Many proposed applications of neural networks in machine learning, cognitive/brain science, and society hinge on the feasibility of inner interpretability via circuit discovery. This calls for empirical and theoretical explorations of viable algorithmic options. Despite advances in the design and testing of heuristics, there are concerns about their scalability and faithfulness at a time when we lack understanding of the complexity properties of the problems they are deployed to solve. To address this, we study circuit discovery with classical and parameterized computational complexity theory: (1) we describe a conceptual scaffolding to reason about circuit finding queries in terms of affordances for description, explanation, prediction and control; (2) we formalize a comprehensive set of queries that capture mechanistic explanation, and propose a formal framework for their analysis; (3) we use it to settle the complexity of many query variants and relaxations of practical interest on multi-layer perceptrons (part of, e.g., transformers). Our findings reveal a challenging complexity landscape. Many queries are intractable (NP-hard, $\Sigma^p_2$-hard), remain fixed-parameter intractable (W[1]-hard) when constraining model/circuit features (e.g., depth), and are inapproximable under additive, multiplicative, and probabilistic approximation schemes. To navigate this landscape, we prove there exist transformations to tackle some of these hard problems (NP- vs. $\Sigma^p_2$-complete) with better-understood heuristics, and prove the tractability (PTIME) or fixed-parameter tractability (FPT) of more modest queries which retain useful affordances. This framework allows us to understand the scope and limits of interpretability queries, explore viable options, and compare their resource demands among existing and future architectures.

该研究针对神经网络内部可解释性中的电路发现问题，填补了理论和算法选项之间的空白。作者提出了一个全面的查询框架，并分析了多层感知器中查询的复杂性，揭示了许多查询的不可解性和近似性限制。研究为理解可解释性查询的范围和局限性提供了新视角，进而促进实现更有效的算法选择。

内部可解释性电路发现的计算复杂性