# A Functional (Monadic) Second-Order Theory of Infinite Trees

## TL;DR

This paper provides a complete axiomatization of Monadic Second-Order Logic over infinite trees, enabling formal proof of Rabin's Tree Theorem through a polynomial-time recognizable deduction system.

## Contribution

It introduces Functional (Monadic) Second-Order Logic (FSO), extending MSO with higher abstraction to facilitate axiomatization and proof of key decidability results.

## Key findings

- Complete axiomatization of MSO over infinite trees
- Formal proof of Rabin's Tree Theorem
- Extension of MSO to FSO for higher abstraction

## Abstract

This paper presents a complete axiomatization of Monadic Second-Order Logic (MSO) over infinite trees. MSO on infinite trees is a rich system, and its decidability ("Rabin's Tree Theorem") is one of the most powerful known results concerning the decidability of logics. By a complete axiomatization we mean a complete deduction system with a polynomial-time recognizable set of axioms. By naive enumeration of formal derivations, this formally gives a proof of Rabin's Tree Theorem. The deduction system consists of the usual rules for second-order logic seen as two-sorted first-order logic, together with the natural adaptation In addition, it contains an axiom scheme expressing the (positional) determinacy of certain parity games. The main difficulty resides in the limited expressive power of the language of MSO. We actually devise an extension of MSO, called Functional (Monadic) Second-Order Logic (FSO), which allows us to uniformly manipulate (hereditarily) finite sets and corresponding labeled trees, and whose language allows for higher abstraction than that of MSO.