When can we say that the language is decidable or undecidable?

A language is said to be decidable if there is a Turing machine that accepts and halts on all inputs in the language and rejects and halts on all inputs not in the language. In other words, a language is decidable if there is an algorithm that can determine whether a given string is in the language or not in finite time.

On the other hand, a language is said to be undecidable if there is no Turing machine that can decide whether an arbitrary string is in the language. This means that there is no algorithm that can determine whether a given string is in the language in finite time.

Here are the steps to determine if a language is decidable or undecidable:

1. Define the language: The first step is to clearly define the language. This includes defining the alphabet of the language and the rules for forming strings in the language.

2. Construct a Turing machine: The next step is to try to construct a Turing machine that accepts and halts on all inputs in the language and rejects and halts on all inputs not in the language. If such a Turing machine can be constructed, then the language is decidable.

3. Test the Turing machine: If a Turing machine has been constructed, it must be tested to ensure that it correctly decides whether an arbitrary string is in the language. This involves running the Turing machine on various inputs and checking that it correctly accepts or rejects each one.

4. If no Turing machine can be constructed or if the Turing machine fails to correctly decide whether an arbitrary string is in the language, then the language is undecidable.