An Infinite Pebble Game and Applications

We generalize the well-known pebble game to infinite dag's, and we use this generalization to give new and shorter proofs of results in different areas of computer science (as diverse as "logic of programs" and "formal language theory"). Our applications here include a proof of a theorem due to Salomaa, asserting the existence of a context-free language with infinite index, and a proof of a theorem due to Tiuryn and Erimbetov, asserting that unbounded memory increases the power of logics of programs. The original proofs by Salomaa, Tiuryn, and Erimbetov, are fairly technical. The proofs by Tiuryn and Erimbetov also involve advanced techniques of model theory, namely, back-and-forth constructions based on a variant of Ehrenfeucht-Fraisse games. By contrast, our proofs are not only shorter, but also elementary. All we need is essentially finite induction and, in the case of the Tiuryn-Erimbetov result, the compactness and completeness of first-order logic.