arXiv subjects

Alejandro Díaz-Caro

Publications and source records attributed to Alejandro Díaz-Caro.

At least 37 records · Page 2Linked to original sources

arXiv2023

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 $A\wedge B$ 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 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 $η$-expansion rules to System I permits to drop this restriction, and yields a strongly normalizing calculus which enjoys the full introduction property.

cs.LO↗

arXiv2023

A New Connective in Natural Deduction, and its Application to Quantum Computing

We investigate an unsuspected connection between logical connectives with non-harmonious deduction rules, such as Prior's tonk, and quantum computing. We argue that these connectives model the information-erasure, the non-reversibility, and the non-determinism that occur, among other places, in quantum measurement. We introduce an intuitionistic propositional logic with a non-harmonious logical connective sup and two interstitial rules, and show that the proof language of this logic forms the core of a quantum programming language.

cs.LO↗

arXiv2022

Quantum Control in the Unitary Sphere: Lambda-S1 and its Categorical Model

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 propose a categorical semantics for it. Such a semantics consists of an adjunction between the category of distributive-action spaces of value distributions (that is, linear combinations of values in the lambda calculus), and the category of sets of value distributions.

cs.LO↗

arXiv2022

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.

cs.LO↗

arXiv2022

A Note on Confluence in Typed Probabilistic Lambda Calculi

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 rewriting, affine variables, and strategies on the overall confluence in this strongly normalizing probabilistic calculus.

cs.LO↗

arXiv2021

Polymorphic System I

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 those of System I.

cs.LO↗

arXiv2021

The Vectorial Lambda Calculus Revisited

We revisit the Vectorial Lambda Calculus, a typed version of Lineal. Vectorial (as well as Lineal) has been originally designed 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 the Subject Reduction property. We prove that our revised Vectorial Lambda Calculus supports the standard version of said property, answering a long standing issue. In addition we also introduce the concept of weight of types and terms, and prove a relation between the weight of terms and of its types.

cs.LO↗

arXiv2020

Functional Pearl: The Distributive $λ$-Calculus

We introduce a simple extension of the $λ$-calculus with pairs---called the distributive $λ$-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 as a simply typed setting. Key features of the untyped calculus are confluence, the absence of clashes of constructs, that is, evaluation never gets stuck, and a leftmost-outermost normalization theorem, obtained with straightforward proofs. With respect to simple types, we show that the new rules satisfy subject reduction if types are considered up to the distributivity isomorphism. The main result is strong normalization for simple types up to distributivity. The proof is a smooth variation over the one for the $λ$-calculus with pairs and simple types.

cs.LO↗

arXiv2020

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.

cs.LO↗

arXiv2019

Runtime Analysis of Quantum Programs: A Formal Approach

In this abstract we study the resource consumption of quantum programs. Specifically, we focus on the expected runtime of programs and, inspired by recent methods for probabilistic programs, we develop a calculus à la weakest precondition to formally and systematically derive the (exact) expected runtime of quantum programs. Notably, the calculus admits a notion of loop runtime invariant that can be readily used to derive upper bounds of their runtime. Finally, we show the applicability of our calculus analyzing the runtime of (a simplified version of) the BB84 quantum key distribution protocol.

cs.LO↗

arXiv2019

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.

cs.LO↗

arXiv2019

Proof Normalisation in a Logic Identifying Isomorphic Propositions

We define a fragment of propositional logic where isomorphic propositions, such as $A\land B$ and $B\land A$, or $A\Rightarrow (B\land C)$ and $(A\Rightarrow B)\land(A\Rightarrow C)$ are identified. We define System I, a proof language for this logic, and prove its normalisation and consistency.

cs.LO↗

arXiv2019

Realizability in the Unitary Sphere

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 quantum lambda-calculi.

cs.LO↗

arXiv2018

Confluence in Probabilistic Rewriting

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 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 $λ_1$, an affine probabilistic $λ$-calculus, and for Q$^*$, a quantum programming language for which a related property has already been proven in the literature.

cs.LO↗

arXiv2017

A lambda calculus for density matrices with classical and probabilistic controls

In this paper we present two flavors of a quantum extension to the lambda calculus. The first one, $λ_ρ$, 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, $λ_ρ^\circ$, take advantage of the density matrices presentation in order to follow the mixed trace of programs in a kind of generalised density matrix. Such a control can be seen as a weaker form of the quantum control and data approach.

cs.LO↗

arXiv2017

The Vectorial $λ$-Calculus

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 terms, can be superposed into linear combinations. We prove that the resulting typed $λ$-calculus is strongly normalising and features weak subject reduction. Finally, we show how to naturally encode matrices and vectors in this typed calculus.

cs.LO↗

arXiv2017

Retractions in Intersection Types

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 be a retract of the second one. Moreover, the characterisation of retraction in the standard intersection types is discussed.

cs.LO↗

arXiv2016

Affine computation and affine automaton

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 recognition modes. We show that, in the cases of bounded and unbounded error, AfAs are more powerful than QFAs and PFAs, and, in the case of nondeterministic computation, AfAs are more powerful than PFAs but equivalent to QFAs.

cs.FL↗