Degrees of extensionality in the theory of B\"ohm trees and Sall\'e's conjecture

The main observational equivalences of the untyped lambda-calculus have been characterized in terms of extensional equalities between B\"ohm trees. It is well known that the lambda-theory H*, arising by taking as observables the head normal forms, equates two lambda-terms whenever their B\"ohm trees are equal up to countably many possibly infinite eta-expansions. Similarly, two lambda-terms are equal in Morris's original observational theory H+, generated by considering as observable the beta-normal forms, whenever their B\"ohm trees are equal up to countably many finite eta-expansions. The lambda-calculus also possesses a strong notion of extensionality called "the omega-rule", which has been the subject of many investigations. It is a longstanding open problem whether the equivalence B-omega obtained by closing the theory of B\"ohm trees under the omega-rule is strictly included in H+, as conjectured by Sall\'e in the seventies. In this paper we demonstrate that the two aforementioned theories actually coincide, thus disproving Sall\'e's conjecture. The proof technique we develop for proving the latter inclusion is general enough to provide as a byproduct a new characterization, based on bounded eta-expansions, of the least extensional equality between B\"ohm trees. Together, these results provide a taxonomy of the different degrees of extensionality in the theory of B\"ohm trees

The compressive response of additively-manufactured hollow truss lattices: an experimental investigation

The mechanical response of additively-manufactured hollow truss lattices is experimentally investigated under quasi-static compression testing. Exploiting the recent developments in the Fusing Deposition Modelling (FDM) technique, two families of lattices have been fabricated, obtained as tessellation in space of octet-truss and diamond unit cells. Four specimens for each family of lattices have been designed with prescribed relative density, selecting different inner-to-outer radius ratios r/R of their hollow struts. Compression experiments prove that mechanical properties and failure mechanisms of hollow truss lattices are significantly dependent on the r/R ratio. In particular, a shift from quasi-brittle to ductile mechanical response at increasing r/R values has been revealed for the octet-truss lattice, leading to a stable collapse mechanism and increased energy absorption capacity. On the other hand, a more compliant behaviour has been observed in the diamond lattice response, with a monotonic improvement of mechanical properties as a function of the r/R ratio. Such results substantiate the potentialities of additively-manufactured hollow lattice structures as an attractive solution when lightweight, resistant and efficient energy absorption materials are required. Graphic Abstract: [Figure not available: see fulltext.

Addressing Machines as models of lambda-calculus

Turing machines and register machines have been used for decades in theoretical computer science as abstract models of computation. Also the $\lambda$-calculus has played a central role in this domain as it allows to focus on the notion of functional computation, based on the substitution mechanism, while abstracting away from implementation details. The present article starts from the observation that the equivalence between these formalisms is based on the Church-Turing Thesis rather than an actual encoding of $\lambda$-terms into Turing (or register) machines. The reason is that these machines are not well-suited for modelling \lam-calculus programs. We study a class of abstract machines that we call \emph{addressing machine} since they are only able to manipulate memory addresses of other machines. The operations performed by these machines are very elementary: load an address in a register, apply a machine to another one via their addresses, and call the address of another machine. We endow addressing machines with an operational semantics based on leftmost reduction and study their behaviour. The set of addresses of these machines can be easily turned into a combinatory algebra. In order to obtain a model of the full untyped $\lambda$-calculus, we need to introduce a rule that bares similarities with the $\omega$-rule and the rule $\zeta_\beta$ from combinatory logic

A lightweight BPMN extension for business process-oriented requirements engineering

Process-oriented requirements engineering approaches are often required to deal with the effective adaptation of existing processes in order to easily introduce new or updated requirements. Such approaches are based on the adoption of widely used notations, such as the one introduced by the Business Process Model and Notation (BPMN) standard. However, BPMN models do not convey enough information on the involved entities and how they interact with process activities, thus leading to ambiguities, as well as to incomplete and inconsistent requirements definitions. This paper proposes an approach that allows stakeholders and software analysts to easily merge and integrate behavioral and data properties in a BPMN model, so as to fully exploit the potential of BPMN without incurring into the aforementioned limitation. The proposed approach introduces a lightweight BPMN extension that specifically addresses the annotation of data properties in terms of constraints, i.e., pre- and post-conditions that the different process activities must satisfy. The visual representation of the annotated model conveys all the information required both by stakeholders, to understand and validate requirements, and by software analysts and developers, to easily map these updates to the corresponding software implementation. The presented approach is illustrated by use of two running examples, which have also been used to carry out a preliminary validation activity

Mechanical response of multistable tensegrity-like lattice chains

Recent developments in the quality and accuracy of additive manufacturing have drawn particular attention to metamaterials characterised by a multistable response to achieve exceptional mechanical properties. This work focuses on the design, fabrication, testing, and simulation of tensegrity-like lattice chains accomplishing a multistable behaviour. The chains are composed of chiral tensegrity-like units featuring a highly nonlinear bistable response with compression-twisting coupling. Different chains are designed by exploiting the chirality of the units and realised by the inverted stereolithography technique. Their mechanical response is experimentally characterised, demonstrating the attainment of the desired multistable behaviour. A predictive semi-analytical model is derived to reconstruct the multistable energy landscape and force-vs.-displacement curve of the whole chain. The presented chains may constitute a flexible platform for programmable materials, potentially extending to modular chains also based on other types of tensegrity-like units

Modeling And Design Of Periodic Lattices With Tensegrity Architecture And Highly Nonlinear Response

In recent years, the nonlinear response of tensegrity systems has attracted increasing attention in the study of mechanical metamaterials. It has been shown in the literature that geometry and prestress of an elastic tensegrity structure can be designed to obtain different behaviors: stiffening, softening, and snap-through behavior in statics; propagation of solitary waves in dynamics. However, the realization of tensegrity systems is challenging, because of their prestressed state and the presence of tension-only cable members. A design method for periodic lattices with null prestress and no cables is here proposed, in which the repeating unit is at, or close to, a tensegrity configuration, maintaining the nonlinear types of response aforementioned. These structures can be realized by conventional additive manufacturing techniques, while the static and dynamic response can be predicted by means of stick-and-spring models

Design of piezoelectric lattice metamaterials

Piezoelectric lattice metamaterials are considered. A computationally-effective homogenisation method is developed based on the recent solution to the Saint-Venant problem for general anisotropic piezoelectric cylinders. A publicly available repository of unit cell topologies is used to identify piezoelectric metamaterials with optimal figures of merit

The Unbuilt Musmeci Parabolic Cross Vault Reinvented as a Dry-Masonry Structure

This paper investigates the unbuilt Musmeci parabolic vault, reinventing the original reinforced concrete structure as a dry-masonry vault. In the framework of rigid no-tension constitutive model with no sliding, the equilibrium analysis is conducted with the aim ofevaluating the design thickness of the masonry vault, respecting the original Musmeci shape. A parametric survey is performed to assess the minimum thickness of the vault, and its structural capacity under spreading supports. Attention is focused on the different collapse mechanisms and the corresponding crack patterns. For a better insight into the behaviour of the parabolic vault, the relevant case of the parabolic arch is first analysed and discussed. The numerical results show the feasibility of the project, with a thickness comparable with that proposed by Musmeci