  In this paper, we introduce a new framework for parameterised counting in
logspace, inspired by the parameterised space bounded models developed by
Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined
the operators paraW and paraBeta for parameterised space complexity classes by
allowing bounded nondeterminism with multiple-read and read-once access,
respectively. Using these operators, they characterised the parameterised
complexity of natural problems on graphs. In the spirit of the operators paraW
and paraBeta by Stockhusen and Tantau, we introduce variants based on
tail-nondeterminism, paraW[1] and paraBeta-Tail. Then, we consider counting
versions of all four operators applied to logspace and obtain several natural
complete problems for the resulting classes: counting of paths in digraphs,
counting first-order models for formulas, and counting graph homomorphisms.
Furthermore, we show that the complexity of a parameterised variant of the
determinant function for (0,1)-matrices is #paraBeta-Tail-L-hard and can be
written as the difference of two functions in #paraBetaTail-L. For example, we
show that the closure of #paraBetaTail-L under parameterised logspace
parsimonious reductions coincides with #paraBeta-L, that is, modulo
parameterised reductions, tail-nondeterminism with read-once access is the same
as read-once nondeterminism. We show that all introduced classes are closed
under addition and multiplication, and those without tail-nondeterminism are
closed under parameterised logspace parsimonious reductions. Finally, we
underline the significance of this topic by providing a promising outlook
showing several open problems and options for further directions of research.
