#  SocREval: Large Language Models with the Socratic Method for Reference-Free
Reasoning Evaluation

Hangfeng He† Hongming Zhang‡ Dan Roth§
†University of Rochester ‡Tencent AI Lab, Seattle §University of Pennsylvania
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

To comprehensively assess the capacity of current models for complex
reasoning, it is crucial to assess their step-by-step reasoning in a scalable
manner. Established reference-based evaluation metrics rely on human-annotated
reasoning chains to assess the model-derived chains. However, such “gold-
standard” human-written reasoning chains may not be unique and their
acquisition is often labor-intensive. Existing reference-free reasoning
metrics eliminate the need for human-crafted reasoning chains as references,
but they typically require fine-tuning on datasets with human-derived
reasoning chains, which complicates the process and raises concerns regarding
generalizability across diverse datasets. To address these challenges, we
harness GPT-4 to automatically evaluate reasoning chain quality, obviating the
need for human-crafted references. Leveraging the Socratic method, we devise
tailored prompts to enhance reference-free reasoning evaluation, which we term
SocREval (Socratic method for Reasoning Evaluation). Empirical results from
four human annotated datasets reveal that SocREval significantly improves
GPT-4’s performance, surpassing existing reference-free and reference-based
reasoning evaluation metrics. Beyond its demonstrated efficacy, our proposed
framework, large language models (LLMs) with the Socratic method, proves to be
both cost-efficient and robust to prompt writing and example selection, as
substantiated by our in-depth analysis.111Our code is publicly available at
https://github.com/HornHehhf/SocREval.

##  1 Introduction

Recent advances in large language models (LLMs) have led to state-of-the-art
results in a plethora of natural language processing (NLP) tasks,
demonstrating the effectiveness of in-context learning without the need for
task-specific training or fine-tuning (OpenAI, 2023; Anil et al., 2023;
Touvron et al., 2023). Despite these impressive achievements, the inherent
reasoning capabilities of LLMs remain notably below human expectations
(Arkoudas, 2023). Although the core of reasoning fundamentally involves
offering justifications, most contemporary evaluations primarily assess a
model’s reasoning capability based on its end-task performance (Huang & Chang,
2022). Such evaluations, focusing solely on the accuracy of the final answer,
neglect the complexities of the underlying reasoning chains. This oversight
inhibits a comprehensive understanding of a model’s reasoning ability and
poses challenges to further advancements in this domain.

To assess the quality of reasoning chains produced by models, a direct and
intuitive approach centers on contrasting these generated chains with human-
constructed ones, termed as reference-based reasoning evaluation (Clinciu et
al., 2021; Welleck et al., 2022; Saparov & He, 2022). However, these
reference-based metrics highly rely on human-constructed reasoning chains,
which are both labor-intensive and costly. Furthermore, “gold-standard”
reasoning chains may not be unique (Dalvi et al., 2021), implying that the
effectiveness of reference-based evaluations can be significantly influenced
by the choice and breadth of human-crafted references. In light of these
challenges, recent research has begun to explore the evaluation of reasoning
chains without necessitating human-annotated references—termed reference-free
reasoning evaluation (Golovneva et al., 2022; Prasad et al., 2023).
Regrettably, these reference-free metrics necessitate the fine-tuning of
models on datasets with human-annotated reasoning chains, which is not only
complicated but also restricts their applicability across diverse datasets.

In this work, we introduce a novel framework that harnesses the capabilities
of LLMs to evaluate the model-generated reasoning chains, without the need of
human-annotated reference chains. The application of the Socratic method has
been demonstrated to enhance the quality of prompts for LLMs in reasoning
tasks (Chang, 2023a; Dong et al., 2023). Building upon this insight, we
further integrate the Socratic method to craft optimized prompts, facilitating
better reference-free reasoning evaluations using LLMs, which we denote as
SocREval \- representing the Socratic method for Reasoning Evaluation.
Specifically, we employ three fundamental strategies from the Socratic
method—Definition, Maieutics, and Dialectic—and their combinations, aiming to
refine the prompting mechanism of LLMs for reference-free reasoning
evaluation.

To verify the efficacy of our proposed framework, LLMs with the Socratic
method, we assessed its correlation with human judgment concerning the overall
quality of reasoning chains produced by LLMs across four diverse datasets from
ROSCOE (Golovneva et al., 2022): GSM8K (Cobbe et al., 2021) for arithmetic
reasoning; e-SNLI (Camburu et al., 2018) for both deductive and commonsense
reasoning; DROP (Dua et al., 2019) for discrete reasoning; and Cosmos QA
(Huang et al., 2019) for commonsense reasoning. Our empirical findings reveal
that GPT-4 exhibits a superior correlation with human judgment in comparison
to existing reference-free reasoning evaluation metrics, notably ROSCOE
(Golovneva et al., 2022) and RECEVAL (Prasad et al., 2023). By leveraging the
Socratic method, SocREval notably improves GPT-4’s correlation coefficient
with human judgment from $0.40$ to a remarkable $0.58$ when assessing the
overall quality of the generated reasoning chains—surpassing even the
performance of ROSCOE when furnished with human-written reasoning chains as
references. A comprehensive analysis underscores the robustness of our
proposed framework in terms of prompt writing and example selection while
highlighting its cost-efficiency.

GSM8K
Question: Janet’s ducks lay 16 eggs per day. She eats three for breakfast
every morning and bakes muffins for her friends every day with four. She sells
the remainder at the farmers’ market daily for $2 per fresh duck egg. How much
in dollars does she make every day at the farmers’ market?
Generated response: Janet eats 3 duck eggs for breakfast and bakes 4 into
muffins so 3 + 4 = $<<$3+4=7$>>$7 duck eggs are used||Each day Janet’s ducks
lay 16 eggs and she uses 7, 16 - 7 = $<<$16-7=9$>>$9 duck eggs are for
sale||She sells her eggs for $2 per egg and has 9 available for sale so 2 * 9
= $$<<$2*9=18$>>$18 per day||A: 18 DROP
Situation (Premise): Hoping to rebound from their loss to the Patriots, the
Raiders stayed at home for a Week 16 duel with the Houston Texans. Oakland
would get the early lead in the first quarter as quarterback JaMarcus Russell
completed a 20-yard touchdown pass to rookie wide receiver Chaz Schilens. The
Texans would respond with fullback Vonta Leach getting a 1-yard touchdown run,
yet the Raiders would answer with kicker Sebastian Janikowski getting a
33-yard and a 30-yard field goal. Houston would tie the game in the second
quarter with kicker Kris Brown getting a 53-yard and a 24-yard field goal.
Oakland would take the lead in the third quarter with wide receiver Johnnie
Lee Higgins catching a 29-yard touchdown pass from Russell, followed up by an
80-yard punt return for a touchdown. The Texans tried to rally in the fourth
quarter as Brown nailed a 40-yard field goal, yet the Raiders’ defense would
shut down any possible attempt.  Who scored the first touchdown of the game?
Claim (Hypothesis): Chaz Schilens
Question: Is the Claim supported by the Situation?
Generated response: Chaz Schilens was the receiver of the touchdown pass from
JaMarcus Russell, so he scored the first touchdown. The answer is Yes. Figure
1: Examples from the GSM8K and DROP datasets illustrating two explanation
paradigms: Explain-then-Predict and Predict-then-Explain. For clarity, we use
“||” to represent the Python newline character “\n”.

###  1.1 Related Work

#### Prompting LLMs with the Socratic method.

The Socratic method’s essence is a sequence of probing questions to elucidate
complex ideas, closely relevant to the LLM prompting techniques. Chang (2023a)
crafted prompt templates utilizing the Socratic method, introducing the
Critical Inquisitive Template (CRIT) for reasoning evaluation and subsequent
Socratic synthesis for decision-making (Chang, 2023b). Dong et al. (2023)
further employed the Socratic method for deeper LLM engagements in intricate
problem-solving. Unlike these endeavors, which emphasize qualitative case
analyses, our work focuses on quantitative experiments for reference-free
reasoning evaluation.

#### Evaluation of reasoning chains.

Evaluating the quality of reasoning chains generated by models has been
traditionally approached by contrasting them with human-generated ones,
referred to as reference-based reasoning evaluation. Conventional natural
language generation (NLG) metrics assess the similarity between such machine-
generated and human-crafted reasoning chains (Celikyilmaz et al., 2020;
Clinciu et al., 2021; Welleck et al., 2022). In contrast, several domain-
specific metrics were tailored for assessing reasoning chains relying on the
specific structure of the dataset (Dalvi et al., 2021; Saparov & He, 2022; Han
et al., 2022). Recently, ROSCOE (Golovneva et al., 2022) pioneered reference-
free reasoning evaluation by introducing metrics grounded in step-by-step
reasoning chains, targeting dimensions such as semantic consistency,
logicality, informativeness, fluency, and factuality. RECEVAL (Prasad et al.,
2023) moved further and centered its evaluation on correctness and
informativeness. Both ROSCOE and RECEVAL align closely with our work’s focus
on reference-free reasoning evaluation. Notably, while both methods
necessitate fine-tuning on datasets with human-derived reasoning chains, our
approach avoids such fine-tuning requirements.

#### Evaluating text generation with LLMs.

The stellar capabilities of LLMs on NLP tasks have propelled their adoption in
evaluating generated text quality. Techniques range from harnessing
conditional generative probabilities (Fu et al., 2023) to leveraging the
prompts tailored for specific evaluation needs (Liu et al., 2023; Lu et al.,
2023). Such methods have been deployed across diverse NLG domains, including
summarization (Gao et al., 2023), machine translation (Kocmi & Federmann,
2023), and more (Wang et al., 2023), with evaluations being both individual
and comparative (Chen et al., 2023; Zheng et al., 2023). In contrast to these
endeavors, our research uses LLMs with the Socratic method to realize
reference-free reasoning evaluation, whereas the aforementioned works target
other text-generation tasks.

##  2 Large Language Models with the Socratic method

Our goal is to evaluate the overall quality of step-by-step reasoning without
using references. This section elucidates our approach, wherein we harness the
capabilities of LLMs, combined with the Socratic method, to undertake this
assessment without human-crafted references.

###  2.1 Prompt Skeleton for Reference-Free Reasoning Evaluation

Two dominant explanation paradigms often incorporate step-by-step reasoning
chains to elucidate final answers: Explain-then-Predict (E-P) and Predict-
then-Explain (P-E) (Ye & Durrett, 2022; Zelikman et al., 2022). Given the
nature of LLMs such as GPT-4, it’s crucial to formulate appropriate prompt
templates to assess the reasoning chains within these paradigms.

