The logical style painting classifier based on Horn clauses and explanations (ℓ-SHE)

This paper presents a logical Style painting classifier based on evaluated Horn clauses, qualitative colour descriptors and Explanations ($\ell $-SHE). Three versions of $\ell $-SHE are defined, using rational Pavelka logic (RPL), and expansions of Gödel logic and product logic with rational constants: RPL, $G(\mathbb{Q})$ and $\sqcap (\mathbb{Q})$, respectively. We introduce a fuzzy representation of the more representative colour traits for the Baroque, the Impressionism and the Post-Impressionism art styles. The $\ell $-SHE algorithm has been implemented in Swi-Prolog and tested on 90 paintings of the QArt-Dataset and on 247 paintings of the Paintings-91-PIB dataset. The percentages of accuracy obtained in the QArt-Dataset for each $\ell $-SHE version are 73.3% (RPL), 65.6% ($G(\mathbb{Q})$) and 68.9% ($\sqcap (\mathbb{Q})$). Regarding the Paintings-91-PIB dataset, the percentages of accuracy obtained for each $\ell $-SHE version are 60.2% (RPL), 48.2% ($G(\mathbb{Q})$) and 57.0% ( $\sqcap (\mathbb{Q})$). Our logic definition for the Baroque style has obtained the highest accuracy in both datasets, for all the $\ell $-SHE versions (the lowest Baroque case gets 85.6$\%$ of accuracy). An important feature of the classifier is that it provides reasons regarding why a painting belongs to a certain style. The classifier also provides reasons about why outliers of one art style may belong to another art style, giving a second classification option depending on its membership degrees to these styles.

Lyndon Interpolation holds for the Prenex ⊃ Prenex Fragment of Gödel Logic

First-order interpolation properties are notoriously hard to determine, even for logics where propositional interpolation is more or less obvious. One of the most prominent examples is first-order G ̈odel logic. Lyndon interpolation is a strengthening of the interpolation property in the sense that propositional variables or predicate symbols are only allowed to occur positively (negatively) in the interpolant if they occur positively (negatively) on both sides of the implication. Note that Lyndon interpolation is difficult to establish for first-order logics as most proof-theoretic methods fail. In this paper we provide general derivability conditions for a first-order logic to admit Lyndon interpolation for the prenex ⊃ prenex fragment and apply the arguments to the prenex ⊃ prenex fragment of first-order Go ̈del logic.