Tracking Equivalent Mechanistic Interpretations Across Neural Networks
Authors: Alan Sun
Mariya Toneva
Venue: ICLR 2026 Poster (2026)
Keywords: mechanistic interpretability
Primary area: interpretability and explainable AI
TL;DR: We introduce interpretive equivalence as a way to formally compare different mechanistic model explanations

Abstract: Mechanistic interpretability (MI) is an emerging framework for interpreting neural networks. Given a task and model, MI aims to discover a succinct algorithmic process, an interpretation, that explains the model's decision process on that task. However, MI is difficult to scale and generalize. This stems in part from two key challenges: there is no precise notion of a valid interpretation; and, generating interpretations is often an ad hoc process. In this paper, we address these challenges by defining and studying the problem of interpretive equivalence: determining whether two different models share a common interpretation, without requiring an explicit description of what that interpretation is. At the core of our approach, we propose and formalize the principle that two interpretations of a model are equivalent if all of their possible implementations are also equivalent. We develop an algorithm to estimate interpretive equivalence and case study its use on Transformer-based models. To analyze our algorithm, we introduce necessary and sufficient conditions for interpretive equivalence based on models' representation similarity. We provide guarantees that simultaneously relate a model's algorithmic interpretations, circuits, and representations. Our framework lays a foundation for the development of more rigorous evaluation methods of MI and automated, generalizable interpretation discovery methods.

Published as a conference paper at ICLR 2026
TRACKING EQUIVALENT MECHANISTIC INTERPRETATIONS ACROSS NEURAL NETWORKS
Alan Sun
Mariya Toneva
Carnegie Mellon University al ansun@andrew. cmu. edu
Max Planck Institute for Software Systems
mtonevaemmpi-sw..
ABSTRACT
Mechanistic interpretability (MI) an emerging framework for interpreting neural networks. Given a task and model, MI aims to discover a succinct algorithmic process, an interpretation, that explains the model's decision process on that task. However, MI is difficult to scale and generalize. This stems in part from two key challenges: there is no precise notion of valid interpretation; and, generating interpretations is often an ad hoc process. In this paper, we address these challenges by defining and studying the problem of interpretive equivalence: determining whether two different models share common interpretation, without requiring an explicit description of what that interpretation is. At the core of our approach, we propose and formalize the principle that two interpretations of model are equivalent if all of their possible implementations are also equivalent. We develop an algorithm to estimate interpretive equivalence and case study its use Transformer-base models. To analyze our algorithm, we introduce necessary and sufficient conditions for interpretive equivalence based on models' representation similarity. We provide guarantees that simultaneously relate model's algorithmic interpretations, circuits, and representations. Our framework lays a foundation for the development of more rigorous evaluation methods of MI and automated, generalizable interpretation discovery methods."
INTRODUCTION
Ensuring the interpretability of deep neural networks is central to concerns around AI safety and trustworthiness. Among many proposed interpretation methods, mechanistic interpretability (MI) has recently emerged as a promising post-hoe interpretability framework? (Olah et al., 2020a; Elhage et 2021; Wang et al., 2022, inter alia). MI generally operates in two stages: (a) identifying minimal subset of the model's computational graph that drives functional behavior; and (b) attaching algorithmic interpretations to each of the recovered mechanisms. In contrast to other attribution- based interpretability methods, MI yields a concrete, human- iimman--tterreetabll algorithmic process that faithfully
describes model behavior (Bereska and Gavves, 2024; Koulogeorge et al., 2025). The resulting processes can be used to better understand a model's inductive biases which may inform further improvements (Geva et al., 2021; Meng al., 2022; Arditi et al., 2024, inter alia).
For fixed task and network, MI approaches can be broadly categorized as either top-down or bottom-up (Vilas et al., 2024). Top-down methods (b (b propose a set of high-level candidate algorithms for the task, and then attempt to align those algorithms to the network. While the alignment step can be automated and made statistically rigorous, proposing candidate algorithms is highly ad hoc. For complex tasks, it is often unclear how even formulate plausible candidates. Additionally, aligument with a proposed algorithm is only a necessary condition for interpretability, and does not guarantee that the model truly implements the intended algorithm (Wu et al., 2023; Geiger et al., 2024; 2025; Sun et al., 2025). In contrast, bottom-up methods (a (a-b) b) first isolate mechanisms-of-inteee within the model (called circuit) then assign interpretations to components within that circuit (Olah et al., 2020a). Although circuit discovery can be rigorously formulated as
Our codebase can be found at https: // thub. com aomallane awwaaaann900//mmmmma 10t04//ineerreeeee -quii An interpretabiliiyy method that does not require any retraining. These methods can be applied in parallel with inference and do not compromise the model's performance.

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Published as a conference paper at ICLR 2026
Interpretations
Circuits
Representations
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Figure 1: A high-level overview of our algorithmic approach to interpretive equivalence. Consider models hn h2 that correspond to possibly unknown interpretations A,, (Left). To determine whether models h1 and h2 are interpretively equivalent, we propose two-step procedure. First, we sample another model that also has interpretation A (Center). Second, we compare the representation similarity (drepe) between h1 4" and h", .h2 (Righ(). If A1, are equivalent, then averaged over all implementations ht,, we should not be able to differentiate diepr (hi,("))")))) and drepr (hh)))))
an optimization problem (Bhaskar al., 2024), creating and assigning meaningful interpretations to circuits typically demand significant manual effort. Moreover, Méloux et al. (2025) has recently shown that neither top-down nor bottom-up approaches are identifiable: there exists many-to-many relationship between high-level algorithms and circuits. This ambiguity makes it hard to verify whether a given interpretation is complete and faithful to the model (Jacovi and Goldberg, 2020). We present detailed discussion of the related literature in Appendix B.
In this paper, we define and study a relaxed, subproblem of MI: interpretive equivalence. Concretely, we seek to determine whether two models implement the same high-level algorithm, without requiring an explicit description of what that algorithm is. Understanding interpretive equivalence can bridge the gap between bottom-up and top-down approaches through reductions. Consider two examples:
Example 1.1 (Reduction to Simpler Models). MI is computationally prohibitive on larger mod- els (Adolfi et al., 2025). To mitigate this, if we could show that small model is interpretively equivalent to large one, then MI analyses on the small model could interpret the larger model.
Example 1.2 (Reduction to Simpler Tasks). MI's scope is limited because interpreting models that solve complex tasks is at least as hard as manually designing an algorithm solve them (Nanda et al., 2022; Zhong et al., 2023). Interpretive equivalence provides a criterion to decompose complex tasks into simpler, interpretively equivalent ones.
At the core of our approach (Figure 1), we propose the principle that two high-level algorithms (henceforth termed interpretations") are equivalent if their implementations are equivalent. Based on this principle, we design algorithm to detect equivalence by measuring representation similarity. We prove that representation similarity is necessary and sufficient to characterize interpretive equivalence. Overall, our contributions span both theory and practice:
We propose an algorithm to compute interpretive equivalence between two models without interpreting them (Algorithm 1).
We demonstrate that Algorithm 1 is well-calibrated on a simple task where the ground truth is known. Then, we show potential to find reductions (Examples 1.1 and 1.2).
"e distinguish between an algorithm and an interpretation. Informally, an algorithm is independent of any computational abstraction. However, we show in Appendix E that this abstraction -aree notion results in pathological equivalence definitions. Thus, we use the terminology interpretation to signify a dependence on chosen level of computational abstraction.
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We specialize the theory of causal abstraction to define interpretations, circuits, and representations (Sections and 5). As byproducts this formality, we obtain a measure of interpretation quality and an alternative characterization of interpretations interventions (Section 5 and Appendix H). We prove necessary and sufficient conditions for models' interpretive equivalence based on their representation similarity. To knowledge, these guarantees are the first relate model's interpretations, circuits, and representations (Section 6).
Throughout our paper, we use examples from language processing and Transformers. However, our framework does not depend on any properties of the input modality.
INTERPRETIVE EQUIVALENCE THROUGH CONGRUENT REPRESENTATIONS
We define mechanistic interpretations A 2 as equivalent they have equivalent implementations. That is, any model which can be interpreted by A₁ must also be interpreted by A2 and vice versa. We justify this principle later, but first we offer an algorithm to approximate this equivalence. This requires clarifying two procedures: (1) how do we enumerate all implementations of A1,?? (2) How do we measure the distance between these sets of implementations?
Enumerating Implementations through Inter- ventions. We call any model that has a mech- anistic interpretation an implementation of A (Definition 5. We take a bottom-up approach to generating implementations. Using causal in- terventions, we identify components the model that causally unrelated the model's behav- ior (Conmy et al., 2023; Goldowsky-Dill et al., 2023) (i.e. they are not a part of the model's circuit). A model's mechanistic interpretations should be invariant perturbation or ablation of these components. Dually, each time we perturb or ablate such a component, we yield a "new" model that is causally equivalent to the original one (and as a corollary, shares the same interpretation). In- tuitively, every implementation of A should be reachable through a causal intervention applied onto h. We take this dual perspective gener- ating implementations by adding, removing, or modifying these unimportant components (Geiger et al.. 2024; Gupta et al., 2024). This procedure denoted as GETIMPL (Algorithm detailed in Appendix C).
1 procedure CONGRUITY(H,
2
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Algorithm I: Two models are congruent if representation similarity alone cannot differ- entiate them. drepe (R1 measures repre- sentation similarity between representations of h2 GETREPRS, GETIMPL. retrieve the hidden representations a given model and respetttiiiiirreeetiiiiilllel respectively.
its implementations,
Representation Similarity. We identify each implementation with their hidden representation spaces (GETREPRS in Algorithm 1). Given two implementations, we measure their distance using linear representation similarity (deepe defined in Definition G.1). Informally, linear representation similarity captures the extent to which the representations of one network can be linearly transformed to the other and vice versa. We notate this REPRDIST (Algorithm
Suppose that the implementation sets of A1,A2 were exactly equal. Let h1 be implemen- tations sampled from A₁ and hn be an implementation sampled from A2. By symmetry, (hdrepe(h,,11) < drepr(h2, hī]] = P(drepr(h),,tt) > (he In this way, the implementations of A1,A2 are congruent with respect diepr-
Combining these concepts, we have our approximation of interpretive equivalence: CONGRUITY.
3 EXPERIMENTS
Herein, we demonstrate three applications of Algorithm 1. First, on toy task where ground-trude interpretations are known, we show that CONGRUITY is well-calibrated (Section 3.1). Next, we apply
For the sake of argument, suppose i.i.d. sampling from uniform distribution A1,A2.
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our framework to pre-trained language models of various sizes (GPT2 (Radford et al., 2019) and Pythia (Biderman et al., 2023)) and demonstrate that CONCRUITY can distinguish between models that exhibit fine-grate fine-grained algorithmic differences across various model scales (Section 3.2). Lastly, show how CONGRUTTY can he used to relate a complex task like next-token prediction to a simpler one such parts-of-speech identification (Section 3.3).
3.1 CALIBRATING CONGRUITY
We consider the task of r-Permutation Detection: determining whether a given sequence of numbers is a permutation of the elements II. n1. For example, when n 3, 3, [3, 1, 2] - True and [1, 2, 2] + False. The task is sufficiently rich to support different interpretations, yet simple enough that solutions can be hard-coded as Transformers. We manually devise six different interpretations solve this task whose procedures we detail in Appendix C. Roughly, their approaches can be organized into two buckets:
1. Sorting-Basse Based (Interpretations 1-4). First sort the list of numbers by ascending (or descending) order, then directly check whether the resulting sequence is equal to 1,2,..., (or n, - 1,...,11 1). Counting- -aatid (Interpretations 5-6). Exploit the fact that the vocabulary contains exactly numbers and use the pigeonhole principle to detect duplicates in the given sequence.
For all experiments, we fix n == 10. Using the Restricted Access Sequence Processing Language (RASP), we hard-code six bidirectional Transformers (of various architectures) to implement each interpretation (Weiss et al., 2021). Then, for each hard-coded RASP Transformer, we leverage a technique of Gupta et al.'s (2024) generate 100 model variants (with different architectures and weight configurations). These variants are constrained to maintain the same underlying interpretation as their hard-coded counterpart, So for each interpretation, we have 100 implementations. In total, we have 6 x 100 models that all achieve 96%+ accuracy on 10-permutation detection. We now directly compute CONGRUITY (Algorithm 1) between pairs of models from our generated implementations. We perform hypothesis testing by bootstrapping a 95% confidence interval over the final output. The results are shown in Figure ((Lef)).
