I can understand that they are not closed under concatenation because without non determinism, PDA cannot decide whether to loop in the first PDA or jump to the next one.But can someone prove this with an example. Also prove that the resulting language cannot be accepted by DPDA
why are deterministic PDAs not closed under concatenation?
automata;pushdown automata
Pick the languages:$L_1 = \{a^ib^jc^k | i \neq j \}$ and $L_2 = \{a^ib^jc^k | j \neq k\}$; both are DCFL and $L_3 = 0L_1 \cup L_2$ is DCFL, too.$L_0 = 0^*$ is DCFL (regular)But $L_{conc} = L_0 \cdot L_3 = 0^* L_3$ is not DCFL.Proof: Suppose that $L_{conc}$ (which is the concatenation of two DCFLs) is DCFL.If we intersect $L_{conc}$ with the regular language $0a^*b^*c^*$, we should get a DCFL language:$L_{conc} \cap \{0a^*b^*c^*\}$ = $0L_1 \cup 0L_2$. suppose $0L_1 \cup 0L_2$ is a DCFL, so $L_1 \cup L_2$ should be a DCFL, too but:$L_1 \cup L_2 = \overline{\overline{L_1} \cap \overline{L_2}} = \overline{\{a^ib^ic^i\}}$ which is not DCFL $\Rightarrow$ contradiction.