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

X-Ray Phase-Contrast Tomography of Renal Ischemia-Reperfusion Damage

Purpose: The aim of the study was to investigate microstructural changes occurring in unilateral renal ischemia-reperfusion injury in a murine animal model using synchrotron radiation. Material and Methods: The effects of renal ischemia-reperfusion were investigated in a murine animal model of unilateral ischemia. Kidney samples were harvested on day 18. Grating-Based Phase-Contrast Imaging (GB-PCI) of the paraffin-embedded kidney samples was performed at a Synchrotron Radiation Facility (beam energy of 19 keV). To obtain phase information, a two-grating Talbot interferometer was used applying the phase stepping technique. The imaging system provided an effective pixel size of 7.5 mu m. The resulting attenuation and differential phase projections were tomographically reconstructed using filtered back-projection. Semi-automated segmentation and volumetry and correlation to histopathology were performed. Results: GB-PCI provided good discrimination of the cortex, outer and inner medulla in non-ischemic control kidneys. Post-ischemic kidneys showed a reduced compartmental differentiation, particularly of the outer stripe of the outer medulla, which could not be differentiated from the inner stripe. Compared to the contralateral kidney, after ischemia a volume loss was detected, while the inner medulla mainly retained its volume (ratio 0.94). Post-ischemic kidneys exhibited severe tissue damage as evidenced by tubular atrophy and dilatation, moderate inflammatory infiltration, loss of brush borders and tubular protein cylinders. Conclusion: In conclusion GB-PCI with synchrotron radiation allows for non-destructive microstructural assessment of parenchymal kidney disease and vessel architecture. If translation to lab-based approaches generates sufficient density resolution, and with a time-optimized image analysis protocol, GB-PCI may ultimately serve as a non-invasive, non-enhanced alternative for imaging of pathological changes of the kidney

Finding Recurrent Sets with Backward Analysis and Trace Partitioning

We propose an abstract-interpretation-based analysis for recurrent sets. A recurrent set is a set of states from which the execution of a program cannot or might not (as in our case) escape. A recurrent set is a part of a program’s nontermination proof (that needs to be complemented by reachability analysis). We find recurrent sets by performing a potentially over-approximate backward analysis that produces an initial candidate. We then perform over-approximate forward analysis on the candidate to check and refine it and ensure soundness. In practice, the analysis relies on trace partitioning that predicts future paths through the program that non-terminating executions will take. Using our technique, we were able to find recurrent sets in many benchmarks found in the literature including some that, to our knowledge, cannot be handled by existing tools. In addition, we note that typically, analyses that search for recurrent sets are applied to linear under-approximations of programs or employ some form of non-approximate numeric reasoning. In contrast, our analysis uses standard abstract-interpretation techniques and is potentially applicable to a larger class of abstract domains (and therefore – programs)

Syntax-Guided Termination Analysis

We present new algorithms for proving program termination and non-termination using syntax-guided synthesis. They exploit the symbolic encoding of programs and automatically construct a formal grammar for symbolic constraints that are used to synthesize either a termination argument or a non-terminating program refinement. The constraints are then added back to the program encoding, and an off-the-shelf constraint solver decides on their fitness and on the progress of the algorithms. The evaluation of our implementation, called Freq-Term, shows that although the formal grammar is limited to the syntax of the program, in the majority of cases our algorithms are effective and fast. Importantly, FreqTerm is competitive with state-of-the-art on a wide range of terminating and non-terminating benchmarks, and it significantly outperforms state-of-the-art on proving non-termination of a class of programs arising from large-scale Event-Condition-Action systems