5,991 research outputs found
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
Constructive contextual modal judgments for reasoning from open assumptions
Dependent type theories using a structural notion of context are largely explored in their applications to programming languages, but less investigated for knowledge representation purposes. In particular, types with modalities are already used for distributed and staged computation. This paper introduces a type system extended with judgmental modalities internalizing epistemically different modes of correctness to explore a calculus of provability from refutable assumptions
A Categorical Normalization Proof for the Modal Lambda-Calculus
We investigate a simply typed modal -calculus,
, due to Pfenning, Wong and Davies, where we define a
well-typed term with respect to a context stack that captures the possible
world semantics in a syntactic way. It provides logical foundation for
multi-staged meta-programming. Our main contribution in this paper is a
normalization by evaluation (NbE) algorithm for which we
prove sound and complete. The NbE algorithm is a moderate extension to the
standard presheaf model of simply typed -calculus. However, central to
the model construction and the NbE algorithm is the observation of Kripke-style
substitutions on context stacks which brings together two previously separate
concepts, structural modal transformations on context stacks and substitutions
for individual assumptions. Moreover, Kripke-style substitutions allow us to
give a formulation for contextual types, which can represent open code in a
meta-programming setting. Our work lays the foundation for extending the
logical foundation by Pfenning, Wong, and Davies towards building a practical,
dependently typed foundation for meta-programming
Positive energy unitary irreducible representations of D=6 conformal supersymmetry
We give a constructive classification of the positive energy (lowest weight)
unitary irreducible representations of the D=6 superconformal algebras
osp(8*/2N). Our results confirm all but one of the conjectures of Minwalla (for
N=1,2) on this classification. Our main tool is the explicit construction of
the norms of the states that has to be checked for positivity. We give also the
reduction of the exceptional UIRs.Comment: 27 pages, TeX with harvmac, amssym.def, amssym.tex; v.2: minor
corrections and references added; v.3: minor corrections; v.4: to appear in
J. Phys.
Modal logics for reasoning about object-based component composition
Component-oriented development of software supports the adaptability and maintainability of large systems, in particular if requirements change over time and parts of a system have to be modified or replaced. The software architecture in such systems can be described by components
and their composition. In order to describe larger architectures, the composition concept becomes crucial. We will present a formal framework for component composition for object-based software development. The deployment of modal logics for defining components and component composition will allow us to reason about and prove properties of components and compositions
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