In this paper we mainly study preservation theorems for two fragments of the infinitary languagesLκκ, withκregular, without the equality symbol: the universal Horn fragment and the universal strict Horn fragment. In particular, whenκisω, we obtain the corresponding theorems for the first-order case.The universal Horn fragment of first-order logic (with equality) has been extensively studied; for references see [10], [7] and [8]. But the universal Horn fragment without equality, used frequently in logic programming, has received much less attention from the model (...) theoretic point of view. At least to our knowledge, the problem of obtaining preservation results for it has not been studied before by model theorists. In spite of this, in the field of abstract algebraic logic we find a theorem which, properly translated, is a preservation result for the strict universal Horn fragment of infinitary languages without equality which, apart from function symbols, have only a unary relation symbol. This theorem is due to J. Czelakowski; see [5], Theorem 6.1, and [6], Theorem 5.1. A. Torrens [12] also has an unpublished result dealing with matrices of sequent calculi which, properly translated, is a preservation result for the strict universal Horn fragment of a first-order language. And in [2] of W. J. Blok and D. Pigozzi we find Corollary 6.3 which properly translated corresponds to our Corollary 19, but for the case of a first-order language that apart from its function symbols has only oneκ-ary relation symbol, and for strict universal Horn sentences. The study of these results is the basis for the present work. In the last part of the paper, Section 4, we will make these connections clear and obtain some of these results from our theorems. In this way we hope to make clear two things: (1) The field of abstract algebraic logic can be seen, in part, as a disguised study of universal Horn logic without equality and so has an added interest. (2) A general study of universal Horn logic without equality from a model theoretic point of view can be of help in the field of abstract algebraic logic. (shrink)

This paper presents a logical Style painting classifier based on evaluated Horn clauses, qualitative colour descriptors and Explanations. Three versions of $\ell $-SHE are defined, using rational Pavelka logic, and expansions of Gödel logic and product logic with rational constants: RPL, $G$ and $\sqcap $, 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%, 65.6% $) and 68.9% $). Regarding the Paintings-91-PIB dataset, the percentages of accuracy obtained for each $\ell $-SHE version are 60.2%, 48.2% $) and 57.0% $). Our logic definition for the Baroque style has obtained the highest accuracy in both datasets, for all the $\ell $-SHE versions. 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. (shrink)

We consider separably closed fields of characteristic $p > 0$ and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the $p$-component functions.

Our work is a contribution to the model theory of fuzzy predicate logics. In this paper we characterize elementary equivalence between models of fuzzy predicate logic using elementary mappings. Refining the method of diagrams we give a solution to an open problem of Hájek and Cintula (J Symb Log 71(3):863–880, 2006, Conjectures 1 and 2). We investigate also the properties of elementary extensions in witnessed and quasi-witnessed theories, generalizing some results of Section 7 of Hájek and Cintula (J Symb Log (...) 71(3):863–880, 2006) and of Section 4 of Cerami and Esteva (Arch Math Log 50(5/6):625–641, 2011) to non-exhaustive models. (shrink)

In Dellunde et al. 997–1015), we determined the complete theory Te of modules of separably closed fields of characteristic p and imperfection degree e, eω{∞}. Here, for 0≠eω, we describe the closed set of the Ziegler spectrum corresponding to Te. Further, we establish a correspondence between certain submodules and n-types and we investigate several notions of dimensions and their relationships with the Lascar rank. Finally, we show that Te has uniform p.p. elimination of imaginaries and deduce uniform weak elimination of (...) imaginaries. (shrink)

This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.

Este artículo aborda el concepto de discriminación algorítmica desde una perspectiva conjunta de la filosofía y la ciencia de la computación, con el propósito de establecer un marco de discusión común para avanzar en el despliegue de las inteligencias artificiales en las sociedades democráticas. Se presenta una definición no normativa de discriminación y se analiza y contextualiza el concepto de algoritmo usando un enfoque intencional, enmarcándolo en el proceso de toma de decisiones e identificando las fuentes de discriminación, así como (...) los conceptos detrás de su cuantificación para terminar exponiendo algunos límites y desafíos. (shrink)

We prove the following preservation theorem for the Horn fragment of Equality-free Logic:Theorem 0.1. For any sentence σ ϵ L, the following are equivalent:i ) σ is preserved under Hs, Hs -1 and PR.i i ) σ is logically equivalent to an equality-free Horn sentence.