12 research outputs found

    On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components

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    For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k. We show that for every k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1 and Pi1). This contrasts with the class of all graphs, where Delta2=Comp(Sigma1,Pi1)

    On closure ordinals for the modal mu-calculus

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    The closure ordinal of a formula of modal mu-calculus mu X phi is the least ordinal kappa, if it exists, such that the denotation of the formula and the kappa-th iteration of the monotone operator induced by phi coincide across all transition systems (finite and infinite). It is known that for every alpha < omega^2 there is a formula phi of modal logic such that mu X phi has closure ordinal alpha (Czarnecki 2010). We prove that the closure ordinals arising from the alternation-free fragment of modal mu-calculus (the syntactic class capturing Sigma_2 cap Pi_2) are bounded by omega^2. In this logic satisfaction can be characterised in terms of the existence of tableaux, trees generated by systematically breaking down formulae into their constituents according to the semantics of the calculus. To obtain optimal upper bounds we utilise the connection between closure ordinals of formulae and embedded order-types of the corresponding tableaux

    On closure ordinals for the modal mu-calculus

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    The closure ordinal of a formula of modal mu-calculus mu X phi is the least ordinal kappa, if it exists, such that the denotation of the formula and the kappa-th iteration of the monotone operator induced by phi coincide across all transition systems (finite and infinite). It is known that for every alpha < omega^2 there is a formula phi of modal logic such that mu X phi has closure ordinal alpha (Czarnecki 2010). We prove that the closure ordinals arising from the alternation-free fragment of modal mu-calculus (the syntactic class capturing Sigma_2 cap Pi_2) are bounded by omega^2. In this logic satisfaction can be characterised in terms of the existence of tableaux, trees generated by systematically breaking down formulae into their constituents according to the semantics of the calculus. To obtain optimal upper bounds we utilise the connection between closure ordinals of formulae and embedded order-types of the corresponding tableaux

    On the Way to Alternating Weak Automata

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    Expressiveness of Monadic Second-Order Logics on Infinite Trees of Arbitrary Branching Degree

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    In this thesis we study the expressive power of variants of monadic second-order logic (MSO) on infinite trees by means of automata. In particular we are interested in weak MSO and well-founded MSO, where the second-order quantifiers range respectively over finite sets and over subsets of well-founded trees. On finitely branching trees, weak and well-founded MSO have the same expressive power and are both strictly weaker than MSO. The associated class of automata (called weak MSO-automata) is a restriction of the class characterizing MSO-expressivity. We show that, on trees with arbitrary branching degree, weak MSO-automata characterize the expressive power of well-founded MSO, which turns out to be incomparable with weak MSO. Indeed, in this generalized setting, weak MSO gives an account of properties of the ‘horizontal dimension’ of trees, which cannot be described by means of MSO or well-founded MSO formulae. In analogy with the result of Janin and Walukiewicz for MSO and the modal μ-calculus, this raises the issue of which modal logic captures the bisimulation-invariant fragment of well-founded MSO and weak MSO. We show that the alternation-free fragment of the modal μ-calculus and the bisimulation-invariant fragment of well-founded MSO have the same expressive power on trees of arbitrary branching degree. We motivate the conjecture that weak MSO modulo bisimulation collapses inside MSO and well-founded MSO

    Computational Techniques to Address the Sign Problem in Non-Relativistic Quantum Thermodynamics

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    Understanding quantum many-body physics is crucial to physical systems throughout condensed matter, high-energy, and nuclear physics, as well as the development of new applications based upon such systems. Stochastic techniques are generally required to study strongly-interacting quantum matter, but are frequently hindered by the sign problem, a signal-to-noise issue which breaks down importance sampling methods for many physical models. This dissertation develops several novel stochastic nonperturbative and semi-analytic perturbative techniques to circumvent the sign problem in the context of non-relativistic quantum gases at finite temperature. These techniques include an extension to hybrid Monte Carlo based on an analytic continuation, complex Langevin, and an automated perturbative expansion of the partition function, all of which use auxiliary field methods. Each technique is used to compute first predictions for thermodynamic equations of state for non-relativistic Fermi gases in spin-balanced and spin-polarized systems for both attractive and repulsive interactions. These results are frequently compared against second- and third-order virial expansions in appropriate limits. The calculation of observables including the density, magnetization, pressure, compressibility, and Tan’s contact are benchmarked in one spatial dimension, and extended to two and three dimensions, including a study of the unitary Fermi gas. The application of convolutional neural networks to improve the efficiency of Monte Carlo methods is also discussed.Doctor of Philosoph

