Blocked All-Pairs Shortest Paths Algorithm on Intel Xeon Phi KNL Processor: A Case Study

Manycores are consolidating in HPC community as a way of improving performance while keeping power efficiency. Knights Landing is the recently released second generation of Intel Xeon Phi architecture. While optimizing applications on CPUs, GPUs and first Xeon Phi's has been largely studied in the last years, the new features in Knights Landing processors require the revision of programming and optimization techniques for these devices. In this work, we selected the Floyd-Warshall algorithm as a representative case study of graph and memory-bound applications. Starting from the default serial version, we show how data, thread and compiler level optimizations help the parallel implementation to reach 338 GFLOPS.Comment: Computer Science - CACIC 2017. Springer Communications in Computer and Information Science, vol 79

Formalizing Size-Optimal Sorting Networks: Extracting a Certified Proof Checker

Since the proof of the four color theorem in 1976, computer-generated proofs have become a reality in mathematics and computer science. During the last decade, we have seen formal proofs using verified proof assistants being used to verify the validity of such proofs. In this paper, we describe a formalized theory of size-optimal sorting networks. From this formalization we extract a certified checker that successfully verifies computer-generated proofs of optimality on up to 8 inputs. The checker relies on an untrusted oracle to shortcut the search for witnesses on more than 1.6 million NP-complete subproblems.Comment: IMADA-preprint-c

Locally Chain-Parsable Languages

If a context-free language enjoys the local parsability property then, no matter how the source string is segmented, each segment can be parsed in- dependently, and an efficient parallel parsing algorithm becomes possible. The new class of locally chain-parsable languages (LCPL), included in deterministic context-free languages, is here defined by means of the chain-driven automa- ton and characterized by decidable properties of grammar derivations. Such au- tomaton decides to reduce or not a factor in a way purely driven by the terminal characters, thus extending the well-known concept of Input-Driven (ID) (visibly) pushdown machines. LCPL extend and improve the practically relevant operator- precedence languages (Floyd), which are known to strictly include the ID lan- guages, and for which a parallel-parser generator exists. Consistently with the classical results for ID, chain-compatible LCPL are closed under reversal and Boolean operations, and language inclusion is decidable

Optimizing a Certified Proof Checker for a Large-Scale Computer-Generated Proof

In recent work, we formalized the theory of optimal-size sorting networks with the goal of extracting a verified checker for the large-scale computer-generated proof that 25 comparisons are optimal when sorting 9 inputs, which required more than a decade of CPU time and produced 27 GB of proof witnesses. The checker uses an untrusted oracle based on these witnesses and is able to verify the smaller case of 8 inputs within a couple of days, but it did not scale to the full proof for 9 inputs. In this paper, we describe several non-trivial optimizations of the algorithm in the checker, obtained by appropriately changing the formalization and capitalizing on the symbiosis with an adequate implementation of the oracle. We provide experimental evidence of orders of magnitude improvements to both runtime and memory footprint for 8 inputs, and actually manage to check the full proof for 9 inputs.Comment: IMADA-preprint-c

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

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

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

Credible Autocoding of Convex Optimization Algorithms

International audienceThe efficiency of modern optimization methods, coupled with increasing computational resources, has led to the possibility of real-time optimization algorithms acting in safety critical roles. There is a considerable body of mathematical proofs on on-line optimization programs which can be leveraged to assist in the development and verification of their implementation. In this paper, we demonstrate how theoretical proofs of real-time optimization algorithms can be used to describe functional properties at the level of the code, thereby making it accessible for the formal methods community. The running example used in this paper is a generic semi-definite programming (SDP) solver. Semi-definite programs can encode a wide variety of optimization problems and can be solved in polynomial time at a given accuracy. We describe a top-to-down approach that transforms a high-level analysis of the algorithm into useful code annotations. We formulate some general remarks about how such a task can be incorporated into a convex programming autocoder. We then take a first step towards the automatic verification of the optimization program by identifying key issues to be adressed in future work

Prevalence and predictors of complementary and alternative medicine use among people with coronary heart disease or at risk for this in the sixth Tromsø study: a comparative analysis using protection motivation theory

Background Engagement in healthy lifestyle behaviors, such as healthy diet and regular physical activity, are known to reduce the risk of developing coronary heart disease (CHD). Complementary and alternative medicine (CAM) is known to be associated with having a healthy lifestyle. The primary aim of this study was to examine the prevalence and predictors of CAM use in CHD patients, and in those without CHD but at risk for developing CHD, using Protection Motivation Theory (PMT) as a guiding conceptual framework. Method Questionnaire data were collected from 12,981 adult participants in the cross-sectional sixth Tromsø Study (2007–8). Eligible for analyses were 11,103 participants who reported whether they had used CAM or not. Of those, 830 participants reported to have or have had CHD (CHD group), 4830 reported to have parents, children or siblings with CHD (no CHD but family risk), while 5443 reported no CHD nor family risk of CHD. We first compared the patterns of CAM use in each group, and then examined the PMT predictors of CAM use. Health vulnerability from the threat appraisal process of PMT was assessed by self-rated health and expectations for future health. Response efficacy from the coping appraisal process of PMT was assessed as preventive health beliefs and health behavior frequency. Results Use of CAM was most commonly seen in people with no CHD themselves, but family risk of developing CHD (35.8%), compared to people already diagnosed with CHD (30.2%) and people with no CHD nor family risk (32.1%). All four of the PMT factors; self-rated health, expectations for future health, preventive health beliefs, and the health behavior index – were predictors for CAM use in the no CHD but family risk group. Conclusion These findings suggest that people use CAM in response to a perceived risk of developing CHD, and to prevent disease and to maintain health

Cellular Automata Applications in Shortest Path Problem

Cellular Automata (CAs) are computational models that can capture the essential features of systems in which global behavior emerges from the collective effect of simple components, which interact locally. During the last decades, CAs have been extensively used for mimicking several natural processes and systems to find fine solutions in many complex hard to solve computer science and engineering problems. Among them, the shortest path problem is one of the most pronounced and highly studied problems that scientists have been trying to tackle by using a plethora of methodologies and even unconventional approaches. The proposed solutions are mainly justified by their ability to provide a correct solution in a better time complexity than the renowned Dijkstra's algorithm. Although there is a wide variety regarding the algorithmic complexity of the algorithms suggested, spanning from simplistic graph traversal algorithms to complex nature inspired and bio-mimicking algorithms, in this chapter we focus on the successful application of CAs to shortest path problem as found in various diverse disciplines like computer science, swarm robotics, computer networks, decision science and biomimicking of biological organisms' behaviour. In particular, an introduction on the first CA-based algorithm tackling the shortest path problem is provided in detail. After the short presentation of shortest path algorithms arriving from the relaxization of the CAs principles, the application of the CA-based shortest path definition on the coordinated motion of swarm robotics is also introduced. Moreover, the CA based application of shortest path finding in computer networks is presented in brief. Finally, a CA that models exactly the behavior of a biological organism, namely the Physarum's behavior, finding the minimum-length path between two points in a labyrinth is given.Comment: To appear in the book: Adamatzky, A (Ed.) Shortest path solvers. From software to wetware. Springer, 201