Definition of the logic replicant

A logic replicant is a tuple \(\langle L, V\rangle\) able to replicate the hidden classification logic of a problem; it receives \(F\in\mathbb{N}^+\) input signals \(x_1\) through \(x_F\) where:

\(L\colon X\to Q\) is a leietanic transformation function that maps every input case \(\boldsymbol{x}\) into a point \(\boldsymbol{q}\) in the \(D\)-dimensional space \(Q\), where

where \(\boldsymbol{\Lambda}_a\) and the other \(\boldsymbol{\Lambda}_s\colon s\in[1,S]\) are \(D\times F\) matrices that multiply the vector instance \(\boldsymbol{x}\). The vector collections \(\boldsymbol{\beta}_{sr}\) and \(\boldsymbol{\xi}_{sr}\) have the same dimension \(D\). The operator \(\circ\) is the Hadamard product, defined for matrices and vectors as entrywise multiplication.

\(V\) is the set of vortices and their respective scalar fields \(U_{ck}\in V\), \(U_{ck}\colon Q\to\mathbb{R}\), \(c\in C\), \(k\in K\subset\mathbb{N}^+\wedge C\cdot K=|V|\), that collectively decide what class to assign to the output of the leietanic function. Every \(c,k\)-th vortex (defined by its scalar field \(U_{ck}\)) is assigned to only one specific output class \(c\), where

is based on three parameters: its centre \(\boldsymbol{v}_{ck}\in\mathbb{R}^D\), its density \(\rho_{ck}\in\mathbb{R}\) and its intensity \(\Phi_{ck}\in\mathbb{R}\).