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Horn-satisfiability or Horn-SAT is the problem of deciding whether a satisfying assignment exists for a Horn formula, a conjunction of clauses each with at most one positive literal (also known as Horn clauses). It is a well-known P-complete problem, which implies that unless P = NC, it is a hard problem to parallelize. In this paper, we empirically show that, under a known simple random model for generating the Horn formula, the ratio of hard-to-parallelize instances (closer to the worst-case behavior) is infinitesimally small. We show that the depth of a parallel algorithm for Horn-SAT is polylogarithmic on average, for almost all instances, while keeping the work linear. This challenges theoreticians and programmers to look beyond worst-case analysis and come up with practical algorithms coupled with respective performance guarantees.

We consider a combinatorial multi-armed bandit problem for maximum value reward function under maximum value and index feedback. This is a new feedback structure that lies in between commonly studied semi-bandit and full-bandit feedback structures. We propose an algorithm and provide a regret bound for problem instances with stochastic arm outcomes according to arbitrary distributions with finite supports. The regret analysis rests on considering an extended set of arms, associated with values and probabilities of arm outcomes, and applying a smoothness condition. Our algorithm achieves a $O((k/\Delta)\log(T))$ distribution-dependent and a $\tilde{O}(\sqrt{T})$ distribution-independent regret where $k$ is the number of arms selected in each round, $\Delta$ is a distribution-dependent reward gap and $T$ is the horizon time. Perhaps surprisingly, the regret bound is comparable to previously-known bound under more informative semi-bandit feedback. We demonstrate the effectiveness of our algorithm through experimental results.

The Functional Machine Calculus (FMC) was recently introduced as a generalization of the lambda-calculus to include higher-order global state, probabilistic and non-deterministic choice, and input and output, while retaining confluence. The calculus can encode both the call-by-name and call-by-value semantics of these effects. This is enabled by two independent generalisations, both natural from the perspective of the FMC's operational semantics, which is given by a simple multi-stack machine. The first generalization decomposes the syntax of the lambda-calculus in a way that allows for sequential composition of terms and the encoding of reduction strategies. Specifically, there exist translations of the call-by-name and call-by-value lambda-calculus which preserve operational semantics. The second parameterizes application and abstraction in terms of 'locations' (corresponding to the multiple stacks of the machine), which gives a unification of the operational semantics, syntax, and reduction rules of the given effects with those of the lambda-calculus. The FMC further comes equipped with a simple type system which restricts and captures the behaviour of effects. This thesis makes two main contributions, showing that two fundamental properties of the lambda-calculus are preserved by the FMC. The first is to show that the categorical semantics of the FMC, modulo an appropriate equational theory, is given by the free Cartesian closed category. The equational theory is validated by a notion of observational equivalence. The second contribution is a proof that typed FMC-terms are strongly normalising. This is an extension (and small simplification) of Gandy's proof for the lambda-calculus, which additionally emphasizes its latent operational intuition.

An Eulerian circuit in a directed graph is one of the most fundamental Graph Theory notions. Detecting if a graph $G$ has a unique Eulerian circuit can be done in polynomial time via the BEST theorem by de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, 1941-1951 (involving counting arborescences), or via a tailored characterization by Pevzner, 1989 (involving computing the intersection graph of simple cycles of $G$), both of which thus rely on overly complex notions for the simpler uniqueness problem. In this paper we give a new linear-time checkable characterization of directed graphs with a unique Eulerian circuit. This is based on a simple condition of when two edges must appear consecutively in all Eulerian circuits, in terms of cut nodes of the underlying undirected graph of $G$. As a by-product, we can also compute in linear-time all maximal $\textit{safe}$ walks appearing in all Eulerian circuits, for which Nagarajan and Pop proposed in 2009 a polynomial-time algorithm based on Pevzner characterization.

This paper delves into stochastic optimization problems that involve Markovian noise. We present a unified approach for the theoretical analysis of first-order gradient methods for stochastic optimization and variational inequalities. Our approach covers scenarios for both non-convex and strongly convex minimization problems. To achieve an optimal (linear) dependence on the mixing time of the underlying noise sequence, we use the randomized batching scheme, which is based on the multilevel Monte Carlo method. Moreover, our technique allows us to eliminate the limiting assumptions of previous research on Markov noise, such as the need for a bounded domain and uniformly bounded stochastic gradients. Our extension to variational inequalities under Markovian noise is original. Additionally, we provide lower bounds that match the oracle complexity of our method in the case of strongly convex optimization problems.

