Non-definability of languages by generalized first-order formulas over (N,+)

TL;DR

This paper proves that for languages with a neutral letter, adding the addition predicate does not increase definability beyond order alone in certain logical frameworks, impacting circuit class separations.

Contribution

It demonstrates that over words with a neutral letter, monoidal quantifier logic with addition collapses to order-only logic, confirming the Crane Beach conjecture in this context.

Findings

Languages with a neutral letter definable with addition are also definable with order only.

FO+MOD[<,+] collapses to FO+MOD[<].

MOD[<,+] collapses to MOD[<] for cyclic groups.

Abstract

We consider first-order logic with monoidal quantifiers over words. We show that all languages with a neutral letter, definable using the addition numerical predicate are also definable with the order predicate as the only numerical predicate. Let S be a subset of monoids. Let LS be the logic closed under quantification over the monoids in S and N be the class of neutral letter languages. Then we show that: LS[<,+] cap N = LS[<] Our result can be interpreted as the Crane Beach conjecture to hold for the logic LS[<,+]. As a corollary of our result we get the result of Roy and Straubing that FO+MOD[<,+] collapses to FO+MOD[<]. For cyclic groups, we answer an open question of Roy and Straubing, proving that MOD[<,+] collapses to MOD[<]. Our result also shows that multiplication is necessary for Barrington's theorem to hold. All these results can be viewed as separation results for very uniform circuit classes. For example we separate FO[<,+]-uniform CC0 from FO[<,+]-uniform ACC0.