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 ny PDA that recognizes A. There are languages A such that both A and compliment(A) are recognizable by a PDA, but the PDA for A is much smaller than the PDA for compliment(A). There are languages A1, A2 such that A1,A2,A1 INTERSECT A_2 are recognizable by a PDA, but the PDA for A1 and A2 are much smaller than the PDA for A1 INTERSECT 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.