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IEEE-conference-template-062824.aux
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\bibcite{hendrycks2021measuringmathematicalproblemsolving}{1}
\bibcite{ahn2024largelanguagemodelsmathematical}{2}
\bibcite{cobbe2021trainingverifierssolvemath}{3}
@@ -45,5 +43,9 @@
\bibcite{gao2023palprogramaidedlanguagemodels}{8}
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IEEE-conference-template-062824.bbl
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@@ -25,45 +26,70 @@
\bibitem{ahn2024largelanguagemodelsmathematical}
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J.~Ahn, R.~Verma, R.~Lou, D.~Liu, R.~Zhang, and W.~Yin, ``Large language models for mathematical reasoning: Progresses and challenges,'' 2024. [Online]. Available: \url{https://arxiv.org/abs/2402.00157}
J.~Ahn, R.~Verma, R.~Lou, D.~Liu, R.~Zhang, and W.~Yin, ``Large language models
for mathematical reasoning: Progresses and challenges,'' 2024. [Online].
Available: \url{https://arxiv.org/abs/2402.00157}
\BIBentrySTDinterwordspacing
\bibitem{cobbe2021trainingverifierssolvemath}
\BIBentryALTinterwordspacing
K.~Cobbe, V.~Kosaraju, M.~Bavarian, M.~Chen, H.~Jun, L.~Kaiser, M.~Plappert, J.~Tworek, J.~Hilton, R.~Nakano, C.~Hesse, and J.~Schulman, ``Training verifiers to solve math word problems,'' 2021. [Online]. Available: \url{https://arxiv.org/abs/2110.14168}
K.~Cobbe, V.~Kosaraju, M.~Bavarian, M.~Chen, H.~Jun, L.~Kaiser, M.~Plappert,
J.~Tworek, J.~Hilton, R.~Nakano, C.~Hesse, and J.~Schulman, ``Training
verifiers to solve math word problems,'' 2021. [Online]. Available:
\url{https://arxiv.org/abs/2110.14168}
\BIBentrySTDinterwordspacing
\bibitem{hoffmann2022trainingcomputeoptimallargelanguage}
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J.~Hoffmann, S.~Borgeaud, A.~Mensch, E.~Buchatskaya, T.~Cai, E.~Rutherford, D.~de~Las~Casas, L.~A. Hendricks, J.~Welbl, A.~Clark, T.~Hennigan, E.~Noland, K.~Millican, G.~van~den Driessche, B.~Damoc, A.~Guy, S.~Osindero, K.~Simonyan, E.~Elsen, J.~W. Rae, O.~Vinyals, and L.~Sifre, ``Training compute-optimal large language models,'' 2022. [Online]. Available: \url{https://arxiv.org/abs/2203.15556}
J.~Hoffmann, S.~Borgeaud, A.~Mensch, E.~Buchatskaya, T.~Cai, E.~Rutherford,
D.~de~Las~Casas, L.~A. Hendricks, J.~Welbl, A.~Clark, T.~Hennigan, E.~Noland,
K.~Millican, G.~van~den Driessche, B.~Damoc, A.~Guy, S.~Osindero,
K.~Simonyan, E.~Elsen, J.~W. Rae, O.~Vinyals, and L.~Sifre, ``Training
compute-optimal large language models,'' 2022. [Online]. Available:
\url{https://arxiv.org/abs/2203.15556}
\BIBentrySTDinterwordspacing
\bibitem{hendrycksmath2021}
D.~Hendrycks, C.~Burns, S.~Kadavath, A.~Arora, S.~Basart, E.~Tang, D.~Song, and J.~Steinhardt, ``Measuring mathematical problem solving with the math dataset,'' \emph{NeurIPS}, 2021.
D.~Hendrycks, C.~Burns, S.~Kadavath, A.~Arora, S.~Basart, E.~Tang, D.~Song, and
J.~Steinhardt, ``Measuring mathematical problem solving with the math
dataset,'' \emph{NeurIPS}, 2021.
