Additional insight into the form of the absorbing state rows can be obtained with
the aid of the following notation. Let mA, A SA' be distinct but otherwise arbitrary
absorbing states of the one-operator Markov chain, let iA e S be the bit-string represented
in nA and let S(iA) S be the set of bit-strings accessible from iA via exactly one bit muta-
tion event (i.e. S(iA)= {i:i, e S,H(il,iA)= 1}). It follows from this definition that
card(S(iA)) = L. Then, for M > 1 let S(nA)' be defined as
S(nA)' = {n:ne S',n(iA)= M-l,n(i,)= ,i1 i S(iA)} CS',
the set of nonabsorbing states adjacent to the absorbing state nA. The restriction on M is
required to ensure that no absorbing state mA is contained in the adjacency set of any
absorbing state nA. S(nA)' includes exactly one distinct element for each i, e S(iA), and
consequently card(S(nA)') = card(S(iA))= L. Also, from the form of S(nA)', it follows that
for M 2 3, S(mA)' and S(nA)' are disjoint if mA and n, are distinct one-operator absorbing
states. Thus, if SA" is defined as
SA"= S(n)'
A SA'
and M > 3, then card(SA")= card(SA') x L = NL. This restriction on M is assumed in all of
the following.
With the aid of the new notation, the element in column n e S(nA)' of row RA in
[P-II can be written as
Vn E S(na)': T(n nA)=P(n I nA) =( M
M-1 (1+o )ML
Moe
S(1 + a)ML
Ma
(1 + a)ML
Thus, Eq. 7.4 can be revised as follows