If we have two TMs that describe a decidable language is the equivalence question still undecidable?
In the field of computational complexity theory, the concept of decidability plays a fundamental role. A language is said to be decidable if there exists a Turing machine (TM) that can determine, for any given input, whether it belongs to the language or not. The decidability of a language is a important property, as it allows us to reason about the language and its properties algorithmically.
The equivalence question for Turing machines is concerned with determining whether two given TMs recognize the same language. Formally, given two TMs M1 and M2, the equivalence question asks whether L(M1) = L(M2), where L(M) represents the language recognized by TM M.
The general problem of determining the equivalence of two TMs is known to be undecidable. This means that there is no algorithm that can always decide whether two arbitrary TMs recognize the same language or not. This result was proven by Alan Turing in his seminal work on computability.
However, it is important to note that this result holds for the general case of arbitrary TMs. In the specific case where both TMs describe decidable languages, the equivalence question becomes decidable. This is because decidable languages are those for which there exists a TM that can decide membership in the language. Therefore, if two TMs describe decidable languages, we can construct a new TM that decides their equivalence.
To illustrate this, let's consider an example. Suppose we have two TMs M1 and M2 that describe decidable languages. We can construct a new TM M that decides their equivalence as follows:
1. Given an input x, simulate M1 on x and M2 on x simultaneously.
2. If M1 accepts x and M2 accepts x, then accept.
3. If M1 rejects x and M2 rejects x, then accept.
4. Otherwise, reject.
By construction, the TM M will accept an input x if and only if both M1 and M2 accept x, or both M1 and M2 reject x. This means that M decides the equivalence of M1 and M2 for any given input x.
While the general problem of determining the equivalence of two arbitrary TMs is undecidable, if the TMs describe decidable languages, the equivalence question becomes decidable. This is because decidable languages can be decided by a TM, allowing us to construct a TM that decides their equivalence. The decidability of the equivalence question for TMs describing decidable languages provides important insights into the computational complexity of these languages.
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