## 6 Implementation and Experimental Results
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{3}{*}{\(\delta\)**(sec)**} & \multirow{3}{*}{**Trans.**} & \multicolumn{4}{c}{**Concrete Blockchain**} & \multirow{3}{*}{**Abstract Blockchain**} & \multirow{3}{*}{**\% of**} \\  & & & \multicolumn{2}{c}{**(\(N_{1}+N_{2}+N_{3}\))**} & & & & **Data Size** & \\ \cline{3-10}  & & & **Data Size** & & **Unit** & **(bytes)** & **Blocks** & **Gas** & **Ratio** \\ \hline \multirow{3}{*}{2} & 0.8 & 1,793,118 & 752 & 290,018,052 & 423,444 & 415 & 64,383,622 & 76.39 \\  & 1 & 2,222,252 & 802 & 376,017,792 & 455,207 & 450 & 68,488,727 & 79.52 \\  & 1.25 & 2,908,553 & 1,122 & 637,439,592 & 642,433 & 603 & 85,578,176 & 77.91 \\ \hline \multirow{3}{*}{3} & 0.8 & 1,758,420 & 718 & 286,315,196 & 371,572 & 364 & 57,597,793 & 78.87 \\  & 1 & 2,238,783 & 837 & 375,500,652 & 446,487 & 442 & 65,240,401 & 80.06 \\  & 1.25 & 2,922,238 & 1,117 & 611,331,736 & 570,512 & 571 & 78,130,294 & 80.48 \\ \hline \multirow{3}{*}{5} & 0.8 & 1,761,710 & 702 & 285,278,812 & 324,078 & 320 & 50,817,916 & 81.6 \\  & 1 & 2,271,361 & 941 & 381,924,264 & 433,224 & 458 & 60,367,976 & 80.93 \\  & 1.25 & 2,856,380 & 1,051 & 533,500,956 & 454,180 & 479 & 62,558,909 & 84.1 \\ \hline \multirow{3}{*}{6} & 0.8 & 1,788,095 & 758 & 286,096,610 & 343,825 & 345 & 51,486,689 & 80.77 \\  & 1 & 2,284,781 & 918 & 386,398,262 & 396,471 & 406 & 54,157,749 & 82.65 \\ \cline{1-1}  & 1.25 & 2,822,752 & 984 & 518,845,796 & 400,365 & 430 & 54,536,705 & 85.82 \\ \hline \multirow{3}{*}{7} & 0.8 & 1,783,445 & 731 & 286,144,944 & 306,870 & 307 & 46,327,839 & 82.79 \\ \cline{1-1}  & 1 & 2,245,741 & 840 & 397,938,268 & 355,289 & 384 & 48,614,951 & 84.18 \\ \cline{1-1}  & 1.25 & 2,790,292 & 934 & 509,479,548 & 356,596 & 378 & 49,091,622 & 87.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Concrete vs. Abstract Blockchain. A sample instance occurred during our experiment is shown in Table 3 which describes how the leader performs the abstraction of local blockchain transactions. The first column of every row represents the list of transactions mined into local blockchain for \(\delta\) = 5 s. The second column represents its equivalent abstract form in the form of interval which is generated by the leader in network \(N_{1}\). Observe that given a list of transactions {\"32134\": 138, \"32135\": 636, \"32136\": 1365\": 1365] and its corresponding abstract form [\"32134\", \"32136\"]: [138, 1365], both the operations Search(636, [138, 1365]) and Search(640, [138, 1365]) result into \"_Maybe_\" according to its definition in Section 5.4. Let us now illustrate the experimental results of the Search operations performed both in the concrete and the abstract blockchains, depicted in Table 4. In the experiment, we have randomly chosen a test dataset containing 500 transaction-records and we have categorized it into two halves: (1) data that exists in the concrete blockchain, and (2) data that does not exist in the concrete blockchain (shown in columns 3, 4, and 5). The result of Search operation on a test dataset by varying \(\delta\) and transaction-rate is shown in columns 6 and 7. The values of \"Exact Discovery Rate (EDR)\" are computed from the transactions representing cases 1 and 2 of Table 1, whereas the values of \"Partial Discovery Rate (PDR)\" are computed from the transactions representing case 3 of Table 1. Observe that, as \(\delta\) and transaction-rate increase, the percentage of EDR decreases due to increasing imprecision in the abstract records. Although the current result is showing high PDR, this can easily be reduced by tuning our abstractions and by reducing wait-time \(\delta\). The use of the Abstract Interpretation framework enables one to use different levels of abstractions, from coarse-grained to fine-grained, according to the need. For example, as per the abstract lattice of interval domain, an integer \(x\) can be abstracted by \([x,x]\) or it can be abstracted by \([\infty,\infty]\).