Along the diagonals of Figure 2(Left), we see significantly high congruence. So when models share the same interpretation, representations across their implementations are approximately equivalent under linear transformation. On the other hand, off-diagonal entries admit low congruence. Thus, representation similarity across set of implementation as whole can differentiate individual interpretations. These empirical findings complement our bounds in Main Results and 2 because they suggest that CONGRUITY both necessary and sufficient to detect interpretive equivalence. Perhaps even more interesting, we observe a breaking row/column in Figure 2, where Interpretations 1 1-4 (the top-left 4 4 square) have an average nithin-group congruence of 0.43. This is larger compared to the average across-group congruence (rows/columns 5-6) with Interpretations 5-6: 0.01. These groupings correspond to our broad algorithmic buckets above and suggest that CONGRUITY could also be adopted graded notion to characterize broad-stroke interpretive differences.
3.2 REDUCTION TO SIMPLER MODELS
We now consider the task of indirect-object identification (IOI) (Wang et al., 2022): given sentence like "When John and Mary went to the store, John gave the drink to " "the model should complete the sentence with "Mary." " Because IOI can be solved algorithmically. it has been studied across many models: GPT2-small/medium and the Pythia class of models. While Merullo et al. (2024) find that GPT2-small and medium use the same circuit, Tigges al. (2024) find that Pythia models across all scales use a consistent but different circuit from GPT2 models (Merullo al. 2024, Appendix C; Tigges al. 2024: Appendix D). We these differences practical testbed for CONGRUITY and explore how it generalizes across both model families and scales (85M to 2.9B parameters).
'Here, Ho : the two models not share same interpretation; and our alternate, H1 : the two models share the same interpretations. We compute a Wald confidence interval with 20 bootstrap samples. 'Notably, Tigges et al. (2024) also find that circuits in larger models require more components, even when the underlying interpretation is constant. Thus, for CONGRUITY to be effective in this setting, it also needs to move beyond identifying architectural similarities.
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Figure 2: (Left) Average congruity between models associated with different interpretations. indicates models have different interpretations: whereas indicates that models have statistically indistinguishable interpretations. (Center) Congruity between GPT2 and Pythia family of models on the IOI task. groups models based on their actual interpretive differences observed by Tigges et al. (2024); Merullo et al. (2024). (Right) Congruity between GPT2 on next-token prediction (for different token types: all tokens, articles, prepositions, punctuation, parentheses, and terminal punctuation) vs. GPT2 on in-context parts-of-speech identification.
For each model, we generate 10 parallel implementations by intervening on the components found not be in the IOI circuit (identified by both Tigges al. 2024 and Merullo et al. 2024). Then, we apply CONGRUITY, each time computing representation similarity with 200 IOI sentences. The results are shown in Figure 2(Center). We find that the Pythia models show high within- group congruence across different scales. GPT2 models exhibit similar behavior: This supports our intuition that interpretive equivalence could be leveraged to reduce the interpretations of complex models into interpretations of smaller ones (Example 1.1). For example, our results affirm that the IOI task, Pythia-2.8h is interpretively equivalent to Pythia- 160M; thus, priori, it suffices to only interpret the latter. Indeed, Tigges et al. (2024) finds that both models share the same interpretation. On average, across-group congruence between Pythia and GPT2 is significantly lower (0.13 versus 0.73 and 0.92) which supports our previous insight that representation similarity provides increased identifiability over interpretations. It is unclear why Pythia- 160M and 410M show increased congruence with GPT2-medium and small, respectively; perhaps, subtle similarities in name-mover heads representations could explain this (Tigges et al. 2024, Section 3.2).
3.3 REDUCTION TO SIMPLER TASKS
Interpreting next-token prediction poses challenge for both bottom-up and top-down MI approaches. For top-down approaches writing down a symbolic, end-to-end algorithm for next-token prediction is difficult. Bottom-up approaches face the opposite problem: circuit discovery may identify the entire model as important, failing to reduce the search space of interpretations. We show here that interpretive equivalence may offer some first-steps towards mechanistically understanding next-token prediction. Concretely, we identify sets of next-tokens for which GPT2's next-token prediction process is interpretively similar to pfrts-of-speech identification (POS7).
POS is a largely syntactic task since it focuses solely on the grammatical role of words rather than their meaning in-context. Thus, we expect POS to be interpretively equivalent the prediction of "syntactic tokens." " For computational ease, we conduct all experiments on GPT2. To discover the POS circuit GPT2, we apply a method of Todd et (2024). We detail this process, along with the circuit we discover in Appendix C. We construct dataset for next-token prediction by uniformly sampling 100 sentences from the C4 dataset (Raffel et al., 2020).
We collate tokens from our next-token prediction sentences into disjoint groups of interest. For each group, we apply CONGRUITY and compute representation similarity between our extracted POS circuit
"Givn noun:; run:verb; quick: fluffy: the model needs to output adjective.
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and the last-token hidden representations of the model. The results across different token groups shown in Figure 2(Right).
As control, we first consider the group of all tokens. We observe nonzero congruence of 0.48 with POS. Next, we consider token groups articles and prepositions. Prediction of these tokens typically appear mid-sentence and depend heavily semantics. 8 Thus, expect that predicting these tokens should yield no more interpretive equivalence to POS than the control. Indeed, we find POS's congruity to articles and prepositions be statistically indistinguishable from the all-token average.
We now heuristically identify sets of "syntactic tokens:"
1. Terminal Punctuation like ".", "?", or "!" mark the end of a sentence. Accurate prediction of these tokens intuitively requires syntactic identification of subject-verb-objee relations and (in)dependent clauses.
2. Closing Brackets/Quotations like "Y" can be implemented as skip-trigrams in one-layer attention Transformers (Elhage et al., 2021), So, we suspect that understanding sentence pragmatics is not needed for their prediction.
We find that both groups yield significantly higher congruence compared our all-token control with medium effect size (measured through Cohen's d). This stands in contrast to general punctuation and opening brackets/quotatiioss. Although these other punctuation groups admit statistically significant congruence with POS, we observe a small effect size (d < 0.3). We hypothesize that the placement of these tokens rely more semantic processes. For example, commas may be placed for emphasis or to add dependent clauses rather than strictly maintain grammatical consistency.
REPRESENTATIONS, CIRCUITS, INTERPRETATIONS AS CAUSAL MODELS
We present framework of interpretive equivalence. This theory grounds algorithmic contributions in Section Whenever possible, we present an informal treatment and defer the precise definitions Appendix D, where also have full glossary and summary of notations. The next sections are organized as follows:
In Section 4, we define circuits, representations, and interpretations (Definitions 4.2, 4.3, and 4.4, respectively).
In Section 5, we define criterion for when two interpretations are equivalent. We also introduce interpretive compression, a measure of interpretation abstraction.
In Section 6, we prove that representation similarity can describe interpretive equivalence (Main Results and 2) and show that CONGRUITY is closely related to these quantities (Main Result 3).
Let be an alphabet and E* be all finite strings formed using E. We define a language model as function h - I We often are interested in h behavior on subset of inputs S C Σ*, rather than on all of Σ* For this reason, we term S a task. While / alone serves as a sufficient functional description, interpretability requires understanding h's underlying computational process. For this, we rely on three behavioral characterizations of neural network at different levels of abstraction: h's circuits, representations, and interpretation.
4.1 CIRCUITS
Definition 4.1 (Deterministic Causal Model, Geiger et al. 2025). A (deterministic) causal model with components is quadruple (V,U,E, where V = (V),..,, Um) is set of hidden variables such that [ = [F] = m, is an input variable, and defines a partial ordering over V. For each vA E V.
V - fa (Pa(
(4.1)
where fa IF and Pa(v:) (v V:u> d.} are the parents of Ug.
*Consider the fragment: "[context] The students are looking [mask]" the next prepositions- ""atttiions""""",""""""""."""""" "into", or are all syntactically sound, but they are semantically ambiguous without further context, Articles are similar: consider, "I need to buy [mask] car." [mask] could be either the indefinite (*aa") definite ("the"), both syntactically valid but the choice requires pragmatic reasoning.

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A deterministic causal model describes a computational graph where hidden variables V (nodes) store latent computation results computed by the functions in F (edges). Since a computation graph is acyclic, the data/compute flow determined by F induces partial ordering over V, >-
For a specific input setting U +- u, we denote by v (u) to be the unique solution: a configuration of the values consistent with both u and (Peters et al., 2017). Likewise, of (u) is the solution of 2y E V to U +- u. In this way, oj is a function that maps from S S+R" R"
There is natural tradeoff' between the collective complexity of V and the complexity of any individual operator in F. On one extreme, any blackbox model /t can be expressed as a trivial causal model with one hidden variable: 11 A )U)). On the other, each v E could be a single neuron (thus [ ~ 10"), while elements in F consist of primitive operations like dot-products. In this way, V determines the level of abstraction of the causal model. We now use this formalism to define circuits.
Circuits form the starting point for almost all bottom-up approaches in MI (Wang et al., 2022; Conmy et al., 2023; Lieberum et al., 2023, inter alia). Intuitively, for some task S, a circuit describes the (minimal) computational pathways used by the model to produce h(S). Throughout our treatment, we do not require minimality as finding minimal circuits has been shown to be an intractable combinatorial optimization problem (Adolfi et al., 2025).
Definition 4.2 (Circuit). An m-circuit of h on S is causal graph (V,U,F,>) with components that satisfies: (1) U is S-valued; (2) Hidden variables are real-valued; (3) There exists a path from U to variable Vout E V such that v (S) hSS).
For any h and m Z+, an m-circuit of h exists by its trivial causal model. Since we do not require minimality, m-circuits are not unique: a joint change of bases to any U2 E V Vout and fú: or, adding a new variable that does not contribute vout results in new causal model that preserves faithfulness to h. We further explore some of these properties in Section 5.
4.2 REPRESENTATIONS
Representations of a neural network are sequence of activation spaces. Increasingly, representations have been understood to encode concepts which deep networks then iteratively refine (Jastrzebski et al., 2018). We formally view representations as abstractions of circuits and define them through causal abstraction (Beckers and Halpern, 2019; Geiger et al., 2025). A causal model K* abstracts K if there exists surjective map from K's variables to K* variables that preserves K's causal relationships. Any map that admits this abstraction is an alignment from to K* (Definition D.3). Essentially, an alignment maps K's "low-level" variables to K*'s "high-level" variables (Rubenstein et al., 2017). Like m-circuits, for a given low-level and high-level causal model pair, the set of alignment maps that exist between them are not unique.
Definition 4.3 (Representations). Let K be an m-circuit of h on a task S. An L-circuit that abstracts K an (L, represettttnnn of K if for each U2 EV. V,
Pa(+x+1) = {. Pa(v1)= = (U), and vz Yout- For each < k < L, there exists linear maps A IR" + R B RE +
Ra where
||Axllop | opp < 1,
| -All <
and |Bahh - of <8
(4.2)
for some norm (to be specified).
Representations have two crucial properties: they compress the complex causal model of K into single chain of length L. and, they have high signal and low noise. By Equation (4.2), each layer's 's hidden representation can identify K"s output behavior (independent of its children). Yet, the representations each layer are simple enough that outputs of h can identify their value (independent of its ancestors). In MI, this latter constraint is often used as a search criterion to localize a network's 's circuit (Belrose et al., 2025). For h, its set possible of circuits explode combinatorially with depth, whereas the set of possible representations only scale quadratically. Therefore, compression through abstract representation gives us tractable staging ground to compare eircuits.
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4.3 INTERPRETATIONS
Interpretations are symbolic, human-underssanddbble descriptions of h that faithfully captures its functional behavior on a task. We formalize them abstractions.
Definition 4.4. Fix a task S. A causal A (V*, U,F", ,>)",>)))))))) abstraccs circuit K & W,U,F.>) an 17-faithful interpretation if A's output approximates K's with error at most 17 across imputs
Notably, our construction does not assume there is distance metric over A's variables nor do we specify the values of these variables. This generality aligns framework with most MI literature where V could take on symbolic values (Hanna et al., 2023; Zhong et al., 2023, inter alia).
For any circuit K, an interpretation always exists through the trivial interpretation: K itself is an interpretation of K (with the identity abstraction; albeit, this is not very useful). Since interpretations depend fixed level of abstraction (V*), they must be unique: by varying V and can produce different interpretations. As a corollary, circuits are not identifiable by their interpretations. This seems problematie, but in Appendix E demonstrate that abstrrattoon-dependett interpretations are necessary for a non-trivial notion of interpretive equivalence.
