No no, we are not yet moving to cuisines and mouth-watering posts and neither are we turning to a food blog. The Rice theorem has nothing to do with sushi, biryani or paella. Our similarity to these is only till the use of the cover image and that's where it ends.
Rice's theorem is a fundamental result in computer science that states that any non-trivial property of the language of a Turing machine is undecidable. In other words, there is no algorithm that can take as input a description of a Turing machine and a property of its language, and correctly output whether or not the property holds for that language.
This theorem has far-reaching consequences: it implies that many properties of programming languages are undecidable, including many that are commonly used in program verification. As a result, Rice's theorem is a fundamental limit on the power of automated program verification.
The theorem is named after Henry Gordon Rice, who first proved it in 1953. Rice's theorem is a generalization of Kurt Gödel's incompleteness theorem to formal languages.
The statement of Rice's theorem is as follows:
Let L be a language recognized by a Turing machine M. If P is any non-trivial property of L, then P is undecidable.
A property P is non-trivial if there exists a language L such that P holds for L but P does not hold for all languages. For example, the property "L is empty" is non-trivial, since the empty language satisfies it but not all languages do.
To prove Rice's theorem, we need to show that any non-trivial property P is undecidable. We do this by showing that there is no algorithm that can take as input a Turing machine M and a property P, and correctly output whether or not P holds for the language of M.Suppose such an algorithm exists. We can use it to construct a Turing machine that decides the halting problem, which is known to be undecidable.
Given a Turing machine M and an input x, the halting problem is to decide whether M halts on x. This is a non-trivial property since there are Turing machines that halt on all inputs and Turing machines that do not halt on any input.Let P be the halting problem.
We can use our algorithm to construct a Turing machine that decides P as follows:
On input M and x, the machine first runs the algorithm with input M and P. If the algorithm outputs "P holds for M", the machine halts. Otherwise, the machine runs M on x. If M halts, the machine outputs "P doesn't hold for M". Otherwise, the machine runs forever.
This machine correctly decides P: if P holds for M, the machine halts; if P doesn't hold for M, the machine doesn't halt. But this contradicts the fact that P is undecidable, which we assumed. Therefore, our assumption must be false, and there is no algorithm that can take as input a Turing machine and a property P, and correctly output whether or not P holds for the language of M.
While you learn this and explore more on YouTube about the topic, a plate of rice to accompany this is highly recommended.
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