Ranking and Repulsing Supermartingales for Reachability in Probabilistic Programs

Computing reachability probabilities is a fundamental problem in the analysis of probabilistic programs. This paper aims at a comprehensive and comparative account on various martingale-based methods for over- and under-approximating reachability probabilities. Based on the existing works that stretch across different communities (formal verification, control theory, etc.), we offer a unifying account. In particular, we emphasize the role of order-theoretic fixed points---a classic topic in computer science---in the analysis of probabilistic programs. This leads us to two new martingale-based techniques, too. We give rigorous proofs for their soundness and completeness. We also make an experimental comparison using our implementation of template-based synthesis algorithms for those martingales

Stochastic Invariants for Probabilistic Termination

Termination is one of the basic liveness properties, and we study the termination problem for probabilistic programs with real-valued variables. Previous works focused on the qualitative problem that asks whether an input program terminates with probability~1 (almost-sure termination). A powerful approach for this qualitative problem is the notion of ranking supermartingales with respect to a given set of invariants. The quantitative problem (probabilistic termination) asks for bounds on the termination probability. A fundamental and conceptual drawback of the existing approaches to address probabilistic termination is that even though the supermartingales consider the probabilistic behavior of the programs, the invariants are obtained completely ignoring the probabilistic aspect. In this work we address the probabilistic termination problem for linear-arithmetic probabilistic programs with nondeterminism. We define the notion of {\em stochastic invariants}, which are constraints along with a probability bound that the constraints hold. We introduce a concept of {\em repulsing supermartingales}. First, we show that repulsing supermartingales can be used to obtain bounds on the probability of the stochastic invariants. Second, we show the effectiveness of repulsing supermartingales in the following three ways: (1)~With a combination of ranking and repulsing supermartingales we can compute lower bounds on the probability of termination; (2)~repulsing supermartingales provide witnesses for refutation of almost-sure termination; and (3)~with a combination of ranking and repulsing supermartingales we can establish persistence properties of probabilistic programs. We also present results on related computational problems and an experimental evaluation of our approach on academic examples.Comment: Full version of a paper published at POPL 2017. 20 page

Electrical activation and electron spin coherence of ultra low dose antimony implants in silicon

We implanted ultra low doses (2x10^11 cm-2) of 121Sb ions into isotopically enriched 28Si and find high degrees of electrical activation and low levels of dopant diffusion after rapid thermal annealing. Pulsed Electron Spin Resonance shows that spin echo decay is sensitive to the dopant depths, and the interface quality. At 5.2 K, a spin decoherence time, T2, of 0.3 ms is found for profiles peaking 50 nm below a Si/SiO2 interface, increasing to 0.75 ms when the surface is passivated with hydrogen. These measurements provide benchmark data for the development of devices in which quantum information is encoded in donor electron spins

Finding polynomial loop invariants for probabilistic programs

Quantitative loop invariants are an essential element in the verification of probabilistic programs. Recently, multivariate Lagrange interpolation has been applied to synthesizing polynomial invariants. In this paper, we propose an alternative approach. First, we fix a polynomial template as a candidate of a loop invariant. Using Stengle's Positivstellensatz and a transformation to a sum-of-squares problem, we find sufficient conditions on the coefficients. Then, we solve a semidefinite programming feasibility problem to synthesize the loop invariants. If the semidefinite program is unfeasible, we backtrack after increasing the degree of the template. Our approach is semi-complete in the sense that it will always lead us to a feasible solution if one exists and numerical errors are small. Experimental results show the efficiency of our approach.Comment: accompanies an ATVA 2017 submissio