5 INTERPRETIVE EQUIVALENCE THROUGH IMPLEMENTATION EQUIVALENCE
Méloux et al. (2025) shows that there exists a many-to- many relationship between circuits and interpretations. This implies that comparing individual circuits and interpretations is ill-defined. Thus, propose that two interpretations are (approximately) equivalent if and only if their implementations are (approximately) equivalent. By examining families of circuits instead of individual ones, we can effectively quotient out the many-to-on mapping from circuits interpretations.
To start, define implementations. Let K (V,U,F,>) be m-circuit of h some task S. Let A A (V* U,F* >) be 17 faithhlu interpretation of K through alignment π SubsetsOf(() V* (see Definition D.3). Notice that for fixed A, π, by carefully varying F we could construct many different circuits (and computation graphs) over (V,U) that all abstract to the same interpretation A under z. Concretely, in our framework, V defines a neural architecture and FF corresponds to many weight configurations that result in the same interpretation A under л. We then define A's implementations as the set of all admissible F' under A, π. This construction mirrors observations that implementations are diffuse and plentiful in the weight space (Lubana et al., 2023; Zhong et al., 2023; Gupta et al., 2024). Precisely,
Definition 5.1 (Implementation), Let II be a class of alignments. F an implementation of there exists π E: II such that A is an f-fatthful interpretation of (V,U,F,44 <F) under π.
By a slight abuse of notation, we denote II IT-(A) as the set of all implementations of A under alignments II. Then, non-empty intersection between I1, and I1,"1 directly corresponds to Méloux et al. 's (2025) non-identifiability result. We assume that any circuit comes with metric attached to its variables, i.c. d: V V R20 This is reasonable, as circuit's ss variables real-valued (Defini- tion 4.2). d naturally induces a pseudometric over II (A), where for F.F II
Circnit
W..UU.F Interpretation
Inteppetive
72
m
Il,"
=(V= ( (S)). (5.1) Given two sets of implementations over (V. we can leverage to quantify (1) the Hausdorff distance between them; and, (2) their diameters. We call the former interperteeee equivalence
d(F.F) a
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compression (see Figure 3). More precisely,
Definition 5.2. For i E (1,2), let A; & U.)) be faitttfful interpretation of K & (V. and IT, be a class of alignments that map from SubsetsOf(() + V/. Then, A1, are e-approximate interpretively equivalent if
dinterp SAneer(A1,A0 é du (AI), <E,
(5.2)
Too details of this construction, refer Definition D.3 and Equation (D.2)
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where du is the Hausdorff distance 10 defined by d (see Equation (5.1)).
Definition 5.3. Let A be 17-aithhuul interpretation of K and IT be class of alignments. Then, the interpretive compression of
(AA, K, IT) _ (iameter([I-
sup d(F.F").
(5.3)
F.F'II (A)
Interpretive equivalence and compression are tightly coupled. To explore their relationship, let us consider two extremes. On one hand, suppose A = K and |IT = 1. Then, there exists exactly one implementation of under II: K. This implies that there no interpretive compression. Further, any interpretation equivalent to A must admit the same exact same weight configuration as K. This effectively creates a bijective mapping between model weights and interpretations. On the other hand, suppose that A is the trivial causal model of h, and let II be all alignments. Since A conveys no information about the computational process of K and places no constraint on alignment, any circuit by Definition 4.2 must be implementation of A. In fact, Sutter al. (2025) show under mild assumptions that this phenomenon persists as long as is the set of all alignments. And so, in this case, interpretive compression is maximal. These examples illustrate duality between equivalence and compression: while more compression shrinks and simplifies the set of possible interpretations, simultaneously enlarges the set of implementations and complicates interpretive equivalence.
Interpretive equivalence, in Definition 5.2, requires that both interpretations originate from the same circuit K. This constraint can be relaxed by augmenting II, such that (æ SubsetsOf((i) + +V,) E IT; Through IT we only consider implementations of A1, on the same neural architecture V1. This creates asymmetry in choosing which circuit to use for comparison (V1 or V2). Our intuition is that we should select the circuit least compression that admits at least one implementation for both interpretations." 1I Our bounds in Section 6 justify this principle.
We describe critical properties of interpretive equivalence and compression Appendix F.
6 REPRESENTATIONAL SIMILARITY AND INTERPRETIVE EQUIVALENCE
Using interpretive compression and representation similarity, we now prove both upper and lower bounds on interpretive equivalence (Main Result and Main Result 2, respectively). The proofs and formalisms are deferred to Appendix G. Throughout this section, we consider the most general setup: for E {1,2), let K; be a circuit model h; over shared task S. Assume each circuit admits an (Li, represeeeennnnnn R; through an alignment map P;. First, we define notion of distance between these two representations.
Definition 6.1 (Representation Similarity). Let Hi be the concatenation of all hidden variables in R; Then, the representation similarity of R1, deepr R2), is the bidirectional linear approximation error between H1, H2 (formally, Definition G.I).
Defining similarity with a strict linear approximation not necessary to develop our theory, but offers attractive computational tradeoffs. This is because we can formulate representation similarity as multi-target linear regression for which there are sharp approximations with small sample complexity. Alternatively, drepr could be defined with arbitrary kernel (Kornblith et al., 2019). We also make the following assumptions: (1) p; is Lipschitz continuous with Lipschitz constant Lip(p:). ) (2) There exists A > 0 such that ||Л -h2l < A over S. (3) The implementation distance d (see Equation (5.1)) is induced by fixed distillation function Φ: for implementations F, F over the same variables V, dF,,F) F) A |yyF - 4./lI. (4) There is a I-Lipschitz map y such that |P 4H
< drepr (R,,Ro) R₂)
and / 4/ < w.
Assumption stabilizes representations such that small change in circuitry implies a proportional representation perturbation. In practice, this is reasonable since representations are usually distilled from neural networks via linear projection or by averaging the activations of several neural compo- nents (Sucholutsky et al., 2023). Assumption 2 guarantees that functional differences between our
The Hausdorff distance between two sets A, is the greatest distance point from A, B needs to travel reach B.A, respectively; d(a, B) & inf b), (A, B) á max(supace d(a, supspp (b,A))- Attached to each V,, V22 are distance metries d ,dz- Alternatively, one could also choose the circuit where
most convenient compute.

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target models are bounded, otherwise their interpretive inequivalence is trivial. Assumptions 3-4 state that our implementation metric only measures distance using a set of observable components. Further, these components recoverable from representations (up to @). In this way, representations can serve as surrogate for distances between circuits.
For i E {1,2), suppose that A; is an 17-faithful interpretation of K;. We first show that representation similarity between K,, K2 upperbounds interpretive equivalence.
Main Result 1. Let II and II, be alignment classes that map K1 's variables into A1, 's variables, respectively. Suppose there exists F E (I,'(A2) such that its representatin under P1 is an (L, ō1)- representation. Then, the approximate interpretive equivalence of A1, under alignments II II.
is
Amterp (Aomp(AA,A))) S SK(A, K1, II, 200 + drepr R1 R2) (2)+ + 82 + A.
(6.1)
(b)
Equation (6.1) explicits two dependencies: (a) measures interpretive compression of A A2 on K When compression is low, circuits retain more information about interpretive differences. And since representations distill circuits, it is intuitive that representation similarity dominates the upperbound in this regime. In contrast, when compression is high, our examples in Section 5 illustrate that when circuits become uninformative, so should representations and their similarities; (b) jointly measures the quality of representations R1 R₂ and their differences. If 8, 82 > 1, representations lose meaningful linear structure even though they sufficiently abstract implementations of A, This bound exposes a tension between the geometric richness of representations and our computational approach to their similarities. For example, representations lying on complex manifolds may not be well-described by our linear formulation of representation similarity (Ansuini et al., 2019; Pimentel et al., 2020). Thus, we hypothesize representations and their similarity criterion need to be simultaneously broadened to achieve more generality. We leave the implications of this for future work. Now, the lowerbound:
Let and II, be an alignment that map Kj 's variables onto A1, 's variables. Suppose A2 are E E-approximate interpretively equivalent under the alignment classes II, II. Then,
Main Result 2.
drepe (R1, (1, - (8) + 82 AA)
(6.2)
Lip(pI) -1
The terms in Equation (6.2) mirror the dynamics of Equation (6.1). the implementation sets are E-close, then there cannot be too much discrepancy between the observed representations (see the numerator), unless can be explained away by the slack terms (8, + 82 + A). Along these lines, Lip(p:) (in the denominator) captures the intuition that when the mapping from circuits to representations unstable, small implementation-lewe differences. Beyond interpretive cquivalence, Main Result 2 has implications for steering (Singh et al., 2024). When the identifiability of representations is low (Lip(p:) >> and diepr R₂) < 1), steering interventions are less likely to induce interpretive change, i.e. significant alterations to the representation space may see little change in model interpretation. In Main Result 3, we show how this directly relates to CONGRUITY. For i E {1,2), let us assume an arbitrary distribution over II, and denote IP A P1 & Pi P & F2 the product distribution.
gaps can lead to larger representational
Main Result 3. Let p be the expected value of REPRDIST in Algorithm F1, lid II, (A1), and F3 ~ I1, (A2) then
p21 1- inf Pn (F) K(A),K,,I)) + P(drepe(Fi, - dinter A2) E]. E>0
(6.3)
This lowerbound demonstrates that REPRDIST (and as a corollary, CONORUITY; see Corollary H.I) accounts for interpretive compression and equivalence. Concretely, it predicts that, in expectation, REPRDIST will be large when I1, y II,' except in two cases: (a) wilhin-interppetttion dispersion high: the representations of two implementations of A₁ can differ more than their interpretive compression; (b) cross cross-internn interpretation collision: implementation of A2 lies close in representation to A₁ despite dimtemp (A A2) > 1. We discuss these results further in Appendix H and also analyze the sample complexity of CONGRUTTY.
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7 CONCLUSION
In this paper, we define and study the problem interpretive equivalence: detecting whether two models share the same mechanistic interpretation without interpreting them. We contribute an algorithm to estimate equivalence and demonstrate its use on toy and pre-trained language models. Then, we provide theoretical foundation ground and explain our algorithm. Our framework and results lay foundation for the development of more rigorous evaluation methods in MI and automated, generalizable interpretation discovery methods.
ACKNOWLEDGEMENTS
Alan is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE2 140739. Alan thanks Büşra Asan and Andrew Koulogeorge for helpful feedback and discussion of early drafts of the work.
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A QUESTIONS AND CLARIFICATIONS
It seems that a prerequisite of CONGRUITY is circuit discovery. Given that circuit discovery is NP-HARD, how can this algorithm be efficient?
A: A circuit is minimal subset of the computation graph that explains the model's functional behavior. Notably, this minimality requirement what makes circuit discovery hard. In CONGRUITY, we do not require circuit in strict sense. Instead, the practitioner needs to identify individual components that are unimportant, i.c. removing does not affect functional behavior. This is more relaxed than performing circuit discovery and applying the complement. For model with m components, existing component scoring methods such as activation/attribttion patching can effectively identify unimportant component sets in 0(1) time (Conmy al., 2023; Nanda, 2023).
In Section 5, it is argued that there could be many interpretations for given circuit. How do we know which (or what level) interpretations are being compared through CONGRUITY, if do not interpret the models first?
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A: The level at which interpretations are being compared in CONGRUITY depends wholly the interventions being performed in GETIMPL. Given an interpretation A of some circuit K, consider the set of A's implementations under some arbitrary alignment class II. We could try and enumerate the set IT (A) naively. In most cases, this is extremely difficult because IT (A) could have complex geometry in the weight space (Lubana et al., 2023). We could identify this by dually enumerating interventions. Concretely, fix some F E _ (A). Construct a set of interventions I such that for each E (A), there is y E I where ((F) = F. In many cases, I' more intuitive than II (A) (Geiger ct al., 2024; Park et al., 2024; Gupta et al., 2024). Alternatively, we could directly define II with Г. This strategy is employed in GETIMPLtO implicitly define the set of interpretations being compared. To our knowledge, there is surprisingly little literature on the nature of these interventions and we believe that our insights in Sections and suggest this may be fruitful direction for future work.
The tasks presented in Section 3 are quite simple. Would linear representation alignment as a similarity metric between representations in CONGRUITY still be effective for more complex tasks?
A: We choose linear representation alignment because of the linear representation hypothesis (Park al., 2024). However, this does not restrict the generality of our framework. This is because the choice of alignment map from circuits to representations, p. is unrestricted. One could indirectly construct more complex drepr by indirectly increasing the complexity p. this case, p can be thought of as constant feature map of arbitrary complexity.
How does the proposed framework allow practitioners to translate interpretive equivalence into actual interpretations?
