Connected Maximum-density Still Life

Problem Description

A still life is a set of connected live cells from a grid that is fixed under the transition function of John Horton Conway 1970's Game of Life: 1. Any cell with 3 neighbours becomes a live cell. 2. Any live cell with 2-3 live neighbours lives on. 3. Any live cell with <2 or >3 live neighbours dies. The Connected Maximum-density Still Life problem is the task of fitting a given region of the grid with a maximally dense still life, i.e., a maximum number of live cells. (Most work has only looked at pseudo still lifes where the connectedness criterion is dropped.)

Predicates

• Input: size/1

• Output: lives/2

Input format

We here consider square shaped regions. The size n of the region is provided by an instance of the predicate size/1. Cells connect horizontally, vertically, and diagonally.

Output format

If a still life exists, a witness containing live cells encoded in the predicate lives/2 has to be provided. lives(X,Y) means the grid cell with the unsigned integer coordinates X,Y (=< n) is a live cell.

Cost function

The cost function is given as the number of non-living cells. This value as to be minimized (consequently maximizing the number of living cells).

Example(s)

The first example encodes the region size three by size(3). The maximum density of a connected still life on a region with size three is six live cells. Thus, a sample output is:

lives(1,2). lives(1,3). lives(2,1). 
lives(2,3). lives(3,1). lives(3,2).

The second example encodes the region size four by size(4). The maximum density of a connected still life on a region with size four is eight live cells. Thus, a sample output is:

lives(1,3). lives(1,4). lives(2,2). lives(2,4).
lives(3,1). lives(3,3). lives(4,1). lives(4,2).

On the other hand, the following output does not represent a still life. In fact, this configuration brings cells outside the region to live, e.g., lives(0,2) and lives(2,0).

lives(1,1). lives(1,2). lives(1,3). lives(1,4).
lives(2,1). lives(2,4). lives(3,1). lives(3,4).
lives(4,1). lives(4,2). lives(4,3).

Additional sample instances: download

Problem Peculiarities

Type: Optimization Competition: Both Complexity: Beyond NP

Notes and Updates

• Feb 8th, 2013 - Problem Description updated

Author(s)

• Author: Christian Drescher
  • Affiliation: NICTA, University of New South Wales, Australia