Modulo quantifiers over functional vocabularies extending addition

TL;DR

This paper investigates the properties of first-order logic with modulo quantifiers over functional vocabularies extending addition, establishing the Crane beach property under certain conditions and exploring implications for circuit complexity classes.

Contribution

It proves the Crane beach property for FO and FOMOD over vocabularies extending addition with a normal form, and connects logical definability to circuit complexity separations.

Findings

FOMOD over addition vocabularies satisfies the Crane beach property under positional normal form.

Certain regular languages are not definable in FOMOD[<,+,2^x], indicating limitations in logical expressiveness.

The results relate logical definability to the separation of circuit complexity classes ACC0 and NC1.

Abstract

We show that first order logic (FO) and first order logic extended with modulo counting quantifiers (FOMOD) over purely functional vocabularies which extend addition, satisfy the Crane beach property (CBP) if the logic satisfies a normal form (called positional normal form). This not only shows why logics over the addition vocabulary have the CBP but also gives new CBP results, for example for the vocabulary which extends addition with the exponentiation function. The above results can also be viewed from the perspective of circuit complexity. Showing the existence of regular languages not definable in FOMOD[<, +, *] is equivalent to the separation of the circuit complexity classes ACC0 and NC1 . Our theorem shows that a weaker logic , namely, FOMOD[<,+,2^x] cannot define all regular languages.