A: Interpretive equivalence alone is not sufficient to identify interpretations. We leave the formal exploration of this for future work. But, as discussed in Examples 1.1 and 1.2, interpretive equívalence potentially simplifies the interpretation workflow.
How would the theoretical framework in Sections and generalize different notions of causal abstraction?
A: The difference between various causal abstraction frameworks is their definition of an alignment map π (see Definition D.3). For example, bijective translations (in contrast to constructive abstractions. the notion of abstraction we use throughout the main text) require that the alignment map be bijective. Our framework can easily accommodate these different notions by simply tightening or loosening the set of permissible alignments IT. Further, assuming all else equal, as IT increases or decreases in size, interpretive compression necessarily increases and decreases, respectively. In this way, the tightness of Main Results I and adjusts to the chosen notion of causal abstraction. On the other hand, Main Results and 2 do not rely on structural properties of II. Potential future work could significantly tighten these bounds by considering sharp restrictions over the allowed alignments.
Why is it that in Main Results 1 and 2, there is no dependence on the ni-fathffllnss of the interpretations that we are comparing?
A: Faithfulness is implicitly encoded into interpretive compression which appears in both Main Results and 2. Let be an 1 -aithfull interpretation of K. Fix some alignment class and consider II (A). As 17 - co, the size of II (A) grows monotonically. To see this, let E > 0 be arbitrary. If A an n-interppetation of K, then A must also be an (1 + -iiipeerpreetaiinn of K. In other words, if the faithfulness constraint is relaxed, then A could be a faithful interpretation for any circuit. This implies that any notion of interpretive equivalence must be trivial since every interpretation corresponds to every circuit.
B RELATED LITERATURE
There has been extensive study in interpreting neural networks (specifically, language models). With respect to our paper, they can be organized into the following three buckets.
Mechanistic Interpretability. Mechanistic Interpretability (MI) can be broadly broken into two distinct phases: first, identifying a minimal subset of the model's computational graph that is responsible for specific behavior-this process is also referred circuit discovery (Olah et al.,
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2020b;a; Conmy al., 2023; Bhaskar al., 2024; Hanna et al., 2024); second, assigning human- interpretable explanations to each of the extracted components this process is generally referred to as mechanistic interpretability (Chan et al., 2022; Wang et al., 2022; Zhong et al., 2023; Hanna et al., 2023; Merullo et al., 2024). This paper focuses on the latter process; thus, when we invoke the term mechanistic interpretabilityy, we are referring specifically this second phase. For completeness, we briefly review circuit discovery as well.
model can be seen as a computational graph (Elhage et al., 2021; Olsson et al., 2022): G = (V.E) with vertices V and edges E C V x V. A circuit is then any binary function f: E + {0, The optimal circuit f satisfies two properties: (1) it minimizes Lee f(e); and (2) when edges not in the circuit E E(e() = O are ablated, the model's 's functional behavior remains unchanged. Although circuit discovery can be concretely formulated as an optimization problem, it is computationally intractable in its naive form (Bhaskar et al., 2024; Adolfi et al., 2025). As result, many relaxations and heuristics have been developed (Conmy et al., 2023; Syed et al., 2023; Nanda, 2023; Hanna et al., 2024; Bhaskar et al., 2024). Nevertheless, there exists a solution set that can be statistically verified (Shi et al., 2024). The process of mechanistic interpretability, which we focus on in this paper, presents different challenges. MI involves iteratively generating hypotheses about the interpretations for the interactions between different components, then testing those hypotheses through carefully crafted interventions (Chan al., 2022; Geiger al., 2024; Méloux et al., 2025; Sun al., 2025, inter alia). MI methods can be further clustered into two approaches: top-down and bottom-up. Top-down approaches start by enumerating hypotheses about the possible algorithms the model could be implementing. Then, they iterate through these hypotheses and isolate those with the closest causal alignment (Wu et al., 2023; Geiger et al., 2024; Bereska and Gavves, 2024; Vilas et al., 2024; Sun et al., 2025). On the other hand, bottom-up up approaches are data-driven. They seek to directly find algorithmic explanations of the model's s mechanistic behaviors by analyzing activations or attention patterns (Nanda et al., 2022; Zhong et al., 2023; Lee et al., 2024; Arditi et al., 2024; Nikankin et al., 2025).
Casual Abstraction. Causal abstraction seeks to formally characterize when a symbolic explanation faithfully explains a date-gener process (in our application, this data getteeenerating process would be model) (Pearl, 2009; Peters et al., 2017; Beckers and Halpern, 2019; Beckers et al., 2020; Otsuka and Saigo, 2022). Causal abstraction as defined in Definition D.3 seeks to align "low-level" models (neural networks) with "high-level" models (symbolic explanations). These theoretical constructions have driven the development of many top-down mechanistic interpretability methods such as Wu al. (2023); Geiger et al. (2024); Sun al. (2025) which directly verify this condition. While causal abstraction provides a necessary condition for a valid interpretation, recent works such as Méloux et al. (2025) argue that alone is insufficient to fully characterize what constitutes a valid interpretation. Practical procedures that stem from this theory also fail to address this. This highlights the need for additional theoretical frameworks to complement causal abstraction.
On the other hand, causal abstraction represents hard equivalence: two models either are are not causal abstractions of each other. The binary nature of this definition fails to capture our intuition that explanations can vary in quality or completeness. Some interpretations may better capture the model's behavior than others, or may only explain a subset of the model's 's functionality. The hard equivalence of causal abstraction makes it difficult to reason about these partial or imperfect explanations. As a result, framemworks such as Beckers et al. (2020); Massidda et al. (2023) soften this criterion by introducing distance metrics the total settings of the "high-level" model. In practice, these approaches face two main drawbacks:
1. It is challenging to define meaningful distance metrics when high-level interpretations involve discrete symbols and symbolic reasoning.
2. Comparing interpretations between models requires fully interpreting each model first, which forces a computationally expensive top-down analysis approach.
Our framework addresses these limitations by focusing on implementations rather than interpretations directly. Since neural network computations are fundamentally real-valued, we can leverage natural distance metrics in their computational space. Additionally, by analyzing families of implementations rather than requiring complete interpretations, we can compare models' interpretive similarity without the overhead of fully interpreting each model first
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Representation Similarity. Representation similarity in neural networks has emerged as fundamen- tal research area wheih addresses how internal representations correspond across different models, domains, and biological systems (Kornblith et al., 2019). We provide brief overview of techniques to measure representation similarity, but encourage the reader to refer to Sucholutsky et al. (2023); Klabunde et al. (2025) for comprehensive survey of the techniques within this field.
Representations of deep neural networks have shown to exhibit an array of powerful properties. They have given insights into training dynamics and the generalization capabilities of models (Tishby et al., 2000; Xu and Raginsky, 2017; Shwartz Ziv and LeCun, 2024) and also provide tractable method to compare the inductive biases between different models (Wang and Isola. 2020; Skean et al., 2025). These representations have also shown to admit rich geometric properties (Tulchinskii et al., 2023). In this paper, we measure representation through learned linear regression. These techniques have been used extensively to compare different neural architecture and even human-language model similarity (Toneva and Wehbe, 2019; Muttenthaler al., 2023).
C EXPERIMENTAL DETAILS
In this section, we detail our experimental methods. We first review constructs of RASP Weiss et al. (2021). Then, we demonstrate that RASP provides an interface to craft interpretations, and through methods like Geiger et (2024); Gupta al. (2024) we can enumerate implementations of these interpretations. Lastly, we describe our experimental hyperparameters.
C.1 RASP INTERPRETATIONS
Background on RASP Programs. The Restricted Access Sequence Programming (RASP) language is functional programming model designed to capture the computational behavior of Transformer architdectures (Weiss et al., 2021). RASP programs have shown use in mechanistic interpretability both effective benchmarking tool for faithfulness (Conmy al., 2023; Hanna et al., 2024) and as a method to develop "inherently" interpretable language models (Friedman et al., 2023). Another line of work uses it (and other similar methods) architecture's s generalizability on a host of tasks (Weiss et al., 2021; Merrill et al., 2022; Giannou al., 2023). In this paper, we focus on RASP's applications in interpretability.
proof technique reason about the Transformer
RASP programs operate on two primary types of variables: s-ops, representing the input sequence, and selectors, corresponding attention matrices. These variables are manipulated through two fundamental instructions: elementwise operations and select- ellet-aggreggte. Elementwise operations simulate computations performed by a multilayer perceptron (MLP), while select-aggregate combines token-level operations, modeling the functionality of attention heads.
Every RASP program equipped with two global variables tokens and indices, essentially primitive s-ops. tokens maps strings into their token representations:
("coe"" = ["c", "o", "d", "e"] indices "ndicess""cod""" ("code") = [o, 1, 2, 3]
On the other hand, indi ces map n-length strings into their indices. That is, a list of [0, I....., .n 1]. Elementwise operations can be computed through composition. That is,
(3 indices) ("code") = [o, 3, 6, 9] (sin(indices) [sin sin(1), sin(2), sin(3)]
Tokens and their indices can also be mixed through selection matrices which are represented through the s-op select. This operations captures the mechanism of the QK-matrix. takes input two sequences K, Q, representing keys and queries respectively, and Boolean predicate p and returns matrix S of size KK [@] such that S.y = p(K,, 2)). Then, the OV-cireuit can be computed through the select-aggee select-aggregate operation, which performs an averaging over an arbitrary sequence with respect the aforementioned selection matrix. For example,
0
07
aggregate
0
0 0 [10
20
30]
[10
15).
1
0
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MLP
Ro
1
MLP
No
h1
h2
h3
hg
988
MLP
RASP Program
Tracr-Compiled Transformer
Implemeniations
Figure 4: Pipeline for generating implementations for our interpretations (i.e. RASP language models). We first construct RASP programs, then using the procedure introduced in Lindner et al. (2023), compile these programs into Transformer models. These Transformers exclusively have hard attention. Moreover, their architecture is minimal (containing only the necessary components fully implement the given RASP program). We then apply the procedure introduced in Gupta et al. (2024) to translate these Craci-Compiled Transformers into "real" Transformers: ones whose weight distribution matches those trained with stochastic gradient descent; these translated models also contain more
The previous example is directly lifted from Lindner et al. (2023).
Compiling RASP Programs. The power of RASP programming lies in its ability to translate any RASP program into a Transformer, a process known as compilation. As described in Lindner et al. (2023), this involves a two-stage approach. First, a computational graph constructed by tracing the s-ops in the program, identifying how these operations interact with and modify the residual stream. Elementwise operations are converted into MLP weights, and individual components are heuristically assigned to Transformer layers. For further details, refer the reader to Lindner et al. (2023).
As observed by Lindner et al. (2023), this compilation through "translation" introduces inefficiencies. Specifically, the heuristic layer-assignment of RASP components results in Transformers that often contain more layers than they need to have. Moreover, since RASP enforces the use of categorical sequences and hard attention (we only allow Boolean predicates) it requires various s-ops to lie orthogonal to each other after embedding as Transformer weights. As a result, this leads to a much larger embedding dimension that is usually observed in actual Transformers (Elhage al., 2022). Thus, Lindner et al. (2023) proposes to compress this dimension through learned projection matrix. The caveat is that this transformation largely not faithful to the original program (measured through cosine similarity of the outputs at individual layers).
Friedman al. (2023) takes a different approach, addressing the inherent difficulty of writing RASP programs. To overcome this challenge, the authors propose a method for directly learning RASP programs. This is achieved by constraining the space of learnable weights to those that compile into valid RASP programs, ensuring outputs with categorical variables and hard attention mechanisms. Optimizing over this constrained hypothesis class performed through a continuous relaxation using the Gumbel distribution (Jang et al., 2017).
RASP Benchmarks. Thurnherr and Scheurer (2024) is dataset of RASP programs that have been generated by GPT-4. It contains 121 RASP programs. Gupta et al. (2024) provides 86 RASP programs and compiled Transformers. The compiled Transformers are claimed to be more realistic than Tracr compiled ones instead of performing compression using a linear projection, they leverage strict interchange intervention training essentially aligning the intervention effects of the compressed and uncompressed model. This is similar in vein to many existing techniques on causal abstraction Otsuka and Saigo (2022); Zennaro (2022); Massidda et al. (2023). In our paper, we leverage the curated dataset Gupta et al. (2024) craft and compose our interpretations.
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C.2 STRICT INTERCHANGE INTERVENTION TRAINING
To evaluate our methods, we need a way verifiably ellicit different mechanisms on the same task. Let us first fix some task. Then, we proceed with the following steps:
Using Friedman et al. (2023), we learn several different explicit Transformer programs (source of randomness). We can check that they are different by looking explicitly at the Transformer programs.
