From IF to BI, A Tale of Dependence and Separation
S. Abramsky, J.A. Vaananen
Abstract:
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their
compositional semantics due to Hodges. We show how Hodges' semantics
can be seen as a special case of a general construction, which
provides a context for a useful completeness theorem with respect to a
wider class of models. We shed some new light on each aspect of the
logic. We show that the natural propositional logic carried by the
semantics is the logic of Bunched Implications due to Pym and O'Hearn,
which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics
of informational dependence, but which we show play a very natural
role, most notably intuitionistic implication. As regards the
quantiers, we show that their interpretation in the Hodges semantics
is forced, in that they are the image under the general construction
of the usual Tarski semantics; this implies that they are adjoints to
substitution, and hence uniquely determined. As for the dependence
predicate, we show that this is definable from a simpler predicate, of
constancy or dependence on nothing. This makes essential use of the
intuitionistic implication. The Armstrong axioms for functional
dependence are then recovered as a standard set of axioms for
intuitionistic implication. We also prove a full abstraction result in
the style of Hodges, in which the intuitionistic implication plays a
very natural role.
Keywords: dependence logic; separation logic; linear logic; intuitionsitic logic