LambdaY-Calculus With Priorities

International audienceThe lambdaY-calculus with priorities is a variant of the simply-typed lambda calculus designed for higher-order model-checking. The higher-order model-checking problem asks if a given parity tree automaton accepts the Böhm tree of a given term of the simply-typed lambda calculus with recursion. We show that this problem can be reduced to the same question but for terms of lambdaY-calculus with priorities and visibly parity automata; a subclass of parity automata. The latter question can be answered by evaluating terms in a simple powerset model with least and greatest fixpoints. We prove that the recognizing power of powerset models and visibly parity automata are the same. So, up to conversion to the lambdaY-calculus with priorities, powerset models with least and greatest fixpoints are indeed the right semantic framework for the model-checking problem. The reduction to lambdaY-calculus with priorities is also efficient algorithmically: it gives an algorithm of the same complexity as direct approaches to the higher-order model-checking problem. This indicates that the task of calculating the value of a term in a powerset model is a central algo-rithmic problem for higher-order model-checking

Using models to model-check recursive schemes

We propose a model-based approach to the model checking problem for recursive schemes. Since simply typed lambda calculus with the fixpoint operator, lambda-Y-calculus, is equivalent to schemes, we propose the use of a model of lambda-Y-calculus to discriminate the terms that satisfy a given property. If a model is finite in every type, this gives a decision procedure. We provide a construction of such a model for every property expressed by automata with trivial acceptance conditions and divergence testing. Such properties pose already interesting challenges for model construction. Moreover, we argue that having models capturing some class of properties has several other virtues in addition to providing decidability of the model-checking problem. As an illustration, we show a very simple construction transforming a scheme to a scheme reflecting a property captured by a given model.Comment: Long version of a paper presented at TLCA 201

A model for divergence insensitive properties of lambdaY-terms

A term of a simply typed λ-calculus with fixpoints can be considered as an abstraction of a higher-order functional program. The result of the computation of a term is its Böhm tree. Given a tree automaton describing a property of Böhm trees, we are interested in constructing a model recognizing the property, in a sense that the value of a term determines if its Böhm tree satisfies the property. We show how to construct models recognizing properties expressed by parity automata that cannot detect divergence. We call them Ω-blind parity automata, as the symbol Ω is used in Böhm trees to represent divergence; an automaton is Ω-blind when it has to accept Ω from every state. The models we construct resemble standard Scott models of latices of monotone functions, but application needs to be modified and the the fixpoint operator should be interpreted as a particular non-extremal fixpoint in a lattice

On the Termination Problem for Probabilistic Higher-Order Recursive Programs

In the last two decades, there has been much progress on model checking of both probabilistic systems and higher-order programs. In spite of the emergence of higher-order probabilistic programming languages, not much has been done to combine those two approaches. In this paper, we initiate a study on the probabilistic higher-order model checking problem, by giving some first theoretical and experimental results. As a first step towards our goal, we introduce PHORS, a probabilistic extension of higher-order recursion schemes (HORS), as a model of probabilistic higher-order programs. The model of PHORS may alternatively be viewed as a higher-order extension of recursive Markov chains. We then investigate the probabilistic termination problem -- or, equivalently, the probabilistic reachability problem. We prove that almost sure termination of order-2 PHORS is undecidable. We also provide a fixpoint characterization of the termination probability of PHORS, and develop a sound (but possibly incomplete) procedure for approximately computing the termination probability. We have implemented the procedure for order-2 PHORSs, and confirmed that the procedure works well through preliminary experiments that are reported at the end of the article

Doctor of Philosophy

dissertationIn the static analysis of functional programs, control- ow analysis (k-CFA) is a classic method of approximating program behavior as a infinite state automata. CFA2 and abstract garbage collection are two recent, yet orthogonal improvements, on k-CFA. CFA2 approximates program behavior as a pushdown system, using summarization for the stack. CFA2 can accurately approximate arbitrarily-deep recursive function calls, whereas k-CFA cannot. Abstract garbage collection removes unreachable values from the store/heap. If unreachable values are not removed from a static analysis, they can become reachable again, which pollutes the final analysis and makes it less precise. Unfortunately, as these two techniques were originally formulated, they are incompatible. CFA2's summarization technique for managing the stack obscures the stack such that abstract garbage collection is unable to examine the stack for reachable values. This dissertation presents introspective pushdown control-flow analysis, which manages the stack explicitly through stack changes (pushes and pops). Because this analysis is able to examine the stack by how it has changed, abstract garbage collection is able to examine the stack for reachable values. Thus, introspective pushdown control-flow analysis merges successfully the benefits of CFA2 and abstract garbage collection to create a more precise static analysis. Additionally, the high-performance computing community has viewed functional programming techniques and tools as lacking the efficiency necessary for their applications. Nebo is a declarative domain-specific language embedded in C++ for discretizing partial differential equations for transport phenomena. For efficient execution, Nebo exploits a version of expression templates, based on the C++ template system, which is a type-less, completely-pure, Turing-complete functional language with burdensome syntax. Nebo's declarative syntax supports functional tools, such as point-wise lifting of complex expressions and functional composition of stencil operators. Nebo's primary abstraction is mathematical assignment, which separates what a calculation does from how that calculation is executed. Currently Nebo supports single-core execution, multicore (thread-based) parallel execution, and GPU execution. With single-core execution, Nebo performs on par with the loops and code that it replaces in Wasatch, a pre-existing high-performance simulation project. With multicore (thread-based) execution, Nebo can linearly scale (with roughly 90% efficiency) up to 6 processors, compared to its single-core execution. Moreover, Nebo's GPU execution can be up to 37x faster than its single-core execution. Finally, Wasatch (the pre-existing high-performance simulation project which uses Nebo) can scale up to 262K cores