Goal-oriented error estimation provides the ability to approximate the discretization error in a chosen functional quantity of interest. Adaptive mesh methods provide the ability to control this discretization error to obtain accurate quantity of interest approximations while still remaining computationally feasible. Traditional discrete goal-oriented error estimates incur linearization errors in their derivation. In this paper, we investigate the role of linearization errors in adaptive goal-oriented error simulations. In particular, we develop a novel two-level goal-oriented error estimate that is free of linearization errors. Additionally, we highlight how linearization errors can facilitate the verification of the adjoint solution used in goal-oriented error estimation. We then verify the newly proposed error estimate by applying it to a model nonlinear problem for several quantities of interest and further highlight its asymptotic effectiveness as mesh sizes are reduced. In an adaptive mesh context, we then compare the newly proposed estimate to a more traditional two-level goal-oriented error estimate. We highlight that accounting for linearization errors in the error estimate can improve its effectiveness in certain situations and demonstrate that localizing linearization errors can lead to more optimal adapted meshes.

News articles are driven by the informational sources journalists use in reporting. Modeling when, how and why sources get used together in stories can help us better understand the information we consume and even help journalists with the task of producing it. In this work, we take steps toward this goal by constructing the largest and widest-ranging annotated dataset, to date, of informational sources used in news writing. We show that our dataset can be used to train high-performing models for information detection and source attribution. We further introduce a novel task, source prediction, to study the compositionality of sources in news articles. We show good performance on this task, which we argue is an important proof for narrative science exploring the internal structure of news articles and aiding in planning-based language generation, and an important step towards a source-recommendation system to aid journalists.

The Stochastic Gradient Langevin Dynamics (SGLD) are popularly used to approximate Bayesian posterior distributions in statistical learning procedures with large-scale data. As opposed to many usual Markov chain Monte Carlo (MCMC) algorithms, SGLD is not stationary with respect to the posterior distribution; two sources of error appear: The first error is introduced by an Euler--Maruyama discretisation of a Langevin diffusion process, the second error comes from the data subsampling that enables its use in large-scale data settings. In this work, we consider an idealised version of SGLD to analyse the method's pure subsampling error that we then see as a best-case error for diffusion-based subsampling MCMC methods. Indeed, we introduce and study the Stochastic Gradient Langevin Diffusion (SGLDiff), a continuous-time Markov process that follows the Langevin diffusion corresponding to a data subset and switches this data subset after exponential waiting times. There, we show that the Wasserstein distance between the posterior and the limiting distribution of SGLDiff is bounded above by a fractional power of the mean waiting time. Importantly, this fractional power does not depend on the dimension of the state space. We bring our results into context with other analyses of SGLD.

Extreme valuation and volatility of cryptocurrencies require investors to diversify often which demands secure exchange protocols. A cross-chain swap protocol allows distrusting parties to securely exchange their assets. However, the current models and protocols assume predefined user preferences for acceptable outcomes. This paper presents a generalized model of swaps that allows each party to specify its preferences on the subsets of its incoming and outgoing assets. It shows that the existing swap protocols are not necessarily a strong Nash equilibrium in this model. It characterizes the class of swap graphs that have protocols that are safe, live and a strong Nash equilibrium, and presents such a protocol for this class. Further, it shows that deciding whether a swap is in this class is NP-hard through a reduction from 3SAT, and further is $\Sigma_2^{\mathsf{P}}$-complete through a reduction from $\exists\forall\mathsf{DNF}$.

A bond in a graph is a minimal nonempty edge-cut. A connected graph $G$ is dual Hamiltonian if the vertex set can be partitioned into two subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are both trees. There is much interest in studying the longest cycles and largest bonds in graphs. H. Wu conjectured that any longest cycle must meet any largest bond in a simple 3-connected graph. In this paper, the author proves that the above conjecture is true for certain classes of 3-connected graphs: Let $G$ be a simple 3-connected graph with $n$ vertices and $m$ edges. Suppose $c(G)$ is the size of a longest cycle, and $c^*(G)$ is the size of a largest bond. Then each longest cycle meets each largest bond if either $c(G) \geq n - 3$ or $c^*(G) \geq m - n - 1$. Sanford determined in her Ph.D. thesis the cycle spectrum of the well-known generalized Petersen graph $P(n, 2)$ ($n$ is odd) and $P(n, 3)$ ($n$ is even). Flynn proved in her honors thesis that any generalized Petersen graph $P(n, k)$ is dual Hamiltonian. The author studies the bond spectrum (called the co-spectrum) of the generalized Petersen graphs and extends Flynn's result by proving that in any generalized Petersen graph $P(n, k)$, $1 \leq k < \frac{n}{2}$, the co-spectrum of $P(n, k)$ is $\{3, 4, 5, ..., n+2\}$.