10 research outputs found
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
Proof Normalisation in a Logic Identifying Isomorphic Propositions
We define a fragment of propositional logic where isomorphic propositions,
such as and , or and
are identified. We define System I, a
proof language for this logic, and prove its normalisation and consistency
A Concrete Categorical Semantics of Lambda-S
Lambda-S is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi. One is to forbid duplication of variables, while the other is to consider all lambda-terms as algebraic linear functions. The type system of Lambda-S have a constructor S such that a type A is considered as the base of a vector space while S(A) is its span. A first semantics of this calculus have been given when first presented, with such an interpretation: superposed types are interpreted as vectors spaces while non-superposed types as their basis. In this paper we give a concrete categorical semantics of Lambda-S, showing that S is interpreted 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.Fil: DÃaz Caro, Alejandro. Universidad Nacional de Quilmes; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; ArgentinaFil: Malherbe, Octavio. Universidad de la República; Urugua
A Quick Overview on the Quantum Control Approach to the Lambda Calculus
In this short overview, we start with the basics of quantum computing,
explaining the difference between the quantum and the classical control
paradigms. We give an overview of the quantum control line of research within
the lambda calculus, ranging from untyped calculi up to categorical and
realisability models. This is a summary of the last 10+ years of research in
this area, starting from Arrighi and Dowek's seminal work until today.Comment: In Proceedings LSFA 2021, arXiv:2204.0341
Extensional proofs in a propositional logic modulo isomorphisms
System I is a proof language for a fragment of propositional logic where
isomorphic propositions, such as and , or
and are made
equal. System I enjoys the strong normalisation property. This is sufficient to
prove the existence of empty types, but not to prove the introduction property
(every closed term in normal form is an introduction). Moreover, a severe
restriction had to be made on the types of the variables in order to obtain the
existence of empty types. We show here that adding -expansion rules to
System I permits to drop this restriction, and yields a strongly normalising
calculus with enjoying the full introduction property.Comment: 15 pages plus references and appendi
The Vectorial -Calculus
We describe a type system for the linear-algebraic -calculus. The
type system accounts for the linear-algebraic aspects of this extension of
-calculus: it is able to statically describe the linear combinations
of terms that will be obtained when reducing the programs. This gives rise to
an original type theory where types, in the same way as terms, can be
superposed into linear combinations. We prove that the resulting typed
-calculus is strongly normalising and features weak subject reduction.
Finally, we show how to naturally encode matrices and vectors in this typed
calculus.Comment: Long and corrected version of arXiv:1012.4032 (EPTCS 88:1-15), to
appear in Information and Computatio
Two linearities for quantum computing in the lambda calculus
We propose a way to unify two approaches of non-cloning in quantum
lambda-calculi: logical and algebraic linearities. 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.Comment: Long journal version of TPNC'17 paper
(doi:10.1007/978-3-319-71069-3_22) extended with third author's
"Licenciatura"'s thesi
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
Call-by-value, call-by-name and the vectorial behaviour of the algebraic \lambda-calculus
We examine the relationship between the algebraic lambda-calculus, a fragment
of the differential lambda-calculus and the linear-algebraic lambda-calculus, a
candidate lambda-calculus for quantum computation. Both calculi are algebraic:
each one is equipped with an additive and a scalar-multiplicative structure,
and their set of terms is closed under linear combinations. However, the two
languages were built using different approaches: the former is a call-by-name
language whereas the latter is call-by-value; the former considers algebraic
equalities whereas the latter approaches them through rewrite rules. In this
paper, we analyse how these different approaches relate to one another. To this
end, we propose four canonical languages based on each of the possible choices:
call-by-name versus call-by-value, algebraic equality versus algebraic
rewriting. We show that the various languages simulate one another. Due to
subtle interaction between beta-reduction and algebraic rewriting, to make the
languages consistent some additional hypotheses such as confluence or
normalisation might be required. We carefully devise the required properties
for each proof, making them general enough to be valid for any sub-language
satisfying the corresponding properties