Final answer:
The statement that a language A is Turing-recognizable if and only if A is many-one reducible to the Halting problem (ATM) is true. To verify this, a computable function must be found that correctly transforms instances of A into instances of ATM and vice versa.
Step-by-step explanation:
The question asks to validate whether the following statement is true or false: a language A is Turing-recognizable if and only if A is many-one reducible to the Halting problem (ATM). This statement is, in fact, true. A language is Turing-recognizable if a Turing machine exists that will accept any string in the language and either reject or run indefinitely on strings not in the language. The Halting problem, or ATM, is a well-known undecidable problem that consists of determining whether a given Turing machine will halt on a particular input.
To show that a language A is Turing-recognizable if it is many-one reducible to ATM, we need to find a computable function that transforms instances of A into instances of ATM, such that the transformed instances are in ATM if and only if the original instances are in A.
Likewise, if A is Turing-recognizable, there exists a Turing machine that recognizes it, and we can construct a reduction from A to ATM by simulating this Turing machine within the reduction procedure. Hence, this establishes the equivalence as stated in the question.