267 research outputs found
Strict Ideal Completions of the Lambda Calculus
The infinitary lambda calculi pioneered by Kennaway et al. extend the basic
lambda calculus by metric completion to infinite terms and reductions.
Depending on the chosen metric, the resulting infinitary calculi exhibit
different notions of strictness. To obtain infinitary normalisation and
infinitary confluence properties for these calculi, Kennaway et al. extend
-reduction with infinitely many `-rules', which contract
meaningless terms directly to . Three of the resulting B\"ohm reduction
calculi have unique infinitary normal forms corresponding to B\"ohm-like trees.
In this paper we develop a corresponding theory of infinitary lambda calculi
based on ideal completion instead of metric completion. We show that each of
our calculi conservatively extends the corresponding metric-based calculus.
Three of our calculi are infinitarily normalising and confluent; their unique
infinitary normal forms are exactly the B\"ohm-like trees of the corresponding
metric-based calculi. Our calculi dispense with the infinitely many
-rules of the metric-based calculi. The fully non-strict calculus (called
) consists of only -reduction, while the other two calculi (called
and ) require two additional rules that precisely state their
strictness properties: (for ) and (for and )
A new coinductive confluence proof for infinitary lambda calculus
We present a new and formal coinductive proof of confluence and normalisation
of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than
previous proofs of this result. The technique of the proof is new, i.e., it is
not merely a coinductive reformulation of any earlier proofs. We formalised the
proof in the Coq proof assistant.Comment: arXiv admin note: text overlap with arXiv:1501.0435
Sequence Types for Hereditary Permutators
The invertible terms in Scott\u27s model D_infty are known as the hereditary permutators. Equivalently, they are terms which are invertible up to beta eta-conversion with respect to the composition of the lambda-terms. Finding a type-theoretic characterization to the set of hereditary permutators was problem # 20 of TLCA list of problems. In 2008, Tatsuta proved that this was not possible with an inductive type system. Building on previous work, we use an infinitary intersection type system based on sequences (i.e., families of types indexed by integers) to characterize hereditary permutators with a unique type. This gives a positive answer to the problem in the coinductive case
Two-dimensional Kripke Semantics II:Stability and Completeness
We revisit the duality between Kripke and algebraic semantics of intuitionistic and intuitionistic modal logic. We find that there is a certain mismatch between the two semantics, which means that not all algebraic models can be embedded into a Kripke model. This leads to an alternative proposal for a relational semantics, the stable semantics. Instead of an arbitrary partial order, the stable semantics requires a distributive lattice of worlds. We constructively show that the stable semantics is exactly as complete as the algebraic semantics. Categorifying these results leads to a 2-duality between two-dimensional stable semantics and categories of product-preserving presheaves, i.e. models of algebraic theories in the style of Lawvere
Processes, Systems \& Tests: Defining Contextual Equivalences
In this position paper, we would like to offer and defend a new template to
study equivalences between programs -- in the particular framework of process
algebras for concurrent computation.We believe that our layered model of
development will clarify the distinction that is too often left implicit
between the tasks and duties of the programmer and of the tester. It will also
enlighten pre-existing issues that have been running across process algebras as
diverse as the calculus of communicating systems, the -calculus -- also
in its distributed version -- or mobile ambients.Our distinction starts by
subdividing the notion of process itself in three conceptually separated
entities, that we call \emph{Processes}, \emph{Systems} and \emph{Tests}.While
the role of what can be observed and the subtleties in the definitions of
congruences have been intensively studied, the fact that \emph{not every
process can be tested}, and that \emph{the tester should have access to a
different set of tools than the programmer} is curiously left out, or at least
not often formally discussed.We argue that this blind spot comes from the
under-specification of contexts -- environments in which comparisons takes
place -- that play multiple distinct roles but supposedly always \enquote{stay
the same}.We illustrate our statement with a simple Java example, the
\enquote{usual} concurrent languages, but also back it up with
-calculus and existing implementations of concurrent languages as
well
Profunctors, Open Maps and Bisimulation
This paper studies fundamental connections between profunctors (i.e., distributors, or bimodules), open maps and bisimulation. In particular, it proves that a colimit preserving functor between presheaf categories (corresponding to a profunctor) preserves open maps and open map bisimulation. Consequently, the composition of profunctors preserves open maps as 2-cells. A guiding idea is the view that profunctors, and colimit preserving functors, are linear maps in a model of classical linear logic. But profunctors, and colimit preserving functors, as linear maps, are too restrictive for many applications. This leads to a study of a range of pseudo-comonads and how non-linear maps in their co-Kleisli bicategories preserve open maps and bisimulation. The pseudo-comonads considered are based on finite colimit completion, ``lifting'', and indexed families. The paper includes an appendix summarising the key results on coends, left Kan extensions and the preservation of colimits. One motivation for this work is that it provides a mathematical framework for extending domain theory and denotational semantics of programming languages to the more intricate models, languages and equivalences found in concurrent computation. But the results are likely to have more general applicability because of the ubiquitous nature of profunctors
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