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Alejandro Díaz-Caro

Alejandro Díaz-Caro
CONICET-UBA ICC & UNQ · Instituto de Investigación en Ciencias de la Computación (CONICET-UBA)

PhD in Theoretical Computer Science

About

62
Publications
2,127
Reads
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248
Citations
Introduction
Alejandro Díaz-Caro currently works at the Computer Science Institute (ICC), a shared institute between CONICET and Buenos Aires University. He also holds a tenure track professorship at the National University of Quilmes. Alejandro does research in Theory of Computation and Quantum Computing.
Additional affiliations
April 2018 - present
Instituto de Ciencias de la Computación (CONICET-UBA)
Position
  • Investigador Asistente
Description
  • At CONICET since July 2016. At ICC since April 2018.
July 2016 - March 2018
CONICET/UNQ
Position
  • Investigador Asistente
Description
  • At CONICET since July 2016. Since April 2018 attached to ICC (FCEN-UBA)
August 2014 - present
National University of Quilmes
Position
  • Profesor Adjunto
Education
October 2008 - September 2011
University of Grenoble
Field of study
  • Theoretical Computer Science
March 2000 - December 2007
Rosario National University
Field of study
  • Theoretical Computer Science

Publications

Publications (62)
Preprint
In a recent paper, the $\mathcal L^{\mathcal S}$-calculus has been defined. It is a proof-language for a significant fragment of intuitionistic linear logic. Its main feature is that the linearity properties can be expressed in its syntax, since it has interstitial logical rules whose proof-terms are a sum and a multiplication by scalar. The calcul...
Preprint
Full-text available
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+ y...
Preprint
Full-text available
We prove a linearity theorem for an extension of linear logic with addition and multiplication by a scalar: the proofs of some propositions in this logic are linear in the algebraic sense. This work is part of a wider research program that aims at defining a logic whose proof language is a quantum programming language.
Chapter
We investigate an unsuspected connection between non-harmonious logical connectives, such as Prior’s tonk, and quantum computing. We argue that non-harmonious connectives model the information erasure, the non-reversibility, and the non-determinism that occur, among other places, in quantum measurement. We introduce a propositional logic with a non...
Preprint
Full-text available
On the topic of probabilistic rewriting, there are several works studying both termination and confluence of different systems. While working with a lambda calculus modelling quantum computation, we found a system with probabilistic rewriting rules and strongly normalizing terms. We examine the effect of small modifications in probabilistic rewriti...
Preprint
Full-text available
System I is a simply-typed lambda calculus with pairs, extended with an equational theory obtained from considering the type isomorphisms as equalities. In this work we propose an extension of System I to polymorphic types, adding the corresponding isomorphisms. We provide non-standard proofs of subject reduction and strong normalisation, extending...
Preprint
Full-text available
We investigate an unsuspected connection between non harmonious logical connectives, such as Prior's tonk, and quantum computing. We defend the idea that non harmonious connectives model the information erasure, the non-reversibility, and the non-determinism that occur, among other places, in quantum measurement. More concretely, we introduce a pro...
Preprint
Full-text available
In a recent paper, a realizability technique has been used to give a semantics of a quantum lambda calculus. Such a technique gives rise to an infinite number of valid typing rules, without giving preference to any subset of those. In this paper, we introduce a valid subset of typing rules, defining an expressive enough quantum calculus. Then, we p...
Article
Full-text available
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 s...
Chapter
Full-text available
We introduce a simple extension of the -calculus with pairs—called the distributive -calculus—obtained by adding a computational interpretation of the valid distributivity isomorphism of simple types. We study the calculus both as an untyped and as a simply typed setting. Key features of the untyped calculus are confluence, the absence of clashes o...
Preprint
Full-text available
We revisit the Vectorial Lambda Calculus, a typed version of Lineal. Vectorial (as well as Lineal) was originally meant for quantum computing, as an extension to System F where linear combinations of lambda terms are also terms and linear combinations of types are also types. In its first presentation, Vectorial only provides a weakened version of...
Preprint
Full-text available
We introduce a simple extension of the $\lambda$-calculus with pairs---called the distributive $\lambda$-calculus---obtained by adding a computational interpretation of the valid distributivity isomorphism $A \Rightarrow (B\wedge C)\ \ \equiv\ \ (A\Rightarrow B) \wedge (A\Rightarrow C)$ of simple types. We study the calculus both as an untyped and...
Preprint
Full-text available
System I is a proof language for a fragment of propositional logic where isomorphic propositions, such as $A\wedge B$ and $B\wedge A$, or $A\Rightarrow(B\wedge C)$ and $(A\Rightarrow B)\wedge(A\Rightarrow C)$ are made equal. System I enjoys the strong normalization property. This is sufficient to prove the existence of empty types, but not to prove...
Preprint
Full-text available
In this extended abstract we provide a first step towards a tool to estimate the resource consumption of programs. We specifically focus on the runtime analysis of programs and, inspired by recent methods for probabilistic programs, we develop a calculus \`a la weakest precondition to formally and systematically derive the (exact) runtime of quantu...
Article
Full-text available
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-calcul...
Article
Full-text available
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...
Conference Paper
Full-text available
In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and show how it extends both to classical and q...
Preprint
Full-text available
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 s...
Preprint
Full-text available
In this paper we present a semantics for a linear algebraic lambda-calculus based on realizability. This semantics characterizes a notion of unitarity in the system, answering a long standing issue. We derive from the semantics a set of typing rules for a simply-typed linear algebraic lambda-calculus, and show how it extends both to classical and q...
Preprint
Full-text available
In this paper we introduce classically time-controlled quantum automata or CTQA, which is a slight but reasonable modification of Moore-Crutchfield quantum finite automata that uses time-dependent evolution operators and a scheduler defining how long each operator will run. Surprisingly enough, time-dependent evolutions provide a significant change...
Preprint
Full-text available
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...
Conference Paper
Full-text available
