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

Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

In this provocative book, Barwise and Perry tackle the slippery subject of \"meaning, \" a subject that has long vexed linguists, language philosophers, and logicians.

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Situations and Attitudes.

Rapid progress in machine learning and artificial intelligence (AI) has brought increasing attention to the potential impacts of AI technologies on society. In this paper we discuss one such potential impact: the problem of accidents in machine learning systems, defined as unintended and harmful behavior that may emerge from poor design of real-world AI systems. We present a list of five practical research problems related to accident risk, categorized according to whether the problem originates from having the wrong objective function ("avoiding side effects" and "avoiding reward hacking"), an objective function that is too expensive to evaluate frequently ("scalable supervision"), or undesirable behavior during the learning process ("safe exploration" and "distributional shift"). We review previous work in these areas as well as suggesting research directions with a focus on relevance to cutting-edge AI systems. Finally, we consider the high-level question of how to think most productively about the safety of forward-looking applications of AI.

Concrete Problems in AI Safety

Deep neural networks have emerged as a widely used and effective means for tackling complex, real-world problems. However, a major obstacle in applying them to safety-critical systems is the great difficulty in providing formal guarantees about their behavior. We present a novel, scalable, and efficient technique for verifying properties of deep neural networks (or providing counter-examples). The technique is based on the simplex method, extended to handle the non-convex Rectified Linear Unit (ReLU) activation function, which is a crucial ingredient in many modern neural networks. The verification procedure tackles neural networks as a whole, without making any simplifying assumptions. We evaluated our technique on a prototype deep neural network implementation of the next-generation airborne collision avoidance system for unmanned aircraft (ACAS Xu). Results show that our technique can successfully prove properties of networks that are an order of magnitude larger than the largest networks verified using existing methods.

Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks

This paper introduces a new proof calculus for differential dynamic logic (dL) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform substitutions make it possible to rely on axioms rather than axiom schemata, substantially simplifying implementations. Instead of nontrivial schema variables and soundness-critical side conditions on the occurrence patterns of variables, the resulting calculus adopts only a finite number of ordinary dL formulas as axioms. The static semantics of differential dynamic logic is captured exclusively in uniform substitutions and bound variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this paper introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivations as first-class axioms in dL.

A Uniform Substitution Calculus for Differential Dynamic Logic

We introduce Open image in new window , which extends differential dynamic logic ( Open image in new window ) for hybrid systems with definite descriptions and tuples, thus enabling its theoretical foundations to catch up with its implementation in the theorem prover Open image in new window . Definite descriptions enable partial, nondifferentiable, and discontinuous terms, which have many examples in applications, such as divisions, nth roots, and absolute values. Tuples enable systems of multiple differential equations, arising in almost every application. Together, definite description and tuples combine to support long-desired features such as vector arithmetic.

dL ι : Definite Descriptions in Differential Dynamic Logic.

Both SAT and #SAT can represent difficult problems in seemingly dissimilar areas such as planning, verification, and probabilistic inference Here, we examine an expressive new language, #∃SAT, that generalizes both of these languages #∃SAT problems require counting the number of satisfiable formulas in a concisely-describable set of existentially quantified, propositional formulas We characterize the expressiveness and worst-case difficulty of #∃SAT by proving it is complete for the complexity class #P , and relating this class to more familiar complexity classes We also experiment with three new general-purpose #∃SAT solvers on a battery of problem distributions including a simple logistics domain Our experiments show that, despite the formidable worst-case complexity of #P , many of the instances can be solved efficiently by noticing and exploiting a particular type of frequent structure

A Generalization of SAT and #SAT for Robust Policy Evaluation (CMU-CS-13-107)

Stability is required for real world controlled systems as it ensures that those systems can tolerate small, real world perturbations around their desired operating states This paper shows how stability for continuous systems modeled by ordinary differential equations (ODEs) can be formally verified in differential dynamic logic (dL) The key insight is to specify ODE stability by suitably nesting the dynamic modalities of dL with first-order logic quantifiers Elucidating the logical structure of stability properties in this way has three key benefits: i) it provides a flexible means of formally specifying various stability properties of interest, ii) it yields rigorous proofs of those stability properties from dL’s axioms with dL’s ODE safety and liveness proof principles, and iii) it enables formal analysis of the relationships between various stability properties which, in turn, inform proofs of those properties These benefits are put into practice through an implementation of stability proofs for several examples in KeYmaera X, a hybrid systems theorem prover based on dL

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Deductive Stability Proofs for Ordinary Differential Equations

The use of deductive techniques, such as theorem provers, has several advantages in safety verification of hybrid systems; however, state-of-the-art theorem provers require manual intervention to handle complex systems. Furthermore, there is often a gap between the type of assistance that a theorem prover requires to make progress on a proof task and the assistance that a system designer is able to provide directly. This paper presents an extension to KeYmaera, a deductive verification tool for differential dynamic logic; the new technique allows local reasoning using system designer intuition about performance within particular modes as part of a proof task. Our approach allows the theorem prover to leverage forward invariants, discovered using numerical techniques, as part of a proof of safety. We introduce a new inference rule into the proof calculus of KeYmaera, the forward invariant cut rule, and we present a methodology to discover useful forward invariants, which are then used with the new cut rule to complete verification tasks. We demonstrate how our new approach can be used to complete verification tasks that lie out of the reach of existing automatic verification approaches using several examples, including one involving an automotive powertrain control system.

André Platzer

Papers

A Uniform Substitution Calculus for Differential Dynamic Logic

dL ι : Definite Descriptions in Differential Dynamic Logic.

A Generalization of SAT and #SAT for Robust Policy Evaluation (CMU-CS-13-107)

Deductive Stability Proofs for Ordinary Differential Equations

Forward invariant cuts to simplify proofs of safety