8,828 research outputs found
Category-theoretical Semantics of the Description Logic ALC (extended version)
Category theory can be used to state formulas in First-Order Logic without
using set membership. Several notable results in logic such as proof of the
continuum hypothesis can be elegantly rewritten in category theory. We propose
in this paper a reformulation of the usual set-theoretical semantics of the
description logic ALC by using categorical language. In this setting, ALC
concepts are represented as objects, concept subsumptions as arrows, and
memberships as logical quantifiers over objects and arrows of categories. Such
a category-theore\-tical semantics provides a more modular representation of
the semantics of and a new way to design algorithms for
reasoning.Comment: 14 page
Categories for Dynamic Epistemic Logic
The primary goal of this paper is to recast the semantics of modal logic, and
dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We
first review the category of relations and categories of Kripke frames, with
particular emphasis on the duality between relations and adjoint homomorphisms.
Using these categories, we then reformulate the semantics of DEL in a more
categorical and algebraic form. Several virtues of the new formulation will be
demonstrated: The DEL idea of updating a model into another is captured
naturally by the categorical perspective -- which emphasizes a family of
objects and structural relationships among them, as opposed to a single object
and structure on it. Also, the categorical semantics of DEL can be merged
straightforwardly with a standard categorical semantics for first-order logic,
providing a semantics for first-order DEL.Comment: In Proceedings TARK 2017, arXiv:1707.0825
Category Theory and Model-Driven Engineering: From Formal Semantics to Design Patterns and Beyond
There is a hidden intrigue in the title. CT is one of the most abstract
mathematical disciplines, sometimes nicknamed "abstract nonsense". MDE is a
recent trend in software development, industrially supported by standards,
tools, and the status of a new "silver bullet". Surprisingly, categorical
patterns turn out to be directly applicable to mathematical modeling of
structures appearing in everyday MDE practice. Model merging, transformation,
synchronization, and other important model management scenarios can be seen as
executions of categorical specifications.
Moreover, the paper aims to elucidate a claim that relationships between CT
and MDE are more complex and richer than is normally assumed for "applied
mathematics". CT provides a toolbox of design patterns and structural
principles of real practical value for MDE. We will present examples of how an
elementary categorical arrangement of a model management scenario reveals
deficiencies in the architecture of modern tools automating the scenario.Comment: In Proceedings ACCAT 2012, arXiv:1208.430
Types and forgetfulness in categorical linguistics and quantum mechanics
The role of types in categorical models of meaning is investigated. A general
scheme for how typed models of meaning may be used to compare sentences,
regardless of their grammatical structure is described, and a toy example is
used as an illustration. Taking as a starting point the question of whether the
evaluation of such a type system 'loses information', we consider the
parametrized typing associated with connectives from this viewpoint.
The answer to this question implies that, within full categorical models of
meaning, the objects associated with types must exhibit a simple but subtle
categorical property known as self-similarity. We investigate the category
theory behind this, with explicit reference to typed systems, and their
monoidal closed structure. We then demonstrate close connections between such
self-similar structures and dagger Frobenius algebras. In particular, we
demonstrate that the categorical structures implied by the polymorphically
typed connectives give rise to a (lax unitless) form of the special forms of
Frobenius algebras known as classical structures, used heavily in abstract
categorical approaches to quantum mechanics.Comment: 37 pages, 4 figure
A categorical analogue of the monoid semiring construction
This paper introduces and studies a categorical analogue of the familiar
monoid semiring construction. By introducing an axiomatisation of summation
that unifies notions of summation from algebraic program semantics with various
notions of summation from the theory of analysis, we demonstrate that the
monoid semiring construction generalises to cases where both the monoid and the
semiring are categories. This construction has many interesting and natural
categorical properties, and natural computational interpretations.Comment: 34 pages, 5 diagram
Generic Trace Semantics via Coinduction
Trace semantics has been defined for various kinds of state-based systems,
notably with different forms of branching such as non-determinism vs.
probability. In this paper we claim to identify one underlying mathematical
structure behind these "trace semantics," namely coinduction in a Kleisli
category. This claim is based on our technical result that, under a suitably
order-enriched setting, a final coalgebra in a Kleisli category is given by an
initial algebra in the category Sets. Formerly the theory of coalgebras has
been employed mostly in Sets where coinduction yields a finer process semantics
of bisimilarity. Therefore this paper extends the application field of
coalgebras, providing a new instance of the principle "process semantics via
coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page
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