#### Explain-then-Predict (E-P).

Within this paradigm, the explanation precedes the final answer, both being
part of the LLM’s generated response. A concrete illustration is provided in
Figure 1. We propose the subsequent prompt template for evaluating reasoning
chains: Instruction \+ Example question + Example generated response + Example
representation + Question + Generated response + Evaluation prompt. The
detailed prompt template can be found in Appendix A.1.

#### Predict-then-Explain (P-E).

Here, the explanation follows the final answer. Notably, the explanation
generated doesn’t influence the final answer in P-E. For specific instances,
refer to Figure 1 (see more in Figure 4 in Appendix A.1). In crafting prompts
for this paradigm, we leverage terminology from the expert annotation user
interface presented by Golovneva et al. (2022), including terms like Situation
(Premise) and Claim (Hypothesis). Our proposed template is: Instruction \+
Example Situation (Premise) + Example Claim (Hypothesis) + Example question +
Example generated response + Example representation + Situation (Premise) +
Claim (Hypothesis) + Question + Generated response + Evaluation prompt. Note
that the question in this template is always ”Is the Claim supported by the
Situation?” which serves as a directive for LLMs to elucidate the provided
Claim (Hypothesis) given the Situation (Premise). Beyond evaluating reasoning
chains for question-answering explanations, this template is also suitable for
assessing reasoning within natural language inference explanations. Detailed
prompt templates are in Appendix A.1.

Despite the distinction between prompt templates for both paradigms, both
employ the identical Instruction and Evaluation prompt. Drawing inspiration
from Golovneva et al. (2022), the Instruction and Evaluation prompt used in
our experiments are described below:

(1) Instruction and Evaluation Prompt for GPT-4:
Instruction: Does the generated response answer the question in a well-
justified manner? Please give me an overall quality score in [1, 2, 3, 4, 5]
(1=incomprehensible and wrong, 5=clear and correct).
(Evaluation prompt) Please give me the overall quality of the generated
response for the question based on the instruction and the format of the
example representation.

While the term “Instruction” is incorporated within the prompt, the term
“Evaluation prompt” is omitted and is solely used to denote the final sentence
of the prompt. Additionally, to ensure LLMs generate the intended output
format, each of our prompt templates consistently integrates one demonstration
example for reference-free reasoning evaluation.

###  2.2 Refinement of Prompts Through the Socratic Method

The Socratic method, characterized by a series of probing questions aimed at
exploring complex ideas, is a foundational approach in teaching and
philosophy, fostering critical thinking and promoting self-discovery. Recent
work has adopted this method to enhance prompts for LLMs, leading to enriched
reasoning capacities (Chang, 2023a; Dong et al., 2023). Chang (2023a)
identifies ten principal strategies within the Socratic method: Definition,
Generalization, Induction, Elenchus, Hypothesis Elimination, Maieutics,
Dialectic, Recollection, Irony, and Analogy. Similar to CRIT as decribed in
(Chang, 2023a), we identify three strategies highly aligned with our use
cases. Diverging from CRIT, we omit the Elenchus strategy from our framework.
This decision stems from our observation that our datasets do not necessitate
obtaining extra evidence beyond the provided context to support the reasoning
chains. Nonetheless, we acknowledge its potential significance for reasoning
tasks lacking adequate context, such as StrategyQA (Geva et al., 2021).
Exploring this further is deferred to future work.

#### Definition strategy.

Socrates frequently employed definitions to elucidate key terminologies. In
reference-free reasoning evaluation, the definition strategy can be used to
refine the comprehension of assessment criteria for LLMs. When incorporating
this strategy into GPT-4, we denote the resultant evaluation metric as
SocREval (Definition). Differences, when compared with the original prompt
(Prompt (2.1)), are highlighted in italicized text in purple.

(2) Instruction and Evaluation Prompt for SocREval (Definition):
Instruction: Does the generated response answer the question in a well-
justified manner? Please give me an overall quality score in [1, 2, 3, 4, 5]
(1=incomprehensible and wrong, 5=clear and correct). Note that you need to
take into account both the explanation and the answer in the generated
response.
(Evaluation prompt) Please give me the overall quality of the generated
response for the question based on the instruction and the format of the
example representation.

#### Maieutics strategy.

Maieutics assists individuals in revealing their inherent knowledge. By
applying maieutics, we prompt LLMs to analyze the quality of reasoning chains
prior to delivering the final score. This bears similarity to chain-of-thought
prompting (Wei et al., 2022; Kojima et al., 2022) with a divergent focus on
reference-free reasoning evaluation. It’s important to differentiate our
approach from the Maieutic prompting in Jung et al. (2022). Although both are
inspired by the Maieutics strategy from the Socratic method, their methodology
induces a tree of explanations through an abductive and recursive manner. With
the integration of the maieutics strategy, the resultant evaluation metric is
labeled SocREval (Maieutics). Any deviations from the original prompt (Prompt
(2.1)) are highlighted in italicized blue.

(3) Instruction and Evaluation Prompt for SocREval (Maieutics):
Instruction: Does the generated response answer the question in a well-
justified manner? Please conduct a qualitative analysis on the generated
response first and then give me an overall quality score in [1, 2, 3, 4, 5]
(1=incomprehensible and wrong, 5=clear and correct) for the given generated
response by taking into account the qualitative analysis.
(Evaluation prompt) Please conduct a qualitative analysis on the generated
response first and then give me the overall quality of the given generated
response for the question by taking into account the qualitative analysis
based on the instruction and the format of the example representation:

#### Dialectic strategy.

The dialectic approach navigates diverse perspectives through constructive
discourse, fostering profound insights into the subject matter. Within the
context of reference-free reasoning evaluation, we harness the dialectic
strategy by prompting LLMs to formulate their own responses to a given
question before evaluating existing reasoning chains. This methodology aligns
with generating “pseudo references” in reference-free summarization
evaluation, as introduced by (Gao et al., 2020; Chen et al., 2021). While
their emphasis is on summarization assessment, ours is dedicated to reasoning
evaluation. The resultant evaluation metric is termed SocREval (Dialectic).
Discrepancies against the original prompt (Prompt (2.1)) are highlighted in
italicized brown.

(4) Instruction and Evaluation Prompt for SocREval (Dialectic):
Instruction: Does the generated response answer the question in a well-
justified manner? Please generate your own response for the question first and
then give me an overall quality score in [1, 2, 3, 4, 5] (1=incomprehensible
and wrong, 5=clear and correct) for the given generated response by taking
into account your own response.
(Evaluation prompt) Please generate your own response for the question first
and then give me the overall quality of the given generated response for the
question by taking into account your own response based on the instruction and
the format of the example representation:

#### Integration of three strategies.

We combine the aforementioned three strategies to devise optimized prompts for
reference-free reasoning evaluation. The resultant evaluation metric is termed
SocREval (All)222Unless otherwise specified, references to SocREval implicitly
denote SocREval (All).. Further explorations of different combinations are
detailed in Appendix A.1.

(5) Instruction and Evaluation Prompt for SocREval (All):
Instruction: Does the generated response answer the question in a well-
justified manner? Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me an overall quality score in [1,
2, 3, 4, 5] (1=incomprehensible and wrong, 5=clear and correct) for the given
generated response by taking into account both your own response and the
qualitative analysis. Note that you need to take into account both the
explanation and the answer in the generated response.
(Evaluation prompt) Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me the overall quality of the
given generated response for the question by taking into account both your own
response and the qualitative analysis based on the instruction and the format
of the example representation:

| GSM8K | e-SNLI | DROP | Cosmos QA | Average [t] [b]
---|---|---|---|---|---
Reference-based ROSCOE | 0.81 | 0.43 | - | - | \- [t] [b]
Reference-free ROSCOE | 0.32 | 0.30 | 0.22 | 0.18 | 0.26 [t]
RECEVAL | 0.36 | - | - | - | \- [b]
GPT-4 | 0.57 | 0.24 | 0.42 | 0.39 | 0.40 [t]
SocREval (Definition) | 0.62 | 0.30 | 0.43 | 0.39 | 0.44
SocREval (Maieutics) | 0.64 | 0.58 | 0.47 | 0.39 | 0.52
SocREval (Dialectic) | 0.74 | 0.31 | 0.43 | 0.43 | 0.48
SocREval (All) or SocREval | 0.82 | 0.58 | 0.49 | 0.42 |  0.58 [b]
|  |  |  |  |

Table 1: Somers’ D correlations between reasoning evaluation metrics and human
judgment. For reference-free evaluation metrics (excluding reference-based
ROSCOE), the top two correlations in each column are distinctly highlighted in
bold and underlined. “-” indicates missing values: Prasad et al. (2023)
assessed RECEVAL exclusively on GSM8K, and only GSM8K and e-SNLI encompass
human-annotated explanations, enabling the acquisition of reference-based
ROSCOE scores. For GPT-4 and SocREval variants, all p-values of the
correlations are below $0.0001$ (****).

##  3 Experiments

In this section, we evaluate the efficacy of our proposed framework, LLMs with
the Socratic Method, in the context of reference-free reasoning evaluation.

#### Datasets.

In our experiments, we utilize four333SemEVAL (Ostermann et al., 2018) was
excluded as Golovneva et al. (2022) did not release their annotations for it
due to approval constraints. human judged datasets from ROSCOE (Golovneva et
al., 2022), including GSM8K (Cobbe et al., 2021) for arithmetic reasoning,
e-SNLI (Camburu et al., 2018) for deductive and commonsense reasoning, DROP
(Dua et al., 2019) for discrete reasoning, and Cosmos QA(Huang et al., 2019)
for commonsense reasoning. In the context of GSM8K, the reasoning chains in
the ROSCOE datasets are derived from the GPT-3 175B Verification model (Cobbe
et al., 2021), leveraging the chain-of-thought prompting approach (Wei et al.,
2022). For the remaining datasets, GPT-3 (Brown et al., 2020) is used to
extract detailed reasoning chains. Notably, GSM8K adopts the Explain-then-
Predict explanatory paradigm, while the others are aligned with the Predict-
then-Explain explanation paradigm. Expert annotators evaluated the reasoning
chains for factors such as overall quality, and examined individual steps,
identifying issues like commonsense errors. For the scope of this study, we
only focus on the overall quality of reasoning chains, placing discussions on
specific error types in Appendix A.2. The datasets comprise human-judged
annotations on reasoning chains for $200$ examples in GSM8K, $151$ in ESNLI,
$210$ in DROP, and $195$ in COSMOS-QA. We refer readers to Golovneva et al.
(2022) for more details.

