# 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 order to prevent illegal modifications of entries written in some format unified by participants, it is proposed to form the logic ledger function \(F_{ldg}\) by product terms, containing pairs of entries obtained at different time moments \(t_{m}\) and \(t_{m-s}\) (see Figure 5). In such a ledger scheme, the function formation responds to the generation of the AGA truth table, where fields of entries may have repeated values if they respond to the same agents, tasks, documents, permissions, plans, and used resources. In order to split such data into separate rows of the MVL ledger function, every entry receives time stamp parameter \(t\) and the number of entry \(m\) in the ledger. External network nodes are to assign random hash values \(h\) in order to exclude the prediction of the ledger function and to simplify its verification, as the uniform distribution of random numbers is appropriate for revealing possible modifications. The last entry obtained at time moment \(t_{m}\) is written as \(\mathbf{e}_{m}~{}=~{}\left\{e_{1,m}~{},\ldots,e_{p,m}\right\}\), where \(m\) is the number of the entry in the ledger and \(p\) is the chosen number of fields in the entry. An earlier obtained entry \(\mathbf{e}_{m-s}~{}=~{}\left\{e_{1,m-s}~{},\ldots,e_{p,m-s}\right\}\) corresponds to the time moment \(t_{m-s}\). In order to confirm authenticity for two entries, the ledger should contain the data of both of them and sets of random keys unique for each of them. In Figure 5 this responds to the shift to right of the position of entry \(\mathbf{e}\) when a new entry arrives. **Definition 1**. : _Logic ledger function or a linked list of logic entries is given within AGA as a hash function_  
\[h^{(m,s)}~{}=~{}F_{ldg}(m,s,~{}t,e_{1,m}~{},\ldots,e_{p,m},~{}e_{1,m-s}~{}, \ldots,e_{p,m-s},~{}~{}h_{1,m},\ldots,h_{Q,m},~{}~{}h_{1,m-s}~{},\ldots,h_{Q,m -s}) \tag{5}\]  
In Equation (5), \(m\) is the number of the last entry in the ledger, \(s\) is the shift of the number for the previous entry, \(t\) denotes the assigned time stamp for the last entry, \(e_{1,m}~{},\ldots,e_{p,m}\) are the parameters of the last entry \(\mathbf{e}_{m}\); \(e_{1,m-s}~{},\ldots,e_{p,m-s}\) correspond to parameters of the earlier received entry \(\mathbf{e}_{m-s}\), \(h_{1,m},\ldots,h_{Q,m}\) are hash values assigned by network verifiers to the last entry \(\mathbf{e}_{m}\), \(h_{1,m-s}~{},\ldots,h_{Q,m-s}\) are hash values of the earlier received entry \(\mathbf{e}_{m-s}\), and \(q\) is the number of a verifier, \(q=1_{m}\), \(Q\). Non-indexed parameters \(m,~{}s,~{}q\) and indexed parameters \(t,~{}e,~{}h\) are natural numbers. For brevity, Equation (5) can be rewritten briefly as:  
\[h^{(m,s)}~{}=~{}F_{ldg}(m,t,\mathbf{e}_{m}~{},~{}\mathbf{e}_{m-s}~{},\mathbf{h}_{m},\mathbf{ h}_{m-s}) \tag{6}\]  
The specifics of data backup for autonomous robotic agents are that the verification of data should include the possibility of its use both by a distant human expert and by a software bot. That is why a logic-linked list should not be accepted only on the procedure of rigid binding of a block of data to the pointer of the next block, as in BC schemes [37; 38; 39], which can be realized by the special field-to-code data pointer just in the entry. Certainly, the assigning of data pointers should be adapted to a specific software platform. Figure 5: Principle of formation of the ledger function with two linked entries 1 and 2 obtained at different time moments. Thus, using the same set of basic rules disclosed by Tables 1, 3 and 4, Equation (5) for the logic ledger can be written according to its truth table. As it was mentioned in the previous section, by default storage functions \(F_{ste}\) and \(F_{sth}\) should have the first row of the ledger filled in by a randomly chosen set of data. The output variable column should contain random hash values for all rows in order to exclude any preliminary prediction of the ledger function.