Goodness-of-fit statistics measure the compatibility of random samples against some theoretical probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2 d − 1 independent ways of defining a cumulative distribution function when d dimensions are involved. In this paper three variations on the Kolmogorov-Smirnov test for multi-dimensional data sets are surveyed: Peacock's test [1] that computes in O(n 3); Fasano and Franceschini's test [2] that computes in O(n 2); Cooke's test that computes in O(n 2). We prove that Cooke's algorithm runs in O(n 2), contrary to his claims that it runs in O(n lg n). We also compare these algorithms with ROOT's version of the Kolmogorov-Smirnov test.

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Abstract. Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2 d -1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n 2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of Ω(n 2 lg n), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test.

Attackers take advantage of any security breach to penetrate an organisation perimeter and exploit hosts as stepping stones to reach valuable assets, deeper in the network. The exploitation of hosts is possible not only when vulnerabilities in commercial off-the-shelf (COTS) software components are present, but also, for example, when an attacker acquires a credential on one host which allows exploiting further hosts on the network. Finding attacks involving the latter case requires the ability to represent dynamic models. In fact, more dynamic aspects are present in the network domain such as attackers accumulate resources (i.e. credentials) along an attack, and users and assets may move from one environment to another, although always constrained by the ruling of the network. In this paper we address these dynamic issues by presenting MsAMS (Multi-step Attack Modelling and Simulation), an implemented framework, based on Mobile Ambients, to discover attacks in networks. The idea of ambients fits naturally into this domain and has the advantage of providing flexibility for modelling. Additionally, the concept of mobility allows the simulation of attackers exploiting opportunities derived either from the exploitation of vulnerable and non-vulnerable hosts, through the acquisition of credentials. It also allows expressing security policies embedded in the rules of the ambients.

Parsing is pervasive in computing and fundamental in several software artifacts. This dissertation reports the first step in our ultimate goal: a formally verified toolset for parsing regular and context free languages based on derivatives. Specifically, we describe the formalization of Brzozowski and Antimirov derivative based algorithms for regular expression parsing, in the dependently typed language Agda. The formalization produces a proof that either an input string matches a given regular expression or that no matching exists. A tool for regular expression based search in the style of the well known GNU Grep has been developed using the certified algorithms. Practical experiments conducted using this tool are reported. ObjectivesThe main objective of this dissertation is to formalize, in the dependently typed language Agda, algorithms for RE parsing using Brzozowski's derivatives and Antimirov's partial derivatives. [9] [3] The certified algorithms should be used in a RE-based text search tool in the style of the well-known GNU Grep. ContributionsOur contributions are:• A formalization of Brzozowski derivatives based RE parsing in Agda. The certified algorithm presented produces as a result either a proof term (parse tree) that is * s = λ: In this case the conclusion follows by rule StarBase.