The Effect of the 2018 Tariffs on European Wine

This paper estimates a vector autoregression model for average wine prices across U.S. cities to assess the impact of tariff changes on the U.K., France, Germany, and Spain after they were enacted in October 2019. It uses impulse response functions to gauge how a one-unit impulse in the per-liter duty rate may effect the average wine price in the U.S. and the quantity of wine from various exporters to the U.S. It finds that a one-unit impulse in the duty rate levied against the bloc of countries impacted by the tariff results in a fall in the quantity of wine imported from those countries and that wine from the bloc of countries is substituted with wine from the top three exporters not included in the bloc

Experimental study and computational modelling of the biomechanical properties of arteries

Biomechanics of artery has been shown to play an important role in the progression of cardiovascular diseases. Thus studies of arterial biomechanics are essential to find the causes and preventions of these diseases. The thesis first introduces work demonstrating the use of the optical flow estimation technique to track the deformation of biological tissue in biomechanical experiments. The study shows that the optical flow technique provides a good estimation of the strains for the constructed deformation images by comparing the results with those of the finite element (FE) models. The optical flow technique was then used on the analysis of the images obtained from the experiments of human coronary arteries with atherosclerosis. Atherosclerosis is a cardiovascular disease in which the rupture of atherosclerotic plaque leads to stroke or myocardial infarction, where the risk of the rupture has been associated with mechanical factors. An approach was developed combining the tensile ring tests, optical flow analysis, and FE modelling to obtain the heterogeneous mechanical properties of atherosclerotic arteries. This approach provides a sensible estimation of the hyperelastic properties of atherosclerotic plaques by matching the forces and strains of the experiments and FE models using an error minimisation approach. The study on the biomechanics of atherosclerotic arteries was then extended to the muscle active properties of the diseased vessels. Previous drug response experiments have shown that human atherosclerotic arteries maintained active forces of more than 70% of those in healthy arteries in response to endothelin-1 (ET-1) despite an extensive thinning of the media smooth muscle layer. FE analyses have been performed to model the passive tensile ring tests and active ET-1 response tests, which demonstrate the potential elevated contractile strains of diseased smooth muscle cells in response to ET-1. The results suggest that adaptation mechanisms occur with atherosclerosis to maintain the distensibility of the diseased artery in physiological condition with the presence of cell activation agents. In addition, a project investigating the role of hyaluronan in systemic inflammation-induced aortic stiffening has been carried out. Systemic inflammation occurs in several diseases such as rheumatoid arthritis which increases aortic stiffness and causes altered distribution and function of arterial hyaluronan. Tensile ring tests were performed on rat aortas that have hyaluronan digested by hyaluronidase and control samples. The results show no significant difference between the elastic modulus of the two groups despite successful digestion of the hyaluronan content in the treated samples in biochemistry assays. A potential mechanism of this contradiction is that the hyaluronan accumulation-induced aortic stiffening is a long term effect in patients and cannot be shown with the methodology of this study. In order to solve the non-physiological loading limitation of the tensile ring tests in the studies above, an inflation test system has been designed for future experiments. This system inflates an artery ring with a balloon tube with the cross-sectional deformation captured using a camera in an inverted position. The system has been tested with a pig aortic ring with the deformation analysed using the optical flow technique. The estimated elastic modulus of the artery was compared to that obtained from a tensile ring test, which shows a good matching between the two values. Overall, the thesis demonstrates experimental and computational approaches to study the biomechanical properties of arteries. The results also provide information on the hyperelastic and active properties of atherosclerotic arteries as well as the effects of hyaluronan digestion on the stiffness of rat aortas