On the Sizes of DPDAs, PDAs, LBAs by Richard Beigel and William Gasarch Abstract 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 Tur- ing 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 A 1 ,A 2 such that A 1 ,A 2 ,A 1 ∩ A 2 are recognizable by a PDA, but the PDA for A 1 and A 2 are much smaller than the PDA for A 1 ∩ A 2 . We investigate these phenomema 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 A n such that there is a small PDA for A n , but any DPDA for A n is large. We look at cases where we can get almost-all results, though with much smaller size differences. 1 Introduction Let DPDA be the set of Deterministic Push Down Automaton, PDA be the set of Push Down Automata, and LBA be the set of Linear Bounded Automata (usually called NSPACE(n)). Let L(DPDA) be the set of languages recognized by DPDAs (similar for L(PDA) and L(LBA)). It is well known that L(DPDA) ⊂ L(PDA) ⊂ L(LBA). Our concern is with the size of the DPDA, PDA, LBA. For example, let A ∈ L(DPDA). Is it possible that there is a PDA for A that is much smaller than any DPDA for A? For all adjacent 1