# Multiple-Valued Logic Modelling for Agents Controlled via Optical Networks
## 3 Results
### MVL Function of Two Entries as a Base Model for the Logic-Linked List
In Table 4, the new entry is tagged by the number \(m\) and the time stamp \(t\). For a simple version of the ledger parameter, \(s\) can be taken as equal to 1 and Equation (5) for \(h^{(m,1)}\) will simply connect the last and the previously received entries such as with blocks of data in BC schemes [37, 38, 39, 53, 54]. According to definitions given in [29] and commented in Tables 1-3, the last row of the truth table shown in Table 4 for the ledger responds to the product term Equation (  
for AGA functions [29]. Further, the definition used is cited from [29] and was commented on in detail in [30]. **Definition 2**. : _Product term \(r_{1}\,*\,X_{1}(a_{1},b_{1})\,*\,\ldots\,*\,X_{n}(a_{n},b_{n})\)_**subsumes** another product term \(r_{2}\,*\,X_{1}(c_{1},d_{1})\,*\,\ldots\,*\,X_{n}(c_{n},d_{n})\), **if and only if** conditions (1) and (2) are true:_  
1. \(r_{1}\leq r_{2}\);
2. \(c_{i}\leq a_{i}\leq b_{i}\leq d_{i}\) for all \(X_{i}\), \(i\ =\ 1,\ldots,n\). Using this definition for Example 1, one can see that \(567\leq 830\) is true and \(c_{i}\ =\ a_{i}\ \leq\ b_{i}\ =\ d_{i}\) for all \(X_{i}\), \(i\ =\ 1,2,3\), respectively. Minimization after illegal replacement of data will provide further that the first product term will subsume the second one and consequently will be deleted:  
\(567\!\!\!\!\!\!\!\!\!\!\!\! Prohibited product term \(pt_{m-1,m}^{-}\) should contain logic constant \(k-1\), which will provide subsuming of any illegally added product terms with smaller constants written into the prohibited part of the truth table. Such a property of AGA is being actively used in its minimization method based on \"don't care states\" [29]. According to it, the idea of minimization in AGA is grounded in the extension of the initially given AGA function. One should initially add unused sets of input variables into the truth table and assign logic constants \(k-1\) to them, thus defining by hand the \"don't care states\". These new product terms will be the subsumed ones for all added fake product terms with smaller constants and equal sets of variables [29], so that one can also exploit this experience of a minimization procedure and intentionally assign constant \(k-1\) to all unused product terms, thus excluding their use by eavesdroppers. In order to obtain formal expression for prohibited product term \(pt_{m-1,m}^{-}\), one can firstly guess the appropriate logic expression, propose the algorithm for its realization, and further prove it. In the further proposed Algorithm 2, a necessary preliminary condition is that one should arrange entries in the truth table according to the sequential enlargement of time variable and entry numbers \(m\). Note that further \(x_{i}\) corresponds to time moment \(t_{m-1}\), and \(\overline{x}_{i}\) responds to \(t_{m}\). Figure 6: Formation of the part prohibited for modifications of the truth table, corresponding to the segment L-, which is complementary to the true segment L+. Proof of correctness: The prohibited product term according to its idea cannot coincide with the true one, thus its Literals parameters are natural numbers and in the formal expression should differ at least for one. Due to the definition of operator Literal [29], always for any input variable \(b_{i}\ \geq a_{i}\). As condition \(b_{i}-a_{i}<2\) is equivavent to \(b_{i}<2+a_{i}\), but always \(b_{i}\geq a_{i}\), then according to Figure 7, possible values of \(b_{i}\) are either \(a_{i}\) or \(a_{i}+1\). Non-coinciding with true ones, the prohibited segment cannot then be inserted. As condition \(b_{i}-a_{i}\geq 2\) responds to condition \(b_{i}\geq a_{i}+2\), a non-coinciding prohibited product term can include at least one possible value, and briefly this can be written as \(X_{i}(a+1,b-1)\). Based on Algorithm 2, it is more appropriate to design the joint segment of the logic function by combining both true entries and the prohibited segment \"between\" them.