116 research outputs found
Functional Kan Simplicial Sets: Non-Constructivity of Exponentiation
Functional Kan simplicial sets are simplicial sets in which the horn-fillers required by the Kan extension condition are given explicitly by functions. We show the non-constructivity of the following basic result: if B and A are functional Kan simplicial sets, then A^B is a Kan simplicial set. This strengthens a similar result for the case of non-functional Kan simplicial sets shown by Bezem, Coquand and Parmann [TLCA 2015, v. 38 of LIPIcs]. Our
result shows that-from a constructive point of view-functional
Kan simplicial sets are, as it stands, unsatisfactory as a model of even simply typed lambda calculus. Our proof is based on a rather involved Kripke countermodel which has been encoded and verified in the Coq proof assistant
Geometric Aspects of Multiagent Systems
Recent advances in Multiagent Systems (MAS) and Epistemic Logic within
Distributed Systems Theory, have used various combinatorial structures that
model both the geometry of the systems and the Kripke model structure of models
for the logic. Examining one of the simpler versions of these models,
interpreted systems, and the related Kripke semantics of the logic (an
epistemic logic with -agents), the similarities with the geometric /
homotopy theoretic structure of groupoid atlases is striking. These latter
objects arise in problems within algebraic K-theory, an area of algebra linked
to the study of decomposition and normal form theorems in linear algebra. They
have a natural well structured notion of path and constructions of path
objects, etc., that yield a rich homotopy theory.Comment: 14 pages, 1 eps figure, prepared for GETCO200
Impure Simplicial Complexes: Complete Axiomatization
Combinatorial topology is used in distributed computing to model concurrency
and asynchrony. The basic structure in combinatorial topology is the simplicial
complex, a collection of subsets called simplices of a set of vertices, closed
under containment. Pure simplicial complexes describe message passing in
asynchronous systems where all processes (agents) are alive, whereas impure
simplicial complexes describe message passing in synchronous systems where
processes may be dead (have crashed). Properties of impure simplicial complexes
can be described in a three-valued multi-agent epistemic logic where the third
value represents formulas that are undefined, e.g., the knowledge and local
propositions of dead agents. In this work we present the axiomatization called
and show that it is sound and complete for the class of
impure complexes. The completeness proof involves the novel construction of the
canonical simplicial model and requires a careful manipulation of undefined
formulas
Simplicial Models for the Epistemic Logic of Faulty Agents
In recent years, several authors have been investigating simplicial models, a
model of epistemic logic based on higher-dimensional structures called
simplicial complexes. In the original formulation, simplicial models were
always assumed to be pure, meaning that all worlds have the same dimension.
This is equivalent to the standard S5n semantics of epistemic logic, based on
Kripke models. By removing the assumption that models must be pure, we can go
beyond the usual Kripke semantics and study epistemic logics where the number
of agents participating in a world can vary. This approach has been developed
in a number of papers, with applications in fault-tolerant distributed
computing where processes may crash during the execution of a system. A
difficulty that arises is that subtle design choices in the definition of
impure simplicial models can result in different axioms of the resulting logic.
In this paper, we classify those design choices systematically, and axiomatize
the corresponding logics. We illustrate them via distributed computing examples
of synchronous systems where processes may crash
Geometric Model Checking of Continuous Space
Topological Spatial Model Checking is a recent paradigm where model checking
techniques are developed for the topological interpretation of Modal Logic. The
Spatial Logic of Closure Spaces, SLCS, extends Modal Logic with reachability
connectives that, in turn, can be used for expressing interesting spatial
properties, such as "being near to" or "being surrounded by". SLCS constitutes
the kernel of a solid logical framework for reasoning about discrete space,
such as graphs and digital images, interpreted as quasi discrete closure
spaces. Following a recently developed geometric semantics of Modal Logic, we
propose an interpretation of SLCS in continuous space, admitting a geometric
spatial model checking procedure, by resorting to models based on polyhedra.
Such representations of space are increasingly relevant in many domains of
application, due to recent developments of 3D scanning and visualisation
techniques that exploit mesh processing. We introduce PolyLogicA, a geometric
spatial model checker for SLCS formulas on polyhedra and demonstrate
feasibility of our approach on two 3D polyhedral models of realistic size.
Finally, we introduce a geometric definition of bisimilarity, proving that it
characterises logical equivalence
Semi-simplicial Set Models for Distributed Knowledge
In recent years, a new class of models for multi-agent epistemic logic has
emerged, based on simplicial complexes. Since then, many variants of these
simplicial models have been investigated, giving rise to different logics and
axiomatizations.
In this paper, we present a further generalization, where a group of agents
may distinguish two worlds, even though each individual agent in the group is
unable to distinguish them. For that purpose, we generalize beyond simplicial
complexes and consider instead simplicial sets. By doing so, we define a new
semantics for epistemic logic with distributed knowledge. As it turns out,
these models are the geometric counterpart of a generalization of Kripke
models, called "pseudo-models". We identify various interesting sub-classes of
these models, encompassing all previously studied variants of simplicial
models; and give a sound and complete axiomatization for each of them
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