S_i \text{ also sufficient (}S_i \text{ is Unnecessary)} ]

The INUS structure of a circuit can be represented as a disjunction of conjunctions (DNF): ( B \iff (c_1 \wedge c_2 \wedge c_3) \vee (c_1 \wedge c_4 \wedge c_5) \vee \ldots ). Each conjunct is a minimal sufficient set. Components appearing in multiple conjuncts are “harder” INUS conditions (more broadly necessary); components in only one conjunct are more replaceable.

Why INUS matters for MI

Standard necessity/sufficiency testing treats each component in isolation: “is head A necessary?” “is set S sufficient?” INUS conditions reveal the combinatorial structure of causation. In the IOI circuit, the name-mover heads are individually insufficient (neither alone produces correct output) but jointly sufficient — and they form one of potentially several sufficient configurations (backup name movers exist). This cannot be captured by per-component scores alone; it requires exhaustive or heuristic search over component combinations.

The INUS framework also predicts ablation sensitivity: components appearing in many minimal sufficient sets are “harder to kill” because alternative pathways exist. Components in only one MSS are critical failure points.

Instruments under A10

C39 — INUS Conditions (39_inus_conditions.py)

Instrument status: This script is a stub awaiting implementation. The directory listing confirms it does not yet exist.

An implementation would require:

1. Exhaustive or heuristic search over component subsets to identify all minimal sufficient sets (those achieving > threshold faithfulness when only they are active).
2. Necessity testing within each sufficient set (ablating one component from the set and checking whether sufficiency breaks).
3. INUS classification for each component: how many MSS it appears in, and whether it is necessary in all of them (making it a full necessary condition) or only some (true INUS).
4. Output: A table mapping each component to its INUS status, the number of minimal sufficient sets it belongs to, and the size of those sets.

The computational cost scales combinatorially with circuit size, so practical implementations would use greedy search or build on A01’s activation patching scores to prune the search space.

Planned usage:

uv run python 39_inus_conditions.py --tasks ioi --n-prompts 40

Reading the scores

Pattern	What it means
Component in all MSS	Fully necessary; not INUS but a genuine necessary cause
Component in multiple but not all MSS	True INUS condition; necessary within its sub-circuit but replaceable
Component in exactly one MSS	Fragile; no backup circuit for this component’s role
Many MSS of similar size	Highly redundant circuit; robust to ablation

Connection to other frameworks

A10 provides the formal vocabulary for interpreting A08’s (PID) redundancy findings: high redundancy between components A and B means they likely appear in alternative minimal sufficient sets (each is INUS). A01 (SCM) measures necessity and sufficiency but does not decompose them into INUS structure. A03 (CATE) identifies heterogeneous effects — components that are INUS (rather than fully necessary) may show high CATE variance because they are only causally relevant when their particular MSS is the active one.