How unprovable is Rabin’s decidability theorem? How unprovable is Rabin’s decidability theorem? Leszek Kołodziejczyk University of Warsaw (based on joint work with Henryk Michalewski) CTFM/Tanaka60 Tokyo, September 2015 1 / 22
How unprovable is Rabin’s decidability theorem? What is Rabin’s decidability theorem? Rabin’s theorem (1969) The monadic second order (MSO) theory of the infinite binary tree in the language with two successors, ❼➌ 0 , 1 ➑ ❅ N , S 0 , S 1 ➁ , is decidable. ▲ Among the most important decidability results in logic. ▲ Unlike other such results (Presburger, RCF, MSO for ❼ N , ❇ ➁ ), seems like it might require strong axioms. ▲ Typical proofs involve a determinacy principle unprovable in Π 1 2 - CA 0 . Question: How much logical strength is needed to prove Rabin’s theorem? 2 / 22
▲ ❼ ➁ ▲ How unprovable is Rabin’s decidability theorem? Executive summary of the talk Rabin’s theorem MSO theory of ❼➌ 0 , 1 ➑ ❅ N , S 0 , S 1 ➁ is decidable. (By undefinability of truth, it’s hard to state this in full in Z 2 . But the interesting phenomena appear already for Π 1 3 fragment of MSO.) 3 / 22
How unprovable is Rabin’s decidability theorem? Executive summary of the talk Rabin’s theorem MSO theory of ❼➌ 0 , 1 ➑ ❅ N , S 0 , S 1 ➁ is decidable. (By undefinability of truth, it’s hard to state this in full in Z 2 . But the interesting phenomena appear already for Π 1 3 fragment of MSO.) Main result: All forms of Rabin’s theorem that can be meaningfully stated in Z 2 are provable in Π 1 3 - CA 0 but not in ∆ 1 3 - CA 0 . Proofs rely on: ▲ well-known results and techniques from automata theory, ▲ work on determinacy principles for Bool ❼ Σ 0 2 ➁ games in Z 2 (MedSalem, Nemoto,Tanaka; Heinatsch, Möllerfeld). 3 / 22
How unprovable is Rabin’s decidability theorem? What can be said in MSO on ➌ 0 , 1 ➑ ❅ N ? MSO: S 0 ❼ v , w ➁ , S 1 ❼ v , w ➁ , v ❃ X , ✥ , ✲ , ✱ , ➜ v , ➜ X (for X unary!). MSO can say: ▲ “ v is an ancestor of w ”: every X containing v and closed under S 0 , S 1 also contains w ”. ▲ A given subset is a path, something happens on all paths etc. ▲ “All open games in Cantor space are determined” (and more!). ▲ Can interpret Presburger arithmetic, using finite sets as numbers. But there is no pairing function, so no chance to get full arithmetic. 4 / 22
How unprovable is Rabin’s decidability theorem? Rabin’s theorem: proof sketch ▲ Work with labelled trees: ❼➌ 0 , 1 ➑ ❅ N , S 0 , S 1 , P a 1 ,..., P a ℓ ➁ where ➌ 0 , 1 ➑ ❅ N � ✯ i P a i (vertex in P a i is “labelled” with letter a i ). ▲ By induction on MSO sentence ϕ , show that ϕ is equivalent on labelled trees to a nondeterministic tree automaton. ▲ The difficult induction step is for ✥ (nondeterminism!). ▲ This step involves a determinacy principle for parity games. ▲ It remains to find decision procedure for “given automaton ❆ , does it accept any tree at all?” This is easy. 5 / 22
How unprovable is Rabin’s decidability theorem? Tree automata: definition A nondeterministic tree automaton ❆ is given by: ▲ set of letters Σ � ➌ a 1 ,..., a n ➑ (the alphabet), ▲ finite set of states Q , ▲ initial state q I ❃ Q , ▲ transition relation ∆ ❜ Q ✕ Σ ✕ Q ✕ Q , ▲ rank function rk ✂ Q � N . Idea (“like finite automata, but on infinite trees”): ▲ Run of ❆ on tree T labels T with states: vertex ❣ gets label q I . ▲ ∆ ❄ ❼ q , a , q 0 , q 1 ➁ means: if run reaches v in state q and reads a , then it can go to v 0 in state q 0 and v 1 in state q 1 simultaneously. ▲ Run is accepting if on each path, liminf of ranks of states is even. ▲ ❆ accepts T if there is an accepting run on T . (Note: this is Σ 1 2 .) 6 / 22
How unprovable is Rabin’s decidability theorem? Tree automata: an example Let ❆ have alphabet ➌ a , b , c ➑ , states q I of rank 2, q b of rank 1, q c of rank 0, and transitions: q I ⑦ b ⑦ c , a q I ⑦ b ⑦ c , c q I ⑦ b ⑦ c , b q I q I q b q b q c q c Then ❆ accepts exactly a tree T iff on each branch there are either infinitely many c ’s or only finitely many b ’s. 7 / 22
How unprovable is Rabin’s decidability theorem? Tree automata: an example (cont’d) ❆ has alphabet ➌ a , b , c ➑ , states q I , q b , q c . c a b c a a b a c c c a a c b ✝ ✝ 8 / 22
How unprovable is Rabin’s decidability theorem? Tree automata: an example (cont’d) ❆ has alphabet ➌ a , b , c ➑ , states q I , q b , q c . c , q I a b c a a b a c c c a a c b ✝ ✝ 8 / 22
How unprovable is Rabin’s decidability theorem? Tree automata: an example (cont’d) ❆ has alphabet ➌ a , b , c ➑ , states q I , q b , q c . q I a , q c b , q c c a a b a c c c a a c b ✝ ✝ 8 / 22