2. Using Gupta et al. (2024) and Geiger et al. (2024) to get different mechanistic realizations of this abstract Transformer program.
To generate different mechanistic instantiations of the same interpretation across architectures, use the following procedure: First, we take a Tracr-Compiles Transformer model and initialize a new random model with at least as many layers and attention heads. Two models share the same interpretation if the CompilC transformer is a causal abstraction of our mechanistic instantiation. We leverage this insight by softening Definition D.3 and incorporating it directly into objective function. For detailed treatment of this approach, we refer readers to Geiger et al. (2024); Gupta al. (2024); Sun et al. (2025).
To ensure diversity in our implementations, we vary the architecture hyperparameters significantly: models contain between 2-6 layers, 2-8 attention heads, and embedding dimensions ranging from 32 to 2048. This creates rich set of instantiations with widely varying model capacities. As a result, many mechanisms these larger models may not contribute the core interpretations, leading to substantial interpretive compression.
C.3 INTERPRETATIONS FOR PERMUTATION DETECTION
To implement this task, we proceed with the following six interpretations, As discussed in Section 3.1, they can roughly divided into two groups: sort- and counting- based.
Interpretation 1. Sort the sequence; compute the difference between each element and the next one; (3) Check if all elements of the sequence are equal.
Interpretation 2. Sort the sequence in descending order; increment each element by its index; check all elements of the sequence are equal.
Interpretation 3. Sort the sequence; interleave the list with the same list in reverse order; sum each number with the number next it; (4) Check all elements of the list are equal.
Interpretation 4. Sort the sequence; check if the list contains alternating even and odd elements.
Interpretation 5. Check if least two elements in the list are equal; sum each element with the next one in the list; check if all elements of the list are equal.
Interpretation Replace each element with the number of elements less than it in the sequence; check if at least two numbers are the same.
For each of the subroutines of these interpretations, Gupta al. (2024) implements RASP programs for them. Thus, we simply compose them together to create the resulting interpretations.
C.4 GENERATING IMPLEMENTATIONS FOR 10I
Herein, we describe how we generate implementations for the 10I task. For each model, we first identify a subset of the model's attention heads that drive functional behavior. We do this through causal mediation analysis (which is well-documented in Wang et al. (2022) and Tigges et al. (2024)). For example, in GPT2-small this corresponds to attention heads (0.1, 0.10. 2.2, 4.11, 5.5, 6.9, etc., formatted [layer index]. [head index]). The complement of this set are then components that are "unused" (see Section 2).
For any input like (a) construct counterfactual input like (b) see below:
(a) When John went to the store with Mary, John gave a bottle of mi to (b) When Mary went to the store with John, Mary gave bottle of mi to
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Layer Index
0.02
Figure 5: Activation patching results for each attention head. The coloring indicates the % of performance this individual attention head contributes to POS performance.
Then, we generate implementations by ablating the unused components. This is done by running the model on inputs like (a) and then for attention head in the ablation subset we patch in the output of the attention head if it had been run on inputs like (b).
C.5 FUNCTION VECTORS AND MARTS-OF-SPEECH IDENTIFICATION
Function vectors have been found to drive in-context learning behavior (Todd et al., 2024). We leverage this concept find the circuit that responsible for in-context parts-of-speech identification. We use the Penn TreeBank dataset and consider the POS subtask. We sample 1000 points.
We create counterfactual inputs by shuffling the in-context labels. For example,
Clean Input: tree: run:verb, quickly:adverb, ire: Corrupted Input: adee:n, run:noun, quickly:verb, ire:
Then, we first run the model on the clean input and store all attention head output activations on the token "ire." " We then run the model on the corrupted input and patch in the stored attention head activations from the "ire" token. Finally, we compute the difference in logit difference loss of the generated next tokens before/after this patching operation. The results for each attention head are shown Figure It seems that attention heads the early/middle layers are most important for POS 10.1, 0.4, 0.6, 4.11, 5.1, 5.5, 6.8, 6.9, 6.10, 7.11).
Then, we compute the function vector by
Σ Σ a(x)), ØEAK=1
(C.I)
where A is the set of attention heads that we have deemed important and xę is our dataset. Essentially, we are averaging the activations across all important attention heads.
D FORMAL DEFINITIONS
We provide glossary (see Table 1 and Table for a summary) for our technical terms and notations in the main text along with their precise, formal definitions.
Table 1: Glossary of terms.
Term
Intuitive Explanation
Formal Definition
Causal/ Cousal/Computation Graph
graph that describes a model's computational dependencies
Definition 4.1
Variables
Nodes in model's computation graph that store latent computation results
Definition 4.1
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Circuit
subset of the model's computational graph that its own can fully recreate the model's input-output behavior
Definition D.1
Mechanistic Explanation
A symbolic algorithm, we can understand, that explains the model's computation
N/A
Causal Abstraction
Mapping from low-level variables higher-level ones that preserve low-level causal relationships causal graph much smaller than the model's computation eraph that explains the model's com- putation graph
Definition D.3
Interpretation
Definition D.4
Representations
causal graph that is chain that abstracts model's circuit
Definition 4.3
Alignment
A mapping between two causal graphs that pre- serves meaningful causal relations between vari- ables
Definition D.3
Intervention
modification to model's computation graph set of alignments that map to the same interpre-
Definition D.2
Alignment Class
Definition 5.1
tation
Implementation
Sets of circuits that share the same interpretation
Definition 5.1 Definition 5.2
Interpretive Equivalence
Measure of distance between of implementa- tions
Interpretive Compression
Measure diameter of a set of implementations
Definition 5.3
Congruity Representation Similarity
Approximation for interpretive equivalence Extent to which one of representations be linearly transformed into the other and vice versa
Definition H.1
Definition G.1
Table 2: Notations used throughout formal treatment. This is partially adapted from both Geiger et al. (2025) and Ravfogel et al. (2025).
Symbol
Meaning
Σ
An alphabet Set all finite strings constructed from
S
A subset of X An interpretation A circuit A representation
A
R K
variables a causal model Input variable to causal model Functions in causal model that determine variable values Partial ordering variables of causal model induced F
U F
(")
Solution of all variables to input u Solution of vy input An alignment between variables two causal models class of alignments
of (u)
(T)
The of implementations of A induced by alignments
dineer
Interpretive equivalence Hausdorff distance Interpretive compression Representation similarity Lipschitz constant of f
dh
K,T,
drep
Lip(S)
iid.
Sample i.i.d from some distribution
Definition D.1 (Cireuit), An m-cireuit of h on S is causal graph with m components that satisfies four properties:
1. Input variable is S-valued: U E
2. Hidden variables are real-valued: for each E V, E R
3. Existence of a terminal output variable: Vout V such that Vout E R2 and there does not exist o' E V where Yout > V
4. Sufficient description of on for each E S, o (s) h(s)
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Remark. For any and m 2 1, m-circuit of / must exist by the trivial causal model. To this, let U e S and Vout é hUU). Then, for any 1 < S m 1, define vy - 1. In other words, construct a causal graph with effectively one variable, letting the other hidden variables be constant functions. Remark. m-circuits are also not unique. For instance, joint change of bases to any uy E V Dour and fi results in a new causal model that preserves functional faithfulness (Park et al., 2024; Geiger et al., 2024).
Definition D.2 (Intervention). Let (V,U,F,>) be causal model. An intervention of v E V, do(ux + f), redefines _ U(,,( for Pa(ux) C {0 E :0> vz). We denote the set of interventions on Ug as I (V1).
In other words, we replace the process used to derive vy 's value, f. with a new function f. Given any circuit, one type of concrete intervention activation patching. For a of hidden variables, we patch its solutions using values derived from some other input (Vig et al., 2020; Wang et al., 2022; Heimersheim and Nanda, 2024, inter alia). Interventions often serve as a check of consistency between circuits and their interpretations (Beckers and Halpern, 2019; Geiger et al., 2021). We define this notion next.
Definition D.3 (Abstraction). Let K₁ a (V),, & (V.0,F,>2 abstracts K1 if there exists surjective mappings SubsetsOf(() :. + (V) that satisfies for all uk E V, ög E V, dodo(ux + f) (vx):
be causal models. K₂
i = π
U (((Pa())).))
(D.1)
DEET ((x))
x(do(12 + f)) do( + (())
(D.2)
We call π an alignment.
Equation (D.1) describes an observational consistency constraint: if K2 abstracts Kj, then by only observing the hidden variables of K1 we can infer all hidden values of K2. Viewed under this lens, the alignment ạ projects the fine- graiind variables V the coarse-graine Equation (D.2) is intervention consistency constraint- stating that π and @ must commute. In words, this means that we apply the intervention do(e) + to yield new causal model and then abstract that causal model through alignment this is equivalent to abstracting the causal model and then applying the transformed intervention on the high-level model. This implics that all of the causal relationships between variables in K2 can be constructed by relationships between variables in K1. Under this light, an interpretation of circuit is simply an abstraction.
ones V. On the other hand,
Definition D.4 (Interpretation). A causal model A (V*, (V*,U,E*,>) that abstracts circuit K A (V,U,F,>) >) is an faitthffll interpretation K if there exists an output variable v E V such that || D for the output variable Vout V.
Remark. Interpretations are abstractions of circuits. However, Definition D.4 does not rely on the fact that circuits have real-valued hidden variables. Since interpretations also have an output variable, one could further construct an interpretation of an interpretation (Proposition F.2). Remark. For any circuit, K is interpretation of itself (under the identity abstraction), albeit this is not very useful.
Remark. We do not require that A's variables be real-valued, nor do require that there exists a metric over the solutions of A (in contrast to K). This is compatible with the conventions in mechanistic interpretability where often interpretation is symbolic description of K that may not necessarily be performing numeric computation.
E INTERPRETATIONS MUST BE ABSTRACTION DEPENDENT
Our notion of an interpretation is dependent on a fixed level of abstraction (the variable set V). This requirement may seem stringent, especially since colloquially think of interpretations (or algorithms) as iestraction-independed entities. Further, abstraction-dependdee interpretations immediately give rise non-identifiabiiity where single interpretation may correspond many
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implementations (at different levels of abstraction) and a single circuit may also correspond many interpretations, each at a different level of abstraction.
In this section, argue that non-identifiability is a necessary price must pay for meaningful comparison between interpretations. Our main result here is that if defined over an existence statement of V interpretive equivalence must collapse functional equivalence.
Theorem E.1. Fori E {1,2}, let C' and h; be a language model such that h (S) = h2())) Suppose that h; admit circuit K; A (V,,,,;,>), Then,
1. For i E {1,2}, there exists an abstraction of K₁ A, 2 (VA.U, >) such that there is bijective translation 12 (Definition E.1) between A1,
2. There is a language model ht with circuit K" such that It (S) = hy) h2(S) and K1 K2 are abstractions of K*
3. If both K1, K2 not minimal (i.e. there exists at least one causal variable such that any intervention does not affect its oulput), then there exists a bijective translation between K2
Since we do not require circuits to be minimal (see Section 4), statement (3) implies that we must restrict the set of permissible alignments. Otherwise, any non-minimal circuit is O-interpretively equivalent to any other non- minimal circuit (see Proposition F.4).
Recall that for any causal model K e (V,U,F,Y] )), given an input U +- u, there exists a solution to each of its variables which we denote as V (u).
an m-circuit, then
(') E E* x R x RM R
(E.1)
For brevity. we slightly abuse the notation and denote V R" x... x R x R In words, bijective translation a causally-consistere bijective mapping between the solutions of two causal models. Essentially, if one model is a bijective translation of another, we can think of it simply as relabeling of the other one.
Definition F.I. For E [1,2], let _ a (V, ) be a causal model. For any u E. V + v bijective translation if satisfies Equation (D.1) and there exists @ I IVv)) I (V2) that satisfies Equation (D.2),
>)
Proposition E.2. For any n,m E Z+, there exists bijection between Rº and I...
We are now ready for the proof of Theorem E.
Proof. We prove all claims constructively.
1. For i E (1,2), let A; be the trivial causal model associated with h; it has two causal variables
U, Vout such that Vout & (U) Consider an alignment map πr such that (TVourt) Vout. =+(U=U and for all other v E Vi, (b) =1 (i.e. π is undefined on other variables in V.). Define the intervention map to be the identity if and only if do( + f) E I (V,) is an intervention on or Vout Then, it follows that the only valid interventions defined by @ are those directly on the input and output variables. It is easy to see that both Equation (D.I) and (D.2) hold. Since h h2, AI constructed in this way are the same causal model. Thus, the identity map forms trivial bijective translation between them.
