Related papers: Effective Bounds for P-Recursive Sequences
With the increasing application of deep learning in mission-critical systems, there is a growing need to obtain formal guarantees about the behaviors of neural networks. Indeed, many approaches for verifying neural networks have been…
We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root. By additionally verifying finitely many initial terms, the positivity can also be resolved. As an…
Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute…
Let $p\geq 3$ be a prime and $n\geq 1$ be an integer. Let $K\subseteq {\mathbb{F}_p}$ denote a fixed subset with $0\in K$. Let $A\subseteq ({\mathbb{F}_p})^n$ be an arbitrary subset such that $$\{…
Linear recurrence equations with constant coefficients define the power series coefficients of rational functions. However, one usually prefers to have an explicit formula for the sequence of coefficients, provided that such a formula is…
The linear complexity of a periodic sequence over $GF(p^m)$ plays an important role in cryptography and communication [12]. In this correspondence, we prove a result which reduces the computation of the linear complexity and minimal…
We consider a class of multivariate recurrences frequently arising in the worst case analysis of Davis-Putnam-style exponential time backtracking algorithms for NP-hard problems. We describe a technique for proving asymptotic upper bounds…
Deep neural networks are revolutionizing the way complex systems are developed. However, these automatically-generated networks are opaque to humans, making it difficult to reason about them and guarantee their correctness. Here, we propose…
For a wide class of integer linear recurrence sequences $\left(u(n)\right)_{n=1}^\infty$, we give an upper bound on the number of $s$-tuples $\left(n_1, \ldots, n_s\right) \in \left(\mathbb Z\cap [M+1,M+ N]\right)^s$ such that the…
In the field of reinforcement learning there has been recent progress towards safety and high-confidence bounds on policy performance. However, to our knowledge, no practical methods exist for determining high-confidence policy performance…
We revisit parallel-innermost term rewriting as a model of parallel computation on inductive data structures and provide a corresponding notion of runtime complexity parametric in the size of the start term. We propose automatic techniques…
The linear complexity (LC) of a sequence has been used as a convenient measure of the randomness of a sequence. Based on the theories of linear complexity, $k$-error linear complexity, the minimum error and the $k$-error linear complexity…
Several algorithms in computer algebra involve the computation of a power series solution of a given ordinary differential equation. Over finite fields, the problem is often lifted in an approximate $p$-adic setting to be well-posed. This…
We consider questions related to the computation of the capacity of codes that avoid forbidden difference patterns. The maximal number of $n$-bit sequences whose pairwise differences do not contain some given forbidden difference patterns…
The standard coherence criterion for lower previsions is expressed using an infinite number of linear constraints. For lower previsions that are essentially defined on some finite set of gambles on a finite possibility space, we present a…
Bounds on the entropy of patterns of sequences generated by independently identically distributed (i.i.d.) sources are derived. A pattern is a sequence of indices that contains all consecutive integer indices in increasing order of first…
Recurrent neural network (RNN)'s architecture is a key factor influencing its performance. We propose algorithms to optimize hidden sizes under running time constraint. We convert the discrete optimization into a subset selection problem.…
The classical approach to measure the expressive power of deep neural networks with piecewise linear activations is based on counting their maximum number of linear regions. This complexity measure is quite relevant to understand general…
We introduce the concept of regular expression characteristics as a unified way to concisely express bounds on time-series constraints. This allows us not only to define time-series constraints in a compositional way, but also to deal with…
In this paper, we present a new technique to obtain upper bounds on undirected unicast network information capacity. Using this technique, we characterize an upper bound, called partition bound, on the symmetric rate of information flow in…