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Please I would like to know why (Q3,S0) is used in Step 1. From k2 structure, it is shown that Q3 has no successor. So was wondering why it appeared there.

in * TF "Emb. Sys. and Rob." by (120 points)
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Look at the definition of the relation H{i+1}:

Instead of 

∀s1'∈S1.∃s2∈S2. (s1,s1')∈R1 ⟹ (s2,s2')∈R2 ⋀ (s1',s2')∈H{i}, 

we can equivalently write  

∀s1'∈S1. (s1,s1')∈R1 ⟹ ∃s2∈S2. (s2,s2')∈R2 ⋀ (s1',s2')∈H{i}.

Now, if we have a state s1 that does not have a successor state, the left part of the implication is false for all states s1'∈S1, and therefore the implication is true. That is the reason why the pair (Q3,S0) survives in all relations H{i}.

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