2,751 research outputs found
A deep quantitative type system
We investigate intersection types and resource lambda-calculus in deep-inference proof theory. We give a unified type system that is parametric in various aspects: it encompasses resource calculi, intersection-typed lambda-calculus, and simply-typed lambda-calculus; it accommodates both idempotence and non-idempotence; it characterizes strong and weak normalization; and it does so while allowing a range of algebraic laws to determine reduction behaviour, for various quantitative effects. We give a parametric resource calculus with explicit sharing, the "collection calculus", as a Curry-Howard interpretation of the type system, that embodies these computational properties
A concrete model for a typed linear algebraic lambda calculus
We give an adequate, concrete, categorical-based model for Lambda-S, which is
a typed version of a linear-algebraic lambda calculus, extended with
measurements. Lambda-S is an extension to first-order lambda calculus unifying
two approaches of non-cloning in quantum lambda-calculi: to forbid duplication
of variables, and to consider all lambda-terms as algebraic linear functions.
The type system of Lambda-S have a superposition constructor S such that a type
A is considered as the base of a vector space while SA is its span. Our model
considers S as the composition of two functors in an adjunction relation
between the category of sets and the category of vector spaces over C. The
right adjoint is a forgetful functor U, which is hidden in the language, and
plays a central role in the computational reasoning.Comment: Extended revisited version of ENTCS 344:83-100, 201
Semantics of a Typed Algebraic Lambda-Calculus
Algebraic lambda-calculi have been studied in various ways, but their
semantics remain mostly untouched. In this paper we propose a semantic analysis
of a general simply-typed lambda-calculus endowed with a structure of vector
space. We sketch the relation with two established vectorial lambda-calculi.
Then we study the problems arising from the addition of a fixed point
combinator and how to modify the equational theory to solve them. We sketch an
algebraic vectorial PCF and its possible denotational interpretations
Introducing a Calculus of Effects and Handlers for Natural Language Semantics
In compositional model-theoretic semantics, researchers assemble
truth-conditions or other kinds of denotations using the lambda calculus. It
was previously observed that the lambda terms and/or the denotations studied
tend to follow the same pattern: they are instances of a monad. In this paper,
we present an extension of the simply-typed lambda calculus that exploits this
uniformity using the recently discovered technique of effect handlers. We prove
that our calculus exhibits some of the key formal properties of the lambda
calculus and we use it to construct a modular semantics for a small fragment
that involves multiple distinct semantic phenomena
Typing Quantum Superpositions and Measurement
We propose a way to unify two approaches of non-cloning in quantum lambda-calculi. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as algebraic-linear functions. We illustrate this idea by defining a quantum extension of first-order simply-typed lambda-calculus, where the type is linear on superposition, while allows cloning base vectors. In addition, we provide an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis.Fil: DĂaz Caro, Alejandro. Universidad Nacional de Quilmes. Departamento de Ciencia y TecnologĂa; Argentina. Consejo Nacional de Investigaciones CientĂficas y TĂ©cnicas; ArgentinaFil: Dowek, Gilles. Institut National de Recherche en Informatique et en Automatique; Franci
Adequate encodings of logical systems in UTT
In this paper, we present an existing and formalized type theory
(UTT) as a logical framework.
We compare the resulting framework with LF and give
the representation of two significant type systems in
the framework: the typed lambda calculus which is closely related
to higher-order logic and a linear type system which is not
possible to encode in LF.Postprint (published version
QPCF: higher order languages and quantum circuits
qPCF is a paradigmatic quantum programming language that ex- tends PCF with
quantum circuits and a quantum co-processor. Quantum circuits are treated as
classical data that can be duplicated and manipulated in flexible ways by means
of a dependent type system. The co-processor is essentially a standard QRAM
device, albeit we avoid to store permanently quantum states in between two
co-processor's calls. Despite its quantum features, qPCF retains the classic
programming approach of PCF. We introduce qPCF syntax, typing rules, and its
operational semantics. We prove fundamental properties of the system, such as
Preservation and Progress Theorems. Moreover, we provide some higher-order
examples of circuit encoding
On Uniquely Closable and Uniquely Typable Skeletons of Lambda Terms
Uniquely closable skeletons of lambda terms are Motzkin-trees that
predetermine the unique closed lambda term that can be obtained by labeling
their leaves with de Bruijn indices. Likewise, uniquely typable skeletons of
closed lambda terms predetermine the unique simply-typed lambda term that can
be obtained by labeling their leaves with de Bruijn indices.
We derive, through a sequence of logic program transformations, efficient
code for their combinatorial generation and study their statistical properties.
As a result, we obtain context-free grammars describing closable and uniquely
closable skeletons of lambda terms, opening the door for their in-depth study
with tools from analytic combinatorics.
Our empirical study of the more difficult case of (uniquely) typable terms
reveals some interesting open problems about their density and asymptotic
behavior.
As a connection between the two classes of terms, we also show that uniquely
typable closed lambda term skeletons of size are in a bijection with
binary trees of size .Comment: Pre-proceedings paper presented at the 27th International Symposium
on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur,
Belgium, 10-12 October 2017 (arXiv:1708.07854
Confluence via strong normalisation in an algebraic \lambda-calculus with rewriting
The linear-algebraic lambda-calculus and the algebraic lambda-calculus are
untyped lambda-calculi extended with arbitrary linear combinations of terms.
The former presents the axioms of linear algebra in the form of a rewrite
system, while the latter uses equalities. When given by rewrites, algebraic
lambda-calculi are not confluent unless further restrictions are added. We
provide a type system for the linear-algebraic lambda-calculus enforcing strong
normalisation, which gives back confluence. The type system allows an abstract
interpretation in System F.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Linearity in the non-deterministic call-by-value setting
We consider the non-deterministic extension of the call-by-value lambda
calculus, which corresponds to the additive fragment of the linear-algebraic
lambda-calculus. We define a fine-grained type system, capturing the right
linearity present in such formalisms. After proving the subject reduction and
the strong normalisation properties, we propose a translation of this calculus
into the System F with pairs, which corresponds to a non linear fragment of
linear logic. The translation provides a deeper understanding of the linearity
in our setting.Comment: 15 pages. To appear in WoLLIC 201
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