Finite Model Reasoning in Expressive Fragments of First-Order Logic

Abstract

Over the past two decades several fragments of first-order logic have been identified and shown to have good computational and algorithmic properties, to a great extent as a result of appropriately describing the image of the standard translation of modal logic to first-order logic. This applies most notably to the guarded fragment, where quantifiers are appropriately relativized by atoms, and the fragment defined by restricting the number of variables to two. The aim of this talk is to review recent work concerning these fragments and their popular extensions. When presenting the material special attention is given to decision procedures for the finite satisfiability problems, as many of the fragments discussed contain infinity axioms. We highlight most effective techniques used in this context, their advantages and limitations. We also mention a few open directions of study.Comment: In Proceedings M4M9 2017, arXiv:1703.0173

An Extension of Trakhtenbrot's Theorem

“…All of the decidable fragments of FO that have been discovered so far are much weaker than FO with respect to expressive power. For instance, already ∀x∀y∀z(R(x, y, z) ∨ R(z, y, x)) is a sentence which is easily seen to be not, for example, expressible in two-variable logic FO 2 , unary negation fragment UNF or fluted logic FL (for definitions of these fragments and for an introduction to the topic of fragments of FO, see [3]). This motivates the following question: can there be a decidable fragment of FO which has the same expressive power as FO?…”