Most Programs Stop Quickly or Never Halt

Since many real-world problems arising in the fields of compiler optimisation, automated software engineering, formal proof systems, and so forth are equivalent to the Halting Problem--the most notorious undecidable problem--there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect non-halting programs. Our approach is to have the probability space extend over both space and time and to consider the probability that a random $N$-bit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k>0, we can effectively compute a time bound T such that the probability that an N-bit program will eventually halt given that it has not halted by T is smaller than 2^{-k}. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that ``long'' runtimes are effectively rare. More formally, the set of times at which an N-bit program can stop after the time 2^{N+constant} has effectively zero density.Comment: Shortened abstract and changed format of references to match Adv. Appl. Math guideline

On the Dimension and Euler characteristic of random graphs

The inductive dimension dim(G) of a finite undirected graph G=(V,E) is a rational number defined inductively as 1 plus the arithmetic mean of the dimensions of the unit spheres dim(S(x)) at vertices x primed by the requirement that the empty graph has dimension -1. We look at the distribution of the random variable "dim" on the Erdos-Renyi probability space G(n,p), where each of the n(n-1)/2 edges appears independently with probability p. We show here that the average dimension E[dim] is a computable polynomial of degree n(n-1)/2 in p. The explicit formulas allow experimentally to explore limiting laws for the dimension of large graphs. We also study the expectation E[X] of the Euler characteristic X, considered as a random variable on G(n,p). We look experimentally at the statistics of curvature K(v) and local dimension dim(v) = 1+dim(S(v)) which satisfy the Gauss-Bonnet formula X(G) = sum K(v) and by definition dim(G) = sum dim(v)/|V|. We also look at the signature functions f(p)=E[dim], g(p)=E[X] and matrix values functions A(p) = Cov[{dim(v),dim(w)], B(p) = Cov[K(v),K(w)] on the probability space G(p) of all subgraphs of a host graph G=(V,E) with the same vertex set V, where each edge is turned on with probability p. If G is the complete graph or a union of cyclic graphs with have explicit formulas for the signature polynomials f and g.Comment: 18 pages, 14 figures, 4 table

Birthday Inequalities, Repulsion, and Hard Spheres

We study a birthday inequality in random geometric graphs: the probability of the empty graph is upper bounded by the product of the probabilities that each edge is absent. We show the birthday inequality holds at low densities, but does not hold in general. We give three different applications of the birthday inequality in statistical physics and combinatorics: we prove lower bounds on the free energy of the hard sphere model and upper bounds on the number of independent sets and matchings of a given size in d-regular graphs. The birthday inequality is implied by a repulsion inequality: the expected volume of the union of spheres of radius r around n randomly placed centers increases if we condition on the event that the centers are at pairwise distance greater than r. Surprisingly we show that the repulsion inequality is not true in general, and in particular that it fails in 24-dimensional Euclidean space: conditioning on the pairwise repulsion of centers of 24-dimensional spheres can decrease the expected volume of their union

The Union of Probabilistic Boxes: Maintaining the Volume

Suppose we have a set of n axis-aligned rectangular boxes in d-space, {B-1, B-2,..., B-n}, where each box B-i is active (or present) with an independent probability pi. We wish to compute the expected volume occupied by the union of all the active boxes. Our main result is a data structure for maintaining the expected volume over a dynamic family of such probabilistic boxes at an amortized cost of O(n((d-1)/2) log n) time per insert or delete. The core problem turns out to be one-dimensional: we present a new data structure called an anonymous segment tree, which allows us to compute the expected length covered by a set of probabilistic segments in logarithmic time per update. Building on this foundation, we then generalize the problem to d dimensions by combining it with the ideas of Overmars and Yap [13]. Surprisingly, while the expected value of the volume can be efficiently maintained, we show that the tail bounds, or the probability distribution, of the volume are intractable-specifically, it is NP-hard to compute the probability that the volume of the union exceeds a given value V even when the dimension is d = 1

