4 research outputs found
Back-and-forth systems for fuzzy first-order models
This paper continues the study of model theory for fuzzy logics by addressing the fundamental issue of classifying models according to their first-order theory. Three different definitions of elementary equivalence for fuzzy first-order models are introduced and separated by suitable counterexamples. We propose several back-and-forth conditions, based both on classical two-sorted structures and on non-classical structures, that are useful to obtain elementary equivalence in particular cases as we illustrate with several example
Syntactic characterizations of classes of first-order structures in mathematical fuzzy logic
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 Los--Tarski and the Chang--Los--Suszko preservation theorems
follow
Homomorphism Preservation Theorems for Many-Valued Structures
A canonical result in model theory is the homomorphism preservation theorem
which states that a first-order formula is preserved under homomorphisms on all
structures if and only if it is equivalent to an existential-positive formula,
standardly proved compactness. Rossman (2008) established that the h.p.t.
remains valid when restricted to finite structures. This is a significant
result in the field of finite model theory, standing in contrast to other
preservation theorems and as an theorem which remains true in the finite but
whose proof uses entirely different methods. It also has importance to the
field of constraint satisfaction due to the equivalence of existential-positive
formulas and unions of conjunctive queries. Adjacently, Dellunde and Vidal
(2019) established that a version of the h.p.t. holds for a collection of
first-order many-valued logics, those whose (possibly infinite) structures are
defined over a fixed finite MTL-chain.
In this paper we unite these two strands, showing how one can extend
Rossman's proof of a finite h.p.t. to a very wide collection of many-valued
predicate logics and simultaneously establishing a finite variant to Dellunde
and Vidal's result, one which not only applies to structures defined over
algebras more general than MTL-chains but also where we allow for those algebra
to vary between models. This investigation provides a starting point in a wider
development of finite model theory for many-valued logics and, just as the
classical finite h.p.t. has implications for constraint satisfaction, the
many-valued finite h.p.t. has implications for valued constraint satisfaction
problems.Comment: 22 page
Back-and-forth systems for fuzzy first-order models
This paper continues the study of model theory for fuzzy logics by addressing the fundamental issue of classifying models according to their first-order theory. Three different definitions of elementary equivalence for fuzzy first-order models are introduced and separated by suitable counterexamples. We propose several back-and-forth conditions, based both on classical two-sorted structures and on non-classical structures, that are useful to obtain elementary equivalence in particular cases as we illustrate with several example