An Algebraic Point of View on the Crane-Beach Conjecture

A letter e ∈ Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting membership in L. The Crane-Beach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in FO is in fact FO[<] definable and is thus a regular, star-free language. More generally, we say that a logic or a computational model has the Crane Beach property if the only languages with neutral letter that it can define/compute are regular. We develop an algebraic point of view on the Crane Beach properties using the program over monoid formalism which has proved of importance in circuit complexity. Using recent communication complexity results we establish a number of Crane Beach results for programs over specific classes of monoids. These can be viewed as Crane Beach theorems for classes of bounded-width branching programs. We also apply this to a standard extension of FO using modular-counting quantifiers and show that the boolean closure of this logic’s Σ1 fragment has the CBP.