Algebra for Infinite Forests with an Application to the Temporal Logic EF

Abstract

Abstract. We define an extension of forest algebra for ω-forests. We show how the standard algebraic notions (free object, syntactic algebra, morphisms, etc.) extend to the infinite case. To prove its usefulness, we use the framework to get an effective characterization of the ω-forest languages that are definable in the temporal logic that uses the operator EF (exists finally).

Regular Languages of Thin Trees

“…-The projective hierarchy of regular languages collapses to the level 1 1 on thin trees (comparing to 1 2 in the case of all trees). -We observe a gap property (see [16]): a regular language of thin trees, treated as a subset of all trees, is either definable in weak MSO logic or 1 1 -complete.…”

“…Each set that is a projection of a Borel set is called analytic. The class of analytic sets is denoted by 1 1 . The superscript 1 means that the class is a part of the projective hierarchy.…”

“…consists of the complements of the sets from 1 i , -1 i+1 consists of the projections of the sets from 1 i , for i < ω.…”

“…The algebras proposed so far are not completely satisfactory. The algebras proposed in [1] can recognise non-regular languages, while the algebras proposed in [2] are not closed under homomorphisms.…”

“…Effective characterizations are technically challenging and in fact there are very few effective characterizations for languages of infinite trees: for languages recognized by top-down deterministic automata one can compute the Wadge degree [15], for arbitrary regular languages one can decide definability in the temporal logic EF [1] or in the topological class of Boolean combinations of open sets [4]. One of the reasons why effective characterizations are so difficult for infinite trees is that, so far, there is no satisfactory algebraic approach to infinite trees, or even a canonical way to present a regular language.…”