Let K & (V,U,F,> >>) such that V V1 U Vour.. Construct F such that it runs K2 in parallel and then averages their output. Coneretely, let v 2a E V be the output variables of V1, V2, respectively. Also let U 2 e V be their input variables. Define
Pa(Ui) = Pa(U2) = {U}
Pa(Vout) = (u )
and their transition functions
femm (( 2 3.) = (fol (Pa(woun))) + Sui (Pa( (Pa(vSu):U))
1
Clearly, K* is a valid circuit for some language model alignments from K* + K,, By the same constructive above for w. K1, K2 must be both be an abstraction for K*
h2. Let ni, T2 be the identity
in general, bijective translation is neither stronger nor weaker than abstraction (Geiger al., 2025).
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3. Let be the identity function that maps oout V - vout E V2. By Definition 4.2, for {1.2). any variable v2 E V {U, v takes on values in R for my E Z By assumption, for both models K1, K2, there exists least one causal variable that is "unused." Denote this variable as Vumused- By Proposition E.2, there exists a bijection that maps a: : {U, Vout, b Pumused) - 0 unused Composing yields bijection from V + > It is easy to that this is bijective translation which satisfies Equation (D.1) and (D.2). By symmetry, the other direction also holds.
F PROPERTIES OF INTERPRETIVE EQUIVALENCE AND COMPRESSION
As discussed Section 5, an interpretation a model need not be a lossless description of that model. For many applications an interpretable explanation is necessarily lossy with respect the learned model. In this sense, an abstraction alone is too restrictive as it requires any interpretation to preserve the exact functional behavior of the model. Herein, we explore some properties of interpretive equivalence, compression and illustrate their similarities and differences with causal abstraction.
We first show that both implementations and interpretations compose. That is, an implementation of an implementation is also an implementation and an interpretation of an interpretation is also an interpretation.
Lemma F.1. For E [1,2,3), let K; 2 (,,U,,;,7)) be causal models. Let 122 be an alignment from K1 - K2 and 222 alignment from K2 + K3. Then, K3 is abstraction of K₁ with an alignment
map N2,3 o 1-2--
Remark F.1 (On Composing Alignments). We are liberally abusing the notation here. Technically, the composition T2.3 T1.2 is not well defined since the alignments have incompatible types: Z1,2 SubsetsOf(V), - V2 and 72.3 SubsetsOf({{)) starting from the causal graph K1 applying the alignment 21,2 to yield new causal graph K2 and then applying the alignment 2 to yield the causal graph K3. This possible because an alignment alone sufficient to induce abstraction (see Geiger et al. 2025, Remark 32 for a construction).
V3. We write N223 4122 to mean the process of
Proof. We first check Equation (D.1), Since 22,3 is an alignment that enables K2 to abstract K1, for any oi E V3.
U¹ N2,
U
By Definition 4.1, 0i (Pa()))) U). alignment map æ2,2, it follows that
Then, because K2 is an abstraction of Ki through the
A(Patot).0)) ))
2i 122
U
U
((f))
((Pa((b))o)) ((Prato2110)) ))
(T2z3 o T1,2)
U
U
(f
= = 2122)
U
A 11
( (())
It remains to show that the mappings between interventions holds. Denote 01.2 : : - (V22 and (02,3 : (Vv2 - I (V;). We claim that Equation (D.2) satisfied with 02.3 o W1.2. Since 00,2 and W2,3 satisfy Equation (D.2),
(203(do ()) do( (2)
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02, ( +- f))) (0 - (m2.3 (1 + f)) d (e (02.3 (02.3001 0123)
where of V2, of E 2 and f are chosen arbitrarily.
Proposition F.2. Let K a (V,U,E,>) be a circuit. Let A a (VA,,,FA,>)) be an ny n-faithful interpretation of K through an alignment n. Let A" & U. be an n-faithful interpretation of A through an alignment π². Then, A is an 2n-faithful interpretation of <through an alignment π n..
Proof. There exists an alignment from - A and an alignment X' - A*. By Lemma F.1, an abstraction of with the alignment map x² o zt. There exists a variable vout VA vd E V such that || b Woull < n7 and a variable Vout E such that || Vour| < 17. By triangle inequality, 10
Vout
2n.
Proposition F.3. Let K & (V.U,F,>) be a circuit. Let A be an n- n-interpretation of K and II be an alignment class that maps K 's variables onto A Let A be n-interpretation of A and II, be are implementations of A
alignment class that maps A' 's variables onto A"'s. ss. Then, Il
Proof. The proof is trivial and directly follows from Proposition F.2.
We relate interpretive equivalence to causal abstraction and show necessary and sufficient conditions for interpretive equivalence using abstractions.
For E [1,2), let 2 (V,U,F,>) be circuits with n-interppeetatoos A, - NYY,U,R (VA,UR Let II, be alignment classes that map Kj'S ss variables onto Aj's 's variables.
Proposition F.4. Suppose A₁ is an abstraction of A and vice versa through alignment maps 21,2, 12,1- A1,A2 are O-interpretively equivalent if π2,1 I1 Il and R222 o I122 III
Proof. We show that IT (Ai) C II, (A2). Take any E IT (A1). By Definition 5.1, there -
By Lemma F.I. Since 221 o E IT," (A2), this implies that F E 11, (A2)- Applying the same argument in the other direction, II2" (A2) IT,," ' Therefore, I1, "A,)(A)) 1152 (A2) and by Definition 5.2, A1,A2 are O-interpretively equivalent.
exists πj E Ih that forms an abstraction from 2221 is an alignment that forms an abstraction from (V.U,F+))
O
Proposition F.5. Ler IT, be closed. Suppose A1.A2 are O-interpretively equivalent. Suppose there exists E II; that each admits section, VAA + SubsetsOf(V.) such that (((s(v)) v. f. any v E 4/. Suppose that there exists some nt πł, such that πι 01 π E Th and xt 082 TT IT Then, A₁ an abstraction of A and vice versa with alignment maps T o1, o5,,7(1 T/ 022
Proof. Since I1 (AA,) is closed, Definition 5.2 implies that I1 (A,) II," (A2). For brevity, we denote T2, - TTJ o1. Note that T2, Vi Vyz easily extend this to subsets V c V by
Usi(t))
n 22,1 (V) T
(F.1)
vev
Now, refer to π2, as Equation (F.1). We first show that π2,1 satisfies Equation (D.1). Fix 02 VA. By Definition 4.1,
02 - (Pa(22)))
(F.2)
By assumption, Æ2,1 2j E 12. Thus, N2,1 ui an alignment map that enables an abstraction from K + A2, we can rewrite Equation (F.2) as
S2 = (72 01)
U
fo, (Pa(()))U)
(F.3)
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(T2,1 (
U
U Sc(Pa()).U)
(F.4)
men (vz) we (mrs
22,1
π
U
(o(P(()))U)
(F.5)
NxExz VEN ) (ww)
22,1
T
(o(Pa()).U)
(F.6)
((22)
22.1
for (Pa(o)))) (00)
U
(F.7)
The last equality follows from the fact that zj is an alignment. Therefore, 22,1 satisfies Equation (D.1). By the same argument, this also holds for π222- It remains check Equation (D.2). Since π²,, alignments, they admit Ø3j, of that satisfy Equation (D.2). Let 22, be sections Φj of. respectively. We claim that @2,1 - wo ot satisfies Equation (D.2). Since 2221 is an alignment, and take E V,
( n (do + f)) do ) + (02,1 (f)()).
(2 f))) do( ( (vx) 02, (o)
T2 (do (1x) 0/ (f)) do( (xx)) 22,1 (o)
do(m (mx) + of E I (V4) and x (vx) E vA. Thus, through relabeling, of a x (vx) and fi (f),
n2,1 (do f()) = do( (1)) if)))
And, Equation (D.2) holds.
G PROOFS FOR REPRESENTATION SIMILARITY AND INTERPRETIVE EQUIVALENCE
Suppose we have two circuits over a shared task S for models h1 h2 defined over different neural architectures. These circuits we notate as K₁ (VI,U,F1,>) and K2 & (V2, U,,2,>) respectively. Let us also assume each circuit admits an (44, 8, representtaion R, UH U,F,U,FF, > through alignment maps Pi- First, define notion of distance between these representations. We note that since the representations themselves are function of the input (recall Definition 4.1), we are essentially defining distance function space.
Definition G.1 (Representation Similarity). Denote by H, å concat (H,") to be the direct sum (concatenation) of all hidden variables in H,, Then, representation similarity between Ri and R₂
drepr R2) A max
inf [|4 21
|HH -H2ll inf
H -BH21l
(G.1)
||B S1
where A, are linear operators and ||-|| is direct-sum norm.
We first show that representation similarity (Definition G.1) is pseudometric. Lemma G.1. For any representattinns R1, R2,Rs. drepr satisfies
drepr (R1, R1)
2. drepe R2) drepr R2)
3.
(r RR) < drepr R1, R2)= R2)
(R2, R32
13Our formulation is inspired by Chan et al.'s 's (2024) notion of representation alignment between language encoders. Although Definition G. not true similarity, this naming stays consistent with the literature (Kornblith al., 2019).
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Proof. The first two properties trivial. So, remains to show triangle inequality. Let H/ be the con- catenation hidden variables in R, Let 21,2, 92.3 be arbitrary operators where 1111 002llpp 12222lopppp I that map H2 - H1 and H3 + H2, respectively. Then,
||t 91.2 02,1//31 = I/ 21, 01. Q1, 0
< [/ 91,2
Q1.2 92.311311
< HH 91.2 + 91 112 93
K HH 91.2 ||2 92.
Taking infimum over 21,2, 92,3 gives the desired result.
Definition G.2. Let (2,d) )) be a (pseudo)metric space. Let S C Z. The diameter of S, denoted
diameter(S) is defined
diameter(S) = sup d(a.b). a
(G.2)
Lemma G.2. Let (Z,d) be a (pseudo)metric space and du be its Hausdorff distance. Then, for A,B C Z and E A,b E B.
dn(A, < diameter(A) diameter d((.b)).
Proof. Fix any d' E A. Then.
inf d(d ,') S d(d' yeB
< d(aa (a) + d(a,b);
< diameter (A) + d(a,b).
By symmetry, for any b' E A, it follows that en d(d', bb) S diameter(B)) + d(a,b) b). Therefore, by the definition of the Hausdorff distance, yield the desired bound.
Definition G.3. Let K & (V,U,Fi,)) K2 _ U. F2. be two arbitrary eircuits defined over variables V. Suppose some pseudometric V x V + R Let R a (H,U, -) be representation spaces and SubsetsOf(V) - HI an alignment map. Denote R, the representations of K, constructed through p. We say that p is Lipschitz continuous with Lipschitz constant Lip(p) )) if
drepr (Ri, S Lip(p)d
(G.3)
Lemma G.3. Fori E [1,2) let be (.. 8)) representatio for models hy Suppose that |/| h2ll A. Then, drepe R2) - 8 +A.
Proof. Let H1 be the concatenation of all hidden variables in R,. Since Ri is (Li, ( there must exist maps A;, such that
A.H h 8; and
H, 1H--BAh B,hil < 8,
To see this, we know that for each hidden layer k, there exists Aie such that || hi| < 8. Then,
4 Σ (Anathx - mi) (Aitths-hi
(G.4) Define single (block diagonal) linear map A; that acts on H, = (Hi,,...,,HH ...L) where
L Σ AipHi - hi 1
Σ A=1
Ai.Hii fiik -Lat:
|AH A L
Σ AmtHiell
With the l. direct-sum norm on 4, and |A S 1. Substituting A; into Equation (G.4), we have that ||AH - hill 8, Repeating same construction for B, yields the "decoder" inequality. Consider infA, ||A H2ll.
inf AnHHi H2ll < BBA - -Hll.
Ao
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< H B2/22l + [|2222 H2ll
< B2 BBlloop (|A)/H -h hill + || -
h221) 82,
- 8 - 82 +A.
The same argument also holds the other direction:
inf
H1
AoH2ll
- H| B1A2H211
Ag
$ || Byh1 I| || B1A2H2ll,
S 61 |Billop (1I/1 1111-h21 h2ll + ///2 22--2222l)
- 0 +
By Definition G.I, drepr R2) 8
G.1 PROOF OF MAIN RESULT
For i E {1,2), suppose that A, is an n:-faithful interpretation of K, Let II, be alignment classes that map Ki 's variables onto AI,A2's 's variables, respectively. Assume that circuits admit L--layered representations through an alignment map Pi. Let F,F E II (A1), then of Σx + IR" for some n E 2 (Definition 4.2).
