728 research outputs found
A New Linear Logic for Deadlock-Free Session-Typed Processes
The π -calculus, viewed as a core concurrent programming language, has been used as the target of much research on type systems for concurrency. In this paper we propose a new type system for deadlock-free session-typed π -calculus processes, by integrating two separate lines of work. The first is the propositions-as-types approach by Caires and Pfenning, which provides a linear logic foundation for session types and guarantees deadlock-freedom by forbidding cyclic process connections. The second is Kobayashi’s approach in which types are annotated with priorities so that the type system can check whether or not processes contain genuine cyclic dependencies between communication operations. We combine these two techniques for the first time, and define a new and more expressive variant of classical linear logic with a proof assignment that gives a session type system with Kobayashi-style priorities. This can be seen in three ways: (i) as a new linear logic in which cyclic structures can be derived and a CYCLE -elimination theorem generalises CUT -elimination; (ii) as a logically-based session type system, which is more expressive than Caires and Pfenning’s; (iii) as a logical foundation for Kobayashi’s system, bringing it into the sphere of the propositions-as-types paradigm
Retracing some paths in categorical semantics: From process-propositions-as-types to categorified reals and computers
The logical parallelism of propositional connectives and type constructors
extends beyond the static realm of predicates, to the dynamic realm of
processes. Understanding the logical parallelism of process propositions and
dynamic types was one of the central problems of the semantics of computation,
albeit not always clear or explicit. It sprung into clarity through the early
work of Samson Abramsky, where the central ideas of denotational semantics and
process calculus were brought together and analyzed by categorical tools, e.g.
in the structure of interaction categories. While some logical structures borne
of dynamics of computation immediately started to emerge, others had to wait,
be it because the underlying logical principles (mainly those arising from
coinduction) were not yet sufficiently well-understood, or simply because the
research community was more interested in other semantical tasks. Looking back,
it seems that the process logic uncovered by those early semantical efforts
might still be starting to emerge and that the vast field of results that have
been obtained in the meantime might be a valley on a tip of an iceberg.
In the present paper, I try to provide a logical overview of the gamut of
interaction categories and to distinguish those that model computation from
those that capture processes in general. The main coinductive constructions
turn out to be of this latter kind, as illustrated towards the end of the paper
by a compact category of all real numbers as processes, computable and
uncomputable, with polarized bisimulations as morphisms. The addition of the
reals arises as the biproduct, real vector spaces are the enriched
bicompletions, and linear algebra arises from the enriched kan extensions. At
the final step, I sketch a structure that characterizes the computable fragment
of categorical semantics.Comment: 63 pages, 40 figures; cut two words from the title, tried to improve
(without lengthening) Sec.8; rewrote a proof in the Appendi
On choice rules in dependent type theory
In a dependent type theory satisfying the propositions as
types correspondence together with the proofs-as-programs paradigm,
the validity of the unique choice rule or even more of the choice rule says
that the extraction of a computable witness from an existential statement
under hypothesis can be performed within the same theory.
Here we show that the unique choice rule, and hence the choice rule,
are not valid both in Coquand\u2019s Calculus of Constructions with indexed
sum types, list types and binary disjoint sums and in its predicative
version implemented in the intensional level of the Minimalist Founda-
tion. This means that in these theories the extraction of computational
witnesses from existential statements must be performed in a more ex-
pressive proofs-as-programs theory
An Algebra of Synchronous Scheduling Interfaces
In this paper we propose an algebra of synchronous scheduling interfaces
which combines the expressiveness of Boolean algebra for logical and functional
behaviour with the min-max-plus arithmetic for quantifying the non-functional
aspects of synchronous interfaces. The interface theory arises from a
realisability interpretation of intuitionistic modal logic (also known as
Curry-Howard-Isomorphism or propositions-as-types principle). The resulting
algebra of interface types aims to provide a general setting for specifying
type-directed and compositional analyses of worst-case scheduling bounds. It
covers synchronous control flow under concurrent, multi-processing or
multi-threading execution and permits precise statements about exactness and
coverage of the analyses supporting a variety of abstractions. The paper
illustrates the expressiveness of the algebra by way of some examples taken
from network flow problems, shortest-path, task scheduling and worst-case
reaction times in synchronous programming.Comment: In Proceedings FIT 2010, arXiv:1101.426
Virtual Evidence: A Constructive Semantics for Classical Logics
This article presents a computational semantics for classical logic using
constructive type theory. Such semantics seems impossible because classical
logic allows the Law of Excluded Middle (LEM), not accepted in constructive
logic since it does not have computational meaning. However, the apparently
oracular powers expressed in the LEM, that for any proposition P either it or
its negation, not P, is true can also be explained in terms of constructive
evidence that does not refer to "oracles for truth." Types with virtual
evidence and the constructive impossibility of negative evidence provide
sufficient semantic grounds for classical truth and have a simple computational
meaning. This idea is formalized using refinement types, a concept of
constructive type theory used since 1984 and explained here. A new axiom
creating virtual evidence fully retains the constructive meaning of the logical
operators in classical contexts.
Key Words: classical logic, constructive logic, intuitionistic logic,
propositions-as-types, constructive type theory, refinement types, double
negation translation, computational content, virtual evidenc
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Classical logic with Mendler induction
We investigate (co-) induction in classical logic under the propositions-as-types paradigm, considering propositional, second-order and (co-) inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-) iterator and show that it is strongly normalizing. We prove this using a reducibility argument
Logic of Probability and Conjecture
I introduce a formalization of probability which takes the concept of 'evidence' as primitive. In parallel to the intuitionistic conception of truth, in which 'proof' is primitive and an assertion A is judged to be true just in case there is a proof witnessing it, here 'evidence' is primitive and A is judged to be probable just in case there is evidence supporting it. I formalize this outlook by representing propositions as types in Martin-Lof type theory (MLTT) and defining a 'probability type' on top of the existing machinery of MLTT, whose inhabitants represent pieces of evidence in favor of a proposition. One upshot of this approach is the potential for a mathematical formalism which treats 'conjectures' as mathematical objects in their own right. Other intuitive properties of evidence occur as theorems in this formalism
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Classical Logic with Mendler Induction
We investigate (co-)induction in Classical Logic under the propositions-as-types paradigm, considering propositional, second-order, and (co-)inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-)iterator that remains strongly normalizing under head reduction. We prove this using a non-constructive realizability argument.This is the author accepted manuscript. The final version is available from Springer via http://dx.doi.org/10.1007/978-3-319-27683-0_
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