How unprovable is Rabin’s decidability theorem? Tree automata: an example (cont’d) ❆ has alphabet ➌ a , b , c ➑ , states q I , q b , q c . q I q c q c c , q I a , q b a , q b b , q I a c c c a a c b ✝ ✝ 8 / 22
How unprovable is Rabin’s decidability theorem? Tree automata: an example (cont’d) ❆ has alphabet ➌ a , b , c ➑ , states q I , q b , q c . q I q c q c q I q I q b q b a , q c c , q c c , q b c , q I a , q I a , q I c , q I b , q b ✝ ✝ 8 / 22
How unprovable is Rabin’s decidability theorem? Parity games: definition For k ❃ N , a parity game with ranks up to k is given by: ▲ finite or countable set V � V 0 ❅ V 1 (the arena, or set of positions), ▲ initial position v 0 ❃ V , ▲ edge relation E ❜ V 2 , ▲ rank function rk ✂ V � ➌ 0 , 1 ,..., k ➑ . Idea: ▲ two players: 0 and 1, ▲ starting in v 0 , move to positions v 1 , v 2 ,... along edges, ▲ player P chooses move from v i iff v i ❃ V P , ▲ player 0 wins iff liminf i � ➟ rk ❼ v i ➁ is even. 9 / 22
How unprovable is Rabin’s decidability theorem? Parity games: an example 2 1 0 2 Here ❬ is player 0 and ❦ is player 1. Game starts in upper left. Player 0 has a winning strategy. 10 / 22
How unprovable is Rabin’s decidability theorem? Parity games: determinacy Observation (in ACA 0 , say): “All parity games are determined” ✕ “All Bool ❼ Σ 0 2 ➁ games are determined”. (Are the Bool ❼ Σ 0 2 ➁ games in Cantor space or Baire space? Doesn’t matter, cf. MedSalem-Nemoto-Tanaka.) Important fact: Parity games enjoy positional (memoryless, forgetful) determinacy: winning strategy can look at current position ignoring earlier ones! 11 / 22
How unprovable is Rabin’s decidability theorem? Rabin’s theorem: proof sketch, revisited ▲ Work with labelled binary trees. ▲ By induction on MSO sentence ϕ , show that ϕ is equivalent to a nondeterministic tree automaton. ▲ The difficult induction step is for ✥ . (The automata are nondeterministic!) ▲ This step involves a determinacy principle for parity games. ▲ It remains to find decision procedure for “given automaton ❆ , does it accept any tree?” This is easy. 12 / 22
How unprovable is Rabin’s decidability theorem? Rabin’s theorem: proof sketch, revisited ▲ Work with labelled binary trees. ▲ By induction on MSO sentence ϕ , show that ϕ is equivalent to a nondeterministic tree automaton. ▲ The difficult induction step is for ✥ . (The complementation theorem for tree automata). ▲ This step involves a determinacy principle for parity games. ▲ It remains to find decision procedure for “given automaton ❆ , does it accept any tree?” This is easy. 12 / 22
How unprovable is Rabin’s decidability theorem? Rabin’s theorem: proof sketch, revisited ▲ Work with labelled binary trees. ▲ By induction on MSO sentence ϕ , show that ϕ is equivalent to a nondeterministic tree automaton. ▲ The difficult induction step is for ✥ . (The complementation theorem for tree automata). ▲ This step involves positional determinacy of parity games. ▲ It remains to find decision procedure for “given automaton ❆ , does it accept any tree?” This is easy. 12 / 22
❼ ➁ ❼ ➁ How unprovable is Rabin’s decidability theorem? Complementation for tree automata Theorem (Rabin) For every tree automaton ❆ there exists a tree automaton ❇ such that for any tree T, ❇ accepts T iff ❆ does not accept T. 13 / 22
How unprovable is Rabin’s decidability theorem? Complementation for tree automata Theorem (Rabin) For every tree automaton ❆ there exists a tree automaton ❇ such that for any tree T, ❇ accepts T iff ❆ does not accept T. Theorem Over ACA 0 , the above complementation theorem: (i) follows from “all parity games are positionally determined”, (ii) implies Bool ❼ Σ 0 2 ➁ - Det (“all Bool ❼ Σ 0 2 ➁ games are determined”). Remark: The exactly equivalent principle is positional determinacy for a certain class of parity games. 13 / 22
How unprovable is Rabin’s decidability theorem? Positional determinacy ✟ complementation Proof sketch: ▲ We formalize a standard proof. ▲ Main observation: “ ❆ accepts T ” is the same as “Player 0 wins in a certain parity game G ❆ , T ” (Automaton-Pathfinder game). ▲ By positional determinacy “ ❆ does not accept T ” is “Player 1 wins in game G ❆ , T using a positional strategy”. ▲ The latter can be translated into a tree automaton. (Translation is nontrivial and relies on complementation for automata on infinite strings, which is provable in ACA 0 .) 14 / 22
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