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Substructural type systems, such as affine (and linear) type systems, are type systems which impose restrictions on copying (and discarding) of variables, and they have found many applications in computer science, including quantum…
Classical programming languages cannot model essential elements of complex systems such as true random number generation. This paper develops a formal programming language called the lambda-q calculus that addresses the fundamental…
We refine a model for linear logic based on two well-known ingredients: games and simulations. We have already shown that usual simulation relations form a sound notion of morphism between games; and that we can interpret all linear logic…
Typically, quantum mechanics is thought of as a linear theory with unitary evolution governed by the Schr\"odinger equation. While this is technically true and useful for a physicist, with regards to computation it is an unfortunately…
We introduce $\mathsf{LEM}$, a type-assignment system for the linear $ \lambda $-calculus that extends second-order $\mathsf{IMLL}_2$, i.e., intuitionistic multiplicative Linear Logic, by means of logical rules that weaken and contract…
Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambda-calculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms…
This paper discusses the no-cloning theorem in a logico-algebraic approach. In this approach, an orthoalgebra is considered as a general structure for propositions in a physical theory. We proved that an orthoalgebra admits cloning…
In this paper, a new calculus on sequences is defined. Also, the $\lambda$-derivative and the $\lambda$-integration are investigated. The fundamental theorem of $\lambda$-calculus is included. A suitable function basis for the…
Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was innovative in many ways, notably as a precursor of linear logic. But it also showed that we could treat our grammatical framework as a logic (as opposed to a…
We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive…
An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between $\lambda$-symmetries and first integrals. This relation further leads to a new $\lambda$-symmetry linearization…
The lambda calculus with constructors is an extension of the lambda calculus with variadic constructors. It decomposes the pattern-matching a la ML into a case analysis on constants and a commutation rule between case and application…
We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated…
A genoid is a category of two objects such that one is the product of itself with the other. A genoid may be viewed as an abstract substitution algebra. It is a remarkable fact that such a simple concept can be applied to present a unified…
We introduce LAM, a subsystem of IMALL2 with restricted additive rules able to manage duplication linearly, called linear additive rules. LAM is presented as the type assignment system for a calculus endowed with copy constructors, which…
We study an untyped lambda calculus with quantum data and classical control. This work stems from previous proposals by Selinger and Valiron and by Van Tonder. We focus on syntax and expressiveness, rather than (denotational) semantics. We…
In this paper we attempt to consider quantum superpositions from the perspective of the logos categorical approach presented in [26]. We will argue that our approach allows us not only to better visualize the structural features of quantum…
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…
This work exploits the logical foundation of session types to determine what kind of type discipline for the pi-calculus can exactly capture, and is captured by, lambda-calculus behaviours. Leveraging the proof theoretic content of the…
We introduce a proof language for Intuitionistic Multiplicative Additive Linear Logic (IMALL), extended with a modality B to capture mixed-state quantum computation. The language supports algebraic constructs such as linear combinations,…