  To provide a dichotomy between those queries that can be made feasible on big
data after appropriate preprocessing and those for which preprocessing does not
help, Fan et al. developed the $\sqcap$-tractability theory. This theory
provides a formal foundation for understanding the tractability of query
classes in the context of big data. Along this line, we introduce a novel
notion of $\sqcap'$-tractability in this paper. Inspired by some technologies
used to deal big data, we place a restriction on preprocessing function, which
limits the function to produce a relatively small database as output, at most
polylog-size of the input database. At the same time, we bound the redundancy
information when re-factorizing data and queries for preprocessing. These
changes aim to make our theory more closely linked to practice. We set two
complexity classes to denote the classes of Boolean queries that are
$\sqcap'$-tractable themselves and that can be made $\sqcap'$-tractable,
respectively. Based on a new factorization in our complexity classes, we
investigate two reductions, which differ from whether allowing re-factorizing
data and query parts. We verify the transitive and compatible properties of the
reductions and analysis the complete problems and sizes of the complexity
classes. We conclude that all PTIME classes of Boolean queries can be made
$\sqcap'$-tractable, similar to that of the $\sqcap$-tractability theory. With
a little surprise, we prove that the set of all $\sqcap'$-tractable queries is
strictly smaller than that of all $\sqcap$-tractable queries, and thus the set
of $\sqcap'$-tractable queries is properly contained in that of PTIME queries.
In this way, we attain a new complexity class inside the complexity class of
PTIME queries.
