Question

Consider the following relations:

3.1 x ≈L y.
3.2 x ∼M y
3.3 x ≡M y

Match each of them with one of the following statements:

a) x and y cause M to end up in the same state
b) {z : xz ∈ L} = {z : yz ∈ L}
c) The same strings are accepted starting at x as at y
d) For all z, zx ∈ L iff zy ∈ L.
e) None of the above

Answer 1

The relations x ≈L y, x ∼M y, and x ≡M y match with statements (d), (c), and (a) respectively, each of which describes a different type of equivalence or indistinguishability in the context of formal language and automata theory. d) For all z, zx ∈ L iff zy ∈ L is correct.

Step-by-step explanation:

We can match the given relations with the appropriate statements as follows:

• 3.1 x ≈L y: This relation suggests that x and y are related in such a way that if you append any string z to x and y, the resulting strings will both be in language L or both not in L. This matches with option (d) For all z, zx ∈ L iff zy ∈ L.
• 3.2 x ∼M y: Here, the relation indicates that x and y are related by machine M in a way that they are equivalent states. This indicates that starting the machine in state x or y with an input string will yield the same result (acceptance or rejection). Therefore, this matches with (c) The same strings are accepted starting at x as at y.
• 3.3 x ≡M y: This relation typically means that x and y are indistinguishable states within machine M; processing any string starting from either state will lead the machine to the same final state. Thus it matches with option (a) x and y cause M to end up in the same state.