For a monoid G, the iterated multiplication problem is the computation of the product of n elements from G. By refining known completeness arguments, we show that as G varies over a natural series of important groups and monoids, the iterated multiplication problems are complete for most natural, low-level complexity classes. The completeness is with respect to "first-order projections" -- low-level reductions that do not obscure the algebraic nature of these problems.

. We refine the techniques of Beigel, Gill, Hertrampf [4] who investigated polynomial time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes MOD k L and demonstrate their significance by proving that all standard problems of linear algebra over the finite rings Z/kZ are complete for these classes. We then define new complexity classes LogFew and LogFewNL and identify them as adequate logspace versions of Few and FewP. We show that LogFewNL is contained in MODZ k L and that LogFew is contained in MOD k L for all k. Also an upper bound for L #L in terms of computation of integer determinants is given from which we conclude that all logspace counting classes are contained in NC 2 . 1 Introduction Valiant [21] defined the class #P of functions f such that there is a nondeterministic polynomial time Turing machine which, on input x, has exactly f(x) accepting computation paths. Many complexity classes in the area betw...

This is a survey of space-bounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic space-bounded computations, and the application of such generators to obtain deterministic simulations.

We show that in the context of nonuniform complexity, nondeterministic logarithmic space bounded computation can be made unambiguous. An analogous result holds for the class of problems reducible to context-free languages. In terms of complexity classes, this can be stated as: NL/poly = UL/poly LogCFL/poly = UAuxPDA(log n; n O(1) )/poly

: We provide a generalization of Datalog based on generalizing databases by adding integer order constraints to relational tuples. For Datalog queries with integer (gap)-order constraints (denoted Datalog !Z ) we show that there is a closed form evaluation. We also show that the tuple recognition problem can be done in PTIME in the size of the generalized database, assuming that the size of the constants in the query is logarithmic in the size of the database. Note that the absence of negation is critical, Datalog : queries with integer order constraints can express any Turing computable function. 1 Introduction In this paper we consider a generalization of Datalog based on the notion of a constraint tuple. The important idea of a constraint tuple comes from constraint logic programming systems, e.g. CLP [14], Prolog III [4], and CHIP [8], and it generalizes the notion of a ground fact. This allows the declarative programming of new applications, including various combinatorial se...

We present in a uniform manner simple two-sorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.

We present a Logspace, many-one reduction from the undirected st-connectivity problem to its complement. This shows that SL = co - SL.

We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 many-one reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.

We discuss the possibility of using the relatively old technique of diagonalization to separate complexity classes, in particular NL from NP. We show several results in this direction. ffl Any nonconstant level of the polynomial-time hierarchy strictly contains NL. ffl SAT is not simultaneously in NL and deterministic n log j n time for any j. ffl On the negative side, we present a relativized world where P = NP but any nonconstant level of the polynomial-time hierarchy differs from P. 1 Introduction Separating complexity classes remains the most important and difficult of problems in theoretical computer science. Circuit complexity and other techniques on finite functions have seen some exciting early successes (see the survey of Boppana and Sipser [BS90]) but have yet to achieve their promise of separating complexity classes above logarithmic space. Other techniques based on logic and geometry also have given us separations only on very restricted models. We should turn back to...

We consider various types of unambiguity for logarithmic space bounded Turing machines and polynomial time bounded log space auxiliary push-down automata. In particular, we introduce the notions of (general), reach, and strong unambiguity. We demonstrate that closure under complement of unambiguous classes implies equivalence of unambiguity and "unambiguous fewness". This, as we will show, applies in the cases of reach and strong unambiguity for logspace. Among the many relations we exhibit, we show that the unambiguous linear contextfree languages, which are not known to be contained in LOGSPACE, nevertheless are contained in strongly unambiguous logspace, and, consequently, in LOGDCFL. In fact, this turns out to be the case for all logspace languages with reach unambiguous fewness. In addition, we show that general unambiguity and fewness of logspace classes can be simulated by reach unambiguity and fewness of auxiliary push-down automata. 1 Introduction Although the pow...