#### Baselines.

We consider two established suites of reasoning evaluation metrics as our
baselines: ROSCOE (Golovneva et al., 2022) and RECEVAL (Prasad et al., 2023).
ROSCOE encompasses a comprehensive set of metrics, assessing attributes such
as semantic consistency, logicality, informativeness, fluency, and factuality
by exploiting the properties of step-by-step rationales. This suite
incorporates both reference-free and reference-based metrics. In contrast,
RECEVAL introduces a collection of purely reference-free metrics specifically
crafted to measure the correctness and informativeness of reasoning chains.
This is achieved by harnessing entailment and pointwise
$\mathcal{V}$-information (Ethayarajh et al., 2022), leveraging granular
claims in reasoning chains called reasoning content units. Considering the
various metrics within reference-free ROSCOE, reference-based ROSCOE444While
our focus is on reference-free reasoning evaluation, and thus we do not
formally consider reference-based ROSCOE as our baseline, we include
comparisons with them for a more comprehensive overview., and RECEVAL, we
present the peak performance from each group of metrics as our baselines,
though the optimal metric for each group often varies across different
datasets.

#### Meta evaluation.

In alignment with ROSCOE and RECEVAL, we employ Somers’ D (Somers, 1962) to
measure the correlation between human judgments and reasoning evaluation
metrics. Specifically, using Kendall’s $\tau$ coefficient, Somers’ D
correlation is articulated as $D(Y|X)=\tau(X,Y)/\tau(X,X)$. Note that the
Somers’ D coefficient is asymmetric, necessitating that the human score is
chosen as the first variable ($X$) and the metric score as the second variable
($Y$). Unless otherwise stated, we adopt practices from ROSCOE and RECEVAL to
normalize scores to the $[0,1]$ range for correlation analysis, even though
Somers’ D is inherently scale-invariant due to its focus on ordinal
associations. A detailed analysis of the meta evaluation is in Section 4.

#### Results.

As shown in Table 1, GPT-4 outperforms existing reference-free reasoning
evaluation metrics, namely reference-free ROSCOE (on average) and RECEVAL.
Note that while both reference-free ROSCOE and RECEVAL necessitate model fine-
tuning on datasets furnished with human-annotated reasoning chains, GPT-4
operates effectively without such fine-tuning, underscoring its effectiveness
in reference-free reasoning evaluation. By integrating three strategies
derived from the Socratic method — namely, Definition, Maieutics, and
Dialectic — our proposed evaluation metric, SocREval, further outperforms
GPT-4. Among these strategies, Maieutics emerges as the most effective on
average. A fusion of the above three strategies not only augments GPT-4’s
performance but also surpasses that of reference-based ROSCOE. On average,
these strategies from the Socratic method amplify GPT-4’s correlation score
from $0.40$ to $0.58$, highlighting the merit of the Socratic method in
crafting suitable prompts for LLMs in reference-free reasoning evaluation.

| GSM8K | e-SNLI | DROP | Cosmos QA | Average [t] [b]
---|---|---|---|---|---
SocREval (All) or SocREval | 0.82 | 0.58 | 0.49 | 0.42 |  0.58 [t]
SocREval (All \- Definition) | 0.83 | 0.51 | 0.49 | 0.42 | 0.56
SocREval (All \- Maieutics) | 0.81 | 0.38 | 0.51 | 0.42 | 0.53
SocREval (All \- Dialectic) | 0.62 | 0.49 | 0.47 | 0.45 | 0.51 [b]
|  |  |  |  |

Table 2: Ablation study of SocREval. For each column, the top two correlations
are distinctly highlighted in bold and underlined. All reported correlations
have p-values less than 0.0001 (****).

| GSM8K | e-SNLI | DROP | Cosmos QA | Average [t] [b]
---|---|---|---|---|---
SocREval (6 different prompts) | 0.80 $\pm$ 0.01 | 0.56 $\pm$ 0.04 | 0.49 $\pm$ 0.01 | 0.42 $\pm$ 0.03 | 0.57 $\pm$ 0.01 [t]
SocREval (6 different examples) | 0.83 $\pm$ 0.02 | 0.54 $\pm$ 0.03 | 0.50 $\pm$ 0.01 | 0.42 $\pm$ 0.02 | 0.57 $\pm$ 0.01 [b]
|  |  |  |  |

Table 3: Assessment of SocREval’s robustness concerning prompt writing and
example selection. The standard deviation of the performance is indicated post
$\pm$ for each configuration. Figure 2: Cost comparison between GPT-4 and
SocREval. For each reasoning evaluation metric across four datasets, the
OpenAI API costs are computed based on respective input (at $\$0.03$ per $1K$
tokens) and output token counts (at $\$0.06$ per $1K$ tokens).

| SD | $r$ | $\rho$ | $\tau$ | ACC | RMSE | MAE | Micro-F1 | Macro-F1 [t] [b]
---|---|---|---|---|---|---|---|---|---
GPT-4 | 0.40 | 0.55 | 0.55 | 0.49 | 0.48 | 1.40 | 0.91 | 0.48 | 0.28 [t]
SocREval | 0.58 | 0.69 | 0.66 | 0.58 | 0.52 | 1.07 | 0.68 | 0.52 |  0.39 [b]
|  |  |  |  |  |  |  |  |

Table 4: Comparison of SocREval and GPT-4 across various meta-evaluation
metrics. In addition to Somers’ D (SD), metrics encompass Pearson’s
Correlation Coefficient ($r$), Spearman’s Rank Correlation Coefficient
($\rho$), Kendall’s $\tau$ ($\tau$), Accuracy (ACC), Root Mean Squared Error
(RMSE), Mean Absolute Error (MAE), Micro F1 Score (Micro-F1), and Macro F1
Score (Macro-F1). It’s crucial to note that for RMSE and MAE, lower values
indicate superior performance, whereas for the remaining metrics, higher
values are desirable. Note that scores for each configuration are computed as
averages over four datasets: GSM8K, e-SNLI, DROP, and Cosmos QA.

(a) Human judgment

(b) GPT-4

(c) SocREval

Figure 3: Distribution of the quality of reasoning chains across questions on
GSM8K, differentiated by answer correctness (correct versus wrong), as
assessed through human judgment, GPT-4, and SocREval. Quality scores span the
discrete range [1, 2, 3, 4, 5], with the orange line representing the median
and the green triangle signifying the mean.

##  4 Analysis

To elucidate the efficacy of the Socratic method in crafting proper prompts
for LLMs within the context of reference-free reasoning evaluation, we
undertake an in-depth analysis of our proposed framework: LLMs with the
Socratic method. For brevity and clarity, our primary emphasis centers on our
best evaluation metric, SocREval (All), often referred to simply as SocREval.

#### Ablation study.

The SocREval approach harnesses three distinct strategies from the Socratic
method—Definition, Maieutics, and Dialectic—to elicit reference-free reasoning
evaluations from LLMs. To investigate the significance of each strategy within
this integration of three strategies, we omit each from SocREval and observe
the consequences. Detailed prompts utilized in the ablation study are in
Appendix A.1. As shown in Table 2, excluding any single strategy consistently
decreases GPT-4’s efficacy. Notably, the Dialectic strategy emerges as the
pivotal one within the integration of three strategies, with its omission
resulting in the most pronounced performance degradation. However, while
Dialectic holds paramount importance within the integration of three
strategies, Maieutics outperforms when strategies are incorporated
individually (see Table 1). This suggests the complex interactions among the
strategies. Delving into the intricate dynamics of how these strategies
interplay in shaping LLM prompts is compelling and warranted for future
exploration.

#### Robustness analysis.

Prompting techniques, in certain contexts, have exhibited sensitivity to their
specific phrasings (Kojima et al., 2022). To discern the influence of prompt
writing on SocREval, we systematically rephrase Prompt (2.2) five times. More
details are in Appendix A.1. By default, we utilize the first example from
each dataset as the demonstration in the prompt. To systematically evaluate
the influence of this demonstration example, we select five distinct examples
at random for each dataset to serve as demonstrations. As shown in Table 3,
the standard deviation for SocREval across the six variations of Prompt (2.2)
as well as the standard deviation encompassing six disparate demonstration
examples both consistently measure at $0.01$, indicating the robustness of
SocREval on prompt writing and example selection.

#### Cost analysis.

The substantial size of LLMs inherently implies considerable operating
expenses that cannot be overlooked. Numerous methodologies have been devised
to enhance LLMs’ capabilities from varied dimensions, but they frequently come
with escalated costs (Wei et al., 2022; Wang et al., 2022; Madaan et al.,
2023). Within this context, we assess the OpenAI API costs tied to SocREval in
comparison to those of GPT-4. Note that these API costs are based on the
number of input and output tokens processed by the LLMs. As illustrated in
Figure 2, SocREval (All) incurs a cost that is less than $2.1$ times that of
GPT-4, while amplifying the reasoning evaluation performance from $0.40$ to
$0.58$. This highlights the cost-efficacy of our proposed framework.

#### Meta evaluation analysis.

To obtain a deeper insight into the superiority of SocREval over GPT-4, we
expand our evaluation scope by incorporating eight supplementary metrics for
meta evaluation beyond just Somers’ D: Pearson’s Correlation Coefficient,
Spearman’s Rank Correlation Coefficient, Kendall’s $\tau$, Accuracy, Root Mean
Squared Error (RMSE), Mean Absolute Error (MAE), Micro F1 Score, and Macro F1
Score. These meta-evaluation metrics are designed to quantify the alignment
between evaluation metrics and human judgment. As shown in Table 4, SocREval
consistently outperforms GPT-4 across all meta-evaluation metrics.
Furthermore, an examination of the confusion matrix (refer to Figure 5 in
Appendix A.2) demonstrates a tendency in GPT-4 to overestimate the quality of
reasoning chains. This overestimation is clearly alleviated in SocREval,
attributing to the incorporation of strategies from the Socratic method.

#### Answer analysis.

When presented with a question, the whole reasoning trajectory comprises both
an explanation and an answer, as shown in the E-P and P-E explanation
paradigms. In this part, we delve deeper into the interplay between answer
accuracy and the overall quality of reasoning chains. We segment our GSM8K
examples into two groups: examples with correct answers and examples with
wrong answers. Within each group, we investigate the distribution of the
overall quality of reasoning chains, as assessed by human judgment, GPT-4, and
SocREval. Figure 3 reveals that, according to human evaluators, explanations
associated with correct answers typically exhibit high quality, whereas those
linked to wrong answers manifest a discernible decline in quality—a trend
aligning with our anticipations. While GPT-4 demonstrates proficiency in
assessing explanation quality for correctly answered instances, it tends to
overestimate the quality for wrongly answered ones. By leveraging the Socratic
method, SocREval markedly mitigates this overestimation tendency. More
analysis can be found in Figure 6 in Appendix A.2.

