Hexagonal Run-Length Zero Capacity Region-Part II: Automated Proofs

The zero capacity region for hexagonal (d,k) run-length constraints is known for many, but not all, d and k . The pairs (d,k) for which it has been unproven whether the capacity is zero or positive consist of: (i) k=d+2 when d\ge 2 ; (ii) k=d+3 when d \ge 1 ; (iii) k=d+4 when either d=4 or d is odd and d \ge 3 ; and (iv) k=d+5 when d=4 . Here, we prove the capacity is zero in case (i) when 2 \le d \le 9 , in case (ii) when 3 \le d \le 11 , and in case (iii) when d \in \{ 4,5,7,9 \} . We also prove the capacity is positive in case (ii) when d \in \{1,2\} , in case (iii) when d = 3 , and in case (iv). The zero capacities for k=d+4 are the first and only known cases equal to zero when k-d > 3 . All of our results are obtained by developing three algorithms that automatically and rigorously assist in proving either the zero or positive capacity results by efficiently searching large numbers of configurations. The proofs involve either upper bounding the number of path