We examine visibly counter languages, which are languages recognized by visibly counter automata (a.k.a. input driven counter automata). We are able to effectively characterize the visibly counter languages in AC^0 and show that they are contained in FO[+].

We show that every language accepted by a nondeterministic auxiliary pushdown automaton in polynomial time (that is, every language in SAC1= Log(CFL)) can be accepted by a symmetric auxiliary pushdown automaton in polynomial time.

Uniformity notions more restrictive than the usual FO[<, +, *]-uniformity = FO[<, Bit]-uniformity are introduced. It is shown that the general framework exhibited by Barrington et al. still holds if the fan-in of the gates in the corresponding circuits is considered.

The class of languages definable by majority quantifiers using the order predicate is investigated. It is shown that the additional use of first order or counting quantifiers does not increase this class. Further on, addition is in this connection a definable numerical predicate, while the converse does not hold. The emptiness problem for this class turns out to be undecidable.

The Cayley group membership problem (CGM) is to input a groupoid (binary algebra) G given as a multiplication table, a subset X of G, and an element t of G, and to determine whether t can be expressed as a product of elements of X. For general groupoids CGM is P-complete, and for associative algebras (semigroups) it is NL-complete. Here we investigate CGM for particular classes of groups. The problem for general groups is in SL (symmetric log space), but any kind of hardness result seems difficult because it would require constructing the entire multiplication table of a group. We introduce the complexity class FOLL, or FO(log log n), of problems solvable by uniform polysize circuit families of unbounded fan-in and depth O(log log n). Since parity is not in FOLL, no problem in FOLL can be complete for any class containing parity, such as NC/sup 1/, L, or SL. But FOLL is not known to be contained even in SL. We show that CGM for cyclic groups is in FOLL/spl cap/L and that CGM for abelian groups is in FOLL. We then partially extend our method to solvable groups, showing that CGM for groups of constant solvability class is in FOLL and that CGM for nilpotent groups can be solved by poly-size circuits of depth O((log log n)/sup 2/). (Thus the latter problem is provably not complete for any class containing parity) We also consider the problem of testing for various properties of a group input as a table: we prove that cyclicity and nilpotency can each be tested in FOLL/spl cap/L. Finally, we examine the implications of our results for the complexity of iterated multiplication, powering, and division of integers, in the context of the recent results of Chiu, Davida, and Litow.

This paper describes the simulation of an S(n) space-bounded deterministic Turing machine by a reversible Turing machine operating in space S(n). It thus answers a question posed by C. Bennett (1989) and refutes the conjecture, made by M. Li and P. Vitanyi (1996), that any reversible simulation of an irreversible computation must obey Bennett's reversible pebble game rules.

The following survey reviews some connections between formal languages and complexity theory. Families of formal languages are treated with complexity theoretical methods. In particular, the concept of unambiguity, common to both areas, is treated in detail. Some complexity theoretical aspects of operations on formal languages are indicated. This picture is completed by taking parallel models into account.