Abstract. 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. 1

Abstract. Following recent works connecting two-variable logic to circuits and monoids, we establish, for numerical predicate sets P satisfying a certain closure property, a one-to-one correspondence between F O[<, P]-uniform linear circuits, two-variable formulas with P predicates, and weak block products of monoids. In particular, we consider the case of linear TC 0, majority quantifiers, and finitely typed monoids. This correspondence will hold for any numerical predicate set which is F O[<]-closed and whose predicates do not depend on the input length. 1

In the late nineteen-eighties much of our research concerned the application of semigroup-theoretic methods to automata and regular languages, and the connection between computational complexity and this algebraic theory of automata. It was during this period that we became aware of the work of Wolfgang Thomas. Thomas had undertaken the study of concatenation hierarchies of star-free regular languages—a subject close to our hearts — by model-theoretic methods. He showed that the levels of the dot-depth hierarchy corresponded precisely to level of the quantifier alternation hierarchy within first-order logic[26], and applied Ehrenfeucht-Fraïssé games to prove that the dot-depth hierarchy was strict [27], a result previously obtained by semigroup-theoretic means [4, 18]. Finite model theory, a subject with which we’d had little prior acquaintance, suddenly appeared as a novel way to think about problems that we had been studying for many years. We were privileged to have been introduced to this field by so distinguished a practitioner as Wolfgang Thomas, and to have then had the opportunity to work together with him. The study of languages defined with modular quantifiers, the subject of the present survey, began with this collaboration.