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Abstract.We refine the techniques of Beigel et al. [4] who investigated polynomial-time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes J///(9~k i~ 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 LogFewJff.5 a and identify them as adequate logspace versions of Few and Few~. We show that LogFewJffY is contained in ~/~(~k,~ 7 and that LogFew is contained in JC/(9~k~ for all k. Also an upper bound for ~e 4~e in terms of computation of integer determinants is given from which we conclude that all logspace counting classes are contained in jff~,2.

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