Beta Reduction is Invariant, Indeed

International audienceSlot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomially 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 answer is completely machine-independent and based over a standard notion in the theory of λ-calculus: the length of a leftmost-outermost derivation to normal form is an invariant cost model. Such a theorem cannot be proved by directly relating λ-calculus with Turing machines or random access machines, because of the size explosion problem: there are terms that in a linear number of steps produce an exponentially long output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modelled after linear logic and proof-nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that LSC is invariant with respect to the λ-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the λ-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction, deemed useful. Useful evaluation avoids those steps that only unshare the output without contributing to β-redexes, i.e., the steps that cause the blow-up in size

An old threat in a new setting: High prevalence of silicosis among jewelry workers

BACKGROUND: Silicosis is caused by inhaling free crystalline silica. Few case reports have addressed the risk of silicosis in the jewelry trade where chalk molds containing a high percentage of silica are used in gold and silver casting. We conducted a cross-sectional study involving 100 goldsmiths exposed to silica. METHODS: All workers replied to a respiratory questionnaire and underwent a full clinical examination, pulmonary function tests wit a study of flow, volumes and diffusion capacity, a chest X-ray and a high-resolution CT scan. RESULTS: High-resolution CT scan visualized signs of silicosis in 23 workers involved in gold and silver casting, confirmed by standard chest X-rays in 10 cases. In the 23 workers with CT evidence of silicosis Total Lung Capacity, FEV1 and the Lung Diffusing Capacity did not differ from the workers without the disease. Pulmonary function tests, including diffusing capacity did not correlate with silica exposure, measured as refractory material used to make the molds. CONCLUSION: In this study we demonstrate that use of chalk molds in casting in jewelry causes high prevalence of silicosis. Lung function test were not able to discriminate between workers with and without silicosis. The composition of the dust could be responsible of the high prevalence observed