How Much Does One Wrong Answer Cost? Error Propagation in Constraint Pruning with Noisy LLM Oracles

Zhengtao Gui; Weining Shen

cs.LGOct 19, 2026

How Much Does One Wrong Answer Cost? Error Propagation in Constraint Pruning with Noisy LLM Oracles

Authors: Zhengtao Gui, Weining Shen

Organizations: University of California, Irvine

Abstract

Constraint pruning saves queries to a large language model (LLM) or another noisy oracle by skipping each yes/no question implied by answers accepted earlier, for example through transitivity. Entity-resolution systems and LLM-based rankers use this idea. However, their analyses typically assume a correct oracle. We argue that pruning should be judged by its risk when the oracle errs, since an accepted wrong answer spreads to every question it implies. When each answer is wrong independently with small probability $\varepsilon$ , the first-order risk is set by how many facts one wrong answer corrupts on average. We consider two kinds of constraints. For implication rules, as in taxonomies, pruning in a random order adds no first-order risk: a wrong answer that would change $k$ facts is accepted with probability $1/k$ , so it costs one fact on average. Rules with $s$ premises break this balance: a wrong answer at a premise shared by $r$ rules can cost order $r^{1-1/s}$ facts. For entity resolution with $m$ records of one entity, the first-order risk equals $\varepsilon$ times the Wiener index (the total pairwise distance) of the spanning tree built by the query order. Pivot orders, which first compare one record with all others, amplify errors by less than 2 and are optimal; random orders by order $m^{1/3}$ ; the common most-similar-first order is the worst in one dimension and, for large $m$ , worse than random in two. With several entities, the cost of wrongly merging two does not depend on how many others exist. Simulations agree with every predicted constant within sampling error.

Figures & tables

Figure 1: Propagation in three synthetic citation records of one publication: one wrong answer produces two wrong pair decisions and saves one query.

Setting	Mean damage (wrong final facts)	Reference
2-CNF, implication rules	$1$ for unfixed facts, $0$ for fixed facts (backbone); ML: $\leq 1$	Cor. 4 ; Prop. 5
Horn hub: $k$ rules, $s\geq 2$ premises	$\sim\Gamma(1+\frac{1}{s})k^{1-1/s}$ at the shared premise; ML: $0$ ( $k\geq 2$ )	Thm. 6 ; Rem. 18
One entity, $m$ records	pivot $2(m-1)/m$ (optimal); random $\Theta(m^{1/3})$ ; similarity-first under Thm. 11 : $\gtrsim(m/\log m)^{1/d}$	Props. 9 , 10 ; Thm. 11
Cross pair, $s$ records per entity	random $g(s)\in[\frac{s-1}{2s-1},13s]$ , conj. $\Theta(s^{2/3})$ ; pivot $(2s-1)/s$	Props. 13 , 14 ; Conj. 15
Total orders	randomized quicksort $\to 2$	Prop. 24

Table 1: Mean single-error damage; skipped errors count as zero. Logical rows use uniform orders; partition rows average over all within- or cross-cluster pairs, giving first-order amplification over raw risk. Optimality is among greedy orders at first order. Conj.: conjectured growth.

Figure 2: Single-answer error propagation (top) and numerical checks (bottom): (A) logical constraints, (B) within-entity query order, (C) cross-entity false merges. Horn example:

k=s=2

a,c_{i}

false,

b_{i}

true. Shading marks affected true clusters, not the final partition. Thick solid: exact formulas; thin connectors: visual guides; dashed: asymptotes (A), conjecture (C); dotted: fitted guide (B). Open markers: simulations

\pm 1

standard error; diamonds: exact enumeration.

$\delta$	Pivot	Random	Random/Pivot
0.01	$0.019\pm 0.002$	$0.082\pm 0.008$	4.2
0.02	$0.039\pm 0.002$	$0.170\pm 0.012$	4.3
0.05	$0.094\pm 0.003$	$0.320\pm 0.010$	3.4
0.10	$0.174\pm 0.004$	$0.458\pm 0.008$	2.6
0.20	$0.335\pm 0.004$	$0.604\pm 0.006$	1.8

Table 2: Fraction of wrongly separated pairs on one cluster of

m=100

records when each answer is flipped to “different” with probability

\delta

. Means

\pm 1

Monte Carlo standard error over 200 runs; Random/Pivot is the ratio of the mean losses.

