Second Order Monadic Theory of One Successor

1. Second Order Monadic Theory of One Successor Automata Seminar Presented By: Tamar Aizikowitz Spring 2007

2. Second Order Monadic Logic • Variables: • Variables over natural numbers: x, y, z… • Variables over sets (functions  → {0,1}): σ, τ, δ… • Constant:The natural number 0 • Successor function:S(x) = x + 1 • Binary Predicates: • σ(x) = 0 • σ(x) = 1

3. Examples of MSO Formulae • σ is a subset of τ : F(σ,τ) = x (σ(x) →τ(x)) • σ is singleton: F(σ) = x (σ(x) y (σ(y) →x=y)) • x < y: F(x,y) = (x=y) σ [σ(y) zz’ (σ(z) S(z’)=z→σ(z’)) →σ(x)]

4. Theorem 1 • Let F(σ1,…,σn) be an MSO formula, then the following infinitry language over the alphabet {0,1}n is ω-regular:L(F) = {σ1(0)σn(0)σ1(k)σn(k)| F(σ1,…,σn)} • Proof: (1)Prove that F can be transformed to normal form (2) Prove that a Büchi automata can be built s.t. it accepts L(F), for all normal form F.

5. Part 1 – Normal Form (1) • Lemma 1: Every formula F(σ1,…,σn,x1,…,xm) is equivalent to an MSO formula of the form Q1QiQi+1Qj G where: (1)G is a formula with no quantifiers (2)Q1Qi are function quantifiers (3)Qi+1Qj are numerical quantifiers

6. Part 1 – Normal Form (2) • Proof of Lemma 1: • Assume F is in prennix normal formQ1QkF’where F’contains no quantifiers. • Qi is out-of-order if it is a number quantifier with a function quantifier after it. • Let Qi be the rightmost out-of-order quantifier. The weight of Qi is the number of function quantifiers that appear after it. • We prove the claim by induction on the number of out-of-order quantifiers and the weight of the rightmost one.

7. Part 1 – Normal Form (3) Proof of Lemma 1 continued… • Assume x quantifier is rightmost out-of-order: • xσ Q1QkHσx Q1QkH • xσ Q1QkHσx Q1QkH • xσ Q1QkHδσ Q1Qkxy(δ(x)=1  (δ(y)=1→ H)) • xσ Q1QkHδσ Q1Qk xy(δ(x)=0  (δ(y)=1 H))

8. Part 1 – Normal Form (4) • Simple structure:xi=1,…,kj=0,…,nσi(x+j)=εij ; εij=0,1 • i.e.x (σ1(x)=ε10    σ1(x+n)=ε1n  ) • Lemma 2: Every formula has an equivalent of the form Q1QkG where Qi are function quantifiers and G is a prepositional combination of simple structures and atomic formulae.

9. Part 2 – Büchi Automata (1) • Lemma 3:A is atomic L(A) is ω-regular. • Proof of Lemma 3:A is of the form σ(x) = 0/1 • “Count” until x • Verify that the value is 0/1 accordingly • Go to (non-)accepting sink state • Lemma 4:B is a basic structure L(B) is ω-regular. • Proof of Lemma 4: • Skip x-1 letters from {1,0}k (“guess x” non-deterministically) • Verify next n+1 letters match εij values • Go to (non-)accepting sink state

10. Part 2 – Büchi Automata (2) • Proof of Theorem 1: Assume F(σ1,…,σn) is in normal form (i.e. Q1QkG). We prove by induction on the number of Boolean connectives in G that L(G) is ω-regular: • Base:G is an atomic formula or a basic structure  the claim follows from Lemmas 3 and 4. • Closure:L(G1G2) = L(G1) L(G2)L(G1G2) = L(G1) L(G2)L(G) = L(G)C the claim follows from the closure properties of ω-regular languages.

11. Part 2 – Büchi Automata (3) Proof of Theorem 1 continued… • Now we prove the claim for F by induction on the number of quantifiers Qi : • Base: no quantifiers  already proven • Closure: L(σiH(σ1,…,σm)) is the language h(L(H)) where h: {0,1}m→({0,1}m-1)* is a homomorphism s.t. h(ε1 εi-1 εi εi+1 εm) = ε1 εi-1 εi+1 εm L(σH) = L(σH)C

12. Decidability of MSO • Corollary 1: An algorithm exists which determines for a given closed formula F whether F is valid. • Proof of Corollary 1: Assume F is of the form Qσ G(σ). Therefore: • If Q =  then F is valid iff L(G)   • If Q =  then F is valid iff L(G) = {0,1}ω, which is equivalent to L(G)C = The claim follows from the fact that emptiness is decidable for Büchi Automata.