Related papers: Nim Fractals
Here, we present a variant of Nim with two piles. In the first pile, we have stones with a weight of 1, and in the second pile, we have stones with a weight of -2. Two Players take turns to take stones from one of the piles, and the total…
We introduce gym-city, a Reinforcement Learning environment that uses SimCity 1's game engine to simulate an urban environment, wherein agents might seek to optimize one or a combination of any number of city-wide metrics, on gameboards of…
We consider Subtraction Nim, where two players have exactly the same options, but which is partizan in the sense that at the game ending, a partizan rule is applied for the decision of the winner. We consider the following example: Let $S$…
We Propose A Novel Automaton Model which uses Arithmetic Operations as the Evolving Rules, each cell has the states of the Natural Numbers k = (N), a radius of r = 1/2 and operates on an arbitrary input size. The Automaton reads an…
We investigate critical properties of a spatial evolutionary game based on the Prisoner's Dilemma. Simulations demonstrate a jump in the component densities accompanied by drastic changes in average sizes of the component clusters. We argue…
We show how to efficiently enumerate a class of finite-memory stochastic processes using the causal representation of epsilon-machines. We characterize epsilon-machines in the language of automata theory and adapt a recent algorithm for…
Universal register machine, a formal model of computation, can be emulated on the array of the Game of Life, a two-dimensional cellular automaton. We perform spectral analysis on the computation dynamical process of the universal register…
We consider Flipping Coins, a partizan version of the impartial game Turning Turtles, played on lines of coins. We show the values of this game are numbers, and these are found by first applying a reduction, then decomposing the position…
In this paper we investigate the problem of automatically naming pieces of assembly code. Where by naming we mean assigning to an assembly function a string of words that would likely be assigned by a human reverse engineer. We formally and…
Ulam words are binary words defined recursively as follows: the length-$1$ Ulam words are $0$ and $1$, and a binary word of length $n$ is Ulam if and only if it is expressible uniquely as a concatenation of two shorter, distinct Ulam words.…
We survey and prove properties a family of recurrences bears in relation to integer representations, compositions, the Pascal triangle, sums of digits, Nim games and Beatty sequences.
A temporal-theoretic formalism for understanding game theory is described where a strict ordering relation on a set of time points $T$ defines a game on $T$. Using this formalism, a proof of Zermelo's Theorem, which states that every finite…
Berlekamp proposed a class of impartial combinatorial games based on the moves of chess pieces on rectangular boards. We generalize impartial chess games by playing them on Young diagrams and obtain results about winning and losing…
We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps $\{\mathcal{F}_k\}_{k\in \mathbb{N}}$ where each $\mathcal{F}_k$ maps $\mathcal{H}(X)\to…
Partial-monitoring games constitute a mathematical framework for sequential decision making problems with imperfect feedback: The learner repeatedly chooses an action, opponent responds with an outcome, and then the learner suffers a loss…
Cellular automata are computers, similar to Turing machines. The main difference is that Turing machines use a one-dimensional tape, whereas cellular automata use a two-dimensional grid. The best-known cellular automaton is the Game of…
A new combinatorial game is given. It generalizes both Substraction and Nim. It is proved the computation of Nash equilibrium points in this new game is NP-hard.
We study a two-dimensional semi-totalistic binary cell-state cellular automaton, which imitates a reversible precipitation in an abstract chemical medium. The systems exhibits a non-trivial growth and nucleation. We demonstrate how basic…
We exhibit a procedure to asymptotically enumerate monotone grid classes of permutations. This is then applied to compute the asymptotic number of permutations in any connected one-corner class. Our strategy consists of enumerating the…
In this paper, we considered impartial games on a simplicial complex. Each vertex of a given simplicial complex acts as a position of an impartial game. Each player in turn chooses a face of the simplicial complex and, for each position on…