Continuous Monitoring of l_p Norms in Data Streams

In insertion-only streaming, one sees a sequence of indices a_1, a_2, ..., a_m in [n]. The stream defines a sequence of m frequency vectors x(1), ..., x(m) each in R^n, where x(t) is the frequency vector of items after seeing the first t indices in the stream. Much work in the streaming literature focuses on estimating some function f(x(m)). Many applications though require obtaining estimates at time t of f(x(t)), for every t in [m]. Naively this guarantee is obtained by devising an algorithm with failure probability less than 1/m, then performing a union bound over all stream updates to guarantee that all m estimates are simultaneously accurate with good probability. When f(x) is some l_p norm of x, recent works have shown that this union bound is wasteful and better space complexity is possible for the continuous monitoring problem, with the strongest known results being for p=2. In this work, we improve the state of the art for all 0<p<2, which we obtain via a novel analysis of Indyk\u27s p-stable sketch

Tight Bounds for Vertex Connectivity in Dynamic Streams

We present a streaming algorithm for the vertex connectivity problem in dynamic streams with a (nearly) optimal space bound: for any $n$-vertex graph $G$ and any integer $k \geq 1$, our algorithm with high probability outputs whether or not $G$ is $k$-vertex-connected in a single pass using $\widetilde{O}(k n)$ space. Our upper bound matches the known $\Omega(k n)$ lower bound for this problem even in insertion-only streams -- which we extend to multi-pass algorithms in this paper -- and closes one of the last remaining gaps in our understanding of dynamic versus insertion-only streams. Our result is obtained via a novel analysis of the previous best dynamic streaming algorithm of Guha, McGregor, and Tench [PODS 2015] who obtained an $\widetilde{O}(k^2 n)$ space algorithm for this problem. This also gives a model-independent algorithm for computing a "certificate" of $k$-vertex-connectivity as a union of $O(k^2\log{n})$ spanning forests, each on a random subset of $O(n/k)$ vertices, which may be of independent interest.Comment: Full version of the paper accepted to SOSA 2023. 15 pages, 3 Figure

Error performance analysis of n-ary Alamouti scheme with signal space diversity.

Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, a high-rate Alamouti scheme with Signal Space Diversity is developed to improve both the spectral efficiency and overall error performance in wireless communication links. This scheme uses high modulation techniques (M-ary quadrature amplitude modulation (M-QAM) and N-ary phase shift keying modulation (N-PSK)). Hence, this dissertation presents the mathematical models, design methodology and theoretical analysis of this high-rate Alamouti scheme with Signal Space Diversity.To improve spectral efficiency in multiple-input multiple-output (MIMO) wireless communications an N-ary Alamouti M-ary quadrature amplitude modulation (M-QAM) scheme is proposed in this thesis. The proposed N-ary Alamouti M-QAM Scheme uses N-ary phase shift keying modulation (NPSK) and M-QAM. The proposed scheme is investigated in Rayleigh fading channels with additive white Gaussian noise (AWGN). Based on union bound a theoretical average bit error probability (ABEP) of the system is formulated. The simulation results validate the theoretical ABEP. Both theoretical results and simulation results show that the proposed scheme improves spectral efficiency by 0.5 bit/sec/Hz in 2 × 4 16-PSK Alamouti 16-QAM system compared to the conventional Alamouti scheme (16-QAM). To further improve the error performance of the proposed N-ary Alamouti M-QAM Scheme an × N-ary Alamouti coded M-QAM scheme with signal space diversity (SSD) is also proposed in this thesis. In this thesis, based on the nearest neighbour (NN) approach a theoretical closed-form expression of the ABEP is further derived in Rayleigh fading channels. Simulation results also validate the theoretical ABEP for N-ary Alamouti M-QAM scheme with SSD. Both theoretical and simulation results further show that the 2 × 4 4-PSK Alamouti 256-QAM scheme with SSD can achieve 0.8 dB gain compared to the 2 × 4 4-PSK Alamouti 256-QAM scheme without SSD