First-order logic with self-reference

TL;DR

This paper introduces an extension of first-order logic with a recursion operator allowing self-reference, explores its semantics, provides a complete natural deduction system, and analyzes decision problems, highlighting its computational properties.

Contribution

It presents a novel logic with recursion, complete proof systems, and complexity results, contrasting with existing fixed-point logics and emphasizing its naturalness.

Findings

Two semantics for the logic are developed.

The validity problem for two-variable fragments is coNEXPTime-complete.

The logic offers a natural approach to recursion and self-reference.

Abstract

We consider an extension of first-order logic with a recursion operator that corresponds to allowing formulas to refer to themselves. We investigate the obtained language under two different systems of semantics, thereby obtaining two closely related but different logics. We provide a natural deduction system that is complete for validities for both of these logics, and we also investigate a range of related basic decision problems. For example, the validity problems of the two-variable fragments of the logics are shown coNexpTime-complete, which is in stark contrast with the high undecidability of two-variable logic extended with least fixed points. We also argue for the naturalness and benefits of the investigated approach to recursion and self-reference by, for example, relating the new logics to Lindstrom's Second Theorem.