24 2. BackgroundFigure 2.8 The paths through the lattice represent the different possible statesequences sT1under the(0, 1, 2)-model assumptions. In the visualized model, a se-quence has always to start in s1. In the localization context, where states representpositions, the sequence can start anywhere.the decomposition of the predecessor states at t- 1 with:[... ( s, t)]=[... ( s, t- 1)][( s, t- 1) ( s, t)]leads to:t- 1Q( t, s):= maxs[ p( xt, s| s) · maxs1t- 1: st- 1= sp( x, s| s - 1)]=1Q( t- 1 ,s)Figure 2.9 The recursion equation Q( t, s) is computed by deciding for the prede-cessor state s with maximum Q( t- 1, s).and can therefore be simplified to:Q( t, s):= max[ p( xt, s| s) · Q( t- 1, s)]sEach local decision s is stored in a backpointer array B( t, s) and is retrievable asfollows:B( t, s):= s= argmax[ p( xt, s| s) · Q( t- 1, s)]s