Wang et al. 2022; Goldowsky-Dill et al. 2023
Path patching recovers a circuit by patching along specific routes. Rather than measuring whether one internal value matters overall, it patches a value only where it flows along a chosen path to the output, so it can separate a direct effect from an indirect one that passes through other components. This is the technique behind the well-known indirect-object-identification circuit in GPT-2, where it isolated exactly which heads send signals to which. We apply it to the VCS by patching along directed edges of the candidate data-flow and comparing the recovered edge set to the true routine from the disassembly. The idea is powerful because it targets the wiring, not just the nodes. On the VCS its limit is recall in a single frame. When we run it as a one-step directed do-operation, it recovers only the paths that carry a surviving signal in that single step. Many true dependencies are re-derived within the frame through other cells, so a one-step path does not carry them and the method misses them. On some games where the direct circuit is genuinely empty it matches perfectly; on others it recovers little. Path patching is thus precise about what it finds but silent about dependencies that flow through intermediate cells, a recall limit we can measure exactly against the known routine.
Two adjacency matrices over the candidate RAM cells: the true data-flow graph and the graph this method recovered. A bright cell (row = cause, column = effect) is an edge; the difference between the two panels is the error. path_circuit_F1_vs_true_routine = 0.000 — this example only (pong, state gameplay(seed=0,prefix=90)+15); the audit aggregate is below.
The score is the F1 of the recovered path or circuit against the true routine's data-flow. We patch along directed edges of the candidate data-flow and compare the recovered edge set to the true routine from the disassembly, taking the harmonic mean of edge precision and recall. The grading is always against the intervention oracle's true wiring. Faithfulness for this method is that circuit F1. Run as a single-frame, one-step directed do-operation, the method recovers only the paths that carry a surviving signal in that single step. Many true dependencies are re-derived within the frame through other cells, so a one-step path does not carry them and recall suffers, which is the main limit the score reflects. On some games the direct circuit is genuinely empty and the method matches it perfectly; on others it recovers little. There is no sprite-position hard-case here, because the output is a circuit, not a coordinate. The grading is always against the oracle, never against another method. The audit box reports the measured triad across the scored games, so the F1 is an average over the scored set. Path patching is precise about the paths it finds but silent about dependencies that flow through intermediate cells.
The score is measured against the §1 intervention oracle — never against another interpretability method. F (faithful) is always vs the oracle; see the execution stack. How each of F / S / M is actually computed for this method (and whether it matches the paper) is in the box just below; the numbers are in the In the audit box under it.
The exact formula this method uses for each score, read from its runner, and
whether it matches the paper's §3 (F ∧ S ∧ M triad) definition.
From the committed audit
fsm_math_phaseC.json.
| Axis | Formula (as computed) | What it measures | Matches §3? |
|---|---|---|---|
| F faithfulness | \(F = F_1^{\mathrm{edge}}(\hat{E}, E^{\star})\) | Edge-F1 of the recovered path circuit against the true direct-path routine (exhaustive edge-restricted path patch). | ✓ matches path_patching.jl (F = clamp(score.f1)); metric_name path_circuit_F1_vs_true_routine. Exactly the paper's circuit-discovery F = edge F1 vs the true data-flow/routine. Record: extra.triad.F / value (pong F=0.0 — no path edges recovered; battery mean F=0.580, n=42). how it's measured — call stack:
|
| S sufficiency | S = null (undefined) | Not scored: the recovered path circuit is a graph structure, not a per-do(u) predictive map. | — n/a path_patching.jl (S = nothing). n/a for a graph-valued explanation is consistent with the paper (S defined for predictive maps). Record: extra.triad.S = null. |
| M minimality | \(M = |U^{\star}| / |\hat{U}| = |E^{\star}| / |\hat{E}| \in (0,1]\) | True-routine edge count over the number of discovered path edges. | ✓ matches path_patching.jl (M = min(1, ntrue/nrec)). This is the paper's |U*|/|U_hat| applied to edges. n/a when |U_hat|=0 discovered edges (emitted as null, e.g. pong M_named_size=0 -> M=null). Record: extra.triad.M, M_true_minimal_size/M_named_size (battery mean M=1.0 over the n=18 games with >=1 discovered edge). how it's measured — call stack:
|
✓ matches = the same quantity as §3; ◐ partial = the same kind of estimator but a differing detail; ✗ does not match = a different quantity (see the note). Definitions: F = agreement with the oracle's true causal effects Δy(u); S = held-out predictive score in [−1, 1]; M = |U*|/|Û| (true-minimal-set size / named-set size).
This is the method's entry in the actual cross-method audit — scored on the paper's correctness triad, each axis a mean over all 42 scored games (42 committed §R records), not the single example shown above. Tradition: causal. The example figure (Pong) is one of those records.