Setting	Wrong	Pivot	Random	Sim.-first
One entity	6.6%	1.23	2.31	0.85
One entity, 30 orders	6.6%	1.37	2.19	0.85
Shuffled, 30 orders	6.6%	1.73	3.55	3.48
Multi-entity blocks	6.3%	1.14	1.46	1.20

Table 3: Amplification on Qwen2.5-1.5B answers for Cora: final wrong pairs per wrong answer. Wrong: answer-error rate. Pivot/random use 200 orders in rows 1 and 4; rows 2–3 use 30 matched orders before and after uniformly redistributing errors within each entity, preserving its error count.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	Reference	Simulated
Conservation, $\mathbb{E}D(a)$	$\mathbf{1}\{a\notin B\}$	$\max\|z\|=2.66$
Horn hub, $s=2$ , $k=4$	$2.032$	$2.095\pm 0.039$
Horn hub, $s=3$ , $k=3$	$2.314$	$2.291\pm 0.031$
$g(3)$ , $K=2,3,5$	$2.0907$	$2.099$ , $2.106$ , $2.083$
$\mathbb{E}\mathrm{FP}/\varepsilon$ , $K=4$ , $s=3$	$56.87$	$58.1\pm 1.1$
$\mathbb{E}\mathrm{FN}/\varepsilon$ , $K=4$ , $s=3$	$72.03$	$71.9\pm 1.0$

Appendix

Table 4: Analytical references and simulated damage. Rows 1–4 use exact values (rounded); rows 5–6 use first-order coefficients. Uncertainty is

\pm 1

Monte Carlo standard error (MCSE);

|z|

is the standardized deviation.

$(s_{A},s_{B})$	$g_{\mathrm{FP}}$	$g_{\mathrm{FN}}$	$g$
$(1,1)$	$1$	$0$	$1$
$(1,2)$	$1$	$1/3$	$4/3$
$(2,2)$	$46/45$	$53/90$	$29/18=1.611111$
$(1,3)$	$1$	$28/45$	$73/45=1.622222$
$(2,3)$	$1.032959$	$0.830710$	$10567/5670=1.863668$
$(3,3)$	$1.053091$	$1.037571$	$313913/150150=2.090663$

Appendix

Table 5: Exact mean damage of one false merge between two clusters of sizes

s_{A}

and

s_{B}

, split into wrongly merged pairs (

g_{\mathrm{FP}}

) and wrongly separated pairs (

g_{\mathrm{FN}}

), with

g=g_{\mathrm{FP}}+g_{\mathrm{FN}}

. The mean is over random query orders, with zero damage when the false pair is skipped. Decimals are rounded to six places.

$s$	Orders	$g(s)$	$s\,\mathbb{P}(q)$	$g/s^{2/3}$
8	40000	$3.879\pm 0.050$	1.176	0.970
16	40000	$5.719\pm 0.112$	1.244	0.901
32	30000	$8.790\pm 0.295$	1.298	0.872
64	16000	$14.20\pm 0.93$	1.340	0.888
128	5000	$22.9\pm 3.9$	1.369	0.901

Appendix

Table 6: Two clusters of size

s

: Monte Carlo estimates of

g(s)

(mean

\pm 1

Monte Carlo standard error over the given number of random orders), the numerically evaluated rescaled probability

s\,\mathbb{P}(q)

that the false pair is queried, and

g/s^{2/3}

$\delta$	Pivot	Random	Random/Pivot
0.01	$0.020\pm 0.001$	$0.106\pm 0.009$	5.4
0.02	$0.039\pm 0.001$	$0.200\pm 0.012$	5.1
0.05	$0.097\pm 0.002$	$0.361\pm 0.010$	3.7
0.10	$0.179\pm 0.003$	$0.502\pm 0.007$	2.8
0.20	$0.337\pm 0.003$	$0.646\pm 0.005$	1.9

Appendix

Table 7: Scale check for finite noise: fraction of wrongly separated pairs in one cluster of

m=200

records. Mean

\pm 1

Monte Carlo standard error over 200 runs per cell; Random/Pivot is the ratio of mean losses.

Wrong	Pivot	Random	Similarity-first
1.1%	1.48	4.85	1.18
2.1%	1.40	3.75	1.05
5.1%	1.26	2.64	0.71
6.6%	1.23	2.31	0.85
10.0%	1.22	1.99	1.21
20.1%	1.20	1.61	1.00

Appendix

Table 8: Amplification in the one-entity setting as the decision threshold changes. Wrong: wrong answers as a percentage of the 62,423 pairs; 6.6% is the threshold 0 of Table 3 .

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