Expressiveness of Monadic Second-Order Logics on Infinite Trees of Arbitrary Branching Degree

In this thesis we study the expressive power of variants of monadic second-order logic (MSO) on infinite trees by means of automata. In particular we are interested in weak MSO and well-founded MSO, where the second-order quantifiers range respectively over finite sets and over subsets of well-founded trees. On finitely branching trees, weak and well-founded MSO have the same expressive power and are both strictly weaker than MSO. The associated class of automata (called weak MSO-automata) is a restriction of the class characterizing MSO-expressivity. We show that, on trees with arbitrary branching degree, weak MSO-automata characterize the expressive power of well-founded MSO, which turns out to be incomparable with weak MSO. Indeed, in this generalized setting, weak MSO gives an account of properties of the ‘horizontal dimension’ of trees, which cannot be described by means of MSO or well-founded MSO formulae. In analogy with the result of Janin and Walukiewicz for MSO and the modal μ-calculus, this raises the issue of which modal logic captures the bisimulation-invariant fragment of well-founded MSO and weak MSO. We show that the alternation-free fragment of the modal μ-calculus and the bisimulation-invariant fragment of well-founded MSO have the same expressive power on trees of arbitrary branching degree. We motivate the conjecture that weak MSO modulo bisimulation collapses inside MSO and well-founded MSO.