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Let fsa and pda be two predicates such that fsa(x) means x is a finite state automaton and pda(y) means that y is a pushdown automaton. Let equivalent be another predicate such that equivalent(a,b) means a and b are equivalent. Which of the following first order logic statements represents the following: Each finite state automaton has an equivalent pushdown automaton.

1. (∀x fsa(x)) ∧ (∃y pda(y) ∧ equivalent(x, y))
2. (∀y (∃x fsa(x) ⇒ pda(y) ∧ equivalent(x, y)))
3. (∀x ∃y (fsa(x) ∧ pda(y) ∧ equivalent(x, y)))
4. (∀x ∃y (fsa(x) ∧ pda(y) ∧ equivalent(x, y)))

Correct answer: (∀x ∃y (fsa(x) ∧ pda(y) ∧ equivalent(x, y)))

Solution

This statement correctly asserts that for every finite state automaton x, there exists a pushdown automaton y such that x and y are equivalent, fulfilling the requirement that each finite state automaton has a corresponding equivalent pushdown automaton.

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