Constructing Sentences

i.    For the three quadruples that define a TM there are corresponding sentences in L that, under the interpretation A, describe the operation of the machine.

Consider each in turn.

a.    Rewrite.

For the quadruple (q_i, S_j, S_k, q_m) that appears in the machine table, include in D the sentence:

"txy [[Q_itx Ù S_jtx] ® [Q_mt'x Ù S_kt'x Ù (y¹x ® (S₀ty ® S₀t'y Ù ... Ù S_rty ® S_rt'y))]]

which is supposed to be interpreted as saying:

'If TM at t is in q_i and on x and scans S_j then at t+1 TM is in q_m and on x and scanning S_k, and all other cells are unaffected.'

b.    Move Right.

For the quadruple (q_i, S_j, R, q_m) that appears in the machine table, include in D the sentence:

"txy [[Q_itx Ù S_jtx] ® [Q_mt'x' Ù S₀ty ® S₀t'y Ù ... Ù S_rty ® S_rt'y]]

which is supposed to be interpreted as saying:

'If TM at t is in q_i and on x and scans S_j then at t+1 TM is in q_m and on x+1 and all cell contents are unaffected.'

c.    Move Left.

For the quadruple (q_i, S_j, L, q_m) that appears in the machine table, include in D the sentence:

"txy [[Q_itx' Ù S_jtx'] ® [Q_mt'x Ù S₀ty ® S₀t'y Ù ... Ù S_rty ® S_rt'y]]

which is supposed to be interpreted as saying:

'If TM at t is in q_i and on x+1 and scans S_j then at t+1 TM is in q_m and on x and all cell contents are unaffected.'

ii.    Include in D a sentence that says that TM starts in q₁ on cell 0 scanning the leftmost of n 1s, and the tape is otherwise empty.

Q₁ee Ù S₁ee' Ù S₁ee Ù ... Ù S₁ee^n-1 Ù "y[[y ¹ e Ù y ¹ e' Ù ... Ù y ¹ e^n-1] ® S₀ey]