Linearization

This library provides predicates for linearization of FASILL programs.

A term or atom is linear if it does not contain multiple occurrences of the same variable. Any term or atom that is not linear is said to be nonlinear. Given a nonlinear atom $A$, the linearization of $A$ is a process by which the structure $\langle A_l, C_l\rangle$ is computed, where: $A_l$ is a linear atom built from $A$ by replacing each one of the $n_i$ multiple occurrences of the same variable $x_i$ by new fresh variables $y_k$ $(1 \leq k \leq n_i)$; and $C_l$ is a set of proximity constrains $x_i \sim y_k$ (with $1 \leq k \leq n_i$). The operator "$s \sim t$" is asserting the similarity of two terms $s$ and $t$. Now, let $R = A \leftarrow \mathcal{B}$ be a rule and $\langle A_l, C_l\rangle$ be the linearization of $A$, where $C_l = \{x_1 \sim y_1, \dots, x_n \sim y_n\}$, $lin(R) = A_l \leftarrow x_1 \sim y_1 \wedge \dots \wedge x_n \sim y_n \wedge \mathcal{B}$. For a set $\Pi$ of rules, $lin(\Pi) = \{lin(R) : R \in \Pi\}$.

Given a FASILL program $\mathcal{P} = \langle\Pi, \mathcal{R}, L\rangle$, where $\Pi$ is a set of rules, $\mathcal{R}$ is a similarity relation, and $L$ is a complete lattice. Let $\lambda \in L$ be a cut value. The set of rules which are similar to the rules in $lin(\Pi)$, for a level $\lambda$, $\mathcal{K}_\lambda(\Pi) = \{H \leftarrow \alpha \wedge \mathcal{B}: H' \leftarrow \mathcal{B} \in lin(\Pi), \mathcal{R}(H,H') = \alpha \geq \lambda\}$ are reflecting the meaning induced by the similarity relation $\mathcal{R}$ into the set of rules $\Pi$. Note that the concept of extended program, $\mathcal{K}_\lambda(\Pi)$, is a purely instrumental notion, which does not take part in the definition of the operational semantics of the language.

For more details see:

• Julián-Iranzo, Pascual, Ginés Moreno, and Jaime Penabad. "Thresholded semantic framework for a fully integrated fuzzy logic language." Journal of logical and algebraic methods in programming 93 (2017): 42-67. https://doi.org/10.1016/j.jlamp.2017.08.002