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The undecidability of basic decision problems for general FIFO machines such as reachability and unboundedness is well-known. In this paper, we provide an underapproximation for the general model by considering only runs that are input-bounded (i.e. the sequence of messages sent through a particular channel belongs to a given bounded language). We prove, by reducing this model to a counter machine with restricted zero tests, that the rational-reachability problem (and by extension, control-state reachability, unboundedness, deadlock, etc.) is decidable. This class of machines subsumes input-letter-bounded machines, flat machines, linear FIFO nets, and monogeneous machines, for which some of these problems were already shown to be decidable. These theoretical results can form the foundations to build a tool to verify general FIFO machines based on the analysis of input-bounded machines.

System-oriented IR evaluations are limited to rather abstract understandings of real user behavior. As a solution, simulating user interactions provides a cost-efficient way to support system-oriented experiments with more realistic directives when no interaction logs are available. While there are several user models for simulated clicks or result list interactions, very few attempts have been made towards query simulations, and it has not been investigated if these can reproduce properties of real queries. In this work, we validate simulated user query variants with the help of TREC test collections in reference to real user queries that were made for the corresponding topics. Besides, we introduce a simple yet effective method that gives better reproductions of real queries than the established methods. Our evaluation framework validates the simulations regarding the retrieval performance, reproducibility of topic score distributions, shared task utility, effort and effect, and query term similarity when compared with real user query variants. While the retrieval effectiveness and statistical properties of the topic score distributions as well as economic aspects are close to that of real queries, it is still challenging to simulate exact term matches and later query reformulations.

In the theory of linear switching systems with discrete time, as in other areas of mathematics, the problem of studying the growth rate of the norms of all possible matrix products $A_{\sigma_{n}}\cdots A_{\sigma_{0}}$ with factors from a set of matrices $\mathscr{A}$ arises. So far, only for a relatively small number of classes of matrices $\mathscr{A}$ has it been possible to accurately describe the sequences of matrices that guarantee the maximum rate of increase of the corresponding norms. Moreover, in almost all cases studied theoretically, the index sequences $\{\sigma_{n}\}$ of matrices maximizing the norms of the corresponding matrix products have been shown to be periodic or so-called Sturmian, which entails a whole set of "good" properties of the sequences $\{A_{\sigma_{n}}\}$, in particular the existence of a limiting frequency of occurrence of each matrix factor $A_{i}\in\mathscr{A}$ in them. In the paper it is shown that this is not always the case: a class of matrices is defined consisting of two $2\times 2$ matrices, similar to rotations in the plane, in which the sequence $\{A_{\sigma_{n}}\}$ maximizing the growth rate of the norms $\|A_{\sigma_{n}}\cdots A_{\sigma_{0}}\|$ is not Sturmian. All considerations are based on numerical modeling and cannot be considered mathematically rigorous in this part; rather, they should be interpreted as a set of questions for further comprehensive theoretical analysis.

We study the realizability problem for Safety LTL, the syntactic fragment of Linear Temporal Logic capturing safe formulas. We show that the problem is EXP-complete, disproving the existing conjecture of 2EXP-completeness. We achieve this by comparing the complexity of Safety LTL with seemingly weaker subfragments. In particular, we show that every formula of Safety LTL can be reduced to an equirealizable formula of the form $\alpha \land \Box \psi$, where $\alpha$ is a present formula over system variables and $\psi$ contains Next as the only temporal operator. The realizability problem for this new fragment, which we call $\mathsf{GX}_{\mathsf{0}}$, is also EXP-complete.

Inferring inductive invariants is one of the main challenges of formal verification. The theory of abstract interpretation provides a rich framework to devise invariant inference algorithms. One of the latest breakthroughs in invariant inference is property-directed reachability (PDR), but the research community views PDR and abstract interpretation as mostly unrelated techniques. This paper shows that, surprisingly, propositional PDR can be formulated as an abstract interpretation algorithm in a logical domain. More precisely, we define a version of PDR, called $\Lambda$-PDR, in which all generalizations of counterexamples are used to strengthen a frame. In this way, there is no need to refine frames after their creation, because all the possible supporting facts are included in advance. We analyze this algorithm using notions from Bshouty's monotone theory, originally developed in the context of exact learning. We show that there is an inherent overapproximation between the algorithm's frames that is related to the monotone theory. We then define a new abstract domain in which the best abstract transformer performs this overapproximation, and show that it captures the invariant inference process, i.e., $\Lambda$-PDR corresponds to Kleene iterations with the best transformer in this abstract domain. We provide some sufficient conditions for when this process converges in a small number of iterations, with sometimes an exponential gap from the number of iterations required for naive exact forward reachability. These results provide a firm theoretical foundation for the benefits of how PDR tackles forward reachability.

Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank. Our proof of the above result builds on the results of Marinari, M\"oller and Mora (1993), and M\"oller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.

A strict bramble of a graph $G$ is a collection of pairwise-intersecting connected subgraphs of $G.$ The order of a strict bramble ${\cal B}$ is the minimum size of a set of vertices intersecting all sets of ${\cal B}.$ The strict bramble number of $G,$ denoted by ${\sf sbn}(G),$ is the maximum order of a strict bramble in $G.$ The strict bramble number of $G$ can be seen as a way to extend the notion of acyclicity, departing from the fact that (non-empty) acyclic graphs are exactly the graphs where every strict bramble has order one. We initiate the study of this graph parameter by providing three alternative definitions, each revealing different structural characteristics. The first is a min-max theorem asserting that ${\sf sbn}(G)$ is equal to the minimum $k$ for which $G$ is a minor of the lexicographic product of a tree and a clique on $k$ vertices (also known as the lexicographic tree product number). The second characterization is in terms of a new variant of a tree decomposition called lenient tree decomposition. We prove that ${\sf sbn}(G)$ is equal to the minimum $k$ for which there exists a lenient tree decomposition of $G$ of width at most $k.$ The third characterization is in terms of extremal graphs. For this, we define, for each $k,$ the concept of a $k$-domino-tree and we prove that every edge-maximal graph of strict bramble number at most $k$ is a $k$-domino-tree. We also identify three graphs that constitute the minor-obstruction set of the class of graphs with strict bramble number at most two. We complete our results by proving that, given some $G$ and $k,$ deciding whether ${\sf sbn}(G) \leq k$ is an ${\sf NP}$-complete problem.

Based on the Bezout approach we propose a simple algorithm to determine the {\tt gcd} of two polynomials which doesn't need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The algorithm needs only $n$ steps for polynomials of degree $n$. Formal manipulations give the discriminant or the resultant for any degree without needing division nor determinant calculation.

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.

Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.