##  5 Discussion

In this work, we introduced a novel approach that harnesses the Socratic
method to craft optimized prompts for LLMs, specifically GPT-4, enabling
enhanced assessment of reasoning chain quality. Distinctively, our framework
avoids the need for model fine-tuning, a prevalent requirement in current
reference-free reasoning evaluation metrics. The efficacy of our approach is
substantiated across four datasets, wherein our proposed reference-free
evaluation metric SocREval demonstrates superior correlation with human
judgments regarding reasoning chains produced by LLMs. Furthermore,
comprehensive analyses underscore the robustness of our method with respect to
prompt writing and example selection, all while maintaining cost-efficiency.

Similar to ROSCOE, we extended our analysis to evaluate the correlation
between SocREval and ten distinct human-annotated error types across the four
datasets. As detailed in Table 12 within Appendix A.2, on average, SocREval
surpasses GPT-4 in almost all error types except the “Hallucination”.
Interestingly, when compared to reference-free ROSCOE, the SocREval only lags
in errors like “Hallucination,” “Redundancy,” and “Grammar.” It is worthwhile
to note that we prompt LLMs to assess the overall quality of reasoning chains
rather than pinpointing specific error types. One prospective avenue might be
directing LLMs to directly identify the presence of particular error types—a
pursuit we postpone for future exploration. Additionally, while we employ
GPT-4 to evaluate the quality of reasoning chains produced by GPT-3 variants,
an intriguing inquiry emerges: Can GPT-4 effectively evaluate reasoning chains
generated by itself? Addressing this necessitates human judgment of reasoning
chains generated by GPT-4, a facet we postpone for future research.

In concluding this study, we anticipate several promising directions for
future exploration. First, we simply select three strategies deeply relevant
to our datasets; however, the expansive potential of the Socratic method’s
strategies deserves a more comprehensive exploration. Moreover, considering
the important role of reasoning chains in advanced prompting techniques—such
as chain of thought, tree of thoughts (Yao et al., 2023), and graph of
thoughts (Besta et al., 2023), we plan to harness our framework for an
enriched understanding of these techniques, subsequently designing better
prompts. Since evaluation plays a pivotal role in feedback mechanisms, a
fusion of our framework with prominent feedback integration methods like SELF-
REFINE (Madaan et al., 2023) and Reflexion (Shinn et al., 2023) could be
useful in improving the reasoning capabilities of LLMs.

## References

  * Anil et al. (2023) Rohan Anil, Andrew M Dai, Orhan Firat, Melvin Johnson, Dmitry Lepikhin, Alexandre Passos, Siamak Shakeri, Emanuel Taropa, Paige Bailey, Zhifeng Chen, et al.  PaLM 2 technical report.  _arXiv preprint arXiv:2305.10403_ , 2023.
  * Arkoudas (2023) Konstantine Arkoudas.  GPT-4 can’t reason.  _arXiv preprint arXiv:2308.03762_ , 2023.
  * Besta et al. (2023) Maciej Besta, Nils Blach, Ales Kubicek, Robert Gerstenberger, Lukas Gianinazzi, Joanna Gajda, Tomasz Lehmann, Michal Podstawski, Hubert Niewiadomski, Piotr Nyczyk, et al.  Graph of thoughts: Solving elaborate problems with large language models.  _arXiv preprint arXiv:2308.09687_ , 2023.
  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.  Language models are few-shot learners.  _Advances in neural information processing systems_ , 33:1877–1901, 2020.
  * Camburu et al. (2018) Oana-Maria Camburu, Tim Rocktäschel, Thomas Lukasiewicz, and Phil Blunsom.  e-SNLI: Natural language inference with natural language explanations.  _Advances in Neural Information Processing Systems_ , 31, 2018.
  * Celikyilmaz et al. (2020) Asli Celikyilmaz, Elizabeth Clark, and Jianfeng Gao.  Evaluation of text generation: A survey.  _arXiv preprint arXiv:2006.14799_ , 2020.
  * Chang (2023a) Edward Y Chang.  Prompting large language models with the socratic method.  In _2023 IEEE 13th Annual Computing and Communication Workshop and Conference (CCWC)_ , pp. 0351–0360. IEEE, 2023a.
  * Chang (2023b) Edward Y Chang.  Socrasynth: Socratic synthesis for reasoning and decision making.  _Stanford University InfoLab Technical Report_ , 2023b.
  * Chen et al. (2021) Wang Chen, Piji Li, and Irwin King.  A training-free and reference-free summarization evaluation metric via centrality-weighted relevance and self-referenced redundancy.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pp. 404–414, 2021.
  * Chen et al. (2023) Yi Chen, Rui Wang, Haiyun Jiang, Shuming Shi, and Ruifeng Xu.  Exploring the use of large language models for reference-free text quality evaluation: A preliminary empirical study.  _arXiv preprint arXiv:2304.00723_ , 2023.
  * Clinciu et al. (2021) Miruna Clinciu, Arash Eshghi, and Helen Hastie.  A study of automatic metrics for the evaluation of natural language explanations.  In _Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume_ , pp. 2376–2387, 2021.
  * Cobbe et al. (2021) Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, et al.  Training verifiers to solve math word problems.  _arXiv preprint arXiv:2110.14168_ , 2021.
  * Dalvi et al. (2021) Bhavana Dalvi, Peter Jansen, Oyvind Tafjord, Zhengnan Xie, Hannah Smith, Leighanna Pipatanangkura, and Peter Clark.  Explaining answers with entailment trees.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , pp. 7358–7370, 2021.
  * Dong et al. (2023) Qingxiu Dong, Li Dong, Ke Xu, Guangyan Zhou, Yaru Hao, Zhifang Sui, and Furu Wei.  Large language model for science: A study on P vs. NP.  _arXiv preprint arXiv:2309.05689_ , 2023.
  * Dua et al. (2019) Dheeru Dua, Yizhong Wang, Pradeep Dasigi, Gabriel Stanovsky, Sameer Singh, and Matt Gardner.  DROP: A reading comprehension benchmark requiring discrete reasoning over paragraphs.  In _Proceedings of NAACL-HLT_ , pp. 2368–2378, 2019.
  * Ethayarajh et al. (2022) Kawin Ethayarajh, Yejin Choi, and Swabha Swayamdipta.  Understanding dataset difficulty with $\mathcal{V}$-usable information.  In _International Conference on Machine Learning_ , pp. 5988–6008. PMLR, 2022.
  * Fu et al. (2023) Jinlan Fu, See-Kiong Ng, Zhengbao Jiang, and Pengfei Liu.  GPTscore: Evaluate as you desire.  _arXiv preprint arXiv:2302.04166_ , 2023.
  * Gao et al. (2023) Mingqi Gao, Jie Ruan, Renliang Sun, Xunjian Yin, Shiping Yang, and Xiaojun Wan.  Human-like summarization evaluation with ChatGPT.  _arXiv preprint arXiv:2304.02554_ , 2023.
  * Gao et al. (2020) Yang Gao, Wei Zhao, and Steffen Eger.  SUPERT: Towards new frontiers in unsupervised evaluation metrics for multi-document summarization.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pp. 1347–1354, 2020.
  * Geva et al. (2021) Mor Geva, Daniel Khashabi, Elad Segal, Tushar Khot, Dan Roth, and Jonathan Berant.  Did aristotle use a laptop? a question answering benchmark with implicit reasoning strategies.  _Transactions of the Association for Computational Linguistics_ , 9:346–361, 2021.
  * Golovneva et al. (2022) Olga Golovneva, Moya Peng Chen, Spencer Poff, Martin Corredor, Luke Zettlemoyer, Maryam Fazel-Zarandi, and Asli Celikyilmaz.  ROSCOE: A suite of metrics for scoring step-by-step reasoning.  In _The Eleventh International Conference on Learning Representations_ , 2022.
  * Han et al. (2022) Simeng Han, Hailey Schoelkopf, Yilun Zhao, Zhenting Qi, Martin Riddell, Luke Benson, Lucy Sun, Ekaterina Zubova, Yujie Qiao, Matthew Burtell, et al.  FOLIO: Natural language reasoning with first-order logic.  _arXiv preprint arXiv:2209.00840_ , 2022.
  * Huang & Chang (2022) Jie Huang and Kevin Chen-Chuan Chang.  Towards reasoning in large language models: A survey.  _arXiv preprint arXiv:2212.10403_ , 2022.
  * Huang et al. (2019) Lifu Huang, Ronan Le Bras, Chandra Bhagavatula, and Yejin Choi.  Cosmos QA: Machine reading comprehension with contextual commonsense reasoning.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pp. 2391–2401, 2019.
  * Jung et al. (2022) Jaehun Jung, Lianhui Qin, Sean Welleck, Faeze Brahman, Chandra Bhagavatula, Ronan Le Bras, and Yejin Choi.  Maieutic prompting: Logically consistent reasoning with recursive explanations.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pp. 1266–1279, 2022.
  * Kocmi & Federmann (2023) Tom Kocmi and Christian Federmann.  Large language models are state-of-the-art evaluators of translation quality.  _arXiv preprint arXiv:2302.14520_ , 2023.
  * Kojima et al. (2022) Takeshi Kojima, Shixiang Shane Gu, Machel Reid, Yutaka Matsuo, and Yusuke Iwasawa.  Large language models are zero-shot reasoners.  _Advances in neural information processing systems_ , 35:22199–22213, 2022.
  * Liu et al. (2023) Yang Liu, Dan Iter, Yichong Xu, Shuohang Wang, Ruochen Xu, and Chenguang Zhu.  G-EVAL: Nlg evaluation using gpt-4 with better human alignment.  _arXiv preprint arXiv:2303.16634_ , 2023.
  * Lu et al. (2023) Qingyu Lu, Baopu Qiu, Liang Ding, Liping Xie, and Dacheng Tao.  Error analysis prompting enables human-like translation evaluation in large language models: A case study on ChatGPT.  _arXiv preprint arXiv:2303.13809_ , 2023.
  * Madaan et al. (2023) Aman Madaan, Niket Tandon, Prakhar Gupta, Skyler Hallinan, Luyu Gao, Sarah Wiegreffe, Uri Alon, Nouha Dziri, Shrimai Prabhumoye, Yiming Yang, et al.  Self-refine: Iterative refinement with self-feedback.  _arXiv preprint arXiv:2303.17651_ , 2023.
  * OpenAI (2023) OpenAI.  GPT-4 technical report.  _arXiv preprint arXiv:2303.08774_ , 2023.
  * Ostermann et al. (2018) Simon Ostermann, Michael Roth, Ashutosh Modi, Stefan Thater, and Manfred Pinkal.  Semeval-2018 task 11: Machine comprehension using commonsense knowledge.  In _Proceedings of the 12th International Workshop on semantic evaluation_ , pp. 747–757, 2018.
  * Prasad et al. (2023) Archiki Prasad, Swarnadeep Saha, Xiang Zhou, and Mohit Bansal.  ReCEval: Evaluating reasoning chains via correctness and informativeness.  _arXiv preprint arXiv:2304.10703_ , 2023.
  * Saparov & He (2022) Abulhair Saparov and He He.  Language models are greedy reasoners: A systematic formal analysis of chain-of-thought.  In _The Eleventh International Conference on Learning Representations_ , 2022.
  * Shinn et al. (2023) Noah Shinn, Federico Cassano, Beck Labash, Ashwin Gopinath, Karthik Narasimhan, and Shunyu Yao.  Reflexion: Language agents with verbal reinforcement learning.  _arXiv preprint arXiv:2303.11366_ , 2023.
  * Somers (1962) Robert H Somers.  A new asymmetric measure of association for ordinal variables.  _American sociological review_ , pp. 799–811, 1962.
  * Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, et al.  Llama 2: Open foundation and fine-tuned chat models.  _arXiv preprint arXiv:2307.09288_ , 2023.
  * Wang et al. (2023) Jiaan Wang, Yunlong Liang, Fandong Meng, Haoxiang Shi, Zhixu Li, Jinan Xu, Jianfeng Qu, and Jie Zhou.  Is ChatGPT a good NLG evaluator? a preliminary study.  _arXiv preprint arXiv:2303.04048_ , 2023.
  * Wang et al. (2022) Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc V Le, Ed H Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou.  Self-consistency improves chain of thought reasoning in language models.  In _The Eleventh International Conference on Learning Representations_ , 2022.
  * Wei et al. (2022) Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al.  Chain-of-thought prompting elicits reasoning in large language models.  _Advances in Neural Information Processing Systems_ , 35:24824–24837, 2022.
  * Welleck et al. (2022) Sean Welleck, Jiacheng Liu, Ximing Lu, Hannaneh Hajishirzi, and Yejin Choi.  Naturalprover: Grounded mathematical proof generation with language models.  _Advances in Neural Information Processing Systems_ , 35:4913–4927, 2022.
  * Yao et al. (2023) Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Thomas L Griffiths, Yuan Cao, and Karthik Narasimhan.  Tree of thoughts: Deliberate problem solving with large language models.  _arXiv preprint arXiv:2305.10601_ , 2023.
  * Ye & Durrett (2022) Xi Ye and Greg Durrett.  The unreliability of explanations in few-shot prompting for textual reasoning.  _Advances in neural information processing systems_ , 35:30378–30392, 2022.
  * Zelikman et al. (2022) Eric Zelikman, Yuhuai Wu, Jesse Mu, and Noah Goodman.  STaR: Bootstrapping reasoning with reasoning.  _Advances in Neural Information Processing Systems_ , 35:15476–15488, 2022.
  * Zheng et al. (2023) Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al.  Judging LLM-as-a-judge with MT-bench and chatbot arena.  _arXiv preprint arXiv:2306.05685_ , 2023.

