5,701 research outputs found
Logical relations for coherence of effect subtyping
A coercion semantics of a programming language with subtyping is typically
defined on typing derivations rather than on typing judgments. To avoid
semantic ambiguity, such a semantics is expected to be coherent, i.e.,
independent of the typing derivation for a given typing judgment. In this
article we present heterogeneous, biorthogonal, step-indexed logical relations
for establishing the coherence of coercion semantics of programming languages
with subtyping. To illustrate the effectiveness of the proof method, we develop
a proof of coherence of a type-directed, selective CPS translation from a typed
call-by-value lambda calculus with delimited continuations and control-effect
subtyping. The article is accompanied by a Coq formalization that relies on a
novel shallow embedding of a logic for reasoning about step-indexing
Tracking Data-Flow with Open Closure Types
Type systems hide data that is captured by function closures in function
types. In most cases this is a beneficial design that favors simplicity and
compositionality. However, some applications require explicit information about
the data that is captured in closures. This paper introduces open closure
types, that is, function types that are decorated with type contexts. They are
used to track data-flow from the environment into the function closure. A
simply-typed lambda calculus is used to study the properties of the type theory
of open closure types. A distinctive feature of this type theory is that an
open closure type of a function can vary in different type contexts. To present
an application of the type theory, it is shown that a type derivation
establishes a simple non-interference property in the sense of information-flow
theory. A publicly available prototype implementation of the system can be used
to experiment with type derivations for example programs.Comment: Logic for Programming Artificial Intelligence and Reasoning (2013
Kripke Semantics for Martin-L\"of's Extensional Type Theory
It is well-known that simple type theory is complete with respect to
non-standard set-valued models. Completeness for standard models only holds
with respect to certain extended classes of models, e.g., the class of
cartesian closed categories. Similarly, dependent type theory is complete for
locally cartesian closed categories. However, it is usually difficult to
establish the coherence of interpretations of dependent type theory, i.e., to
show that the interpretations of equal expressions are indeed equal. Several
classes of models have been used to remedy this problem. We contribute to this
investigation by giving a semantics that is standard, coherent, and
sufficiently general for completeness while remaining relatively easy to
compute with. Our models interpret types of Martin-L\"of's extensional
dependent type theory as sets indexed over posets or, equivalently, as
fibrations over posets. This semantics can be seen as a generalization to
dependent type theory of the interpretation of intuitionistic first-order logic
in Kripke models. This yields a simple coherent model theory, with respect to
which simple and dependent type theory are sound and complete
A dependent nominal type theory
Nominal abstract syntax is an approach to representing names and binding
pioneered by Gabbay and Pitts. So far nominal techniques have mostly been
studied using classical logic or model theory, not type theory. Nominal
extensions to simple, dependent and ML-like polymorphic languages have been
studied, but decidability and normalization results have only been established
for simple nominal type theories. We present a LF-style dependent type theory
extended with name-abstraction types, prove soundness and decidability of
beta-eta-equivalence checking, discuss adequacy and canonical forms via an
example, and discuss extensions such as dependently-typed recursion and
induction principles
Liquid Intersection Types
We present a new type system combining refinement types and the
expressiveness of intersection type discipline. The use of such features makes
it possible to derive more precise types than in the original refinement
system. We have been able to prove several interesting properties for our
system (including subject reduction) and developed an inference algorithm,
which we proved to be sound.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
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