A language is decidable if some Turing machine halts on every input and correctly answers yes or no. If it might loop forever on rejects, it is merely recognizable.
Many problems about automata are decidable: does a DFA accept a given string? Is its language empty? Are two DFAs equivalent? We answer these by building a machine that simulates and analyzes the input automaton.
decidable ⊊ Turing-recognizableThe plan
Establishing decidability is constructive — we exhibit an algorithm. Establishing undecidability requires a cleverer, indirect argument: diagonalization.