\bibitem{besiroglu2024chinchillascalingreplicationattempt}
\BIBentryALTinterwordspacing
T.~Besiroglu, E.~Erdil, M.~Barnett, and J.~You, ``Chinchilla scaling: A replication attempt,'' 2024. [Online]. Available: \url{https://arxiv.org/abs/2404.10102}
T.~Besiroglu, E.~Erdil, M.~Barnett, and J.~You, ``Chinchilla scaling: A
replication attempt,'' 2024. [Online]. Available:
\url{https://arxiv.org/abs/2404.10102}
\BIBentrySTDinterwordspacing
\bibitem{besold2017neuralsymboliclearningreasoningsurvey}
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T.~R. Besold, A.~d'Avila Garcez, S.~Bader, H.~Bowman, P.~Domingos, P.~Hitzler, K.-U. Kuehnberger, L.~C. Lamb, D.~Lowd, P.~M.~V. Lima, L.~de~Penning, G.~Pinkas, H.~Poon, and G.~Zaverucha, ``Neural-symbolic learning and reasoning: A survey and interpretation,'' 2017. [Online]. Available: \url{https://arxiv.org/abs/1711.03902}
T.~R. Besold, A.~d'Avila Garcez, S.~Bader, H.~Bowman, P.~Domingos, P.~Hitzler,
K.-U. Kuehnberger, L.~C. Lamb, D.~Lowd, P.~M.~V. Lima, L.~de~Penning,
G.~Pinkas, H.~Poon, and G.~Zaverucha, ``Neural-symbolic learning and
reasoning: A survey and interpretation,'' 2017. [Online]. Available:
\url{https://arxiv.org/abs/1711.03902}
\BIBentrySTDinterwordspacing
\bibitem{gao2023palprogramaidedlanguagemodels}
\BIBentryALTinterwordspacing
L.~Gao, A.~Madaan, S.~Zhou, U.~Alon, P.~Liu, Y.~Yang, J.~Callan, and G.~Neubig, ``Pal: Program-aided language models,'' 2023. [Online]. Available: \url{https://arxiv.org/abs/2211.10435}
L.~Gao, A.~Madaan, S.~Zhou, U.~Alon, P.~Liu, Y.~Yang, J.~Callan, and G.~Neubig,
``Pal: Program-aided language models,'' 2023. [Online]. Available:
\url{https://arxiv.org/abs/2211.10435}
\BIBentrySTDinterwordspacing
\bibitem{xu2024chatglmmathimprovingmathproblemsolving}
\BIBentryALTinterwordspacing
Y.~Xu, X.~Liu, X.~Liu, Z.~Hou, Y.~Li, X.~Zhang, Z.~Wang, A.~Zeng, Z.~Du, W.~Zhao, J.~Tang, and Y.~Dong, ``Chatglm-math: Improving math problem-solving in large language models with a self-critique pipeline,'' 2024. [Online]. Available: \url{https://arxiv.org/abs/2404.02893}
Y.~Xu, X.~Liu, X.~Liu, Z.~Hou, Y.~Li, X.~Zhang, Z.~Wang, A.~Zeng, Z.~Du,
W.~Zhao, J.~Tang, and Y.~Dong, ``Chatglm-math: Improving math problem-solving
in large language models with a self-critique pipeline,'' 2024. [Online].
Available: \url{https://arxiv.org/abs/2404.02893}
\BIBentrySTDinterwordspacing
\bibitem{petruzzellis2024assessingemergentsymbolicreasoning}
\BIBentryALTinterwordspacing
F.~Petruzzellis, A.~Testolin, and A.~Sperduti, ``Assessing the emergent symbolic reasoning abilities of llama large language models,'' 2024. [Online]. Available: \url{https://arxiv.org/abs/2406.06588}
F.~Petruzzellis, A.~Testolin, and A.~Sperduti, ``Assessing the emergent
symbolic reasoning abilities of llama large language models,'' 2024.
[Online]. Available: \url{https://arxiv.org/abs/2406.06588}
\BIBentrySTDinterwordspacing
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@@ -12,6 +12,10 @@
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18} % or the version you have installed
\begin{document}
@@ -247,48 +251,113 @@ y^{(4)} &= \sigma(y^{(3)}) \\
The trainable parameters \(W_{\text{pre}}, b_{\text{pre}}, W_{\text{post}}, b_{\text{post}}\) are optimized jointly with the pretrained transformer weights using arithmetic-focused datasets, while the mask \(M\) and rule function \(\mathcal{R}\) remain fixed and deterministic. No gradients propagate through \(\mathcal{R}\).
\section{Results}
\subsection{Evaluation Overview}
\subsection{Figures and Tables}\label{FAT}
\paragraph{Positioning Figures and Tables} Place figures and tables at the top and
bottom of columns. Avoid placing them in the middle of columns. Large
figures and tables may span across both columns. Figure captions should be
below the figures; table heads should appear above the tables. Insert
figures and tables after they are cited in the text. Use the abbreviation
``Fig.~\ref{fig}'', even at the beginning of a sentence.
Each model was evaluated on four mathematical reasoning datasets: GSM8K, SVAMP, MAWPS, and AQuA-RAT. The base LLaMA 3.2 model (7B) served as the control. A fine-tuned version was trained on these datasets using standard supervised loss. A third variant introduced rule-based mutations embedded within the MLP layers during training.
Evaluation measured step-wise solution accuracy (defined as correct subexpression resolution), final answer accuracy, and accuracy as a function of problem step length. Confidence intervals were calculated at the 95\% level using bootstrapped resampling. All reported results are averaged across five random seeds, with standard deviation included.
