Rule 110 Turing machines

Given an initial condition for rule 110, the initial condition for the Turing machine shown here is obtained as Prepend[4 list, 0] with 1's on the left and 0's on the right. The Turing machine

{{1, 2} -> {2, 2, -1}, {1, 1} -> {1, 1, -1}, {1, 0} -> {3, 1, 1}, {2, 2} -> {4, 0, -1}, {2, 1} -> {1, 2, -1}, {2, 0} -> {2, 1, -1}, {3, 2} -> {3, 2, 1}, {3, 1} -> {3, 1, 1}, {3, 0} -> {1, 0, -1}, {4, 2} -> {2, 2, 1}, {4, 1} -> {4, 1, 1}, {4, 0} -> {2, 2, -1}}

with s=4 states and k=3 possible colors also emulates rule 110 when started from Prepend[list+1, 1] surrounded by 0's. The s=3, k=4 Turing machine

{{1,0}->{1,2,1},{1,1}->{2,3,1},{1,2}->{1,0,-1},{1,3}->{1,1,-1},{2,0}- >{1,3,1},{2,1}->{3,3,1},{3,0}->{1,3,1},{3,1}->{3,2,1}}

started from Append[list, 0] with 0's on the left and 2's on the right generates a shifted version of rule 110. Note that this Turing machine requires only 8 out of the 12 possible cases in its rules to be specified.