Bounded fixed-point definability and tabular recognition of languages

Abstract

By relating positive inductive definitions to space-bounded computations of alternating Turing machines, Rounds, Comp. Linguistics 14, 1988, has given uniform grammatical characterizations of the EXPTIME and PTIME languages. But his proof gives fairly poor bounds for language recognition with context-free resp. head grammars. We improve Rounds' analysis in two respects: first, we introduce a modified class of language definitions that allow restricted forms of negative inductions, and second, we show how to build table-driven recognizers from such definitions. For a wide and natural class of language definitions we thereby obtain fairly efficient recognizers; we can recognize the boolean closure of context-free resp. head languages in the well-known O(n ³) resp. O(n ⁶) steps on a RAM. Our ‘bounded’ fixed-point formulas apparently can not define an arbitrary PTIME language.

Our method is based on the existence of fixed-points for a class of operators that need neither be monotone nor increasing, but assume a norm or at least a well-founded quasi-ordering on the underlying set.