Computing backwards with Game of Life, part 1: wires and circuits

Conway's Game of Life is a two-dimensional cellular automaton. As a dynamical system, it is well-known to be computationally universal, i.e.\ capable of simulating an arbitrary Turing machine. We show that in a sense taking a single backwards step of Game of Life is a computationally universal process, by constructing patterns whose preimage computation encodes an arbitrary circuit-satisfaction problem, or (equivalently) any tiling problem. As a corollary, we obtain for example that the set of orphans is coNP-complete, exhibit a $6210 \times 37800$-periodic configuration that admits a preimage but no periodic one, and prove that the existence of a preimage for a periodic point is undecidable. Our constructions were obtained by a combination of computer searches and manual design.Comment: 28 pages, 10 figures in main text. 11 pages, 20 figures in appendix. Accompanied by two GitHub repositories containing programs and auxiliary dat

On The Foundations of Digital Games

Computers have lead to a revolution in the games we play, and, following this, an interest for computer-based games has been sparked in research communities. However, this easily leads to the perception of a one-way direction of influence between that the field of game research and computer science. This historical investigation points towards a deep and intertwined relationship between research on games and the development of computers, giving a richer picture of both fields. While doing so, an overview of early game research is presented and an argument made that the distinction between digital games and non-digital games may be counter-productive to game research as a whole

Turing machine universality of the game of life

This project proves universal computation in the Game of Life cellular automaton by using a Turing machine construction.Existing proofs of universality in the Game of Life rely on a counter machine. These machines require complex encoding and decoding of the input and output and the proof of universality for these machines by the Church Turing thesis is that they can perform the equivalent of a Turing machine. A proof based directly on a Turing machine is much more accessible.The computational power available today allows powerful algorithms such as HashLife to calculate the evolution of cellular automata patterns sufficiently fast that an efficient universal Turing machine can be demonstrated in a conveniently short period of time. Such a universal Turing machine is presented here. It is a direct simulation of a Turing machine and the input and output are easily interpreted.In order to achieve full universal behaviour an infinite storage media is required. The storage media used to represent the Turing machine tape is a pair of stacks. One stack representing the Turing tape to the left of the read/write head and one for the Turing tape to the right. Collision based construction techniques have been used to add stack cells to the ends of the stacks continuously.The continuous construction of the stacks is equivalent to the formatting of blank media. This project demonstrates that large areas of a cellular automata can be formatted in real time to perform complex functions

Simulations Between Programs as Cellular Automata

We present cellular automata on appropriate digraphs and show that any covered normal logic program is a cellular automaton. Seeing programs as cellular automata shifts attention from classes of Herbrand models to orbits of Herbrand interpretations. Orbits capture both the declarative, model-theoretic meaning of programs as well as their inferential behavior. Logically and intentionally different programs can produce orbits that simulate each other. Simple examples of such behavior are compellingly exhibited with space-time diagrams of the programs as cellular automata. Construing a program as a cellular automaton leads to a general method for simulating any covered program with a Horn clause program

Implementation of Logical Functions in the Game of Life

The Game of Life cellular automaton is a classical example of a massively parallel collision-based computing device. The automaton exhibits mobile patterns, gliders, and generators of the mobile patterns, glider guns, in its evolution. We show how to construct basic logical perations, AND, OR, NOT in space-time configurations of the cellular automaton. Also decomposition of complicated Boolean functions is discussed. Advantages of our technique are demonstrated on an example of binary adder, realized via collision of glider streams

Ontology in the Game of Life

The game of life is an excellent framework for metaphysical modeling. It can be used to study ontological categories like space, time, causality, persistence, substance, emergence, and supervenience. It is often said that there are many levels of existence in the game of life. Objects like the glider are said to exist on higher levels. Our goal here is to work out a precise formalization of the thesis that there are various levels of existence in the game of life. To formalize this thesis, we develop a set-theoretic construction of the glider. The method of this construction generalizes to other patterns in the game of life. And it can be extended to more realistic physical systems. The result is a highly general method for the set-theoretical construction of substance

Martin Gardner and His Influence on Recreational Math

Recreational mathematics is a relatively new field in the world of mathematics. While sometimes overlooked as frivolous since those who practice it need no advanced knowledge of the subject, recreational mathematics is a perfect transition for people to experience the joy in logically establishing a solution. Martin Gardner recognized that this pattern of proving solutions to questions is how mathematics progresses. From his childhood on, Gardner greatly influenced the mathematical world. Although not a mathematician, he inspired many to pursue careers and make advancements in mathematics during his 25-year career with Scientific American. He encouraged novices to expand their knowledge, enlightened professionals of computer science developments, and established his own proofs