457 research outputs found
Abstraction and Invariance for Algebraically Indexed Types
Reynoldsâ relational parametricity provides a powerful way to reason about programs in terms of invariance under changes of data representation. A dazzling array of applications of Reynoldsâ theory exists, exploiting invariance to yield âfree theoremsâ, non-inhabitation results, and encodings of algebraic datatypes. Outside computer science, invariance is a common theme running through many areas of mathematics and physics. For example, the area of a triangle is unaltered by rotation or flipping. If we scale a triangle, then we scale its area, maintaining an invariant relationship be-tween the two. The transformations under which properties are in-variant are often organised into groups, with the algebraic structure reflecting the composability and invertibility of transformations. In this paper, we investigate programming languages whose types are indexed by algebraic structures such as groups of geometric transformations. Other examples include types indexed by principalsâfor information flow securityâand types indexed by distancesâfor analysis of analytic uniform continuity properties. Following Reynolds, we prove a general Abstraction Theorem that covers all these instances. Consequences of our Abstraction Theorem include free theorems expressing invariance properties of programs, type isomorphisms based on invariance properties, and non-definability results indicating when certain algebraically indexed types are uninhabited or only inhabited by trivial programs. We have fully formalized our framework and most examples in Coq
Conditionals and modularity in general logics
In this work in progress, we discuss independence and interpolation and
related topics for classical, modal, and non-monotonic logics
Process types as a descriptive tool for interaction
We demonstrate a tight relationship between linearly typed Ï-calculi and typed λ-calculi by giving a type-preserving translation from the call-by-value λ”-calculus into a typed Ï-calculus. The λ”-calculus has a particularly simple representation as typed mobile processes. The target calculus is a simple variant of the linear Ï-calculus. We establish full abstraction up to maximally consistent observational congruences in source and target calculi using techniques from games semantics and process calculi
Control in the Ï-calculus
This paper presents a type-preserving translation from the call-by-value -calculus ( v-calculus) into a typed -calculus, and shows full abstraction up to maximally consistent observational congruences in both calculi. The -calculus has a particularly simple representation as typed mobile processes where a unique stateless replicated input is associated to each name. The corresponding -calculus is a proper subset of the linear -calculus, the latter being able to embed the simplytyped -calculus fully abstractly. Strong normalisability of the v-calculus is an immediate consequence of this correspondence and the strong normalisability of the linear -calculus, using the standard argument based on simulation between the v-calculus and its translation. Full abstraction, our main result, is proved via an inverse transformation from the typed -terms which inhabit the encoded v-types into the v-calculus (the so-called de nability argument), using proof techniques from games semantics and process calculi. A tight operational correspondence assisted by the de nability result opens a possibility to use typed -calculi as a tool to investigate and analyse behaviours of various control operators and associated calculi in a uniform setting, possibly integrated with other language primitives and operational structures
Focusing in Asynchronous Games
Game semantics provides an interactive point of view on proofs, which enables
one to describe precisely their dynamical behavior during cut elimination, by
considering formulas as games on which proofs induce strategies. We are
specifically interested here in relating two such semantics of linear logic, of
very different flavor, which both take in account concurrent features of the
proofs: asynchronous games and concurrent games. Interestingly, we show that
associating a concurrent strategy to an asynchronous strategy can be seen as a
semantical counterpart of the focusing property of linear logic
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
From IF to BI: a tale of dependence and separation
We take a fresh look at the logics of informational dependence and
independence of Hintikka and Sandu and Vaananen, and their compositional
semantics due to Hodges. We show how Hodges' semantics can be seen as a special
case of a general construction, which provides a context for a useful
completeness theorem with respect to a wider class of models. We shed some new
light on each aspect of the logic. We show that the natural propositional logic
carried by the semantics is the logic of Bunched Implications due to Pym and
O'Hearn, which combines intuitionistic and multiplicative connectives. This
introduces several new connectives not previously considered in logics of
informational dependence, but which we show play a very natural role, most
notably intuitionistic implication. As regards the quantifiers, we show that
their interpretation in the Hodges semantics is forced, in that they are the
image under the general construction of the usual Tarski semantics; this
implies that they are adjoints to substitution, and hence uniquely determined.
As for the dependence predicate, we show that this is definable from a simpler
predicate, of constancy or dependence on nothing. This makes essential use of
the intuitionistic implication. The Armstrong axioms for functional dependence
are then recovered as a standard set of axioms for intuitionistic implication.
We also prove a full abstraction result in the style of Hodges, in which the
intuitionistic implication plays a very natural r\^ole.Comment: 28 pages, journal versio
Categorical Models for a Semantically Linear Lambda-calculus
This paper is about a categorical approach to model a very simple
Semantically Linear lambda calculus, named Sll-calculus. This is a core
calculus underlying the programming language SlPCF. In particular, in this
work, we introduce the notion of Sll-Category, which is able to describe a very
large class of sound models of Sll-calculus. Sll-Category extends in the
natural way Benton, Bierman, Hyland and de Paiva's Linear Category, in order to
soundly interpret all the constructs of Sll-calculus. This category is general
enough to catch interesting models in Scott Domains and Coherence Spaces
A Fully Abstract Game Semantics for Countable Nondeterminism
The concept of fairness for a concurrent program means that the program must be able to exhibit an unbounded amount of nondeterminism without diverging. Game semantics models of nondeterminism show that this is hard to implement; for example, Harmer and McCusker\u27s model only admits infinite nondeterminism if there is also the possibility of divergence. We solve a long standing problem by giving a fully abstract game semantics for a simple stateful language with a countably infinite nondeterminism primitive. We see that doing so requires us to keep track of infinitary information about strategies, as well as their finite behaviours. The unbounded nondeterminism gives rise to further problems, which can be formalized as a lack of continuity in the language. In order to prove adequacy for our model (which usually requires continuity), we develop a new technique in which we simulate the nondeterminism using a deterministic stateful construction, and then use combinatorial techniques to transfer the result to the nondeterministic language. Lastly, we prove full abstraction for the model; because of the lack of continuity, we cannot deduce this from definability of compact elements in the usual way, and we have to use a stronger universality result instead. We discuss how our techniques yield proofs of adequacy for models of nondeterministic PCF, such as those given by Tsukada and Ong
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