## 2 Methods: MVL Linked List as the Data Protection Model
### Possible Attacks Aimed at MVL Product Terms and the Role of Subsuming  
As cheaters potentially can modify AGA formalisms written in the memory of the agent, then all possible illegal modifications of MVL function are grouped in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**N** & **Attack Type** & **Consequences, Scheme of Protection** \\ \hline \multirow{1}{*}{1.} & & (1) & If \(C^{*}\geq C\) (_e.g._, \(38>12\)), then \\  & & \(PT_{real}+PT_{fake}=\frac{42}{22}+X_{1}(2,4)+X_{2}(75,175)+X_{3}(31,34)+\) \\  & & \(+38*X_{1}(2,4)*X_{2}(75,175)*X_{3}(31,\ 34)\). \\ \cline{3-3}  & Adding of one new PT with & Result: Subsuming of the real PT by the fake one and incorrect logic values for real data of variables \(x_{1}\), \(x_{2}\), \(x_{3}\). \\  & Previously used real & Protection: Monitoring of the set of logic constants C for all PTs. \\  & parameters \(a,b\). & (2) & If \(C^{*}<C\) (_e.g._, \(11<12\)), then \\  & & \(PT_{real}+PT_{fake}=12*X_{1}(2,4)+X_{2}(75,175)*X_{3}(31,\ 34)+\) \\  & & \(+11+X_{1}(\frac{7}{2},4)+X_{2}(75,175)+X_{3}(31,\ 34)\). \\ \cline{3-3}  & Result: Guaranteed subsuming of the fake PT by the real one. No danger! \\ \hline \multirow{1}{*}{2.} & & \(PT_{real}+PT_{fake}=\frac{42}{22}+X_{1}(2,4)+X_{2}(75,175)+X_{3}(31,\ 34)+\) \\  & & \(+12*X_{1}(2,6)*X_{2}(75,177)*X_{3}(31,\ 36)\). \\  & Adding of one new product & Result: Subsuming of the real PT by the fake one, logic value 12 will be incorrectly assigned to the widened set of values for variables \(x_{1}\), \(x_{2}\), \(x_{3}\), but not only to real ones. Protection: \\ \cline{3-3}  & term (PT) with real constant & \(C\) and fake parameters \\ \cline{3-3}  & \(a^{*},b^{*}\). & 1. & Method of blocking logic terms [19] for \(a^{*},b^{*}\) beyond the real band. \\  & & 2. & Monitoring of a set of of real parameters \(a,b\). \\ \cline{3-3}  & Adding of one new product & Result: Simultaneous impact of NNs 1 and 2 given above. \\ \cline{3-3}  & term with fake logic & Protection: \\ \cline{3-3}  & constant \(C\) and fake values & 1. & Method of blocking logic terms for \(a^{*},b^{*}\) beyond the real band. \\  & & 2. & Monitoring of sets of real logic constants C and parameters a,b. \\ \hline \multirow{1}{*}{3.} & & \multirow{1}{*}{Deleting of the PT, i.e.,} & \multirow{1}{*}{Result: Deleting of the whole PT will cause the change in the overall number \(N_{pt}\) of PTs in the MVL function.} \\  & replacement of the logic & Protection: Monitoring of numbers: \(m\) of product terms, real logic constants C, and parameters \(a,\ b\). \\ \cline{3-3}  & constant \(C\) by \(C^{*}=0\) in PT. & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Possible attacks aimed at MVL functions. (Abbrev. PT—tags product term).  
Product terms, subsumed after possible attacks are shown crossed out in Table 2. The analysis of given schemes of attacks aimed at MVL formal expressions shows the necessity to combine at least two basic schemes for data protection. Three types of attacks (NNs 2,3,6 in the Table 2) can be protected by the earlier proposed method of blocking product terms [19], describing all possible intermediate rows of the truth table located between two given rows. However, other types of attacks 1,4,5 need methods to monitor integrity of parameters of MVL functions. But here MVL version of BC-based schemes gives the method to form the protected from modifications linked list with confirmation by distributed external and internal parameters. Advantage of AGA functions calculations is that they are based on logic primitives and always use the only possible algorithm, thus they has no specific vulnerabilities for traditional attacks [12; 18; 19].