What can oracles teach us about the ultimate fate of life?

TL;DR

This paper resolves two long-standing open problems in Conway's Life, introduces the concept of spatiotemporally periodic configurations, and analyzes the automaton's complex topological and computational properties.

Contribution

It provides the first examples of unsynthetizable still lifes, introduces agar configurations, and characterizes the automaton's dynamics and computational complexity.

Findings

Existence of configurations with arbitrary finite predecessors

First example of an unsynthetizable still life

Game of Life's reachability and limit set language are PSPACE-hard

Abstract

We settle two long-standing open problems about Conway's Life, a two-dimensional cellular automaton. We solve the Generalized grandfather problem: for all $n \geq 0$, there exists a configuration that has an $n$th predecessor but not an $(n+1)$st one. We also solve (one interpretation of) the Unique father problem: there exists a finite stable configuration that contains a finite subpattern that has no predecessor patterns except itself. In particular this gives the first example of an unsynthetizable still life. The new key concept is that of a spatiotemporally periodic configuration (agar) which has a unique chain of preimages; we show that this property is semidecidable, and find examples of such agars using a SAT solver. Our results about the topological dynamics of Game of Life are as follows: it never reaches its limit set; its dynamics on its limit set is chain-wandering, in particular it is not topologically transitive and does not have dense periodic points; and the spatial dynamics of its limit set is non-sofic, and does not admit a sublinear gluing radius in the cardinal directions (in particular it is not block-gluing). Our computability results are that Game of Life's reachability problem, as well as the language of its limit set, are PSPACE-hard.