The Crane Beach Conjecture

A language L over an alphabet A is said to have a neutral letter if there is a letter e/spl isin/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 /spl Nscr/ of numerical predicates. We investigate this conjecture in detail, showing that it fails already for /spl Nscr/={+, *}, or possibly stronger for any set /spl Nscr/ that allows counting up to the m times iterated logarithm, 1g/sup (m)/, for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for /spl Nscr/={+}, for the fragment BC(/spl Sigma/) of first-order logic, and for binary alphabets.