Quantitative Regular Expressions for Arrhythmia Detection Algorithms

Motivated by the problem of verifying the correctness of arrhythmia-detection algorithms, we present a formalization of these algorithms in the language of Quantitative Regular Expressions. QREs are a flexible formal language for specifying complex numerical queries over data streams, with provable runtime and memory consumption guarantees. The medical-device algorithms of interest include peak detection (where a peak in a cardiac signal indicates a heartbeat) and various discriminators, each of which uses a feature of the cardiac signal to distinguish fatal from non-fatal arrhythmias. Expressing these algorithms' desired output in current temporal logics, and implementing them via monitor synthesis, is cumbersome, error-prone, computationally expensive, and sometimes infeasible. In contrast, we show that a range of peak detectors (in both the time and wavelet domains) and various discriminators at the heart of today's arrhythmia-detection devices are easily expressible in QREs. The fact that one formalism (QREs) is used to describe the desired end-to-end operation of an arrhythmia detector opens the way to formal analysis and rigorous testing of these detectors' correctness and performance. Such analysis could alleviate the regulatory burden on device developers when modifying their algorithms. The performance of the peak-detection QREs is demonstrated by running them on real patient data, on which they yield results on par with those provided by a cardiologist.Comment: CMSB 2017: 15th Conference on Computational Methods for Systems Biolog

Counterexample-Guided Polynomial Loop Invariant Generation by Lagrange Interpolation

We apply multivariate Lagrange interpolation to synthesize polynomial quantitative loop invariants for probabilistic programs. We reduce the computation of an quantitative loop invariant to solving constraints over program variables and unknown coefficients. Lagrange interpolation allows us to find constraints with less unknown coefficients. Counterexample-guided refinement furthermore generates linear constraints that pinpoint the desired quantitative invariants. We evaluate our technique by several case studies with polynomial quantitative loop invariants in the experiments

PrIC3: Property Directed Reachability for MDPs

IC3 has been a leap forward in symbolic model checking. This paper proposes PrIC3 (pronounced pricy-three), a conservative extension of IC3 to symbolic model checking of MDPs. Our main focus is to develop the theory underlying PrIC3. Alongside, we present a first implementation of PrIC3 including the key ingredients from IC3 such as generalization, repushing, and propagation

Non-polynomial Worst-Case Analysis of Recursive Programs

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201

The complex TIE between macrophages and angiogenesis

Macrophages are primarily known as phagocytic immune cells, but they also play a role in diverse processes, such as morphogenesis, homeostasis and regeneration. In this review, we discuss the influence of macrophages on angiogenesis, the process of new blood vessel formation from the pre-existing vasculature. Macrophages play crucial roles at each step of the angiogenic cascade, starting from new blood vessel sprouting to the remodelling of the vascular plexus and vessel maturation. Macrophages form promising targets for both pro- and anti-angiogenic treatments. However, to target macrophages, we will first need to understand the mechanisms that control the functional plasticity of macrophages during each of the steps of the angiogenic cascade. Here, we review recent insights in this topic. Special attention will be given to the TIE2-expressing macrophage (TEM), which is a subtype of highly angiogenic macrophages that is able to influence angiogenesis via the angiopoietin-TIE pathway

Handwriting performance in the absence of visual control in writer's cramp patients: Initial observations

BACKGROUND: The present study was aimed at investigating the writing parameters of writer's cramp patients and control subjects during handwriting of a test sentence in the absence of visual control. METHODS: Eight right-handed patients with writer's cramp and eight healthy volunteers as age-matched control subjects participated in the study. The experimental task consisted in writing a test sentence repeatedly for fifty times on a pressure-sensitive digital board. The subject did not have visual control on his handwriting. The writing performance was stored on a PC and analyzed off-line. RESULTS: During handwriting all patients developed a typical dystonic limb posture and reported an increase in muscular tension along the experimental session. The patients were significantly slower than the controls, with lower mean vertical pressure of the pen tip on the paper and they could not reach the endmost letter of the sentence in the given time window. No other handwriting parameter differences were found between the two groups. CONCLUSION: Our findings indicate that during writing in the absence of visual feedback writer's cramp patients are slower and could not reach the endmost letter of the test sentence, but their level of automatization is not impaired and writer's cramp handwriting parameters are similar to those of the controls except for even lower vertical pressure of the pen tip on the paper, which is probably due to a changed strategy in such experimental conditions