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Structure and computability of preimages in the Game of Life

Ville Salo, Ilkka Törmä

TL;DR

The paper demonstrates that a single backward step in Conway's Game of Life can encode arbitrary circuit satisfiability by constructing rectangular gadgets (wires, inversions, crosses, and universal gates) whose preimages correspond to satisfying assignments. It introduces a rigorous framework linking two-dimensional CA preimage problems to circuit and tiling problems, and proves strong results: NP- and coNP-type hardness, undecidability results for periodic-preimage questions, and a concrete $6210 \times 37800$ periodic configuration with preimages but no periodic preimage, all built via SAT-based gadget searches. A concrete Jeandel–Rao tile-set realization is provided, together with a formal notion of universality (weak, semiweak, strong) and a demonstration that Life is semiweakly universal as a block map. The work also delivers a practical verification pipeline (hill-climber and genetic algorithms) for gadget construction, and discusses broader implications for reversibility, intrinsic universality, and the potential extension to higher powers or other two-dimensional CA.

Abstract

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 the 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 whose preimage is nonempty but contains no periodic configurations, 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.

Structure and computability of preimages in the Game of Life

TL;DR

The paper demonstrates that a single backward step in Conway's Game of Life can encode arbitrary circuit satisfiability by constructing rectangular gadgets (wires, inversions, crosses, and universal gates) whose preimages correspond to satisfying assignments. It introduces a rigorous framework linking two-dimensional CA preimage problems to circuit and tiling problems, and proves strong results: NP- and coNP-type hardness, undecidability results for periodic-preimage questions, and a concrete periodic configuration with preimages but no periodic preimage, all built via SAT-based gadget searches. A concrete Jeandel–Rao tile-set realization is provided, together with a formal notion of universality (weak, semiweak, strong) and a demonstration that Life is semiweakly universal as a block map. The work also delivers a practical verification pipeline (hill-climber and genetic algorithms) for gadget construction, and discusses broader implications for reversibility, intrinsic universality, and the potential extension to higher powers or other two-dimensional CA.

Abstract

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 the 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 -periodic configuration whose preimage is nonempty but contains no periodic configurations, 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.
Paper Structure (18 sections, 14 theorems, 9 equations, 12 figures)

This paper contains 18 sections, 14 theorems, 9 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Wires
  4. The basic gadgets
  5. Charging a wire: the charger-splitter combo
  6. Shifting and inverting a signal
  7. Wire crossing
  8. A universal logical gate
  9. Constant charger
  10. Composite gadgets and abstract circuits
  11. A concrete example: the Jeandel-Rao tile set
  12. Universality
  13. Corollaries
  14. Implementation and verification of SAT-based searches
  15. The hill-climber
  16. ...and 3 more sections

Key Result

Theorem 1

There exists a substitution $\tau : T \to \{0,1\}^{450 \times 450}$ such that for $x \in X_T$, $\tau(x)$ has a $g$-preimage if and only if $x \in Y_T$.

Figures (12)

  • Figure 1: On the left, the fundamental domain (a $6210 \times 37800$ rectangle) of a periodic configuration whose preimages factor onto scaled tilings of the Jeandel-Rao tile set (thus are aperiodic). On the right, a subpattern roughly corresponding to a single tile type.
  • Figure 2: A horizontal wire and its three charged preimages, plus one uncharged preimage.
  • Figure 3: The affinity of a preimage of a charged wire near the boundary of a gadget.
  • Figure 4: The charger gadget where one output is near and one is far. Specs: $\vdash N; N \in \{0, 1\}/\{\mathrm{N}, \mathrm{F}\}$. The gadget is inside the black rectangle, and we show the beginning of a wire on top.
  • Figure 5: The splitter. Specs: $N, S \vdash ENS; ENS \in \{100, 011\}/\{\mathrm{F} \mathrm{F} \mathrm{N}, \mathrm{N} \mathrm{N} \mathrm{F}\}$.
  • ...and 7 more figures

Theorems & Definitions (33)

  • Example 1
  • Theorem 1
  • Theorem 2
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • ...and 23 more