Date: 2019-03-10

Imagine unicellular alien lifeforms inhabiting the edge of a circle planet, where adjacent cells can spawn a new cell between them if their genomes are compatible. A cell’s genome is chosen from a finite set based on its parents’ genomes. Furthermore, to achieve diversity, a cell’s genome must be incompatible with its parents and grandparents. Cells can also die, which is necessary since cells can’t pass by each other on the planet’s 1-dimensional surface. Given these rules and those for compatibility/spawning, does there exist some number of cells with genomes that can sustain life indefinitely?

**Theorem:** Is it possible to learn this answer? Not from a Turing machine. It’s undecidable!

This behavior is summarized by a partial function $\xi$ that evaluates to $\xi(a,b)=c$ iff adjacent cells with genomes $a$ and $b$ can spawn a cell with genome $c$. $\xi$ has the following properties:

- $\forall a,b: \xi(a,b)=\xi(b,a)$ (commutativity)
- $\vert {\xi(a,b): \xi(a,b) \ne \bot }\vert \in \mathbb{N}$ (finite set of genomes)
- $\forall a,b: \xi(a,(a,b)) = \bot$ (incompatible with parents)
- $\forall a,b,c: \xi(a,\xi(\xi(a,b),c)) = \bot$ (incompatible with grandparents)
- $\xi$ can be nondeterministic, even though we are writing it as a function here

Life on such a planet becomes rather monotonous because a planet’s life can only visit and revisit a finite set of configurations.
To see this, let the *population potential* be defined as the number of compatible pairs.
When a spawn occurs, the planet’s population increases by 1, but its potential population decreases by 1 since the new cell is incompatible with its parents.
In this way, the sum of the planet’s actual and potential populations is non-increasing.
Since the number of genomes is finite as well (no evolution), only a finite set of configurations can be reached.

In monotonous.smt2, we prove that when life is sustained, the sum of a planet’s actual and potential populations must remain constant, therefore each cell that dies must be incompatible with its neighbors, and those neighbors must be compatible with each other.

In unidirectional.smt2, we prove that sustained life can be modeled as each newly-spawned cell replacing its parent. Furthermore, as the filename implies, we prove that the spawns eventually propagate in one direction. Given these two facts, the problem of determining whether a given $\xi$ can yield sustainable life (i.e., the CircLife problem) is equivalent to finding a livelock in an arbitrarily sized unidirectional ring of finite state machines that each have the same transition function $\xi$.

In reduction.smt2, we give a reduction from the Periodic Domino Problem (for NW-deterministic Wang tiles) to our CircLife problem. This is just the key step of the reduction, whereas the rest of the proof is identical to the undecidability proof of livelock detection for unidirectional rings of finite state machines.