Intersection types and (positive) almost-sure termination

Randomized higher-order computation can be seen as being captured by a λ-calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any λ-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight-not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that

On Probabilistic Applicative Bisimulation and Call-by-Value λ\lambda-Calculi (Long Version)

Probabilistic applicative bisimulation is a recently introduced coinductive methodology for program equivalence in a probabilistic, higher-order, setting. In this paper, the technique is applied to a typed, call-by-value, lambda-calculus. Surprisingly, the obtained relation coincides with context equivalence, contrary to what happens when call-by-name evaluation is considered. Even more surprisingly, full-abstraction only holds in a symmetric setting.Comment: 30 page

Experimental tests on shallow foundations of onshore wind turbine towers

The current effort towards the progressive switch from carbon-based to renewable energy production is leading to a relevant spreading of both on- and off-shore wind turbine towers. Regarding reinforced concrete shallow foundations of onshore wind turbine steel towers, possible reductions of reinforcement may increase their sustainability, speed of erection, and competitiveness. The article presents the results of an experimental program carried out at Politecnico di Milano concerning both cyclic and monotonic loading, simulating extreme wind conditions on 1:15 scaled models of wind turbine steel towers connected by stud bolt adapters to reinforced concrete shallow foundations embedded in a sandy soil. Three couples of foundation specimens were tested with different reinforcement layouts: (a) similar to current praxis, (b) without shear reinforcement, and (c) without shear reinforcement and with 50% of ordinary steel rebars replaced by steel fibers. Additional vertical loads were added to the small-scale models in order to ensure similarity in terms of stresses. The test results allowed to (i) characterize the mechanical behavior of the foundation element considering soil-structure interaction under both service and ultimate load conditions, (ii) assess the foundation failure mode, (iii) highlight the role of each typology of reinforcing bars forming the cage, and (iv) provide hints for the optimization of these latter

Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances (Extended Version)

This paper studies the quantitative refinements of Abramsky's applicative similarity and bisimilarity in the context of a generalisation of Fuzz, a call-by-value $\lambda$-calculus with a linear type system that can express programs sensitivity, enriched with algebraic operations \emph{\`a la} Plotkin and Power. To do so a general, abstract framework for studying behavioural relations taking values over quantales is defined according to Lawvere's analysis of generalised metric spaces. Barr's notion of relator (or lax extension) is then extended to quantale-valued relations adapting and extending results from the field of monoidal topology. Abstract notions of quantale-valued effectful applicative similarity and bisimilarity are then defined and proved to be a compatible generalised metric (in the sense of Lawvere) and pseudometric, respectively, under mild conditions

Cosmic ray short burst observed with the Global Muon Detector Network (GMDN) on June 22, 2015

We analyze the short cosmic ray intensity increase ("cosmic ray burst": CRB) on June 22, 2015 utilizing a global network of muon detectors and derive the global anisotropy of cosmic ray intensity and the density (i.e. the omnidirectional intensity) with 10-minute time resolution. We find that the CRB was caused by a local density maximum and an enhanced anisotropy of cosmic rays both of which appeared in association with Earth's crossing of the heliospheric current sheet (HCS). This enhanced anisotropy was normal to the HCS and consistent with a diamagnetic drift arising from the spatial gradient of cosmic ray density, which indicates that cosmic rays were drifting along the HCS from the north of Earth. We also find a significant anisotropy along the HCS, lasting a few hours after the HCS crossing, indicating that cosmic rays penetrated into the inner heliosphere along the HCS. Based on the latest geomagnetic field model, we quantitatively evaluate the reduction of the geomagnetic cut-off rigidity and the variation of the asymptotic viewing direction of cosmic rays due to a major geomagnetic storm which occurred during the CRB and conclude that the CRB is not caused by the geomagnetic storm, but by a rapid change in the cosmic ray anisotropy and density outside the magnetosphere.Comment: accepted for the publication in the Astrophysical Journa

The Power of Non-Determinism in Higher-Order Implicit Complexity

We investigate the power of non-determinism in purely functional programming languages with higher-order types. Specifically, we consider cons-free programs of varying data orders, equipped with explicit non-deterministic choice. Cons-freeness roughly means that data constructors cannot occur in function bodies and all manipulation of storage space thus has to happen indirectly using the call stack. While cons-free programs have previously been used by several authors to characterise complexity classes, the work on non-deterministic programs has almost exclusively considered programs of data order 0. Previous work has shown that adding explicit non-determinism to cons-free programs taking data of order 0 does not increase expressivity; we prove that this - dramatically - is not the case for higher data orders: adding non-determinism to programs with data order at least 1 allows for a characterisation of the entire class of elementary-time decidable sets. Finally we show how, even with non-deterministic choice, the original hierarchy of characterisations is restored by imposing different restrictions.Comment: pre-edition version of a paper accepted for publication at ESOP'1

Quantitative Behavioural Reasoning for Higher-order Effectful Programs: Applicative Distances

International audienceThis paper studies quantitative refinements of Abramsky's applica-tive similarity and bisimilarity in the context of a generalisation of Fuzz, a call-by-value λ-calculus with a linear type system that can express program sensitivity, enriched with algebraic operations à la Plotkin and Power. To do so a general, abstract framework for studying behavioural relations taking values over quantales is introduced according to Lawvere's analysis of generalised metric spaces. Barr's notion of relator (or lax extension) is then extended to quantale-valued relations, adapting and extending results from the field of monoidal topology. Abstract notions of quantale-valued effectful applicative similarity and bisimilarity are then defined and proved to be a compatible generalised metric (in the sense of Lawvere) and pseudometric, respectively, under mild conditions