On the sizes of DPDAs, PDAs, LBAs

There are languages A such that there is a Pushdown Automata (PDA) that recognizes A which is much smaller than any Deterministic Pushdown Automata (DPDA) that recognizes A. There are languages A such that there is a Linear Bounded Automata (Linear Space Turing Machine, henceforth LBA) that recognizes A which is much smaller than any PDA that recognizes A. There are languages A such that both A and A are recognizable by a PDA, but the PDA for A is much smaller than the PDA for A. There are languages A1, A2 such that A1, A2, A1 ∩ A2 are recognizable by a PDA, but the PDA for A1 and A2 are much smaller than the PDA for A1 ∩A2. We investigate these phenomenon and show that, in all these cases, the size difference is captured by a function whose Turing degree is on the second level of the arithmetic hierarchy. Our theorems lead to infinitely-often results. For example: for infinitely many n there exists a language An recognized by a DPDA such that there is a small PDA for An, but any DPDA for An is large. We look at cases where we can get almost-all results, though with much smaller size differences.