Rediscovering pushdown machines

Abstract

We reexamine pushdown machines from the device-oriented viewpoint, promulgated by Floyd, and by adding lookahead. We reprove some well-known results and prove some new results. First, we prove that every pushdown language is context free using two related yet different proof techniques. Second, we prove that every pushdown transduction is context free and, third, we prove that the language {aⁱbⁱ, aⁱb²ⁱ: i≥ 1} is not a deterministic context-free language. In addition, we investigate lookahead deterministic pushdown machines that have only one state and that accept by empty pushdown. We examine their relationship with LL languages and prove that every LL language is a lookahead deterministic pushdown language, the language {aⁱbⁱ, aⁱcⁱ: i≥0} is a. 1-lookahead deterministic pushdown language that is not LL(k), for any k≥1, and the language {aⁱ0aⁱ0, aⁱ1aⁱ1: i≥0} is not a k-lookahead deterministic pushdown language, for any k≥1. The latter result provides an alternative proof that the corresponding language is not LL(k), for any k≥1.