Theoretical foundations of JLNN

JLNN combines classical propositional logic with deep learning using three pillars:

1. Interval truth

Unlike standard fuzzy systems, where the truth value is \(a \in [0, 1]\), JLNN works with the interval:

\[I_a = [L_a, U_a], \quad 0 \leq L_a \leq U_a \leq 1\]

  • Contradiction: If \(L_a > U_a\) occurs during learning, the system detects a logical contradiction.

  • Uncertainty: The width of the interval \(U_a - L_a\) represents the degree of ignorance (epistemic uncertainty).

2. Weighted Łukasiewicz’s logic

The operators (AND, OR, …, IMPLY) are implemented as differentiable functions. For an AND gate with weights \(w_i\) and threshold \(\beta\):

\[L_{out} = \max\left(0, 1 - \sum w_i (1 - L_i) / \beta\right)\]

Weights \(w \geq 1\) allow the model to learn the relevance of individual antecedents.

3. Mapping to graphs

Thanks to the integration with NetworkX, each logical formula is represented as a directed acyclic graph (DAG).

  • Leaves: PredicateNodes (inputs).

  • Internal nodes: Logical Gates (operations).

  • Edges: Flow of truth intervals.

This structure allows you to export the model to .dot format for visualization or to execute graph algorithms directly on top of the logical rule structure.