There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Machine Learning is the study of methods for programming computers to learn. Computers are applied to a wide range of tasks, and for most of these it is relatively easy for programmers to design and implement the necessary software. However, there are many tasks for which this is difficult or impossible. These can be divided into four general categories. First, there are problems for which there exist no human experts. For example, in modern automated manufacturing facilities, there is a need to predict machine failures before they occur by analyzing sensor readings. Because the machines are new, there are no human experts who can be interviewed by a programmer to provide the knowledge necessary to build a computer system. A machine learning system can study recorded data and subsequent machine failures and learn prediction rules. Second, there are problems where human experts exist, but where they are unable to explain their expertise. This is the case in many perceptual tasks, such as speech recognition, hand-writing recognition, and natural language understanding. Virtually all humans exhibit expert-level abilities on these tasks, but none of them can describe the detailed steps that they follow as they perform them. Fortunately, humans can provide machines with examples of the inputs and correct outputs for these tasks, so machine learning algorithms can learn to map the inputs to the outputs. Third, there are problems where phenomena are changing rapidly. In finance, for example, people would like to predict the future behavior of the stock market, of consumer purchases, or of exchange rates. These behaviors change frequently, so that even if a programmer could construct a good predictive computer program, it would need to be rewritten frequently. A learning program can relieve the programmer of this burden by constantly modifying and tuning a set of learned prediction rules. Fourth, there are applications that need to be customized for each computer user separately. Consider, for example, a program to filter unwanted electronic mail messages. Different users will need different filters. It is unreasonable to expect each user to program his or her own rules, and it is infeasible to provide every user with a software engineer to keep the rules up-to-date. A machine learning system can learn which mail messages the user rejects and maintain the filtering rules automatically. Machine learning addresses many of the same research questions as the fields of statistics, data mining, and psychology, but with differences of emphasis. Statistics focuses on understanding the phenomena that have generated the data, often with the goal of testing different hypotheses about those phenomena. Data mining seeks to find patterns in the data that are understandable by people. Psychological studies of human learning aspire to understand the mechanisms underlying the various learning behaviors exhibited by people (concept learning, skill acquisition, strategy change, etc.).

Machine learning

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

ing WS1S Systems to Verify Parameterized Networks . . . . . . . . . . . . 188 Kai Baukus, Saddek Bensalem, Yassine Lakhnech and Karsten Stahl FMona: A Tool for Expressing Validation Techniques over Infinite State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 J.-P. Bodeveix and M. Filali Transitive Closures of Regular Relations for Verifying Infinite-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bengt Jonsson and Marcus Nilsson Diagnostic and Test Generation Using Static Analysis to Improve Automatic Test Generation . . . . . . . . . . . . . 235 Marius Bozga, Jean-Claude Fernandez and Lucian Ghirvu Efficient Diagnostic Generation for Boolean Equation Systems . . . . . . . . . . . . 251 Radu Mateescu Efficient Model-Checking Compositional State Space Generation with Partial Order Reductions for Asynchronous Communicating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Jean-Pierre Krimm and Laurent Mounier Checking for CFFD-Preorder with Tester Processes . . . . . . . . . . . . . . . . . . . . . . . 283 Juhana Helovuo and Antti Valmari Fair Bisimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Thomas A. Henzinger and Sriram K. Rajamani Integrating Low Level Symmetries into Reachability Analysis . . . . . . . . . . . . . 315 Karsten Schmidt Model-Checking Tools Model Checking Support for the ASM High-Level Language . . . . . . . . . . . . . . 331 Giuseppe Del Castillo and Kirsten Winter Table of

Tools and Algorithms for the Construction and Analysis of Systems. Proc. TACAS 2009

The tool Flow* performs Taylor model-based flowpipe construction for non-linear (polynomial) hybrid systems. Flow* combines well-known Taylor model arithmetic techniques for guaranteed approximations of the continuous dynamics in each mode with a combination of approaches for handling mode invariants and discrete transitions. Flow* supports a wide variety of optimizations including adaptive step sizes, adaptive selection of approximation orders and the heuristic selection of template directions for aggregating flowpipes. This paper describes Flow* and demonstrates its performance on a series of non-linear continuous and hybrid system benchmarks. Our comparisons show that Flow* is competitive with other tools.

