10 research outputs found
Isomorphisms of scattered automatic linear orders
We prove that the isomorphism of scattered tree automatic linear orders as
well as the existence of automorphisms of scattered word automatic linear
orders are undecidable. For the existence of automatic automorphisms of word
automatic linear orders, we determine the exact level of undecidability in the
arithmetical hierarchy
Polygraphs: From Rewriting to Higher Categories
Polygraphs are a higher-dimensional generalization of the notion of directed
graph. Based on those as unifying concept, this monograph on polygraphs
revisits the theory of rewriting in the context of strict higher categories,
adopting the abstract point of view offered by homotopical algebra. The first
half explores the theory of polygraphs in low dimensions and its applications
to the computation of the coherence of algebraic structures. It is meant to be
progressive, with little requirements on the background of the reader, apart
from basic category theory, and is illustrated with algorithmic computations on
algebraic structures. The second half introduces and studies the general notion
of n-polygraph, dealing with the homotopy theory of those. It constructs the
folk model structure on the category of strict higher categories and exhibits
polygraphs as cofibrant objects. This allows extending to higher dimensional
structures the coherence results developed in the first half
Ătudes in Homotopical Thinking: Fâ-geometry, Concurrent Computing, and Motivic Measures
This thesis weaves together three papers, each of which provides a use of homotopical intuition in a different field of mathematics. The first applies it to the study of various models of Fâ-geometry, focusing mainly on the Bost-Connes algebra. The second endeavors to compare two homotopical models for concurrent computing before introducing a new one as well. Finally, the last paper provides a construction for obtaining derived motivic measures from an abstract six functors formalism and, in particular, applies this idea to obtain a lift of the Gillet-SoulĂ© motivic measure
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.
Way of the dagger
A dagger category is a category equipped with a functorial way of reversing morphisms,
i.e. a contravariant involutive identity-on-objects endofunctor. Dagger categories
with additional structure have been studied under different names in categorical
quantum mechanics, algebraic field theory and homological algebra, amongst others.
In this thesis we study the dagger in its own right and show how basic category theory
adapts to dagger categories.
We develop a notion of a dagger limit that we show is suitable in the following
ways: it subsumes special cases known from the literature; dagger limits are unique
up to unitary isomorphism; a wide class of dagger limits can be built from a small
selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to
a diagonal functor; dagger limits can be built from ordinary limits in the presence of
polar decomposition; dagger limits commute with dagger colimits in many cases.
Using cofree dagger categories, the theory of dagger limits can be leveraged to
provide an enrichment-free understanding of limit-colimit coincidences in ordinary
category theory. We formalize the concept of an ambilimit, and show that it captures
known cases. As a special case, we show how to define biproducts up to isomorphism
in an arbitrary category without assuming any enrichment. Moreover, the limit-colimit
coincidence from domain theory can be generalized to the unenriched setting and we
show that, under suitable assumptions, a wide class of endofunctors has canonical fixed
points.
The theory of monads on dagger categories works best when all structure respects
the dagger: the monad and adjunctions should preserve the dagger, and the monad and
its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an
adjunction, with extremal solutions given by the categories of Kleisli and Frobenius-
Eilenberg-Moore algebras, which again have a dagger.
We use dagger categories to study reversible computing. Specifically, we model reversible
effects by adapting Hughesâ arrows to dagger arrows and inverse arrows. This
captures several fundamental reversible effects, including serialization and mutable
store computations. Whereas arrows are monoids in the category of profunctors, dagger
arrows are involutive monoids in the category of profunctors, and inverse arrows
satisfy certain additional properties. These semantics inform the design of functional
reversible programs supporting side-effects
Annual Report of the University, 1968-1969, Volumes 1 & 2
At least once every ten years the University of New Mexico has a chance to see itself as others sec it. The opportunity is provided by its accrediting agency. Routinely every decade, the North Central Association of Colleges and Secondary Schools sends a team of scholars and administrators to campus to determine whether the University is maintaining the prerequisites to continuing accreditation as a doctoral degree granting institution. While the examination is scheduled routinely, it is by no means a routine visit. The visitation team probes such areas as curricula, library, finances, administration, day-to-day operations, and long-range plans. Its report, much like that of an auditor, helps provide operational guidelines for succeeding years. The University of New Mexico in 1969 underwent its decennial examination by the North Central Association. The team of visitors prepared a comprehensive report touching on many areas vital to the University\u27s future. Findings of the committee and the University\u27s responses to them serve as the basis for this annual report of the President