DAL LAGO , Ugo

We introduce a linear infinitary λ-calculus, called Λ∞, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively. The obtained calculus embeds the infinitary applica-tive λ-calculus and is universal for computations over infinite strings. What is...

Applicative bisimulation is a coinductive technique to check program equivalence in higher-order functional languages. It is known to be sound — and sometimes complete — with respect to context equivalence. In this paper we show that applicative bisimulation also works when the underlying language of programs takes the form of a linear...

Applicative bisimiliarity is a coinductively-defined program equivalence in which programs are tested as argument-passing processes. Starting with the seminal work by Abramsky, applicative bisimiliarity has been proved to be a powerful technique for higher-order program equivalence. Recently, applicative bisimiliarity has also been generalised...

This paper introduces a new methodology for the complexity analysis of higher-order functional programs, which is based on three ingredients: a powerful type system for size analysis and a sound type inference procedure for it, a ticking monadic transformation, and constraint solving. Noticeably, the presented methodology can be fully...

Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomial overhead in time. Is λ-calculus a reasonable machine? Is there a way to measure the computational complexity of a λ-term? This paper presents the first complete positive answer to this long-standing problem. Moreover , our...

In this work we study randomised reduction strategies-a notion already known in the context of abstract reduction systems-for the λ-calculus. We develop a simple framework that allows us to prove a randomised strategy to be positive almost-surely normalising. Then we propose a simple example of randomised strategy for the λ-calculus that has...

Polynomial interpretations and their generalizations like quasi-interpretations have been used in the setting of first-order functional languages to design criteria ensuring statically some complexity bounds on programs [10]. This fits in the area of implicit computational complexity, which aims at giving machine-free characterizations of...

We define two intersection type systems for the pure, un-typed, probabilistic λ-calculus, and prove that type derivations precisely reflect the probability of convergence of the underlying term. We first define a simple system of oracle intersection types in which derivations are annotated by binary strings and the probability of termination...

Graded modal types systems and coeffects are becoming a standard formalism to deal with context-dependent, usage-sensitive computations, especially when combined with computational effects. From a semantic perspective, effectful and coeffectful languages have been studied mostly by means of denotational semantics and almost nothing has been...

Terms of Church's λ-calculus can be considered equivalent along many different definitions, but context equivalence is certainly the most direct and universally accepted one. If the underlying calculus becomes probabilistic, however, equivalence is too discriminating: terms which have totally unrelated behaviours are treated the same as terms...

Interactive behaviors are ubiquitous in modern cryptography, but are also present in λ-calculi, in the form of higher-order constructions. Traditionally, however, typed λ-calculi simply do not fit well into cryptography , being both deterministic and too powerful as for the complexity of functions they can express. We study interaction in a...

We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed value has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified...

Normal form bisimulation, also known as open bisimulation, is a coinductive technique for higher-order program equivalence in which programs are compared by looking at their essentially infinitary tree-like normal forms, i.e. at their Böhm or Lévy-Longo trees. The technique has been shown to be useful not only when proving metatheorems about...

Semantics is traditionally concerned with program equivalence, in which all pairs of programs which are not equivalent are treated the same, and simply dubbed as incomparable. In recent years, various forms of program metrics have been introduced such that the distance between non-equivalent programs is measured as an element of an appropriate...

We investigate program equivalence for linear higher-order (sequential) languages endowed with primitives for computational effects. More specifically, we study operationally-based notions of program equivalence for a linear λ-calculus with explicit copying and algebraic effects à la Plotkin and Power. Such a calculus makes explicit the...

A system of session types is introduced as induced by a Curry Howard correspondence applied to bounded linear logic, suitably extended with probabilistic choice operators and ground types. The resulting system satisfies some expected properties, like subject reduction and progress, but also unexpected ones, like a polynomial bound on the time...

We study the deep relations existing between differential logical relations and incremental computing, by showing how self-differences in the former precisely correspond to derivatives in the latter. We also show how differential logical relations can be seen as a powerful metatheoretical tool in the analysis of incremental computations,...

In any setting in which observable properties have a quantitative flavor, it is natural to compare computational objects by way of metrics rather than equivalences or partial orders. This holds, in particular , for probabilistic higher-order programs. A natural notion of comparison , then, becomes context distance, the metric analogue of...

We introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer,...

We introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer,...

We consider the problem of supporting sublinear space programming in a functional programming language. Writing programs with sublinear space usage often requires one to use special implementation techniques for otherwise easy tasks, e.g. one cannot compose functions directly for lack of space for the intermediate result, but must instead...

We present RSLR, an implicit higher-order characterization of the class PP of those problems which can be decided in probabilistic polynomial time with error probability smaller than 1 /2. Analogously, a (less implicit) characterization of the class BPP can be obtained. RSLR is an extension of Hofmann's SLR with a probabilistic primitive, which...

We give a geometry of interaction model for a typed λ-calculus endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The model is based on the category of measurable spaces and partial measurable functions, and is proved adequate...

We study how the adoption of an evaluation mechanism with sharing and memoization impacts the class of functions which can be computed in polynomial time. We first show how a natural cost model in which lookup for an already computed result has no cost is indeed invariant. As a corollary, we then prove that the most general notion of ramified...

In the realm of quantum computing, circuit description languages represent a valid alternative to traditional QRAM-style languages. They indeed allow for finer control over the output circuit, without sacrificing flexibility nor modularity. We introduce a generalization of the paradigmatic lambda-calculus Proto-Quipper-M, which models the core...