/pdf/flow-an-analyzer-for-non-linear-hybrid-systems-1tru390muh.pdf

Flow*: an analyzer for non-linear hybrid systems

S-TaLiRo is a Matlab (TM) toolbox that searches for trajectories of minimal robustness in Simulink/Stateflow diagrams. It can analyze arbitrary Simulink models or user defined functions that model the system. At the heart of the tool, we use randomized testing based on stochastic optimization techniques including Monte-Carlo methods and Ant-Colony Optimization. Among the advantages of the toolbox is the seamless integration inside the Matlab environment, which is widely used in the industry for model-based development of control software.We present the architecture of S-TALIRO and its working on an application example.

/pdf/s-taliro-a-tool-for-temporal-logic-falsification-for-hybrid-okmcw9d0mt.pdf

S-taliro: a tool for temporal logic falsification for hybrid systems

We present a new method for the generation of linear invariants which reduces the problem to a non-linear constraint solving problem. Our method, based on Farkas’ Lemma, synthesizes linear invariants by extracting non-linear constraints on the coefficients of a target invariant from a program. These constraints guarantee that the linear invariant is inductive. We then apply existing techniques, including specialized quantifier elimination methods over the reals, to solve these non-linear constraints. Our method has the advantage of being complete for inductive invariants. To our knowledge, this is the first sound and complete technique for generating inductive invariants of this form. We illustrate the practicality of our method on several examples, including cases in which traditional methods based on abstract interpretation with widening fail to generate sufficiently strong invariants.

/pdf/linear-invariant-generation-using-non-linear-constraint-2vk9cuowff.pdf

Linear Invariant Generation Using Non-linear Constraint Solving

We present a new technique for the generation of non-linear (algebraic) invariants of a program. Our technique uses the theory of ideals over polynomial rings to reduce the non-linear invariant generation problem to a numerical constraint solving problem. So far, the literature on invariant generation has been focussed on the construction of linear invariants for linear programs. Consequently, there has been little progress toward non-linear invariant generation. In this paper, we demonstrate a technique that encodes the conditions for a given template assertion being an invariant into a set of constraints, such that all the solutions to these constraints correspond to non-linear (algebraic) loop invariants of the program. We discuss some trade-offs between the completeness of the technique and the tractability of the constraint-solving problem generated. The application of the technique is demonstrated on a few examples.

/pdf/non-linear-loop-invariant-generation-using-grobner-bases-3as1p339iw.pdf

Non-linear loop invariant generation using Gröbner bases

Given a neural network (NN) and a set of possible inputs to the network described by polyhedral constraints, we aim to compute a safe over-approximation of the set of possible output values. This operation is a fundamental primitive enabling the formal analysis of neural networks that are extensively used in a variety of machine learning tasks such as perception and control of autonomous systems. Increasingly, they are deployed in high-assurance applications, leading to a compelling use case for formal verification approaches. In this paper, we present an efficient range estimation algorithm that iterates between an expensive global combinatorial search using mixed-integer linear programming problems, and a relatively inexpensive local optimization that repeatedly seeks a local optimum of the function represented by the NN. We implement our approach and compare it with Reluplex, a recently proposed solver for deep neural networks. We demonstrate applications of our approach to computing flowpipes for neural network-based feedback controllers. We show that the use of local search in conjunction with mixed-integer linear programming solvers effectively reduces the combinatorial search over possible combinations of active neurons in the network by pruning away suboptimal nodes.

https://par.nsf.gov/servlets/purl/10075836

Sriram Sankaranarayanan

Papers

Flow*: an analyzer for non-linear hybrid systems

S-taliro: a tool for temporal logic falsification for hybrid systems

Linear Invariant Generation Using Non-linear Constraint Solving

Non-linear loop invariant generation using Gröbner bases

Output Range Analysis for Deep Feedforward Neural Networks