232 research outputs found
Comonadic Notions of Computation
AbstractWe argue that symmetric (semi)monoidal comonads provide a means to structure context-dependent notions of computation such as notions of dataflow computation (computation on streams) and of tree relabelling as in attribute evaluation. We propose a generic semantics for extensions of simply typed lambda calculus with context-dependent operations analogous to the Moggi-style semantics for effectful languages based on strong monads. This continues the work in the early 90s by Brookes, Geva and Van Stone on the use of computational comonads in intensional semantics
Modalities, Cohesion, and Information Flow
It is informally understood that the purpose of modal type constructors in
programming calculi is to control the flow of information between types. In
order to lend rigorous support to this idea, we study the category of
classified sets, a variant of a denotational semantics for information flow
proposed by Abadi et al. We use classified sets to prove multiple
noninterference theorems for modalities of a monadic and comonadic flavour. The
common machinery behind our theorems stems from the the fact that classified
sets are a (weak) model of Lawvere's theory of axiomatic cohesion. In the
process, we show how cohesion can be used for reasoning about multi-modal
settings. This leads to the conclusion that cohesion is a particularly useful
setting for the study of both information flow, but also modalities in type
theory and programming languages at large
Structure and Power: an emerging landscape
In this paper, we give an overview of some recent work on applying tools from
category theory in finite model theory, descriptive complexity, constraint
satisfaction, and combinatorics. The motivations for this work come from
Computer Science, but there may also be something of interest for model
theorists and other logicians.
The basic setting involves studying the category of relational structures via
a resource-indexed family of adjunctions with some process category - which
unfolds relational structures into treelike forms, allowing natural resource
parameters to be assigned to these unfoldings. One basic instance of this
scheme allows us to recover, in a purely structural, syntax-free way: the
Ehrenfeucht-Fraisse~game; the quantifier rank fragments of first-order logic;
the equivalences on structures induced by (i) the quantifier rank fragments,
(ii) the restriction of this fragment to the existential positive part, and
(iii) the extension with counting quantifiers; and the combinatorial parameter
of tree-depth (Nesetril and Ossona de Mendez). Another instance recovers the
k-pebble game, the finite-variable fragments, the corresponding equivalences,
and the combinatorial parameter of treewidth. Other instances cover modal,
guarded and hybrid fragments, generalized quantifiers, and a wide range of
combinatorial parameters. This whole scheme has been axiomatized in a very
general setting, of arboreal categories and arboreal covers.
Beyond this basic level, a landscape is beginning to emerge, in which
structural features of the resource categories, adjunctions and comonads are
reflected in degrees of logical and computational tractability of the
corresponding languages. Examples include semantic characterisation and
preservation theorems, and Lovasz-type results on counting homomorphisms.Comment: To appear in special issue for Trakhtenbrot centenary of Fundamenta
Informaticae vol. 186 no 1-
Terminal semantics for codata types in intensional Martin-L\"of type theory
In this work, we study the notions of relative comonad and comodule over a
relative comonad, and use these notions to give a terminal coalgebra semantics
for the coinductive type families of streams and of infinite triangular
matrices, respectively, in intensional Martin-L\"of type theory. Our results
are mechanized in the proof assistant Coq.Comment: 14 pages, ancillary files contain formalized proof in the proof
assistant Coq; v2: 20 pages, title and abstract changed, give a terminal
semantics for streams as well as for matrices, Coq proof files updated
accordingl
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