Abstract. We present a quantitative analysis of various (syntactic and behavioral) properties of random λ-terms. Our main results show that asymptotically, almost all terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the λ-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator.
International audienceLambda calculus is the basis of functional programming and higher order proof assistants. However, little is known about combinatorial properties of lambda terms, in particular, about their asymptotic distribution and random generation. This paper tries to answer questions like: How many terms of a given size are there? What is a ''typical'' structure of a simply typable term? Despite their ostensible simplicity, these questions still remain unanswered, whereas solutions to such problems are essential for testing compilers and optimizing programs whose expected efficiency depends on the size of terms. Our approach toward the afore-mentioned problems may be later extended to any language with bound variables, i.e., with scopes and declarations. This paper presents two complementary approaches: one, theoretical, uses complex analysis and generating functions, the other, experimental, is based on a generator of lambda-terms. Thanks to de Bruijn indices, we provide three families of formulas for the number of closed lambda terms of a given size and we give four relations between these numbers which have interesting combinatorial interpretations. As a by-product of the counting formulas, we design an algorithm for generating lambda terms. Performed tests provide us with experimental data, like the average depth of bound variables and the average number of head lambdas. We also create random generators for various sorts of terms. Thereafter, we conduct experiments that answer questions like: What is the ratio of simply typable terms among all terms? (Very small!) How are simply typable lambda terms distributed among all lambda terms? (A typable term almost always starts with an abstraction.) In this paper, abstractions and applications have size 1 and variables have size 0
We study the sequences of numbers corresponding to lambda terms of given sizes, where the size is this of lambda terms with de Bruijn indices in a very natural model where all the operators have size 1. For plain lambda terms, the sequence corresponds to two families of binary trees for which we exhibit bijections. We study also the distribution of normal forms, head normal forms and strongly normalizing terms. In particular we show that strongly normalizing terms are of density 0 among plain terms.
Although the risk of aneurysm rupture after EVAR is low, all patients treated endovascularly should be routinely monitored, in order to select cases with potential endoleaks or stentgraft migration which may lead to fatal complications. When rupture occurs open aneurysmectomy is feasible, although it requires careful management in these high-risk patients.
Background: The peritoneal dialysis (PD) urgent-start pathway, without typical 2-week break-in period, was meant for late-referral patients able and prone to join PD-first program, with its main advantages such as: keeping the vascular system intact, preserving their residual renal function and retaining life-style flexibility. We compared the short- and long-term outcomes of consecutive 35 patients after urgent- and 94 patients after the planned start of PD as the first choice.Methods: The study included all incident end-stage renal disease patients starting PD program between January 2005 and December 2015, classified into two groups: those with urgent (unplanned) and those with elective (planned) start. Urgent PD was initiated as an overnight automatic procedure (APD) with dwell volume gradually increased, and after 2–3 weeks, target PD method was established.Results: The mean time between catheter implantation and PD start was 3.5 ± 2.3 in urgent and 16.2 ± 1.7 days in planned-start groups (p < 0.00001). 51% of the patients in the urgent-start group required PD during first 48 h after catheter insertion. Mean follow-up of 17.6 ± 11.09 months (median: 19.0) was in the urgent-start group and 28.6 ± 26.6 months (median: 19.5) in the planned-start group. The early mechanical complications were observed more often in the urgent-start group (29 vs. 4%, p = 0.00005). The only significant predictors of early mechanical complications were serum albumin (p = 0.02) and time between the catheter insertion and PD start. The first year patient survival and technique survival censored for death and kidney transplantation were not significantly different between groups. In Cox proportional analysis the independent risk factors for patient survival as well as for method and patient survival appeared Charlson Comorbidity Index CCI (HR 1.4; p = 0.01 and 1.24; p = 0.02) and time from catheter implantation to PD start with HR 5.11; p = 0.03 and 4.29; p = 0.04 for <2 days, while time >14 days lost its predictive value (p = 0.07).Conclusion: Peritoneal dialysis may be a feasible and safe alternative to HD in patients who need to start dialysis urgently without established dialysis access, with an acceptable complications rates, as well as patient and technique survival.
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