Theoretical Motivation

1) Setup & notation

Given an issue description $d$ and repository $R$, let $L$ be the universe of candidate locations (e.g., all functions/methods) with $N = |L|$. Let $\mathcal{G}\subseteq L$ be the ground-truth edit set. Any localization system returns a ranked list of locations under a budget $k_{\text{ret}}$; the returned set is $L_{\text{ret}}$ with $|L_{\text{ret}}|=k_{\text{ret}}$.

In the test-driven pipeline, we retrieve a fixed-size set of tests $\hat{\mathcal{T}}_d$, $|\hat{\mathcal{T}}_d|=k_{\text{test}}$. Each test $t$ has a dynamic coverage set $\text{Cov}(t) \subseteq L$. The trace-constrained subspace is their union:

$$L_{\text{cov}} = \bigcup_{t\in \hat{\mathcal{T}}_d}\text{Cov}(t), \qquad N_{\text{cov}} = |L_{\text{cov}}|.$$

2) Entropy proxy (uncertainty as combinatorial ambiguity)

We approximate uncertainty with Hartley entropy: $$H(\mathcal{C}) \approx \log_2|\mathcal{C}|,$$ capturing the ambiguity of selecting correct edits from a candidate set. We do not estimate the intractable distribution $P(L\mid d)$; instead, we quantify relative uncertainty reduction induced by hierarchical retrieval and trace constraints.

3) Direct retrieval uncertainty

A direct method ranks the entire space $L$ and returns $L_{\text{ret}}$. Let $g_{\text{ret}} = |\mathcal{G}\cap L_{\text{ret}}|$ denote how many ground-truth edits fall inside the returned set. We use a partition-based proxy that explicitly accounts for whether true edits fall inside or outside $L_{\text{ret}}$:

$$ \begin{aligned} H_{\text{direct}} ={}& 1[g_{\text{ret}} > 0]\cdot \Big(-\log_2 \frac{g_{\text{ret}}}{|L_{\text{ret}}|}\Big) + 1[|\mathcal{G}|-g_{\text{ret}} > 0]\cdot \Big(-\log_2 \frac{|\mathcal{G}|-g_{\text{ret}}}{N-|L_{\text{ret}}|}\Big). \end{aligned} $$

4) Test-driven (indirect) uncertainty

Test-driven localization decomposes the search into two stages: (i) retrieving requirement-relevant tests, and (ii) localizing within a trace-constrained subspace. We model uncertainty as the sum of stage-wise entropies, making realistic failure modes explicit.

Stage 1 — Test selection entropy

A retrieved test is effective if it covers at least one ground-truth location: $\text{Cov}(t)\cap \mathcal{G}\neq\emptyset$. Let $n$ be the number of effective tests in $\hat{\mathcal{T}}_d$.

$$H_{\text{stage1}} = 1[n>0]\cdot \Big(-\log_2 \frac{n}{k_{\text{test}}}\Big).$$

Stage 2 — Trace-constrained localization entropy

We partition the global space into three disjoint regions induced by the trace subspace and the returned top-$k_{\text{ret}}$ locations. Critically, in the test-driven pipeline, $L_{\text{ret}}\subseteq L_{\text{cov}}$.

$$S_{\text{ret}} = L_{\text{ret}}, \quad S_{\text{cov}} = L_{\text{cov}} \setminus L_{\text{ret}}, \quad S_{\text{excl}} = L\setminus L_{\text{cov}}.$$

Let $N_i=|S_i|$ and $g_i=|\mathcal{G}\cap S_i|$ for $i\in\{\text{ret},\text{cov},\text{excl}\}$. Define:

$$H_{\text{stage2}} = \sum_{i\in\{\text{ret},\text{cov},\text{excl}\},\, g_i>0} \Big(-\log_2 \frac{g_i}{N_i}\Big).$$

The hierarchical uncertainty is: $$H_{\text{indirect}} = H_{\text{stage1}} + H_{\text{stage2}}.$$

5) Partition diagram (Venn)

Figure: Venn diagram illustrating hierarchical search space partition.

6) Entropy gain & design implications

For each issue, define empirical uncertainty reduction: $$\Delta H = H_{\text{direct}} - H_{\text{indirect}}.$$ A positive $\Delta H$ indicates that introducing tests and traces reduces localization ambiguity compared to direct matching under the same return budget $k_{\text{ret}}$.

Design implications (how this motivates IssueExec)

7) Case studies

This page prioritizes case-driven interpretability. We provide representative mini cases with intermediate quantities (set sizes and overlaps) to illustrate how $\Delta H$ is computed and why it becomes positive/negative. Full narratives and raw artifacts are linked from each case.

H_direct   = 18.105949419261542
      H_stage1   = 0.3219280948873623
      H_stage2   = 2.321928094887362
      H_indirect = 2.6438561897747244
      ΔH         = 15.462093229486818

H_direct   = 12.703471046071947
    H_stage1   = 2.321928094887362
    H_stage2   = 11.934796592178403
    H_indirect = 14.256724687065764
    ΔH         = -1.5532536409938178

- Selected tests (k_test=5):
      xarray/tests/test_dataset.py::test_differentiate
      xarray/tests/test_dataset.py::test_integrate
      xarray/tests/test_dataset.py::TestDataset.test_expand_dims_mixed_int_and_coords
      xarray/tests/test_concat.py::test_concat_merge_single_non_dim_coord
      xarray/tests/test_dataset.py::TestDataset.test_dataset_diff_n2

    - Union coverage size:
      N_cov = 271  (L_sub_size)

    - GT locations (|G|=4):
      xarray/core/dataarray.py::DataArray.integrate
      xarray/core/dataset.py::Dataset.integrate
      xarray/core/dataarray.py::DataArray.map_blocks
      xarray/core/dataarray.py::DataArray.unify_chunks

    - Uncovered GT (g_ext=2):
      xarray/core/dataarray.py::DataArray.map_blocks
      xarray/core/dataarray.py::DataArray.unify_chunks