89 research outputs found
A Graph Model for Imperative Computation
Scott's graph model is a lambda-algebra based on the observation that
continuous endofunctions on the lattice of sets of natural numbers can be
represented via their graphs. A graph is a relation mapping finite sets of
input values to output values.
We consider a similar model based on relations whose input values are finite
sequences rather than sets. This alteration means that we are taking into
account the order in which observations are made. This new notion of graph
gives rise to a model of affine lambda-calculus that admits an interpretation
of imperative constructs including variable assignment, dereferencing and
allocation.
Extending this untyped model, we construct a category that provides a model
of typed higher-order imperative computation with an affine type system. An
appropriate language of this kind is Reynolds's Syntactic Control of
Interference. Our model turns out to be fully abstract for this language. At a
concrete level, it is the same as Reddy's object spaces model, which was the
first "state-free" model of a higher-order imperative programming language and
an important precursor of games models. The graph model can therefore be seen
as a universal domain for Reddy's model
Composing dinatural transformations: Towards a calculus of substitution
Dinatural transformations, which generalise the ubiquitous natural
transformations to the case where the domain and codomain functors are of mixed
variance, fail to compose in general; this has been known since they were
discovered by Dubuc and Street in 1970. Many ad hoc solutions to this
remarkable shortcoming have been found, but a general theory of
compositionality was missing until Petric, in 2003, introduced the concept of
g-dinatural transformations, that is, dinatural transformations together with
an appropriate graph: he showed how acyclicity of the composite graph of two
arbitrary dinatural transformations is a sufficient and essentially necessary
condition for the composite transformation to be in turn dinatural. Here we
propose an alternative, semantic rather than syntactic, proof of Petric's
theorem, which the authors independently rediscovered with no knowledge of its
prior existence; we then use it to define a generalised functor category, whose
objects are functors of mixed variance in many variables, and whose morphisms
are transformations that happen to be dinatural only in some of their
variables. We also define a notion of horizontal composition for dinatural
transformations, extending the well known version for natural transformations,
and prove it is associative and unitary. Horizontal composition embodies
substitution of functors into transformations and vice-versa, and is
intuitively reflected from the string-diagram point of view by substitution of
graphs into graphs
On Compositionality of Dinatural Transformations
Natural transformations are ubiquitous in mathematics, logic and computer science. For operations of mixed variance, such as currying and evaluation in the lambda-calculus, Eilenberg and Kelly\u27s notion of extranatural transformation, and often the even more general dinatural transformation, is required. Unfortunately dinaturals are not closed under composition except in special circumstances. This paper presents a new sufficient condition for composability.
We propose a generalised notion of dinatural transformation in many variables, and extend the Eilenberg-Kelly account of composition for extranaturals to these transformations. Our main result is that a composition of dinatural transformations which creates no cyclic connections between arguments yields a dinatural transformation.
We also extend the classical notion of horizontal composition to our generalized dinaturals and demonstrate that it is associative and has identities
Modelling local variables: possible worlds and object spaces
AbstractLocal variables in imperative languages have been given denotational semantics in at least two fundamentally different ways. One is by use of functor categories, focusing on the idea of possible worlds. The other might be termed event-based, exemplified by Reddy's object spaces and models based on game semantics. O'Hearn and Reddy have related the two approaches by giving functor category models whose worlds are object spaces, then showing that their model is fully abstract for Idealised Algol programs up to order two. But the category of object spaces is not small, and so in order to construct a functor category that is locally small, and hence Cartesian closed, they need to work with a restricted collection of object spaces. This weakens the connection between the object spaces model and the functor-category model: the Yoneda embedding no longer provides a full embedding of the original category of object spaces into the functor-category. Moreoever the choice of the restricted collection of object spaces is ad hoc. In this paper, we refine the approach by proving that the finite objects form a small dense subcategory of a simplified object-spaces model. The functor category over these finite objects is therefore locally small and Cartesian closed, and contains the object-spaces category as a full subcategory. All this work is necessarily enriched in Cpo. We further refine their full abstraction result by showing that full abstraction fails at order three
A graphical foundation for interleaving in game semantics
In 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notion they called a {multimap}-schedule, and the similar notion of ⊗-schedule, both structures describing interleavings of plays in games. Their definition was combinatorial in nature, but researchers often draw pictures when describing schedules in practice. Moreover, several proofs of key properties, such as that the composition of {multimap}-schedules is associative, involve cumbersome combinatorial detail, whereas in terms of pictures the proof is straightforward, reflecting the geometry of the plane. Here, we give a geometric formulation of {multimap}-schedules and ⊗-schedules, prove that they are isomorphic to Harmer et al.'s definitions, and illustrate their value by giving such geometric proofs. Harmer et al.'s notions may be combined to describe plays in multi-component games, and researchers have similarly developed intuitive graphical representations of plays in these games. We give a characterisation of these diagrams and explicitly describe how they relate to the underlying schedules, finally using this relation to provide new, intuitive proofs of key categorical properties
Understanding game semantics through coherence spaces
AbstractGame Semantics has successfully provided fully abstract models for a variety of programming languages not possible using other denotational approaches. Although it is a flexible and accurate way to give semantics to a language, its underlying mathematics is awkward. For example, the proofs that strategies compose associatively and maintain properties imposed on them such as innocence are intricate and require a lot of attention. This work aims at beginning to provide a more elegant and uniform mathematical ground for Game Semantics. Our quest is to find mathematical entities that will retain the properties that make games an accurate way to give semantics to programs, yet that are simple and familiar to work with. Our main result is a full, faithful strong monoidal embedding of a category of games into a category of coherence spaces, where composition is simple composition of relations
A graphical foundation for schedules
AbstractIn 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notion they called a schedule. Their definition was combinatorial in nature, but researchers often draw pictures when describing schedules in practice. Moreover, a proof that the composition of schedules is associative involves cumbersome combinatorial detail, whereas in terms of pictures the proof is straightforward, reflecting the geometry of the plane. Here, we give a geometric formulation of schedule, prove that it is equivalent to Harmer et al.ʼs definition, and illustrate its value by giving a proof of associativity of composition
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