  Graph-based frames have been introduced as a logical framework which
internalizes an inherent boundary to knowability. They also support the
interpretation of lattice-based (modal) logics as hyper-constructive logics of
evidential reasoning. Conceptually, the present paper proposes graph-based
frames as a formal framework suitable for generalizing Pawlak's rough set
theory to a setting in which inherent limits to knowability need to be
considered. Technically, the present paper establishes systematic connections
between the first-order correspondents of Sahlqvist modal reduction principles
on Kripke frames, and on the more general relational environments of
graph-based and polarity-based frames. This work is part of a research line
aiming at: (a) comparing and inter-relating the various (first-order)
conditions corresponding to a given (modal) axiom in different relational
semantics (b) recognizing when first-order sentences in the
frame-correspondence languages of different relational structures encode the
same modal content (c) meaningfully transferring relational properties across
different semantic contexts. The present paper develops these results for the
graph-based semantics, polarity-based semantics, and all Sahlqvist modal
reduction principles. As an application, we study well known modal axioms in
rough set theory on graph-based frames and show that, although these axioms
correspond to different first-order conditions on graph-based frames, their
intuitive meaning is retained.This allows us to introduce the notion of
hyperconstructivist approximation spaces as the subclass of graph-based frames
defined by the first-order conditions corresponding to the same modal axioms
defining classical generalized approximation spaces, and to transfer the
properties and the intuitive understanding of different approximation spaces to
graph-based frames.
