A finite-state machine
A logic replicant can replicate any deterministic finite-state machine, and therefore any combinational logic.
In plain words
A finite-state machine is any rule you can write down as a lookup table: given the current state and the input symbol, the table says what the next state and the output are. Traffic lights, vending machines and every digital circuit can be described this way.
The logic replicant can imitate any such table. Number the input symbols and the states, lay them out as a grid, and put one vortex on every grid cell, each labelled with the answer from the table. At its own cell a vortex's influence is exactly 1, while every other vortex is at least one step away and contributes at most one half, so the right answer always wins by a clear margin. Because these machines can express any combinational logic, so can the replicant, at least in theory. In practice most real problems have continuous inputs, which is why the paper validates the model on those too.
Deterministic finite-state machines
A deterministic finite-state machine is a quintuple \(\langle I,O,S,\delta,\lambda\rangle\) where \(I\) is the set of input symbols, \(O\) the set of output symbols, \(S\) the set of states, \(\delta\colon S\times I\mapsto S\) the transition function and \(\lambda\colon S\times I\mapsto O\) the output function.
Intuitively, and oversimplifying, being a complete DFSM means the logic replicant can learn any problem whose inputs are a finite number of discrete variables and whose output is a finite number of classes.
Construction
Since the replicant accepts any finite number of real variables, each input symbol \(\iota\in I\) can be encoded as an integer in a feature \(x_I\in[1,\dots,|I|]\), and the current state as an integer feature \(x_S\in[1,\dots,|S|]\). The two functions \(\delta\) and \(\lambda\) are joined into a single transition-output function \(\omega\colon S\times I\mapsto S\times O\), and every pair \((s,o)\in S\times O\) becomes one output class of the replicant, so there are \(|S|\times|O|\) classes. For a two-dimensional \(Q\), take the identity transformation:
Because both features are discrete, the possible values of \(L\) form an \(|S|\times|I|\) grid of integer points. Place one vortex on each grid point, with centre \(\boldsymbol{v}_i=(u_{iI},u_{iS})\), intensity \(\Phi_i=1\) and density \(\rho_i=1\). At its own grid point a vortex's field equals exactly 1, while every other vortex is at distance at least 1 and so contributes at most \(1/2\). The greatest field is therefore always the vortex sitting on the input, which is assigned to the class \((s,o)=\omega(s,\iota)\). Instead of two units for the two logics \(\delta\) and \(\lambda\), the replicant learns a single logic \(\omega\).
In this construction \(L\) does not even need to be leietanic, although in general that property is extremely convenient. The argument extends to any number of dimensions, including a one-dimensional \(L\), without loss of rigour.
Implications
DFSMs can reproduce any combinational logic, which is one of the kinds of logic the replicant is meant to replicate. In theory, then, a logic replicant can learn any combinational logic given a sufficient number of parameters. Real-world problems often have continuous rather than discrete features; the experiments validate that the same model handles those too.