Decidability of S2S
Christian Kissig

Abstract:
Leibniz hoped for a calculus of truth in a universal sense. Godel
showed with his Incompleteness Theorem that such a calculus can not
exist. Godels result, however, left open which particular logics are
(un)decidable. For instance, it was not until the 1970s that the
undecidability of Hilbert's Tenth Problem was shown.

Besides decidability, researchers were interested in model checking
properties of infinite structures using finite automata. In particular
Elgot and Buchi pursued the question which properties can be expressed
and verified by automata. In 1962, Elgot and Buchi showed that finite
state deterministic automata and monadic second-order logic
interpreted on finite words, which is the weak monadic second-order
logic WS1S, are equally expressive.  Buchi showed the same for the
monadic second-order logic S1S of infinite words.  Thereby he obtained
an effective translation of formulas of S1S into non-deterministic
word automata with Buchi acceptance condition.  McNaughton generalised
Buchi's method and obtained an effective translation of monadic
second-order logic S2S of binary trees into finite automata.

Using the connection between logics and automata exhibited by Buchi
and McNaughton, Rabin proved the decidability of S2S by reducing the
satisfiability problem to the non-emptiness problem for
non-deterministic binary tree automata with Rabin acceptance
condition, which was known to be effectively solvable. From this
result follow immediately various decidability results such as for
Presburger arithmetic, for the monadic second-order logic of trees
with arbitrary ( but countable ) branching, and for the monadic
second-order logic of countable linear orderings. The most important
contribution, however, is seen in the application of automata theory
to logic.

The proof itself has since been considered hard to comprehend by many
scholars, which lead to various improvements. In 1982, Gurevich and
Harrington gave a reduction of S2S to non-deterministic automata with
Muller acceptance condition. In 1995,Muller and Schupp refined the
reduction. At the heart of their argument is the non-determinisation
of alternating automata. Alternating automata are closer to classical
logics with negation like S2S than non-deterministic automata.

The behaviour of automata operating on infinite input is given in
terms of acceptance games. For Rabin and Muller automata, these
acceptance games are not historyfree, but bounded memory
determined. In order to win a play of such an acceptance game the
players need to have a bounded memory of the play. To quantify the
memory needed, Gurevich and Harrington introduced the latest
appearance record (LAR) which Muller and Schupp adapted as the index
appearance record (IAR) for a complex memory framework. However, for
parity automata, acceptance games are historyfree determined which
follows from a result for parity graph games shown independently by
Emerson and Jutla, and Mostowski. We elaborate on determinacy of
parity graph games in Appendix B. For the proof that acceptance games
for parity automata are historyfree determined we refer to the
literature.

Both infinite -words and -labelled binary trees are instances of
coalgebras for the respective functors \Sigma × (-) taking objects X to
\Sigma × X and \Sigma × ((-) × (-)) taking objects X to \Sigma × (X × X).
Based on this observation, Venema introduced automata recognising
general F-coalgebras for functors F preserving weak pullbacks.  The
class of alternating F-coalgebra automata has been shown to be closed
under union, existential projection and non-determinisation.

In this text we give a comprehensive proof of Rabin's Theorem majorly.
In Chapter 2 we introduce monadic second-order logic S2S in a minimal
representation and alternating automata. We prove the class of
alternating binary tree automata closed under complementation in
Chapter 3. In Chapter 4 we prove that alternating and
non-deterministic binary tree automata are equally expressive by
reducing the non-determinisation of alternating binary tree automata
to the determinisation of non-deterministic word automata as shown by
Safra. In Chapter 5 we define the translation of formulas of S2S into
alternating binary tree automata.  In Chapter 7 we give a
game-theoretical solution to the non-emptiness problem for alternating
binary tree automata

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