Languages Deened with Modular Counting Quantiiers

We prove that a regular language deened by a boolean combination of generalized 1-sentences built using modular counting quan-tiiers can be deened by a boolean combination of 1-sentences in which only regular numerical predicates appear. The same statement, with \1" replaced by \\rst-order" is equivalent to the conjecture that the non-uniform circuit complexity class ACC is strictly contained in NC 1 : The argument introduces some new techniques, based on a combination of semigroup theory and Ramsey theory, which may shed some light on the general case. 1 Background 1.1 Lower bounds questions for small-depth circuit families This paper was motivated by some open problems about the computational power of families of boolean circuits. As it turns out, we will not mention circuits at all after this introductory section. Nonetheless, our main result represents a positive contribution toward the resolution of these problems. For the moment, we deene a circuit with n inputs to be a directed acyclic graph with 2n source nodes (labeled 0; 1; x 1 ; x 1 ; : : : ; x n ; x n) and a single sink node, with all the nodes other than the sources labeled AND or OR. The size of the circuit is the number of nodes, and the depth of the circuit is the length of the longest path from a source to the sink. A circuit C computes a function f C : f0; 1g n ! f0; 1g as follows. Given a 1 a n 2 f0; 1g n ; we assign the value a i to the node x i ; and 1 ? a i to the node x i : Each node other than a source node is assigned the conjunction or disjunction of the values of its predecessor nodes, depending on whether the node is labeled AND or OR: Since there are no cycles in the graph, this procedure assigns a well-deened boolean value to every node in the circuit; f C (a 1 a n) is the value assigned to the sink node. Ordinarily we talk about the behavior of families of circuits, consisting of one circuit for each input length n: A family fC n g n0 of circuits recognizes the language fw 2 f0; 1g : f C jwj (w) = 1g: