Related papers: Collatz convergence is a Hydra game
Online game playing algorithms produce high-quality strategies with a fraction of memory and computation required by their offline alternatives. Continual Resolving (CR) is a recent theoretically sound approach to online game playing that…
Coinduction refers to both a technique for the definition of infinite streams, so-called codata, and a technique for proving the equality of coinductively specified codata. This article first reviews coinduction in declarative programming.…
Counters that hold natural numbers are ubiquitous in modeling and verifying software systems; for example, they model dynamic creation and use of resources in concurrent programs. Unfortunately, such discrete counters often lead to…
We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either 'win' or 'lose'. Molina and…
In this paper we are shown the following facts: The probability of increased $ A_{k}=P(T^{k} (x_{0})>T^{k-1} (x_{0})) $, and the probability of decrease $B_{k}=P(T^{k} (x_{0})<T^{k-1} (x_{0}))$ in step $ k $ of a Collataz procedure…
We introduce the \emph{Collatz conjecture} and its history. Some definition that this conjecture has, will be expressed and with these we try to explain some good lemma to justify the main properties of the \emph{Collatz conjecture}. With…
One-dimensional billiard, i.e. a chain of colliding particles with equal masses, is well-known example of completely integrable system. Billiards with different particles are generically not integrable, but still exhibit divergence of a…
Coalition formation over graphs is a well studied class of games whose players are vertices and feasible coalitions must be connected subgraphs. In this setting, the existence and computation of equilibria, under various notions of…
We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in…
Non-Hermitian quantum systems exhibit unique properties and hold significant promise for diverse applications, yet their dynamical simulation poses a particular challenge due to intrinsic openness and non-unitary evolution. Here, we…
We describe a new algorithm for verifying the Collatz conjecture for all n < 2^N for some fixed N. The algorithm takes less than twice as long to verify convergence for all n < 2^{N+1} as it does to verify convergence for all n < 2^N. We…
We study variants of the well-known Collatz graph, by considering the action of the 3n+1 function on congruence classes. For moduli equal to powers of 2, these graphs are shown to be isomorphic to binary De Bruijn graphs. Unlike the Collatz…
Suppose we are given a system of coupled oscillators on an unknown graph along with the trajectory of the system during some period. Can we predict whether the system will eventually synchronize? Even with a known underlying graph…
Automata operating on infinite objects feature prominently in the theory of the modal $\mu$-calculus. One such application concerns the tableau games introduced by Niwi\'{n}ski & Walukiewicz, of which the winning condition for infinite…
We present a novel coalgebraic formulation of infinite extensive games. We define both the game trees and the strategy profiles by possibly infinite systems of corecursive equations. Certain strategy profiles are proved to be subgame…
The Kaczmarz algorithm (KA) is a popular method for solving a system of linear equations. In this note we derive a new exponential convergence result for the KA. The key allowing us to establish the new result is to rewrite the KA in such a…
We present Mortar, a system for autonomously evolving game mechanics for automatic game design. Game mechanics define the rules and interactions that govern gameplay, and designing them manually is a time-consuming and expert-driven…
Games on recursive game graphs can be used to reason about the control flow of sequential programs with recursion. In games over recursive game graphs, the most natural notion of strategy is the modular strategy, i.e., a strategy that is…
In this paper, we consider concave continuous-kernel games characterized by monotonicity properties and propose discounted mirror descent-type dynamics. We introduce two classes of dynamics whereby the associated mirror map is constructed…
Enormous successes have been made by quantum algorithms during the last decade. In this paper, we combine the quantum game with the problem of data clustering, and then develop a quantum-game-based clustering algorithm, in which data points…