Turing Machines and Decidability

1. Consider the following simple programming language P — which can be used to write
   programs to decide membership in languages L ⊆ Σ
   ⋆
   . As usual, it is assumed that
   ⊔ ∈/ Σ.
   Programs written in this language access a single infinite array A (with registers, or
   “containers” A[i] for i ∈ N) that can store symbols in an alphabet Γ resembling the tape
   alphabet of a Turing machine: Σ ⊆ Γ, ⊔ ∈ Γ, and Γ can also include a finite number of
   additional symbols that are not in Σ ∪ {⊔}.
   When a program is executed on an input string
   ω = σ0σ1 . . . σn−1 ∈ Σ
   ⋆
   with length n, the register A[i] initially contains σi
   , for 0 ≤ i ≤ n − 1, and A[i] contains ⊔
   if i ≥ n.
   The program also access a single integer variable i, whose initial value is 0 and whose
   value is never negative.

Sample Solution