86 research outputs found
Reduction Monads and Their Signatures
International audienc
Innocent strategies as presheaves and interactive equivalences for CCS
Seeking a general framework for reasoning about and comparing programming
languages, we derive a new view of Milner's CCS. We construct a category E of
plays, and a subcategory V of views. We argue that presheaves on V adequately
represent innocent strategies, in the sense of game semantics. We then equip
innocent strategies with a simple notion of interaction. This results in an
interpretation of CCS.
Based on this, we propose a notion of interactive equivalence for innocent
strategies, which is close in spirit to Beffara's interpretation of testing
equivalences in concurrency theory. In this framework we prove that the
analogues of fair and must testing equivalences coincide, while they differ in
the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014
Families of Symmetries as Efficient Models of Resource Binding
AbstractCalculi that feature resource-allocating constructs (e.g. the pi-calculus or the fusion calculus) require special kinds of models. The best-known ones are presheaves and nominal sets. But named sets have the advantage of being finite in a wide range of cases where the other two are infinite. The three models are equivalent. Finiteness of named sets is strictly related to the notion of finite support in nominal sets and the corresponding presheaves. We show that named sets are generalisd by the categorical model of families, that is, free coproduct completions, indexed by symmetries, and explain how locality of interfaces gives good computational properties to families. We generalise previous equivalence results by introducing a notion of minimal support in presheaf categories indexed over small categories of monos. Functors and categories of coalgebras may be defined over families. We show that the final coalgebra has the greatest possible symmetry up-to bisimilarity, which can be computed by iteration along the terminal sequence, thanks to finiteness of the representation
A categorical framework for congruence of applicative bisimilarity in higher-order languages
Applicative bisimilarity is a coinductive characterisation of observational
equivalence in call-by-name lambda-calculus, introduced by Abramsky (1990).
Howe (1996) gave a direct proof that it is a congruence, and generalised the
result to all languages complying with a suitable format. We propose a
categorical framework for specifying operational semantics, in which we prove
that (an abstract analogue of) applicative bisimilarity is automatically a
congruence. Example instances include standard applicative bisimilarity in
call-by-name, call-by-value, and call-by-name non-deterministic
-calculus, and more generally all languages complying with a variant
of Howe's format
Categorical Term Rewriting: Monads and Modularity
Laboratory for Foundations of Computer ScienceTerm rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, that is, if the components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking.
This thesis posits that part of the problem is the usual, concrete and syntax-oriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from the term structure, such as substitutions, layers, redexes etc. Drawing on the concepts of category theory, such a semantics is proposed, based on the concept of a monad, generalising the very elegant treatment of equational presentations in category theory. The theoretical basis of this work is the theory of enriched monads.
It is shown how structuring operations are modelled on the level of monads, and that the semantics is compositional (it preserves the structuring operations). Modularity results can now be obtained directly at the level of combining monads without recourse to the syntax at all. As an application and demonstration of the usefulness of this approach, two modularity results for the disjoint union of two term rewriting systems are proven, the modularity of confluence (Toyama's theorem) and the modularity of strong normalization for a particular class of term rewriting systems (non-collapsing term rewriting systems). The proofs in the categorical setting provide a mild generalisation of these results
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems
Structure and semantics
Algebraic theories describe mathematical structures that are defined in terms of operations
and equations, and are extremely important throughout mathematics. Many generalisations
of the classical notion of an algebraic theory have sprung up for use in different mathematical
contexts; some examples include Lawvere theories, monads, PROPs and operads. The first
central notion of this thesis is a common generalisation of these, which we call a proto-theory.
The purpose of an algebraic theory is to describe its models, which are structures in which
each of the abstract operations of the theory is given a concrete interpretation such that the
equations of the theory hold. The process of going from a theory to its models is called
semantics, and is encapsulated in a semantics functor. In order to define a model of a theory in
a given category, it is necessary to have some structure that relates the arities of the operations in
the theory with the objects of the category. This leads to the second central notion of this thesis,
that of an interpretation of arities, or aritation for short. We show that any aritation gives rise
to a semantics functor from the appropriate category of proto-theories, and that this functor
has a left adjoint called the structure functor, giving rise to a structure{semantics adjunction.
Furthermore, we show that the usual semantics for many existing notions of algebraic theory
arises in this way by choosing an appropriate aritation.
Another aim of this thesis is to find a convenient category of monads in the following sense.
Every right adjoint into a category gives rise to a monad on that category, and in fact some
functors that are not right adjoints do too, namely their codensity monads. This is the structure
part of the structure{semantics adjunction for monads. However, the fact that not every functor
has a codensity monad means that the structure functor is not defined on the category of all
functors into the base category, but only on a full subcategory of it.
This deficiency is solved when passing to general proto-theories with a canonical choice of
aritation whose structure{semantics adjunction restricts to the usual one for monads. However,
this comes at a cost: the semantics functor for general proto-theories is not full and faithful,
unlike the one for monads. The condition that a semantics functor be full and faithful can be
thought of as a kind of completeness theorem | it says that no information is lost when passing
from a theory to its models. It is therefore desirable to retain this property of the semantics of
monads if possible.
The goal then, is to find a notion of algebraic theory that generalises monads for which
the semantics functor is full and faithful with a left adjoint; equivalently the semantics functor
should exhibit the category of theories as a re
ective subcategory of the category of all functors
into the base category. We achieve this (for well-behaved base categories) with a special kind of
proto-theory enriched in topological spaces, which we call a complete topological proto-theory.
We also pursue an analogy between the theory of proto-theories and that of groups. Under
this analogy, monads correspond to finite groups, and complete topological proto-theories
correspond to profinite groups. We give several characterisations of complete topological proto-theories
in terms of monads, mirroring characterisations of profinite groups in terms of finite
groups
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems
Stabilized profunctors and stable species of structures
We introduce a bicategorical model of linear logic which is a novel variation
of the bicategory of groupoids, profunctors, and natural transformations. Our
model is obtained by endowing groupoids with additional structure, called a
kit, to stabilize the profunctors by controlling the freeness of the groupoid
action on profunctor elements.
The theory of generalized species of structures, based on profunctors, is
refined to a new theory of \emph{stable species} of structures between
groupoids with Boolean kits. Generalized species are in correspondence with
analytic functors between presheaf categories; in our refined model, stable
species are shown to be in correspondence with restrictions of analytic
functors, which we characterize as being stable, to full subcategories of
stabilized presheaves. Our motivating example is the class of finitary
polynomial functors between categories of indexed sets, also known as normal
functors, that arises from kits enforcing free actions.
We show that the bicategory of groupoids with Boolean kits, stable species,
and natural transformations is cartesian closed. This makes essential use of
the logical structure of Boolean kits and explains the well-known failure of
cartesian closure for the bicategory of finitary polynomial functors between
categories of set-indexed families and cartesian natural transformations. The
paper additionally develops the model of classical linear logic underlying the
cartesian closed structure and clarifies the connection to stable domain
theory.Comment: FSCD 2022 special issue of Logical Methods in Computer Science, minor
changes (incorporated reviewers comments
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