Safe Recursion on Notation into a Light Logic by Levels

We embed Safe Recursion on Notation (SRN) into Light Affine Logic by Levels (LALL), derived from the logic L4. LALL is an intuitionistic deductive system, with a polynomial time cut elimination strategy. The embedding allows to represent every term t of SRN as a family of proof nets |t|^l in LALL. Every proof net |t|^l in the family simulates t on arguments whose bit length is bounded by the integer l. The embedding is based on two crucial features. One is the recursive type in LALL that encodes Scott binary numerals, i.e. Scott words, as proof nets. Scott words represent the arguments of t in place of the more standard Church binary numerals. Also, the embedding exploits the "fuzzy" borders of paragraph boxes that LALL inherits from L4 to "freely" duplicate the arguments, especially the safe ones, of t. Finally, the type of |t|^l depends on the number of composition and recursion schemes used to define t, namely the structural complexity of t. Moreover, the size of |t|^l is a polynomial in l, whose degree depends on the structural complexity of t. So, this work makes closer both the predicative recursive theoretic principles SRN relies on, and the proof theoretic one, called /stratification/, at the base of Light Linear Logic

Quantum Programming Made Easy

We present IQu, namely a quantum programming language that extends Reynold's Idealized Algol, the paradigmatic core of Algol-like languages. IQu combines imperative programming with high-order features, mediated by a simple type theory. IQu mildly merges its quantum features with the classical programming style that we can experiment through Idealized Algol, the aim being to ease a transition towards the quantum programming world. The proposed extension is done along two main directions. First, IQu makes the access to quantum co-processors by means of quantum stores. Second, IQu includes some support for the direct manipulation of quantum circuits, in accordance with recent trends in the development of quantum programming languages. Finally, we show that IQu is quite effective in expressing well-known quantum algorithms.Comment: In Proceedings Linearity-TLLA 2018, arXiv:1904.0615

A Certified Study of a Reversible Programming Language

We advance in the study of the semantics of Janus, a C-like reversible programming language. Our study makes utterly explicit some backward and forward evaluation symmetries. We want to deepen mathematical knowledge about the foundations and design principles of reversible computing and programming languages. We formalize a big-step operational semantics and a denotational semantics of Janus. We show a full abstraction result between the operational and denotational semantics. Last, we certify our results by means of the proof assistant Matita