Parameterized Regular Expressions and Their Languages

• The rest of the symbols of the first configuration are not blank symbols:e 3 k+1 = #·[0]·∆·&···[k −1]·∆·&·(∆\{%})∗ ·(0 | 1) n ·(Γ∪(Γ×Q)\{B})·∆ ∗Expression e 4 describes words whose final configuration is not in a final state:e 4 = ∆ ∗ ·#·(∆\{%}) ∗ ·(Γ×(Q\{qm }) )·(∆\{%}) ∗ ·%Finally, we describe expression e 5 . Intuitively, it describes words that feature two subsequentconfigurations that are not consistent with each other. More precisely, it is the unionof the following expressions, stating that:• A cell not pointed by the head changed its content from one configuration to thesubsequent one:e 5 1 = ⋃ a∈Γ∆ ∗ ·x 1···x n ·a·&·(∆\{%}) ∗ ·%·#·((0|1) n ·(Γ∪(Γ×Q))·& )∗ ·x 1···x n ·((Γ\{a})∪((Γ\{a})×Q) )·∆ ∗• A configuration that is not final features a pair in Q × Σ for which no transition isdefined (the symbol # states the configuration is not the final one):e 5 2 =⋃{(a,q) | δ(q,a) is not defined}∆ ∗ ·(a,q)·∆ ∗ ·#·∆ ∗• The change of state does not agree with δ:e 5 3 =⋃∆ ∗ ·(a,q)·(∆\{%}) ∗ ·%·{(a,q) | δ(q,a)=(q ′ ,a ′ ,{L,R})}#·(∆\{%}) ∗ ·(Γ×(Q\{q ′ }))·∆ ∗• The symbol written in a given step does not agree with δ:e 5 4 =⋃∆ ∗ ·y 1···y n ·(a,q)·(∆\{%}) ∗ ·%·{(a,q) | δ(q,a)=(q ′ ,a ′ ,{L,R})}(∆\{%}) ∗ ·y 1···y n ·(Γ\{a ′ })·∆ ∗• The movement of the head does not agree with δ:e 5 5 =⋃∆ ∗ ·z 1···z n ·(a,q)·(∆\{%}) ∗ ·%·{(a,q) | δ(q,a)=(q ′ ,a ′ ,R)}(∆\{%}) ∗ ·z 1···z n ·a ′ ·&·(ε| ((0|1) n ·Γ·(∆\{%})) ∗)·%·∆ ∗30