The game of life is one of the most famous cellular automata in 2D. It has a variety of objects, some of them are moving like gliders, some have an oscillating behavior and others do not change at all, i.e. are still lives.
I am interested in the computational complexity of finding still lives. Especially in a restricted setting: by adding constraints to fix some cells to be dead or alive. In this setting, it becomes an extension problem: Starting from an initial pattern and asking whether it can be extended to form a still life.
So far, I was able to prove the NP-completeness of the following decision problem (by reducing planar circuit SAT to it):
Problem: STILL-LIFE
Input: A pair of numbers $(n,m)$ denoting the size of a rectangular grid $G$ with $n$ rows and $m$ columns. Two disjoint subsets $S_{\mathrm{dead}},S_{\mathrm{alive}} \subseteq G$.
Output: Does there exists a pattern $p:G\to\{\mathrm{dead},\mathrm{alive}\}$ s.t. $p$ is a still life and $p$ is dead on $S_{\mathrm{dead}}$ and alive on $S_{\mathrm{alive}}$.
Now I am interested in the following questions:
Question 1+2: Let $S_{\mathrm{dead}} \cup S_{\mathrm{alive}}$ be a 1D subset of $k$ cells in $G$ i.e. we fix $k$ consecutive cells (either vertical or horizontal [see figure]). What is the computational complexity of STILL-LIFE-1D in that case? What about the more general case if $S_{\mathrm{dead}} \cup S_{\mathrm{alive}}$ is a rectangle?
The same can be asked for the infinite setting when the grid covers the whole plane ($G = \mathbb{Z}^2$). But I want to reformulate it a bit:
Question 3+4: Does every 1D pattern of cells appear as a cross-section of a still life? Here the input is a 1D pattern of length $k$ and the question to decide is whether there exists a still life extension or not (Note: There is no restriction on the size of the pattern). Is this question decidable? Same question for a rectangular initial pattern.
