137 research outputs found
The probability of non-confluent systems
We show how to provide a structure of probability space to the set of
execution traces on a non-confluent abstract rewrite system, by defining a
variant of a Lebesgue measure on the space of traces. Then, we show how to use
this probability space to transform a non-deterministic calculus into a
probabilistic one. We use as example Lambda+, a recently introduced calculus
defined through type isomorphisms.Comment: In Proceedings DCM 2013, arXiv:1403.768
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 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
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
Confluence in Probabilistic Rewriting
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of uniqueness of normal forms for them. To provide a tractable proof method for it, we define a property of distribution confluence which is shown to imply the desired uniqueness (even for infinite sequences of reduction) and further properties. We then carry over several criteria from the classical case, such as Newman's lemma, to simplify proving confluence in concrete languages. Using these criteria, we obtain simple proofs of confluence for λ1, an affine probabilistic λ-calculus, and for Q*, a quantum programming language for which a related property has already been proven in the literature.Fil: DÃaz Caro, Alejandro. Universidad Nacional de Quilmes; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; ArgentinaFil: MartÃnez, Guido. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentin
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 categorical construction for the computational definition of vector spaces
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 has a constructor S such that a type A is considered as the base of a vector space while S(A) is its span. Lambda-S can also be seen as a language for the computational manipulation of vector spaces: The vector spaces axioms are given as a rewrite system, describing the computational steps to be performed. In this paper we give an abstract categorical semantics of Lambda-S∗ (a fragment of Lambda-S), showing that S can be interpreted as the composition of two functors in an adjunction relation between a Cartesian category and an additive symmetric monoidal category. 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 Republica. Facultad de IngenierÃa; Urugua
Classically Time-Controlled Quantum Automata: Definition and Properties
In this paper we introduce classically time-controlled quantum automata or
CTQA, which is a reasonable modification of Moore-Crutchfield quantum finite
automata that uses time-dependent evolution and a "scheduler" defining how long
each Hamiltonian will run. Surprisingly enough, time-dependent evolution
provides a significant change in the computational power of quantum automata
with respect to a discrete quantum model. Indeed, we show that if a scheduler
is not computationally restricted, then a CTQA can decide the Halting problem.
In order to unearth the computational capabilities of CTQAs we study the case
of a computationally restricted scheduler. In particular we showed that
depending on the type of restriction imposed on the scheduler, a CTQA can (i)
recognize non-regular languages with cut-point, even in the presence of
Karp-Lipton advice, and (ii) recognize non-regular languages with
bounded-error. Furthermore, we study the closure of concatenation and union of
languages by introducing a new model of Moore-Crutchfield quantum finite
automata with a rotating tape head. CTQA presents itself as a new model of
computation that provides a different approach to a formal study of "classical
control, quantum data" schemes in quantum computing.Comment: Long revisited version of LNCS 11324:266-278, 2018 (TPNC 2018
- …