In this paper we present two flavors of a quantum extension to the lambda calculus. The first one, λρ, follows the approach of classical control/quantum data, where the quantum data is represented by density matrices. We provide an interpretation for programs as density matrices and functions upon them. The second one, λ • ρ , takes advantage of th...
Conference Paper
Full-text available
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of unicity 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 unicity and further properties. We then carry over several criteria from the...
Article
Full-text available
(TO APPEAR IN TPNC 2017) We study a purely functional quantum extension of lambda calculus, that is, an extension of lambda calculus to express some quantum features, where the quantum memory is abstracted out. This calculus is a typed extension of the first-order linear-algebraic lambda-calculus. The type is linear on superpositions, so to forbid...
Article
Full-text available
Driven by the interest of reasoning about probabilistic programming languages, we set out to study a notion of unicity 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 unicity (even for infinite sequences of reduction) and further properties. We...
Article
We describe a type system for the linear-algebraic λ-calculus. The type system accounts for the linear-algebraic aspects of this extension of λ-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 term...
Article
Full-text available
This paper deals with retraction - intended as isomorphic embedding - in intersection types building left and right inverses as terms of a lambda calculus with a bottom constant. The main result is a necessary and sufficient condition two strict intersection types must satisfy in order to assure the existence of two terms showing the first type to...
Conference Paper
Full-text available
We introduce a quantum-like classical computational model, called affine computation, as a generalization of probabilistic computation. After giving the basics of affine computation, we define affine finite automata (AfA) and compare it with quantum and probabilistic finite automata (QFA and PFA, respectively) with respect to three basic language r...
Article
Full-text available
We propose an implementation of lambda+, a recently introduced simply typed lambda-calculus with pairs where isomorphic types are made equal. The rewrite system of lambda+ is a rewrite system modulo an equivalence relation, which makes its implementation non-trivial. We also extend lambda+ with natural numbers and general recursion and use Beki\'c'...
Article
Full-text available
We define a simply typed, non-deterministic lambda-calculus where isomorphic types are equated. To this end, an equivalence relation is settled at the term level. We then provide a proof of strong normalisation modulo equivalence. Such a proof is a non-trivial adaptation of the reducibility method.
Article
Full-text available
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...
Article
Full-text available
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 recent...
Technical Report
Full-text available
We provide a proof of strong normalisation for lambda+, a recently introduced, explicitly typed, non-deterministic lambda-calculus where isomorphic propositions are identified. Such a proof is a non-trivial adaptation of the reducibility candidates technique.
Article
We define an equivalence relation on propositions and a proof system where equivalent propositions have the same proofs. The system obtained this way resembles several known non-deterministic and algebraic lambda-calculi.
Conference Paper
Full-text available
We consider the call-by-value lambda-calculus extended with a may-convergent non-deterministic choice and a must-convergent parallel composition. Inspired by recent works on the relational semantics of linear logic and non-idempotent intersection types, we endow this calculus with a type system based on the so-called Girard's second translation of...
Article
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as...
Conference Paper
Full-text available
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...
Article
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...
Article
Full-text available
The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend the lambda-calculus with the possibility of making arbitrary linear combinations of terms. In this paper we provide a fine-grained, System F-like type system for the linear-algebraic lambda-calculus. We show that this "scalar" type system enjoys both the subject-reduction...
Thesis
Full-text available
The objective of this thesis is to develop a type theory for the linear-algebraic λ-calculus, an extension of λ-calculus motivated by quantum computing. This algebraic extension encompass all the terms of λ-calculus together with their linear combinations, so if t and r are two terms, so is α.t + β.r, with α and β being scalars from a given ring. T...
Conference Paper
Full-text available
The objective of this thesis is to develop a type theory for the linear-algebraic λ-calculus, an extension of λ-calculus motivated by quantum computing. This algebraic extension encompass all the terms of λ-calculus together with their linear combinations, so if t and r are two terms, so is α.t + β.r, with α and β being scalars from a given ring. T...
Conference Paper
Full-text available
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as...
Article
Full-text available
This paper demonstrates how to add a measurement operator to quantum λ-calculi. A proof of the consistency of the semantics is given through a proof of confluence presented in a sufficiently general way to allow this technique to be used for other languages. The method described here may be applied to probabilistic rewrite systems in general, and t...
Article
Full-text available
The Linear-Algebraic λ-Calculus [Arrighi, P. and G. Dowek, Linear-algebraic λ-calculus: higher-order, encodings and confluence, Lecture Notes in Computer Science (RTA'08) 5117 (2008), pp. 17–31] extends the λ-calculus with the possibility of making arbitrary linear combinations of terms α.t+β.u. Since one can express fixed points over sums in this...
Conference Paper
Full-text available
Conference Paper
Full-text available
In this paper we define the confluent additive fragment of the linear-algebraic lambda-calculus. We also define a fine-grained type system which includes sums of types as a reflection of those in the terms. After proving the subject reduction and strong normalisation properties, we study the role of sums within the calculus by interpreting our syst...
Conference Paper
Full-text available
We examine the relationship between the algebraic lambda-calculus Lalg, a fragment of the differential lambda-calculus, and the linear-algebraic lambda-calculus Llin, a candidate lambda-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and the set of terms is c...
Conference Paper
Full-text available
We examine the relationship between the algebraic lambda-calculus λalg, a fragment of the differential lambda-calculus, and the linear-algebraic lambda-calculus λlin, a candidate lambda-calculus for quantum computation. Both calculi are algebraic: each one is equipped with an additive and a scalar-multiplicative structure, and the set of terms is c...

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