A Dual Graph Translation of a Problem in ‘Life’1

Conway’s game of Life provides interesting problems in which modelling issues in constraint programming can be explored. The problem of finding a maximum density stable pattern (‘still-life’) is discussed. A formulation of this problem as a constraint satisfaction problem with 0-1 variables and non-binary constraints is compared with its dual graph translation into a binary CSP. The success of the dual translation is surprising, since it has as many variables as the non-binary CSP and very large domains. An important factor is the identification of many redundant constraints: it is shown that these can safely be removed if arc consistency is maintained during search. The dual variables represent 3 3 squares, and further improvements are found by basing both the search and bounds on the cost of completing a partial solution on a partition into such squares.