Assumption G.4. Let D: Th and assume the implementation distance d in the form
d(F,F) A | - -Dvf 1I.
for some function norm.
c is distillation map that compiles all of the activations of VF into tractable representation. In sense, + exposes the measurable part of VF_ Denote Rş to be the representations of constructed through the alignment map Pj. Since RF a deterministic causal model (Definition 4.3), we denote H Σ" + I to be the solution of all variables in Rp given some input setting. In other words, HH the concatenation of all hidden representations of Ry. This the same construction used in Definition G.I.
Assumption G.5. Assume there exists a 1-Lipschitz map V: RR R² such that
|H - पH SHFf-YHffI S diepe (Rr, Rp)
and
ΦVF WH w.
(G.5)
In other words, the measurable part of our circuit is approximately determined by its representation. Next, we assume that there exists least one implementation (A2) that admits a reasonable representation under Pl. Since we only enforce that p1 is valid abstraction map for K₁, without this assumption, one could construct an adversarial implementation of A2 whose causal variables are completely discarded by Pl.
Assumption G.6. There exists F* E (A2) such that its representation under Pl, Rps, is an (L1, 8 -eprrssentation.
We are now ready to prove the main result.
Main Result. Let II and II, be alignment classes that map Kj 's variables into A1,A2's 's variables, respectively. Then, the approximate interpretive equivalence of A1, under alignments II III"'
dinterp (2), < K(AI,K,,II) Ki, +20 drep R2)
A.
(G.6)
Proof. By Lemma G.2, for any F II [A1), (A2),
dinter (A1 A2) < x(A1, Ki IT) x(A2, II. +
So it remains to upper bound d(F,F) for single pair (F.F). Let choose F to be the circuit Ki which has representations R1. By Assumption G.6, choose F E IA2) that admits an (44,8))= representation (which we denote as Rp) under the alignment P1. By Assumption G.4,
d(F, (F.FF) A 1
ΦVF
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We add and subtract WHA THE
d(F.F) - | - |H |+ ||P wPA/ ||+ ||TTT -DVPI
S 200 + ||P1I" 411
Since V 1-L.isschizz |Аम H By triangle inequality,
THI || S (R1, Rp), where Rp the causal model associated with
depp (R,, Rp) drepr (Re + deepr R2, Ry);
S drepr R2) + 8, + 82 +A.
The last inequality follows from Assumption G.6 and Lemma G.3. Putting it all together, dinterp (A1, S x(A1, 11) x(A2, I1.) 20 dreps R2) P + 82 + A.
G.2 PROOF oF MAIN RESULT 2
For i E (1,2), let us denote A; as n:-faithful interpretations of K; Main Result. Let II and II, be an alignment that map 's variables onto A1,A2 variables. Suppose A1,A2 are E-approximate interpretively equivalent under the alignment classes II,, II. Then,
(R R2) S e(Lip(p)) + 1) 8, + + +A.
(G.7)
Proof. Since A,,A2 are e-interpretively equivalent under the alignment cases II, there exists 1y (A2) such that d(Hi, FE) Let R2, be the representations of Hi, F3 under the alignment Pl, respectively. Then, PI Lipschitz continuous implies that
drepe RR) S Lip(p))d(Fi,F)) F)) s Lip(pi)e.
(G.8)
Since R₁ is an representation of h1, there exists A1. Bi with || Aallop..BB1, S such that |AIH - h1 5 and
H| - B h1 S 8,, where H1 is the concatenation of all variables in R1- drep (R1, R3) - Lip(p)) implies there exists A', B' such that
| Hall s Lip(p))e and
|| - B¹ H3 S Lip(pi)e,
where Hi is the concatenation of all variables in Ry We now bound the "encoder" and "decoder" error of R2 separately:
A h2 s |A - H1) |A - -ill +|/ hall.
|AAllop AA'z Hi|l +
s Lip(pi)E +A.
And,
HH BB < ||A - B'H 18HH B,h1)] ||||++||BBB (1 h22ll,
< Lip(p))) +A.
Therefore, R3 is a (Lq,Lip(p))) + 8, + rreressseenn of h2. Directly applying Lemma G.3, diepe (R2, R2) (Lip(p))
82 + A. By Lemma G.1,
drepr Rí, R2) < drepr (Ri, R3) + drepe (2, <
+8+
Where is faithfulness in all of this? Unintuitively, neither Main Result nor 2 explicitly contain the faithfulness of A,,A2 (7), 72, respectively). Our insight is that faithfulness is implicitly encoded into the alignment classes II. This is because any valid alignment must be consistent with the faithfulness of A₁ (see Definition 5.1). In this way, 77 determine which alignment classes non-empty. This why alignment classes form the crux our constructions, because they describe both structural and behavioral constraints over abstractions.
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H COMPUTING INTERPRETIVE EQUIVALENCE
For i E {1,2), let A, be interpretation and K; be circuit. Let IT, be an alignment class that maps K/'s variables into A variables. We define probability spaces (II;" "PP)). Suppose P1-P2 are alignment maps.
Definition H.1 (Congruity). Denote Fi Lid PI, F2 Lid Pz. Let R1, R2, R2 be their repre- sentations, respectively, under the alignment maps P1.P2.
Pi - Pidrepe Rī) < dieps R2)
P2 a Pidrepr R2) < deep deepr(R2, (R2, R1)),
(H.1)
where the joint distribution over II, (Ai) x (A,'(A2) The congruity between I1, (A1), (A2) is 1
|P +
In one direction (say P1). we sample two implementations from II (Ai) and one from I1," (A2). p is the probability that the two A representations are more similar to each other than either does to the A2 implementation. If IT (A1) n2 (A2) and Pi P2, then pt = 1/2 and congruity is 1, In other words, A2, are completely congruous with respect their representations.
H.I PROOF OF MAIN RESULT 3
For clarity, the result is restated.
Main Result. Let p be the expected value of REPaDIST in Algorithm Fi, lid. PI, and Fg and IP2. Let Ri,Ri,R, be their representations under alignment maps P1,P2. Then,
p z 1 inf (P(dreps (R1, Rī) x(A1, + +e) P(drepe RR) dinterp (2) -e)) (H.2)
es0
PIIPI and R₂ iid. P2IP2, where p,lP, the pushforward of P, under p₁. Consider the event drepr (RI, RT) diepr(R) RR), J. For arbitrary E > O, this event occurs with probability I when drepr (R1, R2) > KAA,, 1 K1, H1) + E and drepe (R1, (R1,Rs) R3) < dinter (A1, Therefore,
Proof. Let R bid.
p 1 P(drer R R1) x(A1, K1 (T)) + PP(drepr (R1, R2) dinterp (A1, E), since 0 was arbitrary, we yield the desired result.
(H.3)
Corollary H.1. Let Fi iid PI, F2, iid IPz. Let Ò be the congruity between II," (A1),11, (A2). Define
inf F(FF)
P (R) K(A1, II1) +
e>0
2F (R1, R2) < dimer (A1 A - El)
(H.4)
+ (IP (reppr (eoprr(R2.2 R2, R3) k(A2, K2, 112) + +8])
Then,
2
S(H
(H.5)
when ((F.FFF
Proof. By Definition H.I, C(p1.P2) e 1pl P2
1/. We break Č into two cases:
(2 (pi+ p2) pI + p2
if < P1 + p2 2, i < P1 + <
C(p1.p2)
Since C(P) +22 = ((2 P1 - P2). any lower (resp. upper) bound proved one side transfers to lower (resp. upper) bound on the reflected side. Thus, we separately bound each case. Consider first 11 P1 + p2 2. Then,
C(11, 2 (p1 + P2),
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$ inf Pidrepr (R), xI,I, + 820
2TP diept (R1, R2) < dinterp (2)) =I) (P [ddeppp (R), > x(A2, (2, =1)]
where the last inequality follows from Equation (H.2). Now the case 0 S P1 P P2 1
P2)),P2)= = P1 +
2 2- - inf Pidrepp (R1, e>0
+
21P 21 < dinterp (Ai A2) - E]) (P [P]rrepr (R2, K2, +E))
H.2 INTERVANTION-IAPLLEEENTTA DUALITY
Our results in the previous sections show that estimating both the representation similarity and interpretive compression crucial to understanding interpretive equivalence. In general, enumerating the set of implementations for given interpretation is hard. Herein, we make connection between implementations and interventions. Specifically, we show that it is possible to view interventions as instantiations of new implementations. With this perspective, we bound the sample complexity (in interventions) interpretive compression estimation.
First, given set of interventions, we define a notion of closure for alignment classes.
Definition H.2. Let & (V.U,F,>) be some circuit and A be an n-faithful interpretation of K. Also, let II be an alignment class that maps K's variables onto A's. Consider a set of interventions over all variables in V which we denote (V) (see Definition D.2). We say that II closed under ((V) if for each implementation K* E (()), there exists E of (V) such that K* do(V + f).
Remark. Any alignment class II is closed under natural set of interventions. Simply, given circuits K"K* E II (A), this is the set of interventions that replace K's transition functions, F, with all of K* transition functions, F*.
(A),,
Remark. Dually, instead of explicitly enumerating an alignment class, we could instead specify set of interventions under which K's interpretation is preserved. This implicitly induces an alignment class. With this method, one only needs to argue that the interventions are interpretation-pree prreeervn.........g..... This is the method we use in GETIMPl to construct implementation sets.
Definition H.3. Let C be some set and d : C x C + IR=00 be some distance metric over C. Let (Xm } C C. Then, the empirical diameter of is maxisi< (x,,x)))
Proposition H.2. Lei : X V + R20 be a distance metric belween circuits. Let K. be a circuit, A be some 1) p-interpretation of K & (V,U,E,>), and II be an alignment class. Let I (V) be a set of interventions on K under which II is closed. Let P. be a distribution over I. LetC=N(IT (A)-N(((I)A))))))))) eA)) be the covering number of the implementation space II (A). Define
Pmin
14JSC Pl(1 E (((LEI())) (V) do(V + 1) BB}]
min
where {Bi} are balls of radius 2/2 that cover II (A). Then, for
log(C)-Int)
Pmin
have that TP[ (P(K(A,KII)) Ryd (I, <e) > 1 8 where & is the empirical diameter of III -(A) computed from m sampled interventions.
Proof. From Definition H.3, P[x(A,K,II) 2 (A, =1 Consider the event (AAK, (AA, K, KIIT) > E. This occurs only if we draw m interventions {I]...., 3m] where there exists at least one ball B, where there does not exist some lj such that do( <- 4) E B,. This event occurs with probability most (I - Pmin) By the Poisson approximation, we have that (I Pmin) " e mPmin. Setting this less than d and rearranging, we yield the result.