##  Appendix A Appendix

In this section, we delve deeper into the experimental details and present
additional results. Further information can be found in our code.

###  A.1 Experimental Details

#### Examples in e-SNLI and Cosmos QA.

In addition to the examples from GSM8K and DROP depicted in Figure 1, we
present examples from e-SNLI and Cosmos QA in Figure 4.

#### Detailed prompting templates.

For SocREval, detailed prompting templates across the four datasets—GSM8K,
e-SNLI, DROP, and Cosmos QA—are described in Tables 5-8. Detailed prompting
templates for SocREval with specific strategies, such as SocREval (Definition)
and SocREval (All - Definition), can be readily derived by adjusting the
Instruction, Evaluation prompt, and Example representation as appropriate.

#### Prompting configurations in the ablation study.

For the ablation study, the specific instructions and evaluation prompts
employed with SocREval (All - Definition), SocREval (All - Maieutics), and
SocREval (All - Dialectic) are detailed in Prompts (A.2)-(A.2).

#### Prompt variants in the robustness analysis.

Building on the discussions in Section 4, for an in-depth robustness
examination related to prompt writing for SocREval, we have rephrased Prompt
(2.2) five times. These five distinct variations of the original Prompt (2.2)
are shown in Prompts (A.2)-(A.2).

###  A.2 Additional Results

#### Extended meta evaluation analysis.

To better understand the advantages of SocREval over GPT-4, we transitioned
the initial five-class classification into both three-class and two-class
classification. Within the three-class classification, the original score of
$1$ is remapped to $-1$ (strongly negative), scores $2$ to $4$ are assigned to
$0$ (moderately negative), and the score of $5$ is designated as $1$ (strongly
positive). For the two-class classification, scores from $1$ to $4$ are
remapped to $0$ (negative), while a score of $5$ corresponds to $1$
(positive). Similar to Section 4, meta evaluation is carried out for both two-
class and three-class classification employing nine distinct metrics. As shown
in Tables 9 and 10, it is evident that SocREval systematically surpasses GPT-4
across all meta-evaluation metrics for both three-class and two-class
classification. Delving deeper, we analyzed the confusion matrices for GPT-4
and SocREval across the four datasets: GSM8K, e-SNLI, DROP, and Cosmos QA.
Figure 5 reveals that while GPT-4 tends to overestimate the quality of
reasoning chains, SocREval successfully mitigates this bias by integrating
strategies from the Socratic method.

#### Extended answer analysis.

Building upon the discussion in Section 4, we provide an in-depth
visualization of the quality distributions for reasoning chains related to
questions in the GSM8K dataset, as presented in Figure 6.

#### Error type analysis.

Drawing inspiration from ROSCOE and RECEVAL, we delve deeper into an analysis
focused on specific error types555While some error types such as “Coherence”
are evaluated at the chain-level, others like “Missing step” are assessed at
the step-level. Consistent with the scoring setup in ROSCOE and RECEVAL, we
aggregate step-level error scores using a “min” operation to derive the error
score for the entire reasoning chain, based on the intuition that the quality
of a reasoning chain is anchored by its weakest step. within reasoning chains.
Initially, we compute the correlation between human-annotated errors and the
overall quality of reasoning chains as judged by humans. As evident in Table
11, “Missing Step” and “Coherence” notably register the strongest correlations
with the overall quality amongst the ten annotated error categories. This
suggests these two errors heavily influence human assessment of reasoning
chain quality. Additionally, we examine the correlation of both GPT-4 and
SocREval with ten human-annotated error types across the four datasets. As
depicted in Table 12, SocREval shows superior performance over GPT-4 in all
error categories with the exception of ”Hallucination”. Furthermore, SocREval
outshines ROSCOE in all error types except “Hallucination”, “Redundancy”, and
“Grammar”. For a granular breakdown of correlations specific to each dataset,
readers are directed to Tables 13-16.

t e-SNLI
Situation (Premise): Two women are embracing while holding to go packages.
Claim (Hypothesis): Two woman are holding packages.
Question: Is the Claim supported by the Situation?
Generated response: The two women are most likely embracing because they are
either friends or family. If they were just holding packages, there would be
no need for them to embrace. The answer is Yes. Cosmos QA
Situation (Premise): “Do i need to go for a legal divorce ? I wanted to marry
a woman but she is not in the same religion , so i am not concern of the
marriage inside church . I will do the marriage registered with the girl who i
am going to get married . But legally will there be any complication , like if
the other woman comes back one day , will the girl who i am going to get
married now will be in trouble or Is there any complication ?  Why is this
person asking about divorce ?
Claim (Hypothesis): He wants to get married to a different person .
Question: Is the Claim supported by the Situation?
Generated response: He wants to get married to a different person means he is
not married to the person he wants to get married to. The answer is Yes.
Figure 4: Examples from the e-SNLI and Cosmos QA datasets.

Instruction: Does the generated response answer the question in a well-
justified manner? Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me an overall quality score in [1,
2, 3, 4, 5] (1=incomprehensible and wrong, 5=clear and correct) for the given
generated response by taking into account both your own response and the
qualitative analysis. Note that you need to take into account both the
explanation and the answer in the generated response. [t]
---
Example question: Janet’s ducks lay 16 eggs per day. She eats three for
breakfast every morning and bakes muffins for her friends every day with four.
She sells the remainder at the farmers’ market daily for $2 per fresh duck
egg. How much in dollars does she make every day at the farmers’ market?
Example generated response: Janet eats 3 duck eggs for breakfast and bakes 4
into muffins so 3 + 4 = $<<$3+4=7$>>$7 duck eggs are used||Each day Janet’s
ducks lay 16 eggs and she uses 7, 16 - 7 = $<<$16-7=9$>>$9 duck eggs are for
sale||She sells her eggs for $2 per egg and has 9 available for sale so 2 * 9
= $$<<$2*9=18$>>$18 per day||A: 18
Example representation: {“own response”: “Janet uses a total of 3 (for
breakfast) + 4 (for muffins) = 7 duck eggs each day. Given that her ducks lay
16 eggs per day, she is left with 16 - 7 = 9 eggs to sell at the farmers’
market. Each of these eggs sells for $2, so Janet makes a total of 9 * $2 =
$18 at the farmers’ market each day.”, “qualitative analysis”: “The generated
response is correct and complete. It precisely calculates the number of eggs
that Janet consumes each day and subtracts that from the total number of eggs
laid. It then multiplies the remaining eggs by the selling price to get the
total amount that Janet makes each day. The response is also clear and easy to
follow, with each step of the calculation laid out explicitly.”, “overall
quality”: 5}
Question: A robe takes 2 bolts of blue fiber and half that much white fiber.
How many bolts in total does it take?
Generated response: It takes 2 bolts of blue and half that much white so it
takes 2*1/2 = $<<$2*1/2=1.0$>>$1 bolt of white||It takes 1 bolt of white and
it takes 2 bolts of blue so it takes 1+2 = $<<$1+2=3$>>$3 bolts in total||A: 3
(Evaluation prompt) Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me the overall quality of the
given generated response for the question by taking into account both your own
response and the qualitative analysis based on the instruction and the format
of the example representation: [b]

Table 5: The full prompt for SocREval on GSM8K. While elements like
“Instruction” are embedded within the prompt, the term “Evaluation prompt” is
exclusive and simply represents the last sentence in the prompt. For clarity,
besides the demonstration example in the prompt, an additional example is
provided. The notation “||” is employed to symbolize the Python newline
character “\n”.

