1,341 research outputs found
Stone-Type Dualities for Separation Logics
Stone-type duality theorems, which relate algebraic and
relational/topological models, are important tools in logic because -- in
addition to elegant abstraction -- they strengthen soundness and completeness
to a categorical equivalence, yielding a framework through which both algebraic
and topological methods can be brought to bear on a logic. We give a systematic
treatment of Stone-type duality for the structures that interpret bunched
logics, starting with the weakest systems, recovering the familiar BI and
Boolean BI (BBI), and extending to both classical and intuitionistic Separation
Logic. We demonstrate the uniformity and modularity of this analysis by
additionally capturing the bunched logics obtained by extending BI and BBI with
modalities and multiplicative connectives corresponding to disjunction,
negation and falsum. This includes the logic of separating modalities (LSM), De
Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics
extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as
corollaries soundness and completeness theorems for the specific Kripke-style
models of these logics as presented in the literature: for DMBI, the
sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene
BI (connecting our work to Concurrent Separation Logic), this is the first time
soundness and completeness theorems have been proved. We thus obtain a
comprehensive semantic account of the multiplicative variants of all standard
propositional connectives in the bunched logic setting. This approach
synthesises a variety of techniques from modal, substructural and categorical
logic and contextualizes the "resource semantics" interpretation underpinning
Separation Logic amongst them
Dual-Context Calculi for Modal Logic
We present natural deduction systems and associated modal lambda calculi for
the necessity fragments of the normal modal logics K, T, K4, GL and S4. These
systems are in the dual-context style: they feature two distinct zones of
assumptions, one of which can be thought as modal, and the other as
intuitionistic. We show that these calculi have their roots in in sequent
calculi. We then investigate their metatheory, equip them with a confluent and
strongly normalizing notion of reduction, and show that they coincide with the
usual Hilbert systems up to provability. Finally, we investigate a categorical
semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see
arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089
Lewis meets Brouwer: constructive strict implication
C. I. Lewis invented modern modal logic as a theory of "strict implication".
Over the classical propositional calculus one can as well work with the unary
box connective. Intuitionistically, however, the strict implication has greater
expressive power than the box and allows to make distinctions invisible in the
ordinary syntax. In particular, the logic determined by the most popular
semantics of intuitionistic K becomes a proper extension of the minimal normal
logic of the binary connective. Even an extension of this minimal logic with
the "strength" axiom, classically near-trivial, preserves the distinction
between the binary and the unary setting. In fact, this distinction and the
strong constructive strict implication itself has been also discovered by the
functional programming community in their study of "arrows" as contrasted with
"idioms". Our particular focus is on arithmetical interpretations of the
intuitionistic strict implication in terms of preservativity in extensions of
Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years
later
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