F faithfulness (scored vs the oracle for every method) · S sufficiency (held-out predictive score in [−1, 1]; a negative value means the explanation predicts held-out interventions worse than the unperturbed baseline; reported for the predictive methods across all three phases where the calibration/held-out split is defined) · M minimality (true-minimal-set / named-set; where the method names a cause set) — n/a otherwise, per the paper's F ∧ S ∧ M triad.
Source: leaderboard.json · the whole leaderboard is on the
methods page and the
Paper 2 audit.
This method's faithfulness on each of the 42 scored games (all-regime F; this method has no content-vs-position split (see the note below)). Click a header to
sort. Every number is read from site_data.json
(methods.path_patching.per_game).
content-F and position-F are n/a for this method. The content-vs-position split applies only to pixel-attribution methods — those that produce an attribution map over the framebuffer, whose score can be read separately on the smooth content outputs and the discrete position outputs (where the naive gradient is provably zero). This method does not attribute to a pixel output; it is graded by its own fidelity metric (path_circuit_F1_vs_true_routine), reported as the F column, so there is no content or position regime to separate.
M is n/a on 24 of these games. Minimality is M = |U*|/|Û|, the size of the true minimal cause set over the size of the set the method actually names. On a game where the method names nothing (|Û| = 0 — it discovers no circuit, its attribution map is all-zero, or it decodes no cell), that ratio is undefined, so M is left blank rather than scored. The aggregate M is the mean over the 18 games where the method did name a cause set.
| Game | F (all) | content-F | position-F | S | M | Note |
|---|---|---|---|---|---|---|
| Bank Heist | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Beam Rider | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Boxing | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Centipede | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Double Dunk | 1.000 | n/a | n/a | n/a | 1.000 | recovers the true causes |
| Fishing Derby | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Freeway | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Gopher | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Ice Hockey | 1.000 | n/a | n/a | n/a | 1.000 | recovers the true causes |
| Kangaroo | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Krull | 1.000 | n/a | n/a | n/a | 1.000 | recovers the true causes |
| Kung-Fu Master | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Name This Game | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Phoenix | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Q*bert | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Venture | 1.000 | n/a | n/a | n/a | 1.000 | recovers the true causes |
| Video Pinball | 1.000 | n/a | n/a | n/a | 1.000 | recovers the true causes |
| Yars' Revenge | 1.000 | n/a | n/a | n/a | n/a | recovers the true causes |
| Tennis | 0.941 | n/a | n/a | n/a | 1.000 | recovers the true causes |
| Riverraid | 0.909 | n/a | n/a | n/a | 1.000 | recovers the true causes |
| Bowling | 0.800 | n/a | n/a | n/a | 1.000 | |
| Montezuma's Revenge | 0.800 | n/a | n/a | n/a | 1.000 | |
| Carnival | 0.667 | n/a | n/a | n/a | 1.000 | |
| Hero | 0.400 | n/a | n/a | n/a | 1.000 | |
| Road Runner | 0.333 | n/a | n/a | n/a | 1.000 | |
| Berzerk | 0.298 | n/a | n/a | n/a | 1.000 | |
| Chopper Command | 0.286 | n/a | n/a | n/a | 1.000 | |
| Space Invaders | 0.286 | n/a | n/a | n/a | 1.000 | |
| Private Eye | 0.222 | n/a | n/a | n/a | 1.000 | |
| Demon Attack | 0.200 | n/a | n/a | n/a | 1.000 | |
| Seaquest | 0.200 | n/a | n/a | n/a | 1.000 | |
| Air Raid | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Alien | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Assault | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Atlantis | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Breakout | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Frostbite | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Jamesbond | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Ms. Pac-Man | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Pacman | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Pitfall | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
| Pong | 0.000 | n/a | n/a | n/a | n/a | no true-cause signal |
The path from this method's runner (run_game) into the bit-exact VCS substrate, and the
computation it involves. Every step links to the exact source on main. From
callstack_phaseC.json.
env_reset!(env; boot_noop_steps = 60, boot_reset_steps = 4) jutarienv_step!(env::StellaEnvironment, action::Integer) jutaribuild_shared_testbed(game; ... intervene_ram = intervene_ram!, run_intervention = run_intervention, soft_ram_peek = soft_ram_peek, ...) runnersnapshot(env::StellaEnvironment, frame::Integer) jutaripath_patch(checkpoint, clean_next, nodes, sender_i, receiver_j, sender_value, rest_tail; hop_action) -> intervene_ram!; env_step! runner intervene_ram! @ path_patching.jl:314 & :319, env_step! @ :316 & :322node_dataflow_graph(checkpoint, tail, nodes, at_target) -> intervene_ram! runner intervene_ram! @ :272recovered_path_circuit(checkpoint, clean_next, nodes, ...) -> edge_fires -> path_patch runner true_routine_graph @ :353, score_circuit @ :424Scored against the exact intervention oracle Δy(u): run_intervention(checkpoint, actions, target_frame, horizon, cause).
The figure is generated from the committed record by
docs/gen_method_figures.py; the game frame
and each RAM cell's screen footprint are produced by
render_scenes.jl /
cell_footprints.jl.