Instruction: Does the generated response answer the question in a well-
justified manner? Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me an overall quality score in [1,
2, 3, 4, 5] (1=incomprehensible and wrong, 5=clear and correct) for the given
generated response by taking into account both your own response and the
qualitative analysis. Note that you need to take into account both the
explanation and the answer in the generated response. [t]
---
Example Situation (Premise): Two women are embracing while holding to go
packages.
Example Claim (Hypothesis): Two woman are holding packages.
Example question: Is the Claim supported by the Situation?
Example generated response: The two women are most likely embracing because
they are either friends or family. If they were just holding packages, there
would be no need for them to embrace. The answer is Yes.
Example representation: {“own response”: “Yes, the Claim is supported by the
Situation. The Situation states that two women are ‘embracing while holding to
go packages,’ which means they are holding packages. Therefore, the Claim is
accurate.”, “qualitative analysis”: “The generated response starts by making
an assumption about why the two women might be embracing, suggesting they
could be friends or family. While this could be true, it is not directly
relevant to the question asked. The core of the question is about the Claim’s
accuracy given the Situation. The latter part of the generated response does
correctly determine that the answer is ‘Yes.’ However, the earlier assumption
makes the response longer and potentially more confusing.”, “overall quality”:
4}
Situation (Premise): Two women are embracing while holding to go packages.
Claim (Hypothesis): The men are fighting outside a deli.
Question: Is the Claim supported by the Situation?
Generated response: Two women are embracing while holding to go packages means
they are friends or family. They are not fighting. The answer is No.
(Evaluation prompt) Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me the overall quality of the
given generated response for the question by taking into account both your own
response and the qualitative analysis based on the instruction and the format
of the example representation: [b]

Table 6: The full prompt for SocREval on e-SNLI. While elements like
“Instruction” are embedded within the prompt, the term “Evaluation prompt” is
exclusive and simply represents the last sentence in the prompt. For clarity,
besides the demonstration example in the prompt, an additional example is
provided.

Instruction: Does the generated response answer the question in a well-
justified manner? Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me an overall quality score in [1,
2, 3, 4, 5] (1=incomprehensible and wrong, 5=clear and correct) for the given
generated response by taking into account both your own response and the
qualitative analysis. Note that you need to take into account both the
explanation and the answer in the generated response. [t]
---
Example Situation (Premise): Hoping to rebound from their loss to the
Patriots, the Raiders stayed at home for a Week 16 duel with the Houston
Texans. Oakland would get the early lead in the first quarter as quarterback
JaMarcus Russell completed a 20-yard touchdown pass to rookie wide receiver
Chaz Schilens. The Texans would respond with fullback Vonta Leach getting a
1-yard touchdown run, yet the Raiders would answer with kicker Sebastian
Janikowski getting a 33-yard and a 30-yard field goal. Houston would tie the
game in the second quarter with kicker Kris Brown getting a 53-yard and a
24-yard field goal. Oakland would take the lead in the third quarter with wide
receiver Johnnie Lee Higgins catching a 29-yard touchdown pass from Russell,
followed up by an 80-yard punt return for a touchdown. The Texans tried to
rally in the fourth quarter as Brown nailed a 40-yard field goal, yet the
Raiders’ defense would shut down any possible attempt. Who scored the first
touchdown of the game?
Example Claim (Hypothesis): Chaz Schilens
Example question: Is the Claim supported by the Situation?
Example generated response: Chaz Schilens was the receiver of the touchdown
pass from JaMarcus Russell, so he scored the first touchdown. The answer is
Yes.
Example representation: {“own response”: “Yes, the Claim is supported by the
Situation as Chaz Schilens caught a 20-yard touchdown pass from JaMarcus
Russell, making him the scorer of the first touchdown.”, “qualitative
analysis”: “Both the generated response and my own response acknowledge that
Chaz Schilens caught the first touchdown of the game from a pass by JaMarcus
Russell. The generated response provides a concise and clear explanation that
supports the answer.”, “overall quality”: 5}
Situation (Premise): Hoping to rebound from their loss to the Patriots, the
Raiders stayed at home for a Week 16 duel with the Houston Texans. Oakland
would get the early lead in the first quarter as quarterback JaMarcus Russell
completed a 20-yard touchdown pass to rookie wide receiver Chaz Schilens. The
Texans would respond with fullback Vonta Leach getting a 1-yard touchdown run,
yet the Raiders would answer with kicker Sebastian Janikowski getting a
33-yard and a 30-yard field goal. Houston would tie the game in the second
quarter with kicker Kris Brown getting a 53-yard and a 24-yard field goal.
Oakland would take the lead in the third quarter with wide receiver Johnnie
Lee Higgins catching a 29-yard touchdown pass from Russell, followed up by an
80-yard punt return for a touchdown. The Texans tried to rally in the fourth
quarter as Brown nailed a 40-yard field goal, yet the Raiders’ defense would
shut down any possible attempt. How many field goals did Kris Brown kick?
Claim (Hypothesis): 3
Question: Is the Claim supported by the Situation?
Generated response: Kris Brown kicked 4 field goals. The answer is No.
(Evaluation prompt) Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me the overall quality of the
given generated response for the question by taking into account both your own
response and the qualitative analysis based on the instruction and the format
of the example representation: [b]

Table 7: The full prompt for SocREval on DROP. While elements like
“Instruction” are embedded within the prompt, the term “Evaluation prompt” is
exclusive and simply represents the last sentence in the prompt. For clarity,
besides the demonstration example in the prompt, an additional example is
provided.

Instruction: Does the generated response answer the question in a well-
justified manner? Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me an overall quality score in [1,
2, 3, 4, 5] (1=incomprehensible and wrong, 5=clear and correct) for the given
generated response by taking into account both your own response and the
qualitative analysis. Note that you need to take into account both the
explanation and the answer in the generated response. [t]
---
Example Situation (Premise): Do i need to go for a legal divorce ? I wanted to
marry a woman but she is not in the same religion , so i am not concern of the
marriage inside church . I will do the marriage registered with the girl who i
am going to get married . But legally will there be any complication , like if
the other woman comes back one day , will the girl who i am going to get
married now will be in trouble or Is there any complication ? Why is this
person asking about divorce ?
Example Claim (Hypothesis): He wants to get married to a different person .
Example question: Is the Claim supported by the Situation?
Example generated response: He wants to get married to a different person
means he is not married to the person he wants to get married to. The answer
is Yes.
Example representation: {“own response”: “Yes, the Situation suggests that the
person wants to get married to a different woman than the one he may have been
previously involved with.”, “qualitative analysis”: “The generated response
made an assumption that ‘He wants to get married to a different person means
he is not married to the person he wants to get married to.’ While this may be
true, it’s not directly stated in the Situation. The Situation does not
provide information about the current marital status of the person in
question. However, the underlying premise that he wants to get married to a
different person is accurate based on the context. The generated response does
conclude correctly that the answer is ‘Yes,’ meaning the Claim is supported by
the Situation.”, “overall quality”: 3}
Situation (Premise): Do i need to go for a legal divorce ? I wanted to marry a
woman but she is not in the same religion , so i am not concern of the
marriage inside church . I will do the marriage registered with the girl who i
am going to get married . But legally will there be any complication , like if
the other woman comes back one day , will the girl who i am going to get
married now will be in trouble or Is there any complication ? Why is he
worried about getting married ?
Claim (Hypothesis): He was married before and she might come back one day .
Question: Is the Claim supported by the Situation?
Generated response: There might be a complication if the woman he was married
to came back one day, due to the fact he is not married to her in a religious
context, so she could come back and claim him as her husband. The answer is
Yes.
(Evaluation prompt) Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me the overall quality of the
given generated response for the question by taking into account both your own
response and the qualitative analysis based on the instruction and the format
of the example representation: [b]

Table 8: The full prompt for SocREval on Cosmos QA. While elements like
“Instruction” are embedded within the prompt, the term “Evaluation prompt” is
exclusive and simply represents the last sentence in the prompt. For clarity,
besides the demonstration example in the prompt, an additional example is
provided.

(6) Instruction and Evaluation Prompt for SocREval (All - Definition):
Instruction: Does the generated response answer the question in a well-
justified manner? Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me an overall quality score in [1,
2, 3, 4, 5] (1=incomprehensible and wrong, 5=clear and correct) for the given
generated response by taking into account both your own response and the
qualitative analysis.
(Evaluation prompt) Please generate your own response for the question first,
then conduct a qualitative analysis on the generated response by taking into
account your own response, and finally give me the overall quality of the
given generated response for the question by taking into account both your own
response and the qualitative analysis based on the instruction and the format
of the example representation:

(7) Instruction and Evaluation Prompt for SocREval (All - Maieutics):
Instruction: Does the generated response answer the question in a well-
justified manner? Please generate your own response for the question first and
then give me an overall quality score in [1, 2, 3, 4, 5] (1=incomprehensible
and wrong, 5=clear and correct) for the given generated response by taking
into account your own response. Note that you need to take into account both
the explanation and the answer in the generated response.
(Evaluation prompt) Please generate your own response for the question first
and then give me the overall quality of the given generated response for the
question by taking into account your own response based on the instruction and
the format of the example representation:

(8) Instruction and Evaluation Prompt for SocREval (All - Dialectic):
Instruction: Does the generated response answer the question in a well-
justified manner? Please conduct a qualitative analysis on the generated
response first and then give me an overall quality score in [1, 2, 3, 4, 5]
(1=incomprehensible and wrong, 5=clear and correct) for the given generated
response by taking into account the qualitative analysis. Note that you need
to take into account both the explanation and the answer in the generated
response.
(Evaluation prompt) Please conduct a qualitative analysis on the generated
response first and then give me the overall quality of the given generated
response for the question by taking into account the qualitative analysis
based on the instruction and the format of the example representation:

(9) Instruction and Evaluation Prompt for SocREval (Variant I):
Instruction: Does the generated response aptly address the question in a well-
reasoned way? First, create your own response to the question. Following this,
conduct a qualitative analysis of the generated response, considering your own
response. Conclude by providing an overall quality score ranging from [1, 2,
3, 4, 5] (1= incomprehensible and incorrect, 5= clear and accurate) for the
given generated response, factoring in both your own response and the
qualitative analysis. It’s imperative to consider both the explanation and the
answer present in the generated response.
(Evaluation prompt) Firstly, formulate your own response to the question.
Then, undertake a qualitative analysis of the generated response, using your
own response as a reference. Lastly, provide an assessment of the overall
quality of the presented generated response by weighing both your own response
and the qualitative analysis, ensuring adherence to the provided instruction
and the format of the example representation:

