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

We combine first-order dynamic logic for reasoning about possible behaviour of hybrid systems with temporal logic for reasoning about the temporal behaviour during their operation. Our logic supports verification of hybrid programs with first-order definable flows and provides a uniform treatment of discrete and continuous evolution. For our combined logic, we generalise the semantics of dynamic modalities to refer to hybrid traces instead of final states. Further, we prove that this gives a conservative extension of dynamic logic. On this basis, we provide a modular verification calculus that reduces correctness of temporal behaviour of hybrid systems to non-temporal reasoning. Using this calculus, we analyse safety invariants in a train control system and symbolically synthesise parametric safety constraints.

/pdf/a-temporal-dynamic-logic-for-verifying-hybrid-system-3kt9g1422r.pdf

A Temporal Dynamic Logic for Verifying Hybrid System Invariants

Intelligent vehicle systems have interesting prospects for solving inefficiencies and risks in ground transportation, e.g., by making cars aware of their environment and regulating speed intelligently. If the computer control technology reacts fast enough, intelligent control can be used to increase the density of cars on the streets. The technology may also help prevent crashes at intersections, which cost the US $97 Billion in the year 2000. The crucial prerequisite for intelligent vehicle control, however, is that it must be correct, for it may otherwise do more harm than good. Formal verification techniques provide the best reliability guarantees but have had difficulties in the past with scaling to such complex systems. We report our successes with a logical approach to hybrid systems verification, which can capture discrete control decisions and continuous driving dynamics. We present a model for the interaction of two cars and a traffic light at a two lane intersection and verify with a formal proof that our system always ensures collision freedom and that our controller always prevents cars from running red lights.

/pdf/safe-intersections-at-the-crossing-of-hybrid-systems-and-jrvcsddeka.pdf

Safe intersections: At the crossing of hybrid systems and verification

This article proves the completeness of an axiomatization for differential equation invariants described by Noetherian functions. First, the differential equation axioms of differential dynamic logic are shown to be complete for reasoning about analytic invariants. Completeness crucially exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. Cleverly chosen differential ghosts are the proof-theoretical counterpart of dark matter. They create a new hypothetical state, whose relationship to the original state variables satisfies invariants that did not exist before. The reflection of these new invariants in the original system then enables its analysis.An extended axiomatization with existence and uniqueness axioms is complete for all local progress properties, and, with a real induction axiom, is complete for all semianalytic invariants. This parsimonious axiomatization serves as the logical foundation for reasoning about invariants of differential equations. Indeed, it is precisely this logical treatment that enables the generalization of completeness to the Noetherian case.

https://dl.acm.org/doi/pdf/10.1145/3380825

Differential Equation Invariance Axiomatization

We introduce differential refinement logic (dRL), a logic with first-class support for refinement relations on hybrid systems, and a proof calculus for verifying such relations. dRL simultaneously solves several seemingly different challenges common in theorem proving for hybrid systems: 1. When hybrid systems are complicated, it is useful to prove properties about simpler and related subsystems before tackling the system as a whole. 2. Some models of hybrid systems can be implementation-specific. Verification can be aided by abstracting the system down to the core components necessary for safety, but only if the relations between the abstraction and the original system can be guaranteed. 3. One approach to taming the complexities of hybrid systems is to start with a simplified version of the system and iteratively expand it. However, this approach can be costly, since every iteration has to be proved safe from scratch, unless refinement relations can be leveraged in the proof. 4. When proofs become large, it is difficult to maintain a modular or comprehensible proof structure. By using a refinement relation to arrange proofs hierarchically according to the structure of natural subsystems, we can increase the readability and modularity of the resulting proof. dRL extends an existing specification and verification language for hybrid systems (differential dynamic logic, dL) by adding a refinement relation to directly compare hybrid systems. This paper gives a syntax, semantics, and proof calculus for dRL. We demonstrate its usefulness with examples where using refinement results in easier and better-structured proofs.

https://dl.acm.org/doi/pdf/10.1145/2933575.2934555

Differential Refinement Logic

Hybrid systems have a complete axiomatization in differential dynamic logic relative to continuous systems. They also have a complete axiomatization relative to discrete systems. Moreover, there is a constructive reduction of properties of hybrid systems to corresponding properties of continuous systems or to corresponding properties of discrete systems. We briefly summarize and discuss some of the implications of these results.

https://download.e-bookshelf.de/download/0000/0149/09/L-G-0000014909-0002347546.pdf

André Platzer

Papers

A Temporal Dynamic Logic for Verifying Hybrid System Invariants

Safe intersections: At the crossing of hybrid systems and verification

Differential Equation Invariance Axiomatization

Differential Refinement Logic

Logical analysis of hybrid systems: a complete answer to a complexity challenge