Algorithmic properties of first-order modal logics of finite Kripke frames in restricted languages

We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $\textbf{GL}$, $\textbf{Grz}$ and $\textbf{KTB}$—among them, $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{K4}$ and $\textbf{S4}$.

Frames with desired angle properties

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] The purpose of this dissertation is to study frames with desired angle properties. More precisely, we study the subspace packing problem, harmonic biangular tight frames, and regular two-distance sets. After a brief review of finite frame theory in Chapter 1, we study the subspace packing problem with respect to the chordal distance in Chapter 2. We show that a solution to this problem is necessarily a fusion frame for the underlying space. We then continue to exploit the idea of using maximal sets of mutually unbiased bases and block designs to construct several infinite families of solutions to the problem. In Chapter 3, motivated by the characterization of harmonic equiangular tight frames in terms of difference sets, we study harmonic biangular tight frames and their connection with other combinatorial structures including divisible difference sets, partial difference sets, and Gaussian difference sets. Chapter 4 is dedicated to studying spherical two-distance sets. These are sets of unit vectors for which the inner products admit two different values. In this chapter, we investigate a special case of such sets when we require one inner product value to appear the same number of times in each row of the Gram matrix. We call this type of set a regular two-distance set. We present various properties of such sets as well as focus on the case where the sets form tight frames for the underlying space. Several connections among regular two-distance sets, equiangular lines, and quasi-symmetric designs are also discussed.

A Dynamic Access Probability Adjustment Strategy for Coded Random Access Schemes

In this paper, a dynamic access probability adjustment strategy for coded random accessschemes based on successive interference cancellation (SIC) is proposed. The developed protocolconsists of judiciously tuning the access probability, therefore controlling the number of transmittingusers, in order to resolve medium access control (MAC) layer congestion states in high load conditions.The protocol is comprised of two steps: Estimation of the number of transmitting users during thecurrent MAC frame and adjustment of the access probability to the subsequent MAC frame, based onthe performed estimation. The estimation algorithm exploits a posteriori information, i.e., availableinformation at the end of the SIC process, in particular it relies on both the frame configuration(residual number of collision slots) and the recovered users configuration (vector of recovered users)to effectively reduce mean-square error (MSE). During the access probability adjustment phase, atarget load threshold is employed, tailored to the packet loss rate in the finite frame length case.Simulation results revealed that the developed estimator was able to achieve remarkable performanceowing to the information gathered from the SIC procedure. It also illustrated how the proposeddynamic access probability strategy can resolve congestion states efficiently.

Quantization for low-rank matrix recovery

We study Sigma–Delta $(\varSigma\!\varDelta) $ quantization methods coupled with appropriate reconstruction algorithms for digitizing randomly sampled low-rank matrices. We show that the reconstruction error associated with our methods decays polynomially with the oversampling factor, and we leverage our results to obtain root-exponential accuracy by optimizing over the choice of quantization scheme. Additionally, we show that a random encoding scheme, applied to the quantized measurements, yields a near-optimal exponential bit rate. As an added benefit, our schemes are robust both to noise and to deviations from the low-rank assumption. In short, we provide a full generalization of analogous results, obtained in the classical setup of band-limited function acquisition, and more recently, in the finite frame and compressed sensing setups to the case of low-rank matrices sampled with sub-Gaussian linear operators. Finally, we believe our techniques for generalizing results from the compressed sensing setup to the analogous low-rank matrix setup is applicable to other quantization schemes.

Parseval transforms for finite frames

The relationship between frames and Parseval frames is an important topic in frame theory. In this paper, we investigate Parseval transforms, which are linear transforms turning general finite frames into Parseval frames. We introduce two classes of transforms in terms of the right regular and left Parseval transform matrices (RRPTMs and LPTMs). We give representations of all the RRPTMs and LPTMs of any finite frame. Two important LPTMs are discussed in this paper, the canonical LPTM (square root of the inverse frame operator) and the RGS matrix, which are obtained by using row’s Gram–Schmidt orthogonalization. We also investigate the relationship between the Parseval frames generated by these two LPTMs. Meanwhile, for RRPTMs, we verify the existence of invertible RRPTMs for any given finite frame. Finally, we discuss the existence of block diagonal RRPTMs by taking the graph structure of the frame elements into consideration.