On Multiphase-Linear Ranking Functions

Multiphase ranking functions ($\mathit{M{\Phi}RFs}$) were proposed as a means to prove the termination of a loop in which the computation progresses through a number of "phases", and the progress of each phase is described by a different linear ranking function. Our work provides new insights regarding such functions for loops described by a conjunction of linear constraints (single-path loops). We provide a complete polynomial-time solution to the problem of existence and of synthesis of $\mathit{M{\Phi}RF}$ of bounded depth (number of phases), when variables range over rational or real numbers; a complete solution for the (harder) case that variables are integer, with a matching lower-bound proof, showing that the problem is coNP-complete; and a new theorem which bounds the number of iterations for loops with $\mathit{M{\Phi}RFs}$. Surprisingly, the bound is linear, even when the variables involved change in non-linear way. We also consider a type of lexicographic ranking functions, $\mathit{LLRFs}$, more expressive than types of lexicographic functions for which complete solutions have been given so far. We prove that for the above type of loops, lexicographic functions can be reduced to $\mathit{M{\Phi}RFs}$, and thus the questions of complexity of detection and synthesis, and of resulting iteration bounds, are also answered for this class.Comment: typos correcte

Complexity of Bradley-Manna-Sipma Lexicographic Ranking Functions

In this paper we turn the spotlight on a class of lexicographic ranking functions introduced by Bradley, Manna and Sipma in a seminal CAV 2005 paper, and establish for the first time the complexity of some problems involving the inference of such functions for linear-constraint loops (without precondition). We show that finding such a function, if one exists, can be done in polynomial time in a way which is sound and complete when the variables range over the rationals (or reals). We show that when variables range over the integers, the problem is harder -- deciding the existence of a ranking function is coNP-complete. Next, we study the problem of minimizing the number of components in the ranking function (a.k.a. the dimension). This number is interesting in contexts like computing iteration bounds and loop parallelization. Surprisingly, and unlike the situation for some other classes of lexicographic ranking functions, we find that even deciding whether a two-component ranking function exists is harder than the unrestricted problem: NP-complete over the rationals and $\Sigma^P_2$-complete over the integers.Comment: Technical report for a corresponding CAV'15 pape

Temperature Sensitivity of Mineral Soil Carbon Decomposition in Shrub and Graminoid Tundra, West Greenland

Background: Shrub expansion is transforming Arctic tundra landscapes, but the impact on the large pool of carbon stored in high-latitude soils is poorly understood. Soil carbon decomposition is a potentially important source of greenhouse gases, which could create a positive feedback to atmospheric temperature. Decomposition is temperature sensitive, but the response to temperature can be altered by environmental variables. We focus on mineral soils, which can comprise a substantial part of the near-surface carbon stock at the landscape scale and have physiochemical characteristics that influence temperature sensitivity. We conducted a soil incubation experiment to measure carbon dioxide (CO2) emissions from tundra soils collected from west Greenland at two depths of mineral soils (0-20 cm and 20-40 cm below the surface organic horizon) incubated at five temperatures (4, 8, 12, 16, 24 °C) and two moisture levels (40 % and 60 % water holding capacity). We used an information theoretic model comparison approach to evaluate temperature, moisture and depth effects, and associated interactions, on carbon losses through respiration and to determine the temperature sensitivity of decomposition in shrub- and graminoid-dominated soils. Results: We measured ecologically important differences in heterotrophic respiration and temperature sensitivity of decomposition between vegetation types. Graminoid soils had 1.8 times higher cumulative respiration and higher temperature sensitivity (expressed as Q-10) in the shallow depths (Q-10graminoid = 2.3, Q-10shrub = 1.8) compared to shrub soils. Higher Q-10 in graminoid soils was also observed for the initial incubation measurements (Q-10graminoid = 2.4, Q-10shrub = 1.9). Cumulative respiration was also higher for shallow soils, increased with moisture level, and had a temperature-depth interaction. Increasing soil moisture had a positive effect on temperature sensitivity in graminoid soils, but not in shrub soils. Conclusion: Mineral soil associated with graminoid-dominated vegetation had greater carbon losses from decomposition and a higher temperature sensitivity than shrub-dominated soils. An extrapolation of our incubation study suggests that organic carbon decomposition in western Greenland soils will likely increase with warming and with an increase in soil moisture content. Our results indicate that landscape level changes in vegetation and soil heterogeneity are important for understanding climate feedbacks between tundra and the atmosphere

