On the Remarkable Features of Binding Forms

TL;DR

This paper introduces a new classification of logical sentences based on binding forms, revealing a hierarchy of first-order fragments with decidability properties, and provides insights into the robustness of modal logic and its extensions.

Contribution

It proposes a novel classification framework based on binding forms, establishing a hierarchy of fragments and analyzing their decidability and model-theoretic properties.

Findings

Established a hierarchy of first-order fragments based on binding forms.

Proved the less expressive fragment has the finite-model property.

Showed the satisfiability problem for the fragment is in PSpace.

Abstract

Hilbert's Entscheidungsproblem has given rise to a broad and productive line of research in mathematical logic, where the classification process of decidable classes of first-order sentences represent only one of the remarkable results. According to the criteria used to identify the particular classes of interest, this process was declined into several research programs, of which some of the most deeply investigated are the ones classifying sentences in prenex normal form in base of their prefix vocabulary. Unfortunately, almost all of these approaches did not shed any light on the reasons why modal logic is so robustly decidable. Trying to answer to this question, Andreka, van Benthem, and Nemeti introduced the guarded fragment of first-order logic, which generalizes the modal framework by essentially retaining all its fundamental properties. They started, so, a completely new research program based on the way quantifications can be relativized. Although this approach succeeded in its original task, we cannot consider it satisfactory in spotting the reasons why some complex extensions of modal logic are well-behaved. In particular, by just using the related results, we are not able to derive the decidability of multi-agent logics for strategic abilities. In this paper, aiming to lay the foundation for a more thorough understanding of some of these decidability questions, we introduce a new kind of classification based on the binding forms that are admitted in a sentence, i.e., on the way the arguments of a relation can be bound to a variable. We describe a hierarchy of first-order fragments based on the Boolean combinations of these forms, showing that the less expressive one is already incomparable with the guarded logic and related extensions. We also prove, via a new model-theoretic technique, that it enjoys the finite-model property and a PSpace satisfiability problem.