The circuit complexity class DLOGTIME-uniform AC^0 is known to be a modest subclass of DLOGTIME-uniform TC^0. The weakness of AC^0 is caused by the fact that AC^0 is not closed under restricting AC^0-computable queries into simple subsequences of the input. Analogously, in descriptive complexity, the logics corresponding to DLOGTIME-uniform AC^0 do not have the relativization property and hence they are not regular. This weakness of DLOGTIME-uniform AC^0 has been elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington, Immerman, Lautemann, Schweikardt and Th{\'e}rien) was that if a language L has a neutral letter, then L can be defined in first-order logic with the collection of all numerical built-in relations, if and only if L can be already defined in FO with order. In the first part of this article we consider logics in the range of AC^0 and TC^0. First we formulate a combinatorial criterion for a cardinality quantifier C_S implying that all languages in DLOGTIME-uniform TC^0 can be defined in FO(C_S). For instance, this criterion is satisfied by C_S if S is the range of some polynomial with positive integer coefficients of degree at least two. In the second part of the paper we first adapt the key properties of abstract logics to accommodate built-in relations. Then we define the regular interior R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane Beach Conjecture can be interpreted as a statement concerning the regular interior of first-order logic with built-in relations B. We show that if B={+}, or B contains only unary relations besides the order, then R-int(FO_B) collapses to FO with order. In contrast, our results imply that if B contains the order and the range of a polynomial of degree at least two, then R-cl(FO_B) includes all languages in DLOGTIME-uniform TC^0.

翻译：电路复杂性类DLOGTIME一致AC⁰被认为是DLOGTIME一致TC⁰的一个适度子类。AC⁰的弱点源于其无法在限制AC⁰可计算查询至输入的简单子序列时保持封闭性。类似地，在描述复杂性理论中，与DLOGTIME一致AC⁰对应的逻辑不具有相对化性质，因此不具有正则性。DLOGTIME一致AC⁰的这一弱点已在"Crane Beach猜想"的研究脉络中得到深入阐述。该猜想（后被Barrington、Immerman、Lautemann、Schweikardt和Thérien证否）指出：若语言L具有中性字母，则当且仅当L可在带序的一阶逻辑中定义时，L才能在所有数值内置关系的一阶逻辑中定义。本文第一部分研究AC⁰与TC⁰范畴内的逻辑。首先，我们为基数量词C_S提出一个组合判别准则，该准则可确保所有DLOGTIME一致TC⁰中的语言均可在带C_S的一阶逻辑中定义。例如，当S是某个次数≥2、系数为正整数的多项式的值域时，C_S满足该准则。第二部分中，我们首先调整抽象逻辑的关键性质以适应内置关系。随后定义逻辑L的正则内部R-int(L)与正则闭包R-cl(L)，并证明Crane Beach猜想可被解释为关于带内置关系B的一阶逻辑的正则内部的断言。我们证明：若B={+}，或B仅包含序关系及一元关系，则R-int(FO_B)退化为带序的一阶逻辑。相反地，我们的结果表明：若B包含序关系及某个次数≥2的多项式的值域，则R-cl(FO_B)包含所有DLOGTIME一致TC⁰中的语言。