Properties of Binary Transitive Closure Logics over Trees

Binary transitive closure logic (FO∗ for short) is the extension of first-order predicate logic by a transitive closure operator of binary relations. Deterministic binary transitive closure logic (FOD∗) is the restriction of FO∗ to deterministic transitive closures. It is known that these logics are more powerful than FO on arbitrary structures and on finite ordered trees. It is also known that they are at most as powerful as monadic second-order logic (MSO) on arbitrary structures and on finite trees. We will study the expressive power of FO∗ and FOD∗ on trees to show that several MSO properties can be expressed in FOD∗ (and hence FO∗). The following results will be shown. . A linear order can be defined on the nodes of a tree. . The class EVEN of trees with an even number of nodes can be defined. . On arbitrary structures with a tree signature, the classes of trees and finite trees can be defined. . There is a tree language definable in FOD∗ that cannot be recognised by any tree walking automaton. . FO∗ is strictly more powerful than tree walking automata. These results imply that FOD∗ and FO∗ are neither compact nor do they have the Löwenheim-Skolem-Upward property.