(10) Instruction and Evaluation Prompt for SocREval (Variant II):
Instruction: Does the generated response aptly address the question with a
well-substantiated justification? First, formulate your own response to the
question. Subsequently, perform a qualitative analysis of the generated
response, considering your own response. Finally, assign an overall quality
score ranging from [1, 2, 3, 4, 5] (1= incomprehensible and incorrect, 5=
lucid and accurate) for the given generated response. This score should
reflect both your own response and the qualitative analysis. Ensure you
consider both the explanation and the answer in the generated response.
(Evaluation prompt) First, craft your own response to the question. Next,
undertake a qualitative analysis of the presented generated response,
referencing your own response for context. Conclude by rating the overall
quality of the generated response, integrating both your own response and the
qualitative analysis, as guided by the instruction and the format of the
example representation:

(11) Instruction and Evaluation Prompt for SocREval (Variant III):
Instruction: Does the generated response adequately address the question?
First, formulate your own response to the question. Following this, perform a
qualitative analysis of the generated response, using your own response as a
reference. Conclude by assigning an overall quality score ranging from [1, 2,
3, 4, 5] (1 being incomprehensible and wrong, 5 being clear and correct) to
the provided generated response. This score should consider both the clarity
of the explanation and the accuracy of the answer in the generated response.
(Evaluation prompt) Begin by crafting your own response to the question. Next,
undertake a qualitative analysis of the generated response, comparing it
against your own response. Finally, rate the overall quality of the generated
response, referencing both your own response and the qualitative analysis.
Ensure your evaluation adheres to the given instruction and the format of the
example representation:

(12) Instruction and Evaluation Prompt for SocREval (Variant IV):
Instruction: Does the generated response adequately address the question with
sound justification? First, provide your own response to the question.
Subsequently, perform a qualitative analysis of the generated response,
comparing it with your own response. Conclude by assigning an overall quality
score ranging from [1, 2, 3, 4, 5] (1 = incomprehensible and wrong, 5 = clear
and correct) to the generated response. This score should reflect both the
clarity of the explanation and the accuracy of the answer in the generated
response.
(Evaluation prompt) Begin by crafting your own response to the question. Then,
execute a qualitative analysis of the generated response, using your own
response as a benchmark. Finally, rate the overall quality of the provided
generated response on a scale of [1 to 5], considering both your own response
and the qualitative analysis, following the instruction provided and the
format of the example representation:

(13) Instruction and Evaluation Prompt for SocREval (Variant V):
Instruction: Does the generated response adequately address the question with
valid justification? First, produce your own response to the question.
Following this, conduct a qualitative analysis of the generated response,
using your own response as a reference. Lastly, provide an overall quality
score ranging from [1, 2, 3, 4, 5] (1=incomprehensible and wrong, 5=clear and
correct) for the generated response. This score should consider both your own
response and the qualitative analysis. It’s important to assess both the
explanation and the answer present in the generated response.
(Evaluation prompt) Begin by crafting your own response to the question. Then,
perform a qualitative analysis of the generated response, referencing your own
response. Conclude by assigning an overall quality rating to the provided
generated response. This rating should factor in both your own response and
the results of your qualitative analysis, following the provided instruction
and the format of the example representation:

| SD | $r$ | $\rho$ | $\tau$ | ACC | RMSE | MAE | Micro-F1 | Macro-F1 [t] [b]
---|---|---|---|---|---|---|---|---|---
GPT-4 | 0.37 | 0.45 | 0.45 | 0.43 | 0.56 | 0.74 | 0.48 | 0.56 | 0.46 [t]
SocREval | 0.53 | 0.56 | 0.57 | 0.54 | 0.66 | 0.60 | 0.35 | 0.66 |  0.57 [b]
|  |  |  |  |  |  |  |  |

Table 9: Comparison between SocREval and GPT-4 over diverse meta-evaluation
metrics in the context of three-class classification. Specifically, we map the
original score of $1$ to $-1$ (strongly negative), scores $2$ through $4$ to
$0$ (moderately negative), and the original score of $5$ to $1$ (strongly
positive). In addition to Somers’ D (SD), metrics encompass Pearson’s
Correlation Coefficient ($r$), Spearman’s Rank Correlation Coefficient
($\rho$), Kendall’s $\tau$ ($\tau$), Accuracy (ACC), Root Mean Squared Error
(RMSE), Mean Absolute Error (MAE), Micro F1 Score (Micro-F1), and Macro F1
Score (Macro-F1). It’s crucial to note that for RMSE and MAE, lower values
indicate superior performance, whereas for the remaining metrics, higher
values are desirable. Scores for each configuration are computed as averages
over four datasets: GSM8K, e-SNLI, DROP, and Cosmos QA.

| SD | $r$ | $\rho$ | $\tau$ | ACC | RMSE | MAE | Micro-F1 | Macro-F1 [t] [b]
---|---|---|---|---|---|---|---|---|---
GPT-4 | 0.37 | 0.40 | 0.40 | 0.40 | 0.69 | 0.55 | 0.31 | 0.69 | 0.64 [t]
SocREval | 0.51 | 0.52 | 0.52 | 0.52 | 0.80 | 0.42 | 0.20 | 0.80 |  0.75 [b]
|  |  |  |  |  |  |  |  |

Table 10: Comparison between SocREval and GPT-4 over diverse meta-evaluation
metrics in the context of two-class classification. Specifically, original
scores ranging from $1$ to $4$ are mapped to $0$ (negative), while the
original score of $5$ is mapped to $1$ (positive). In addition to Somers’ D
(SD), metrics encompass Pearson’s Correlation Coefficient ($r$), Spearman’s
Rank Correlation Coefficient ($\rho$), Kendall’s $\tau$ ($\tau$), Accuracy
(ACC), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Micro F1
Score (Micro-F1), and Macro F1 Score (Macro-F1). It’s crucial to note that for
RMSE and MAE, lower values indicate superior performance, whereas for the
remaining metrics, higher values are desirable. Scores for each configuration
are computed as averages over four datasets: GSM8K, e-SNLI, DROP, and Cosmos
QA.

(a) GPT-4 on GSM8K

(b) SocREval on GSM8K

(c) GPT-4 on e-SNLI

(d) SocREval on e-SNLI

(e) GPT-4 on DROP

(f) SocREval on DROP

(g) GPT-4 on Cosmos QA

(h) SocREval on Cosmos QA

Figure 5: Confusion matrices for GPT-4 and SocREval evaluated across four
datasets: GSM8K, e-SNLI, DROP, and Cosmos QA.

(a) Human judgment (wrong)

(b) GPT-4 (wrong)

(c) SocREval (wrong)

(d) Human judgment (correct)

(e) GPT-4 (correct)

(f) SocREval (correct)

Figure 6: Detailed distributions of the quality of reasoning chains across
questions on GSM8K, differentiated by answer correctness (correct versus
wrong), as assessed through human judgment, GPT-4, and SocREval.

| GSM8K | e-SNLI | DROP | Cosmos QA | Average [t] [b]
---|---|---|---|---|---
COH | 0.73 | 0.15 | 0.33 | 0.56 |  0.44 [t]
COMMON | 0.53 | 0.07 | 0.01 | 0.05 | 0.17
FACT | 0.50 | 0.22 | 0.19 | 0.07 | 0.25
HALL | 0.10 | 0.04 | 0.04 | -0.01 | 0.04
RED | 0.10 | 0.14 | 0.13 | 0.04 | 0.10
REP | 0.02 | - | 0.01 | 0.00 | -
LOGIC | 0.53 | 0.18 | 0.12 | 0.07 | 0.23
MATH | 0.06 | 0.00 | 0.01 | - | -
GRAM | -0.01 | -0.01 | 0.02 | -0.02 | -0.01
MISS | 0.73 | 0.36 | 0.50 | 0.35 |  0.49 [b]
|  |  |  |  |

Table 11: Somers’ D correlations between human-annotated errors and human
judgment of overall quality of reasoning chains. Evaluated errors encompass:
Coherency (COH), Commonsense (COMMON), Factuality (FACT), Hallucination
(HALL), Redundancy (RED), Repetition (REP), Logic (LOGIC), Arithmetic (MATH),
Grammar (GRAM), and Missing Step (MISS). The highest and second-highest
correlations for each column are indicated in bold and underlined,
respectively. A ’-’ symbol indicates no correlation observed.

| QUAL | COH | COMMON | FACT | HALL | RED | REP | LOGIC | MATH | GRAM | MISS [t] [b]
---|---|---|---|---|---|---|---|---|---|---|---
Reference-free ROSCOE | 0.26 | 0.31 | 0.40 | 0.40 | 0.54 | 0.69 | - | 0.35 | - | 0.44 | 0.33[t]
RECEVAL | - | - | - | - | - | - | - | - | - | - | -
GPT-4 | 0.40 | 0.41 | 0.34 | 0.50 | 0.47 | 0.26 | - | 0.25 | - | -0.07 | 0.31
SocREval (Definition) | 0.44 | 0.44 | 0.40 | 0.46 | 0.43 | 0.25 | - | 0.28 | - | -0.04 | 0.40
SocREval (Maieutics) | 0.52 | 0.44 | 0.28 | 0.44 | 0.43 | 0.37 | - | 0.31 | - | -0.03 | 0.47
SocREval (Dialectic) | 0.48 | 0.42 | 0.35 | 0.54 | 0.38 | 0.32 | - | 0.33 | - | -0.06 | 0.37
SocREval (All - Dialectic) | 0.51 | 0.45 | 0.28 | 0.42 | 0.44 | 0.39 | - | 0.33 | - | 0.06 | 0.51
SocREval (All - Maieutics) | 0.53 | 0.48 | 0.34 | 0.53 | 0.48 | 0.39 | - | 0.36 | - | -0.06 | 0.47
SocREval (All - Definition) | 0.56 | 0.47 | 0.34 | 0.57 | 0.41 | 0.36 | - | 0.42 | - | 0.02 | 0.46
SocREval (All) or SocREval | 0.58 | 0.51 | 0.40 | 0.53 | 0.43 | 0.34 | - | 0.36 | - | -0.03 |  0.52 [b]
|  |  |  |  |  |  |  |  |  |  |

Table 12: Average correlation analysis, computed across four datasets (GSM8K,
e-SNLI, DROP, and Cosmos QA), between reasoning evaluation metrics and various
aspects of human judgment. Beyond the overall quality (QUAL), ten specific
error types are considered: Coherency (COH), Commonsense (COMMON), Factuality
(FACT), Hallucination (HALL), Redundancy (RED), Repetition (REP), Logic
(LOGIC), Arithmetic (MATH), Grammar (GRAM), and Missing Step (MISS). For each
column, the highest and second-highest correlations are denoted in bold and
underlined, respectively. The ’-’ symbol signifies an absence of observed
correlation.

