Regular Representations of Uniform TC^0

The circuit complexity class DLOGTIME-uniform AC⁰ is known to be a modest subclass of DLOGTIME-uniform TC^0. The weakness of AC⁰ is caused by the fact that AC⁰ 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 AC⁰ do not have the relativization property and hence they are not regular. This weakness of DLOGTIME-uniform AC⁰ 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 AC⁰ and TC^0. First we formulate a combinatorial criterion for a cardinality quantifier CS implying that all languages in DLOGTIME-uniform TC⁰ can be defined in FO(CS). For instance, this criterion is satisfied by CS 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(FOB) 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.