The Crane Beach Conjecture

A language L over an alphabet A is said to have a neutral letter if there is a letter e∈A such that inserting or deleting e´s from any word in A* does not change its membership (or non-membership) in L. The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order then it is not definable in first-order. Logic with any set 𝒩 of numerical predicates. We investigate this conjecture in detail, showing that it fails already for 𝒩={+, *}, or possibly stronger for any set 𝒩 that allows counting up to the m times iterated logarithm, 1g^(m), for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for 𝒩={+}, for the fragment BC(Σ) of first-order logic, and for binary alphabets