\subsection{Accuracy Comparison Across Models}
\begin{table}[htbp]
\caption{Table Type Styles}
\caption{Final Answer Accuracy (\%) Across Benchmarks}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\begin{tabular}{|l|c|c|c|}
\hline
\textbf{Table}&\multicolumn{3}{|c|}{\textbf{Table Column Head}} \\
\cline{2-4}
\textbf{Head} & \textbf{\textit{Table column subhead}}& \textbf{\textit{Subhead}}& \textbf{\textit{Subhead}} \\
\textbf{Dataset} & \textbf{Base} & \textbf{Fine-tuned} & \textbf{Mutated} \\
\hline
copy& More table copy$^{\mathrm{a}}$& & \\
GSM8K & 58.4 $\pm$ 0.9 & 65.2 $\pm$ 0.7 & \textbf{74.8 $\pm$ 0.6} \\
SVAMP & 54.7 $\pm$ 1.1 & 61.8 $\pm$ 0.9 & \textbf{70.5 $\pm$ 0.5} \\
MAWPS & 60.1 $\pm$ 1.2 & 66.3 $\pm$ 1.0 & \textbf{75.0 $\pm$ 0.8} \\
AQuA-RAT & 48.2 $\pm$ 0.8 & 55.9 $\pm$ 1.1 & \textbf{64.4 $\pm$ 0.7} \\
\hline
\textbf{Average} & 55.4 & 62.3 & \textbf{71.2} \\
\hline
\multicolumn{4}{l}{$^{\mathrm{a}}$Sample of a Table footnote.}
\end{tabular}
\label{tab1}
\label{tab:benchmark-accuracy}
\end{center}
\end{table}
The rule-mutated model consistently outperformed both the base and fine-tuned variants. Statistical testing via two-tailed Welch's t-tests yielded $p$-values less than $10^{-7}$ across all datasets, establishing significance beyond 4$\sigma$. Cohen’s $d$ values ranged from 1.4 to 2.1, indicating large effect sizes.
\subsection{Generalization to Multi-step Reasoning}
\begin{figure}[htbp]
\centerline{\includegraphics{fig1.png}}
\caption{Example of a figure caption.}
\label{fig}
\centering
\begin{tikzpicture}
\begin{axis}[
width=0.45\textwidth,
height=6cm,
xlabel={Solution Step Count},
ylabel={Step-wise Accuracy (\%)},
xmin=1, xmax=10,
ymin=40, ymax=90,
legend style={at={(0.5,-0.2)},anchor=north,legend columns=-1},
ymajorgrids=true,
xmajorgrids=true,
grid style=dashed,
]
\addplot[
color=blue,
mark=square,
error bars/.cd,
y dir=both,y explicit,
]
coordinates {
(1,85) +- (0,1.2)
(2,80) +- (0,1.0)
(3,76) +- (0,1.1)
(4,72) +- (0,1.3)
(5,68) +- (0,1.5)
(6,63) +- (0,1.4)
(7,58) +- (0,1.6)
(8,53) +- (0,1.5)
(9,50) +- (0,1.7)
(10,48) +- (0,1.6)
};
\addlegendentry{Fine-tuned LLaMA}
\addplot[
color=red,
mark=*,
error bars/.cd,
y dir=both,y explicit,
]
coordinates {
(1,86) +- (0,0.9)
(2,83) +- (0,1.0)
(3,81) +- (0,0.8)
(4,79) +- (0,0.9)
(5,77) +- (0,1.0)
(6,75) +- (0,1.2)
(7,72) +- (0,1.1)
(8,70) +- (0,1.0)
(9,68) +- (0,1.1)
(10,66) +- (0,1.0)
};
\addlegendentry{Mutated Architecture}
\end{axis}
\end{tikzpicture}
\caption{Step-wise solution accuracy across increasing solution step counts. Rule-mutated model generalizes significantly better to long-horizon reasoning. Error bars represent 95\% confidence intervals.}
\label{fig:step-accuracy}
\end{figure}
Figure Labels: Use 8 point Times New Roman for Figure labels. Use words
rather than symbols or abbreviations when writing Figure axis labels to
avoid confusing the reader. As an example, write the quantity
``Magnetization'', or ``Magnetization, M'', not just ``M''. If including
units in the label, present them within parentheses. Do not label axes only
with units. In the example, write ``Magnetization (A/m)'' or ``Magnetization
\{A[m(1)]\}'', not just ``A/m''. Do not label axes with a ratio of
quantities and units. For example, write ``Temperature (K)'', not
``Temperature/K''.
Accuracy dropped with increasing step depth for all models. However, the mutated model exhibited substantially improved generalization beyond 6-step reasoning. For example, at 10-step problems, the mutated model retained 66\% accuracy compared to 48\% in the fine-tuned baseline.
\subsection{Summary of Statistical Measures}
Effect sizes exceeded thresholds for practical significance on all metrics:
\begin{itemize}
\item \textbf{Paired $t$-tests}: $p < 10^{-7}$ for all comparisons to baseline.
\item \textbf{Effect sizes (Cohen’s $d$)}: Ranged from 1.4 to 2.1.
\item \textbf{Standard deviation control}: Maintained within 1.2\% across all seeds.
\end{itemize}
These metrics confirm that the observed gains are statistically significant and attributable to the structural changes in the mutated architecture.
\bibliographystyle{IEEEtran}
\bibliography{references}