3,189 research outputs found
Reachability problems for PAMs
Piecewise affine maps (PAMs) are frequently used as a reference model to show
the openness of the reachability questions in other systems. The reachability
problem for one-dimentional PAM is still open even if we define it with only
two intervals. As the main contribution of this paper we introduce new
techniques for solving reachability problems based on p-adic norms and weights
as well as showing decidability for two classes of maps. Then we show the
connections between topological properties for PAM's orbits, reachability
problems and representation of numbers in a rational base system. Finally we
show a particular instance where the uniform distribution of the original orbit
may not remain uniform or even dense after making regular shifts and taking a
fractional part in that sequence.Comment: 16 page
A General Approach to Proving Properties of Fibonacci Representations via Automata Theory
We provide a method, based on automata theory, to mechanically prove the
correctness of many numeration systems based on Fibonacci numbers. With it,
long case-based and induction-based proofs of correctness can be replaced by
simply constructing a regular expression (or finite automaton) specifying the
rules for valid representations, followed by a short computation. Examples of
the systems that can be handled using our technique include Brown's lazy
representation (1965), the far-difference representation developed by Alpert
(2009), and three representations proposed by Hajnal (2023). We also provide
three additional systems and prove their validity.Comment: In Proceedings AFL 2023, arXiv:2309.0112
The target discounted-sum problem
The target discounted-sum problem is the following: Given a rational discount factor 0 < λ < 1 and three rational values a, b, and t, does there exist a finite or an infinite sequence w ε(a, b)∗ or w ε(a, b)w, such that Σ|w| i=0 w(i)λi equals t? The problem turns out to relate to many fields of mathematics and computer science, and its decidability question is surprisingly hard to solve. We solve the finite version of the problem, and show the hardness of the infinite version, linking it to various areas and open problems in mathematics and computer science: β-expansions, discounted-sum automata, piecewise affine maps, and generalizations of the Cantor set. We provide some partial results to the infinite version, among which are solutions to its restriction to eventually-periodic sequences and to the cases that λ λ 1/2 or λ = 1/n, for every n ε N. We use our results for solving some open problems on discounted-sum automata, among which are the exact-value problem for nondeterministic automata over finite words and the universality and inclusion problems for functional automata
Empirical Bounds on Linear Regions of Deep Rectifier Networks
We can compare the expressiveness of neural networks that use rectified
linear units (ReLUs) by the number of linear regions, which reflect the number
of pieces of the piecewise linear functions modeled by such networks. However,
enumerating these regions is prohibitive and the known analytical bounds are
identical for networks with same dimensions. In this work, we approximate the
number of linear regions through empirical bounds based on features of the
trained network and probabilistic inference. Our first contribution is a method
to sample the activation patterns defined by ReLUs using universal hash
functions. This method is based on a Mixed-Integer Linear Programming (MILP)
formulation of the network and an algorithm for probabilistic lower bounds of
MILP solution sets that we call MIPBound, which is considerably faster than
exact counting and reaches values in similar orders of magnitude. Our second
contribution is a tighter activation-based bound for the maximum number of
linear regions, which is particularly stronger in networks with narrow layers.
Combined, these bounds yield a fast proxy for the number of linear regions of a
deep neural network.Comment: AAAI 202
Dynamical behavior of alternate base expansions
peer reviewedWe generalize the greedy and lazy β-transformations for a real base β to
the setting of alternate bases β = (β0, . . . , βp−1), which were recently introduced by the
first and second authors as a particular case of Cantor bases. As in the real base case,
these new transformations, denoted Tβ and Lβ respectively, can be iterated in order to
generate the digits of the greedy and lazy β-expansions of real numbers. The aim of this
paper is to describe the dynamical behaviors of Tβ and Lβ. We first prove the existence
of a unique absolutely continuous (with respect to an extended Lebesgue measure, called
the p-Lebesgue measure) Tβ-invariant measure. We then show that this unique measure
is in fact equivalent to the p-Lebesgue measure and that the corresponding dynamical
system is ergodic and has entropy 1/p log(βp−1 · · · β0). We then express the density of this p
measure and compute the frequencies of letters in the greedy β-expansions. We obtain the dynamical properties of Lβ by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the β- shift. Finally, we show that the β-expansions can be seen as (βp−1 · · · β0 )-representations over general digit sets and we compare both frameworks
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