Question

I want to know if these statements are true/false, and also the reasons if they are false.

6. All computer programs written in modern languages can
be implemented by some Turing machines.
7. It is possible to automatically translating programs written
in C into an equivalent Turing machine.
8. The number of undecidable problems is countably infinite.
9. There is a problem solvable by Turing machines with two
tapes but unsolvable by Turing machines with a single
tape.
10. Every Turing machine can be represented by an expression
in lambda calculus.

Answer 1

6. All computer programs written in modern languages can be implemented by some Turing machines - TRUE (modern languages are turing complete)

7. It is possible to automatically translate programs written in C into an equivalent Turing machine - FALSE

Conversion of programming language statements to Turing machine is possible manually but not automatically.

8. The number of undecidable problems is countably infinite - TRUE

9. There is a problem solvable by Turing machines with two tapes but unsolvable by Turing machines with a single tape - FALSE

Turing machine with more than 1 tape is called multi-tape turing machine. In Multi-tape, they have multiple numbers of tapes and a separate head is used to access each tape. Multi-tape Turing machine can be converted to single tape Turing machines. A problem implemented on multi-tape can also be implemented in single tape.

10. Every Turing machine can be represented by an expression in lambda calculus - FALSE

some statements in realizability theory can be realized by turing machine but not by lambda calculus one such example is:

∀f:nat→nat ∃e ∀n ∃k (T(e,n,k)∧U(k,f(n)))