## 5 Conclusions  
In this study, the effect of nano-MgO on the durability of normal concrete under freeze-thaw conditions is investigated. A total of 98 cubic 10 \(\times\) 10 \(\times\) 10 concrete samples are created for the compressive strength test, while 78 cylindrical concrete 10 \(\times\) 20 samples are considered for the tensile strength and permeability tests. All of the samples are subjected to 300 freeze-thaw cycles. Variables of the study are the water-to-binder ratio (0.44, 0.49, 0.62), percentage of nano-MgO (0, 0.5%, 1%, 1.5%), curing period (7, 28, 56 days), and percentage of superplasticizer (0, 0.5%, 1%). The results are summarized below.  
By comparing the results of the compressive strength for the three water-to-binder ratios, the maximum compressive strength belongs to the water-to-binder ratio of 0.44 with samples containing 1% nano-MgO that were cured for 56 days. This amount of nano-MgO can enhance the compressive strength to 63 MPa, which is 9.12% more than the control samples (57.7 MPa) on average. For the water-to-binder ratio of 0.49, the best compressive strength is gained by the sample with 1.5% nano-MgO (57.4 MPa), which is 17% more than the control samples (49 MPa). The least positive effect of nano-MgO on the compressive strength of concrete samples occurs in samples with a water-to-binder ratio of 0.62 with 0.5% nano-MgO (2.6% increase). The best results of tensile strength and permeability can be gained by the same amounts of nano-MgO. As an example, the addition of 1% nano-MgO to the mixture with a water-to-binder ratio of 0.44 can increase the tensile strength 10.6% more than the control sample and decrease the permeability by 63% in comparison to the control samples. The same trend of effect is gained from the permeability test results.  
For assessment of the results numerically and proposing a mathematical formulation for predicting the mechanical properties of concrete with nano-MgO, the GEP method is applied. Different models are examined, and the best models are chosen regarding statistical results. For the compressive and tensile strengths, GEP1 is selected as the best model in which the amount of R\({}^{2}\) is 0.98 and 0.97 for training and test data, respectively. For permeability, GEP6 is introduced as the best model, while it has an R\({}^{2}\) of 0.96 and 0.95 for the training and testing data, respectively. GEP can provide a simple and very useful formulation that can be used easily. As a result of the acceptable accuracy of the models provided by GEP, they are applicable in further designation and investigation programs, and it is useful for the prediction of the mechanical properties of the concrete with nano-MgO in cold regions.  
The authors confirm contribution to the paper as follows: Study conception and design: A.F.A.; S.T.; M.Y. Data collection: M.Y., A.F.A. Analysis and interpretation of results: A.F.A.; S.T.; M.Y.; Draft manuscript preparation: M.Y., A.F.A., S.T., A.H.G. All authors have read and agreed to the published version of the manuscript.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**GEP Model** & **R\({}^{2}\)** & \begin{tabular}{c} **Root Mean Square Error** \\ **(RMSE) (Fitness Value)** \\ \end{tabular} \\ \hline \multirow{2}{*}{GEP1 for compressive strength} & \begin{tabular}{c} Training = 0.98 \\ Testing = 0.98 \\ \end{tabular} & \begin{tabular}{c} Training = 24.16 \\ Testing = 24.28 \\ \end{tabular} \\ \hline \multirow{2}{*}{GEP1 for tensile strength} & \begin{tabular}{c} Training = 0.97 \\ Testing = 0.96 \\ \end{tabular} & \begin{tabular}{c} Training = 227.1 \\ Testing = 225.25 \\ \end{tabular} \\ \hline \multirow{2}{*}{GEP6 for permeability} & \
\begin{tabular}{c} Training = 0.96 \\ Testing = 0.95 \\ \end{tabular} &
\begin{tabular}{c} Training = 558.14 \\ Testing = 528.16 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 11: Statistical errors for best models of GEP.