Lindstrom theorems for fragments of first-order logic

TL;DR

This paper establishes Lindström theorems for various fragments of first-order logic, including k-variable fragments and modal logics, using modified proof techniques and semantic preservation arguments.

Contribution

It provides the first Lindström theorems for several important fragments of first-order logic, expanding the scope of model-theoretic characterizations beyond full first-order logic.

Findings

Lindström theorems for k-variable fragments with k>2

Lindström theorems for Tarski's relation algebra and graded modal logic

Semantic preservation theorems derived from the results

Abstract

Lindstr\"om theorems characterize logics in terms of model-theoretic conditions such as Compactness and the L\"owenheim-Skolem property. Most existing characterizations of this kind concern extensions of first-order logic. But on the other hand, many logics relevant to computer science are fragments or extensions of fragments of first-order logic, e.g., k-variable logics and various modal logics. Finding Lindstr\"om theorems for these languages can be challenging, as most known techniques rely on coding arguments that seem to require the full expressive power of first-order logic. In this paper, we provide Lindstr\"om theorems for several fragments of first-order logic, including the k-variable fragments for k>2, Tarski's relation algebra, graded modal logic, and the binary guarded fragment. We use two different proof techniques. One is a modification of the original Lindstr\"om proof. The other involves the modal concepts of bisimulation, tree unraveling, and finite depth. Our results also imply semantic preservation theorems.