| QUAL | COH | COMMON | FACT | HALL | RED | REP | LOGIC | MATH | GRAM | MISS [t] [b]
---|---|---|---|---|---|---|---|---|---|---|---
Reference-free ROSCOE | 0.32 | 0.30 | 0.28 | 0.35 | 0.50 | 0.64 | 0.98 | 0.22 | 0.56 | 0.34 | 0.35 [t]
RECEVAL | 0.36 | 0.31 | 0.21 | 0.37 | 0.28 | 0.55 | 0.87 | 0.25 | 0.32 | - | -
GPT-4 | 0.57 | 0.47 | 0.51 | 0.51 | 0.61 | 0.33 | 0.82 | 0.54 | 0.27 | -0.16 | 0.66
SocREval (Definition) | 0.62 | 0.53 | 0.60 | 0.57 | 0.70 | 0.33 | 0.71 | 0.65 | 0.22 | -0.17 | 0.74
SocREval (Maieutics) | 0.64 | 0.54 | 0.67 | 0.54 | 0.63 | 0.41 | 0.77 | 0.66 | 0.31 | -0.02 | 0.75
SocREval (Dialectic) | 0.74 | 0.63 | 0.71 | 0.65 | 0.44 | 0.45 | 0.78 | 0.76 | 0.45 | -0.30 | 0.86
SocREval (All - Dialectic) | 0.62 | 0.52 | 0.64 | 0.56 | 0.55 | 0.41 | 0.82 | 0.64 | 0.39 | -0.00 | 0.75
SocREval (All - Maieutics) | 0.81 | 0.67 | 0.78 | 0.72 | 0.69 | 0.47 | 0.80 | 0.80 | 0.44 | -0.32 | 0.92
SocREval (All - Definition) | 0.83 | 0.68 | 0.79 | 0.70 | 0.60 | 0.46 | 0.68 | 0.78 | 0.46 | -0.31 | 0.91
SocREval (All) or SocREval | 0.82 | 0.72 | 0.78 | 0.71 | 0.63 | 0.44 | 0.84 | 0.76 | 0.50 | -0.24 |  0.92 [b]
|  |  |  |  |  |  |  |  |  |  |

Table 13: Correlation analysis between reasoning evaluation metrics and
diverse facets of human judgment on GSM8K. Beyond the overall quality (QUAL),
ten specific error types are also considered: Coherency (COH), Commonsense
(COMMON), Factuality (FACT), Hallucination (HALL), Redundancy (RED),
Repetition (REP), Logic (LOGIC), Arithmetic (MATH), Grammar (GRAM), and
Missing Step (MISS). For each column, the highest and second-highest
correlations are denoted in bold and underlined, respectively. The ’-’ symbol
signifies an absence of observed correlation.

| QUAL | COH | COMMON | FACT | HALL | RED | REP | LOGIC | MATH | GRAM | MISS [t] [b]
---|---|---|---|---|---|---|---|---|---|---|---
Reference-free ROSCOE | 0.30 | 0.35 | 0.40 | 0.43 | 0.80 | 0.62 | - | 0.53 | 0.60 | 0.41 |  0.60 [t]
RECEVAL | - | - | - | - | - | - | - | - | - | - | -
GPT-4 | 0.24 | 0.42 | 0.43 | 0.65 | 0.33 | 0.07 | - | 0.29 | -0.15 | -0.16 | -0.04
SocREval (Definition) | 0.30 | 0.44 | 0.38 | 0.61 | 0.29 | 0.08 | - | 0.23 | -0.21 | -0.05 | 0.17
SocREval (Maieutics) | 0.58 | 0.34 | 0.07 | 0.55 | 0.24 | 0.35 | - | 0.43 | 0.27 | -0.25 | 0.45
SocREval (Dialectic) | 0.31 | 0.33 | 0.24 | 0.70 | 0.32 | 0.15 | - | 0.39 | -0.21 | -0.04 | -0.04
SocREval (All - Dialectic) | 0.49 | 0.42 | 0.06 | 0.36 | 0.35 | 0.34 | - | 0.47 | -0.47 | -0.03 | 0.50
SocREval (All - Maieutics) | 0.38 | 0.39 | 0.14 | 0.55 | 0.44 | 0.31 | - | 0.44 | -0.34 | -0.35 | 0.22
SocREval (All - Definition) | 0.51 | 0.39 | 0.18 | 0.75 | 0.35 | 0.42 | - | 0.56 | 0.56 | 0.12 | 0.16
SocREval (All) or SocREval | 0.58 | 0.49 | 0.34 | 0.65 | 0.46 | 0.26 | - | 0.47 | 0.00 | -0.05 | 0.41 [b]
|  |  |  |  |  |  |  |  |  |  |

Table 14: Correlation analysis between reasoning evaluation metrics and
diverse facets of human judgment on e-SNLI. Beyond the overall quality (QUAL),
ten specific error types are also considered: Coherency (COH), Commonsense
(COMMON), Factuality (FACT), Hallucination (HALL), Redundancy (RED),
Repetition (REP), Logic (LOGIC), Arithmetic (MATH), Grammar (GRAM), and
Missing Step (MISS). For each column, the highest and second-highest
correlations are denoted in bold and underlined, respectively. The ’-’ symbol
signifies an absence of observed correlation.

| QUAL | COH | COMMON | FACT | HALL | RED | REP | LOGIC | MATH | GRAM | MISS [t] [b]
---|---|---|---|---|---|---|---|---|---|---|---
Reference-free ROSCOE | 0.22 | 0.40 | 0.64 | 0.51 | 0.54 | 0.82 | 0.95 | 0.29 | 0.85 | 0.45 | 0.21[t]
RECEVAL | - | - | - | - | - | - | - | - | - | - | -
GPT-4 | 0.42 | 0.44 | 0.49 | 0.55 | 0.54 | 0.27 | -0.25 | 0.22 | 0.16 | 0.14 | 0.33
SocREval (Definition) | 0.43 | 0.45 | 0.51 | 0.48 | 0.48 | 0.25 | -0.27 | 0.20 | 0.14 | 0.13 | 0.35
SocREval (Maieutics) | 0.47 | 0.53 | 0.40 | 0.47 | 0.50 | 0.40 | 0.61 | 0.13 | 0.22 | 0.23 | 0.37
SocREval (Dialectic) | 0.43 | 0.40 | 0.41 | 0.64 | 0.58 | 0.28 | -0.28 | 0.12 | 0.29 | 0.30 | 0.31
SocREval (All - Dialectic) | 0.47 | 0.46 | 0.42 | 0.53 | 0.43 | 0.39 | 0.44 | 0.18 | 0.24 | 0.31 | 0.39
SocREval (All - Maieutics) | 0.51 | 0.56 | 0.39 | 0.60 | 0.57 | 0.40 | 0.50 | 0.14 | 0.26 | 0.42 | 0.36
SocREval (All - Definition) | 0.49 | 0.47 | 0.39 | 0.61 | 0.53 | 0.33 | 0.02 | 0.22 | 0.29 | 0.28 | 0.39
SocREval (All) or SocREval | 0.49 | 0.51 | 0.43 | 0.56 | 0.47 | 0.36 | 0.53 | 0.18 | 0.18 | 0.38 |  0.42 [b]
|  |  |  |  |  |  |  |  |  |  |

Table 15: Correlation analysis between reasoning evaluation metrics and
diverse facets of human judgment on DROP. Beyond the overall quality (QUAL),
ten specific error types are also considered: Coherency (COH), Commonsense
(COMMON), Factuality (FACT), Hallucination (HALL), Redundancy (RED),
Repetition (REP), Logic (LOGIC), Arithmetic (MATH), Grammar (GRAM), and
Missing Step (MISS). For each column, the highest and second-highest
correlations are denoted in bold and underlined, respectively. The ’-’ symbol
signifies an absence of observed correlation.

| QUAL | COH | COMMON | FACT | HALL | RED | REP | LOGIC | MATH | GRAM | MISS [t] [b]
---|---|---|---|---|---|---|---|---|---|---|---
Reference-free ROSCOE | 0.18 | 0.19 | 0.28 | 0.29 | 0.32 | 0.66 | 0.67 | 0.35 | - | 0.56 | 0.15[t]
RECEVAL | - | - | - | - | - | - | - | - | - | - | -
GPT-4 | 0.39 | 0.30 | -0.06 | 0.29 | 0.39 | 0.39 | -0.34 | -0.04 | - | -0.09 | 0.28
SocREval (Definition) | 0.39 | 0.35 | 0.10 | 0.19 | 0.24 | 0.36 | -0.36 | 0.05 | - | -0.05 | 0.33
SocREval (Maieutics) | 0.39 | 0.34 | -0.02 | 0.19 | 0.36 | 0.31 | -0.28 | 0.03 | - | -0.06 | 0.30
SocREval (Dialectic) | 0.43 | 0.32 | 0.05 | 0.19 | 0.18 | 0.39 | -0.32 | 0.05 | - | -0.20 | 0.35
SocREval (All - Dialectic) | 0.45 | 0.40 | -0.00 | 0.24 | 0.43 | 0.41 | -0.40 | 0.03 | - | -0.05 | 0.39
SocREval (All - Maieutics) | 0.42 | 0.33 | 0.04 | 0.25 | 0.22 | 0.38 | -0.18 | 0.08 | - | 0.01 | 0.36
SocREval (All - Definition) | 0.42 | 0.34 | 0.00 | 0.22 | 0.15 | 0.23 | -0.13 | 0.11 | - | -0.02 | 0.36
SocREval (All) or SocREval | 0.42 | 0.32 | 0.06 | 0.21 | 0.14 | 0.30 | -0.38 | 0.03 | - | -0.20 | 0.33 [b]
|  |  |  |  |  |  |  |  |  |  |

Table 16: Correlation analysis between reasoning evaluation metrics and
diverse facets of human judgment on Cosmos QA. Beyond the overall quality
(QUAL), ten specific error types are considered: Coherency (COH), Commonsense
(COMMON), Factuality (FACT), Hallucination (HALL), Redundancy (RED),
Repetition (REP), Logic (LOGIC), Arithmetic (MATH), Grammar (GRAM), and
Missing Step (MISS). For each column, the highest and second-highest
correlations are denoted in bold and underlined, respectively. The ’-’ symbol
signifies an absence of observed correlation.