    The Power of the Weak

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    A landmark result in the study of logics for formal verification is Janin and Walukiewicz’s theorem, stating that the modal μ-calculus (μML) is equivalent modulo bisimilarity to standard monadic second-order logic (here abbreviated as SMSO) over the class of labelled transition systems (LTSs for short). Our work proves two results of the same kind, one for the alternation-free or noetherian fragment μNML of μML on the modal side and one for WMSO, weak monadic second-order logic, on the second-order side. In the setting of binary trees, with explicit functions accessing the left and right successor of a node, it was known that WMSO is equivalent to the appropriate version of alternation-free μ-calculus. Our analysis shows that the picture changes radically once we consider, as Janin and Walukiewicz did, the standard modal μ-calculus, interpreted over arbitrary LTSs. The first theorem that we prove is that, over LTSs, μNML is equivalent modulo bisimilarity to noetherian MSO (NMSO), a newly introduced variant of SMSO where second-order quantification ranges over “conversely well-founded” subsets only. Our second theorem starts from WMSO and proves it equivalent modulo bisimilarity to a fragment of μNML defined by a notion of continuity. Analogously to Janin and Walukiewicz’s result, our proofs are automata-theoretic in nature: As another contribution, we introduce classes of parity automata characterising the expressiveness of WMSO and NMSO (on tree models) and of μCML and μNML (for all transition systems)

    Feature Selection and Classifier Development for Radio Frequency Device Identification

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    The proliferation of simple and low-cost devices, such as IEEE 802.15.4 ZigBee and Z-Wave, in Critical Infrastructure (CI) increases security concerns. Radio Frequency Distinct Native Attribute (RF-DNA) Fingerprinting facilitates biometric-like identification of electronic devices emissions from variances in device hardware. Developing reliable classifier models using RF-DNA fingerprints is thus important for device discrimination to enable reliable Device Classification (a one-to-many looks most like assessment) and Device ID Verification (a one-to-one looks how much like assessment). AFITs prior RF-DNA work focused on Multiple Discriminant Analysis/Maximum Likelihood (MDA/ML) and Generalized Relevance Learning Vector Quantized Improved (GRLVQI) classifiers. This work 1) introduces a new GRLVQI-Distance (GRLVQI-D) classifier that extends prior GRLVQI work by supporting alternative distance measures, 2) formalizes a framework for selecting competing distance measures for GRLVQI-D, 3) introducing response surface methods for optimizing GRLVQI and GRLVQI-D algorithm settings, 4) develops an MDA-based Loadings Fusion (MLF) Dimensional Reduction Analysis (DRA) method for improved classifier-based feature selection, 5) introduces the F-test as a DRA method for RF-DNA fingerprints, 6) provides a phenomenological understanding of test statistics and p-values, with KS-test and F-test statistic values being superior to p-values for DRA, and 7) introduces quantitative dimensionality assessment methods for DRA subset selection

    LQG/LTR Tilt and Tip Control for the Starfire Optical Range 3.5-meter Telescope\u27s Adaptive Optics System

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    The Air Force Research Laboratory has sponsored research on the tracking control loop portion of the adaptive optics system in the Starfire Optical Range 3.5-meter telescope at Kirtland Air Force Base. The control loop includes two steering mirrors (Coarse Steering Mirror and Fine Steering Mirror) used to remove wavefront tilt and tip phase distortion from light entering the telescope. The objective of this research is to design a single Linear Quadratic Gaussian controller to control both steering mirrors in order to eliminate wavefront tilt and tip distortions induced by the earth\u27s atmosphere, and to evaluate the stability robustness and performance of the controller through simulation and Monte Carlo analysis. Controller design elements and simulation parameters are varied to examine and compare resulting performance and robustness characteristics. The controller design is limited to the use of linear models even though components within the control loop have some nonlinear characteristics; however, consideration has been given to the nonlinear aspects of the system via the simulation environment in order to observe the linear controller in a near-real-world environment
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