Decision Procedure for Entailment of Symbolic Heaps with Arrays

This paper gives a decision procedure for the validity of en- tailment of symbolic heaps in separation logic with Presburger arithmetic and arrays. The correctness of the decision procedure is proved under the condition that sizes of arrays in the succedent are not existentially bound. This condition is independent of the condition proposed by the CADE-2017 paper by Brotherston et al, namely, one of them does not imply the other. For improving efficiency of the decision procedure, some techniques are also presented. The main idea of the decision procedure is a novel translation of an entailment of symbolic heaps into a formula in Presburger arithmetic, and to combine it with an external SMT solver. This paper also gives experimental results by an implementation, which shows that the decision procedure works efficiently enough to use

Match play performance characteristics that predict post-match creatine kinase responses in professional rugby union players

Background: Rugby union players can take several days to fully recover from competition. Muscle damage induced during the match has a major role in player recovery; however the specific characteristics of match play that predict post-match muscle damage remains unclear. We examined the relationships between a marker of muscle damage and performance characteristics associated with physical contacts and high-speed movement in professional rugby union players. Methods: Twenty-eight professional rugby union players (15 forwards, 13 backs) participated in this study. Data were obtained from 4 European Cup games, with blood samples collected 2 h pre, and 16 and 40 h post-match, and were subsequently analysed for creatine kinase (CK). Relationships between changes in CK concentrations and number of physical contacts and high-speed running markers, derived from performance analysis and global positioning system (GPS) data, were assessed. Results: Moderate and moderate-large effect-size correlations were identified between contact statistics from performance analysis and changes in CK at 16 and 40 h post match in forwards and backs, respectively (e.g. backs; total impacts vs. ΔCK (r = 0.638, p < 0.01) and Δ% CK (r = 0.454, p < 0.05) 40 h post-match). Furthermore, moderate effect-size correlations were found between measures of high-speed running and sprinting, and changes in CK at 16 and 40 h post-match within the backs (e.g. high-speed running distance vs. ΔCK (r = 0.434, p = 0.056) and Δ% CK (r = 0.437, p = 0.054) 40 hrs post-match). Conclusions: Our data demonstrate that muscle damage induced by professional rugby union match play is to some extent predicted by the number of physical contacts induced during performance. Furthermore, we show for the first time that muscle damage in backs players is predicted by high-speed running measures derived from GPS. These data increase the understanding of the causes of muscle damage in rugby union; performance markers could potentially be used to tailor individual recovery strategies and subsequent training following rugby union competition

Unrestricted Termination and Non-Termination Arguments for Bit-Vector Programs

Proving program termination is typically done by finding a well-founded ranking function for the program states. Existing termination provers typically find ranking functions using either linear algebra or templates. As such they are often restricted to finding linear ranking functions over mathematical integers. This class of functions is insufficient for proving termination of many terminating programs, and furthermore a termination argument for a program operating on mathematical integers does not always lead to a termination argument for the same program operating on fixed-width machine integers. We propose a termination analysis able to generate nonlinear, lexicographic ranking functions and nonlinear recurrence sets that are correct for fixed-width machine arithmetic and floating-point arithmetic Our technique is based on a reduction from program \emph{termination} to second-order \emph{satisfaction}. We provide formulations for termination and non-termination in a fragment of second-order logic with restricted quantification which is decidable over finite domains. The resulted technique is a sound and complete analysis for the termination of finite-state programs with fixed-width integers and IEEE floating-point arithmetic

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