738 research outputs found
Enriched Stone-type dualities
A common feature of many duality results is that the involved equivalence
functors are liftings of hom-functors into the two-element space resp. lattice.
Due to this fact, we can only expect dualities for categories cogenerated by
the two-element set with an appropriate structure. A prime example of such a
situation is Stone's duality theorem for Boolean algebras and Boolean
spaces,the latter being precisely those compact Hausdorff spaces which are
cogenerated by the two-element discrete space. In this paper we aim for a
systematic way of extending this duality theorem to categories including all
compact Hausdorff spaces. To achieve this goal, we combine duality theory and
quantale-enriched category theory. Our main idea is that, when passing from the
two-element discrete space to a cogenerator of the category of compact
Hausdorff spaces, all other involved structures should be substituted by
corresponding enriched versions. Accordingly, we work with the unit interval
and present duality theory for ordered and metric compact Hausdorff
spaces and (suitably defined) finitely cocomplete categories enriched in
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
Graded Quivers, Generalized Dimer Models and Toric Geometry
The open string sector of the topological B-model model on CY -folds
is described by -graded quivers with superpotentials. This correspondence
extends to general the well known connection between CY -folds and
gauge theories on the worldvolume of D-branes for . We
introduce -dimers, which fully encode the -graded quivers and their
superpotentials, in the case in which the CY -folds are toric.
Generalizing the well known cases, -dimers significantly simplify
the connection between geometry and -graded quivers. A key result of this
paper is the generalization of the concept of perfect matching, which plays a
central role in this map, to arbitrary . We also introduce a simplified
algorithm for the computation of perfect matchings, which generalizes the
Kasteleyn matrix approach to any . We illustrate these new tools with a few
infinite families of CY singularities.Comment: 54 pages, 6 figure
Efficient Parallel Path Checking for Linear-Time Temporal Logic With Past and Bounds
Path checking, the special case of the model checking problem where the model
under consideration is a single path, plays an important role in monitoring,
testing, and verification. We prove that for linear-time temporal logic (LTL),
path checking can be efficiently parallelized. In addition to the core logic,
we consider the extensions of LTL with bounded-future (BLTL) and past-time
(LTL+Past) operators. Even though both extensions improve the succinctness of
the logic exponentially, path checking remains efficiently parallelizable: Our
algorithm for LTL, LTL+Past, and BLTL+Past is in AC^1(logDCFL) \subseteq NC
The Tutte Polynomial of a Morphism of Matroids 5. Derivatives as Generating Functions of Tutte Activities
We show that in an ordered matroid the partial derivative
\partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the
generating function of activities of subsets with corank p and nullity q. More
generally, this property holds for the 3-variable Tutte polynomial of a matroid
perspective.Comment: 28 pages, 3 figures, 5 table
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