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Reviews:
[{"review_id": "Yvufs5XJgu", "reviewer": "Reviewer_K2kq", "summary": "This work introduces the framework of interpretive equivalence (IE) in neural networks, which formalizes what it means for two neural networks to \"do the same thing\". The authors define notions of interpretation (high-level algorithmic process) and implementation (possible low-level realizations). Using the principle that models with non-distinguishable implementations must contain the same interpretation, the authors develop algorithms to check for interpretive equivalence, relying on the concept of ambiguity (are two implementations statistically different?). They validate their algorithms on several real-world models (Pythia, GPT-2) and tasks (permutation detection, IOI, POS/next token prediction) and ground their framework in the theoretical foundations of causal abstraction.", "questions": "- How sensitive is Ambiguity to the number of implementations (n in Alg. 1) and the methods used to generate them (perturbations, deletions, etc.)? Have the authors explored how this scales with model size?\n- How should one interpret graded values of the Ambiguity score?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- This work focuses on an important and only recently acknowledged challenge in MI, which is the lack of rigor and solid theoretical foundations. The framework of interpretive equivalence bridges that gap by helping formalize what it means for two models to be equivalent in the algorithmic sense. The authors move away from most previous approaches that focus on a single interpretation (sometimes hard to define) and now focus on the much larger set of its implementations, which should make future automated discovery methods much more robust.\n- The theoretical foundations were not checked down to the details, but they seem rigorous and strong. The authors build upon the framework of causal abstraction used in previous works and successfully formalize the concepts of interpretations, circuits and representations. In addition, the main theoretical results (1 and 2) provide necessary and sufficient bounds that connect IE to representation similarity, which is an important contribution. This brings much-needed mathematical precision to a field that often relies too heavily on empirical heuristics.\n- The experimental validation is well-designed and compelling, progressing from toy tasks to real-world applications. The authors first demonstrate that their Ambiguity metric is well-calibrated using hard-coded RASP programs with known different interpretations on a toy task (permutation detection), then show the utility of their framework in comparing different models (GPT-2 and Pythia) on the IOI task. This confirms known results but shows the potential of IE to reduce the interpretation of large LLMs to smaller equivalent ones. Finally, the experiment in 3.3 relates next-token prediction (a complex task) to POS identification (which is much more well-understood), which shows that IE can be used to decompose complex problems into simpler ones.\n- This is a rather complex work at the intersection of theory and practice, and effort has been put into making it rather accessible (presence of a glossary, high-level features).", "weaknesses": "- The success of the proposed Algorithm 1 seems to critically rely on the GetImpl procedure, which samples implementations from a model. This procedure is described at a very high level and could use more details. In particular, it is not clear how feasible and costly this procedure is for large LLMs. The quality of the Ambiguity score depends on how well these samples represent the actual set of all possible implementations. The paper does not discuss this limitation or the sensitivity of Ambiguity to the sampling process.\n- The authors choose a metric based on linear transformability (drepr) for representation similarity. They mention that representations may however lie on complex and non-linear manifolds. This choice is common but could limit the framework's ability to detect more complex relationships. A discussion of how the framework would work with other types of similarity metrics would be useful.\n- Ambiguity is not a binary score, but a graded one. It is not entirely clear how MI researchers should interpret these values. For example, what does an ambiguity score of 0.5 mean? Do the models share 50% of their algorithm? Considering the practical goal of the framework, the paper could include a discussion on how to choose proper thresholds or generally interpret these continuous scores.\n- The introduced framework assumes that high representational similarity is evidence of interpretive equivalence, but other factors could lead to representational similarity (architectural biases, similarity in training data). The framework does not explicitly account for this.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ehTMZj54fV", "reviewer": "Reviewer_Td1h", "summary": "The aim of mechanistic interpretability (MI) is to discover the algorithms that are implemented by trained neural networks to perform tasks. This is a difficult problem, and one of the difficulties is that it is unclear what the right target of MI is: it is relatively rare that behaviour can be captured by “circuits” whose function is completely understood, if by “circuit” one means neurons and edges between them. Indeed many of the most promising approaches to MI now replace neurons by other degrees of freedom, such as SAE features.\n\nThe authors of the present paper propose a way of sidestepping this issue by trying to quantify when two neural networks compute in the same way, rather than trying to directly characterise how they compute. This is an interesting idea, which can be viewed as an evolution of the existing literature on representational similarity across models (see e.g. Klabunde et al “Similarity of Neural Network Models: A Survey of Functional and Representational Measures”).\n\nThe contribution of the paper comes in two parts: a theoretical part and an empirical part. In the theoretical part the authors propose precise definitions of various terms such as representation, interpretation, circuit and alignment, and prove some simple facts about their key measure of “ambiguity” and representation distance. In the empirical part they test their notion of ambiguity on three main settings: RASP programs, the IOI circuit in GPT2 and Pythia models, and GPT2 between different tokens.", "questions": "* 260: what are these interventions?  \n* 288: what does “yield the same interpretation” mean?  \n* 156: we sample implementations, but according to what distribution? It seems in practice what we do is just what Algorithm 1 says?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "* The idea of quantifying representational similarity across a circuit, and the associated metric of ambiguity, is an interesting approach that sidesteps some of the intractable parts of MI  \n* The paper makes a serious attempt to ground the somewhat informal terms used in MI in the precise language of casual graphs  \n* Includes an interesting baseline in RASP programs and applications to models up to 2.8B parameters, showing the feasibility of applying the method at scale  \n* There is a significant amount of new terminology in this paper, but the authors do provide a detailed glossary which I appreciated", "weaknesses": "Minor weaknesses:\n\n* I think the related work is not sufficient; in particular, there are not enough references to the existing literature on representational similarity across models which this paper seems to be extending in various ways. Maybe the authors could take a look at Klabunde et al “Similarity of Neural Network Models: A Survey of Functional and Representational Measures”.  \n* I found the leftmost plot in Fig 2 to be a bit misleading, in the sense that the color coding here seems to refer to some threshold that is not specified rather than e.g. linearly representing the ambiguity as in the middle plot.  \n* In several places, including (5.1), distance is defined between sets of variables where it seems that what is intended is some average over distances on actual inputs.\n\nMajor weaknesses:\n\n* I am not convinced of the importance of the theoretical component of this paper. Could the authors elaborate more on the relation between the empirical and theoretical contributions?  \n* In my view the main empirical contribution is the middle plot of Fig 2\\. However I do not have enough details to evaluate how significant I find these ambiguity figures. At first glance the high ambiguity between Pythia models is interesting, but what are the precise circuits being used here? What are the ablations that are performed? It seems that the meaning of the ReprDist metric depends quite heavily on the details and strength of the ablation and I did not find these details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1w7iTqpaju", "reviewer": "Reviewer_pr5o", "summary": "The paper introduces a novel framework for determining interpretive equivalence between neural networks, whether two models implement the same mechanistic interpretation without requiring explicit interpretation of either model. \nThe core insight is nice and sidesteps identifiability difficulties in MI by solving a relaxed problem: two interpretations are equivalent if and only if the set of possible implementations are similar. The authors operationalize this through Algorithm 1 (AMBIGUITY), which leverages representation similarity across sampled implementations to estimate equivalence.", "questions": "What are the convergence properties of Algorithm 1?\nHow would standard representation comparison methods fare for the tasks in Figure 2.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "Novel problem formulation: The shift from \"interpret each model\" to \"determine if models share interpretations\" is valuable and practical as a conceptual idea.\nTheoretical grounding: Main Results 1-3 provide some theoretical guarantees via lower and upper-bounds connecting representational similarity with interpretative equivalence.\nThe advantage of the algorithm 1 is that it is scalable and applied to some standard LLMs.\nI appreciate that the authors put effort in the presentation of the work and the concepts.", "weaknesses": "Despite the framework being stated in very general terms with causal abstraction as its main language, it seems that Algorithm 1 is fairly tied to circuit (with GetImpl doing circuit manipulations basically).\n\nThe paper acknowledges that the linearity assumption is used and could be studied further in future work but this remains a fairly strong and important assumption. \nIn particular, the findings of Figure 2 do not really prove the usefulness of the approach, to produce the same kind of heatmaps it is enough to distinguish model class (on the left panel) or model family (on the center panel), which intuitively is easy to do by looking at their representation, and most distances between representations would produce such heatmaps without making claims about interpretative equivalence. I think it would be more convincing to show cases where \"standard (CCA-like)\" representational similarity do not give the correct answer, and AMBIGUITY does. The POS tagging experiment is already a bit more convincing as it already compare against an alternative hypothesis.\n\nI fear it might be easy to engineer counter-examples for Algorithm 1:\n- false positives: It would be two neural networks whose representations are close to be linearly mappable to each other but who do different things (maybe even beyond different implementation of the same behavior, different behavior altogether). Experiment idea: Start with a network and task, train another neural network adversarially to behave maximally differently from the first one but constrained or regularized to enforce linearly mappable representations. Failure to produce such adversarial examples would be a strong demonstration of Algorithm 1's robustness to false positive.\n- false negatives: It would be two neural network who do implement the same thing but it is not visible by linear inspection of their representations. Experiment Idea: if you apply invertible non-linear transforms at each layer of network to produce a new network, linearity is lost but functionality is unchanged. \n\nConvergence and stability of Algorithm 1:\nI think the paper could engage more with the convergence of algorithm 1, it depends on finite sampling of the number of repeats n in AMBIGUITY. It would be nice to see convergence rates and stability (as measured by some notion of variance of the estimate)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8jWQw2dWPf", "reviewer": "Reviewer_DWxu", "summary": "- This paper studies the problem of “interpretive equivalence”, which is how to determine whether two models share the same interpretation of a behavior (and by extension implementation). They develop algorithms to measure the similarity of two interpretations of a model based on their representations, and evaluate their algorithms/theory on three tasks using both toy transformers and pretrained language models.\n____\n- The paper seems like a solid contribution towards making the evaluation of neural network interpretations more rigorous, and for evaluating the equivalence of interpretations across models.", "questions": "Overall, I think the paper is pretty good. My main concerns are listed in the \"weaknesses\" section, and addressing those would help me feel more confident in the paper. Below, I share minor comments/questions.\n\n- I’m not sure what the text at the top of Figure 1 is trying to communicate, can you elaborate?\n- Line 284: When you say “GPT2” is this GPT2-xl or GPT2-small?\n- Line 281: “interpretive similar” seems to be a new term. Is it just a weaker form of interpretive equivalence? It feels natural, but you might want to define it somewhere.\n- It’s kind of surprising that the diagonal for Figure 2b is not higher than some off-diagonals (particularly for the gpt2 family) - any ideas what's going on there?\n- Another further validation of this method might be to compute ambiguity for individual components with specific hypothesized roles, like induction heads. Have you tested whether your ambiguity score yields high values when only measuring it on component subsets (like induction heads on repeated text snippets)?\n\nMinor Typos:\n- Line 215 (footnote): “the two models do not the same interpretation” -> missing “share” (ie. do not __share__ the same…)\n- Line 357: “abstrations” -> abstractions\n- Line 411: “On the other,” -> On the other hand?\n- Line 423: “simiarlity” -> similarity\n- Line 946: “framemworks” -> frameworks\n\nPapers:\n- You may also be interested in [Miller et al.]’s analysis of evaluating and comparing circuits, which seems related to the problem you’re proposing and addressing here.\n\n___\nMiller et al. [Transformer Circuit Metrics are Not Robust](https://openreview.net/forum?id=zSf8PJyQb2)", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- The paper is well written and its connection to previous work is evidently thought-through throughout the paper.\n- There is a nice mix of theory and practice - providing intuition and showing the framework being applied to both toy settings with known ground truth and slightly more complex settings that have some level of prior understanding.\n- The evidence that this framework can distinguish between implementations on the toy task n-Permutation setting is promising (Section 3.1, Figure 2(left)).", "weaknesses": "- The argument that interpretive equivalence allows us to study simpler/smaller models (Line 90) seems like a nice implication of this framework when the ambiguity between two models is high. However, I have two concerns with this: \n    - The first is that it seems desirable that ambiguity between the same model usually close to 1 for the same task, but it sometimes is not (e.g. GPT2-M w/ itself on IOI task). Does this mean the perturbation of GETIMPL is too aggressive, or what else might cause this?\n    - Second, this is somewhat at odds with evidence from other work that some interesting capabilities seem to appear only in larger models (see e.g. [Brown et al], [Wei et al], [Schaeffer et al]). A recent example that comes to mind from the mechanistic interpretability literature is [Prakash et al] who note “we do not examine smaller models, as they are unable to coherently solve the … task”. How does your framework incorporate how well a model can solve a task? Do you have some performance threshold you decide before you choose to use its representations to compare to another model? While the tasks analyzed here are simple and I’d expect all the models you analyze can do them, it's unclear how model performance is factored in when understanding the similarity of implementation for more complex tasks, which seems like a current limitation of the proposed method.\n\n- The GETIMPL procedure in Algorithm 1 lacks some preciseness that would help clarify my understanding of the Ambiguity computation. As I understand it, the GETIMPL is designed to produce a modified model with an equivalent implementation/interpretation. However, what gets returned seems to be the exact copy of the model you start with, rather than a modified version. It would be helpful to clarify whether the operations in GETIMPL are in fact supposed to be producing a modified copy of the model (with an equivalent interpretation/implementation), or whether it does actual return the same $h_{\\theta}$ that is provided as an input.\n\n- It's unclear to me how to interpret a middle value of \"ambiguity\". Does this mean that _some_ of the implementation is similar? What exactly does that even mean? While not having needing a specific interpretation to base the metric in can be a strength of the method in some ways (e.g. when ambiguity is high, or prior work has come up with \"similar\" interpretations, confidence can be high that the ambiguity score is meaningful) it seems like this can also be a limitation as without an interpretation, an ambiguity of \"0.4\" seems hard to get much out of. I could imagine applying targeted version of this same algorithm to validate specific functional roles of various components (measuring whether \"mover heads\" perform similar roles, or \"induction heads\" act similarly across models, etc.), but this requires a prior interpretation.\n\n___\n- Brown et al. [Language models are few-shot learners](https://papers.nips.cc/paper/2020/hash/1457c0d6bfcb4967418bfb8ac142f64a-Abstract.html)\n\n- Wei et al. [Emergent Abilities of Large Language Models](https://openreview.net/forum?id=yzkSU5zdwD)\n\n- Schaffer et al [Are Emergent Abilities of Large Language Models a Mirage?](https://openreview.net/forum?id=ITw9edRDlD)\n\n- Prakash et al. [Language Models use Lookbacks to Track Beliefs](https://arxiv.org/abs/2505.14685)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)