Schema Revisited

Two schema problems from the 1970s are examined, monadic recursion schemes and first-order recursion schemas. Research on these problems halted because they were shown to be equivalent to the problem of decidability of language equivalence between DPDA (deterministic pushdown automata). Recently a decidability proof for equivalence of DPDA was given by Sénizergues [10, 11], which therefore also solves the schema problems. However Sénizergues proof is quite formidable. A simplification of the proof was presented by the author [13] using ideas from concurrency theory (for showing decidability of bismilarity [9, 12]) and crucial insights from Sénizergues’s intricate proof. In this abstract we concentrate on first-order schemes and we outline a solution, based on the DPDA equivalence proof, which is reasonably close to its original formulation. We make use of Courcelle’s work [1, 2], which shows how to reduce this schema problem to decidability of language equivalence between strict deterministic grammars. And the proof in [13] of decidability of DPDA equivalence proceeds via (a small extension of) these grammars.