1. 1 Finite Model Theory Lecture 10 Second Order Logic

2. 2 Outline Chapter 7 in the textbook: SO, MSO, 9 SO, 9 MSO Games for SO Reachability Buchi’s theorem

3. 3 Second Order Logic Add second order quantifiers: 9 X.  or 8 X.  All 2nd order quantifiers can be done before the 1st order quantifiers [ why ?] Hence: Q 1 X 1. … Q m X m. Q 1 x 1 … Q n x n. , where  is quantifier free

4. 4 Fragments MSO = X 1, … X m are all unary relations 9 SO = Q 1, …, Q m are all existential quantifiers 9 MSO = [ what is that ? ] 9 MSO is also called monadic NP

5. 5 Games for MSO The MSO game is the following. Spoiler may choose between point move and set move: Point move Spoiler chooses a structure A or B and places a pebble on one of them. Duplicator has to reply in the other structure. Set move Spoiler chooses a structure A or B and a subset of that structure. Duplicator has to reply in the other structure.

6. 6 Games for MSO Theorem The duplicator has a winning strategy for k moves if A and B are indistinguishable in MSO[k] [ What is MSO[k] ? ] Both statement and proof are almost identical to the first order case.

7. 7 EVEN  MSO Proposition EVEN is not expressible in MSO Proof: Will show that if  = ; and |A|, |B| ¸ 2 k then duplicator has a winning strategy in k moves. We only need to show how the duplicator replies to set moves by the spoiler [why ?]

8. 8 EVEN  MSO So let spoiler choose U µ A. –|U| · 2 k-1. Pick any V µ B s.t. |V| = |U| –|A-U| · 2 k-1. Pick any V µ B s.t. |V| = |U| –|U| > 2 k-1 and |A-U| > 2 k-1. We pick a V s.t. |V| > 2 k-1 and |A-V| > 2 k-1. By induction duplicator has two winning strategies: –on U, V –on A-U, A-V Combine the strategy to get a winning strategy on A, B. [ how ? ]

9. 9 EVEN 2 MSO + < Why ?

10. 10 MSO Games Very hard to prove winning strategies for duplicator I don’t know of any other application of bare-bones MSO games !

11. 11 9 MSO Two problems: Connectivity: given a graph G, is it fully connected ? Reachability: given a graph G and two constants s, t, is there a path from s to t ? Both are expressible in 8 MSO [ how ??? ] But are they expressible in 9 MSO ?

12. 12 9 MSO Reachability: Try this:  = 9 X.  Where  says: – s, t 2 X –Every x 2 X has one incoming edge (except t) –Every x 2 X has one outgoing edge (except s)

13. 13 9 MSO For an undirected graph G: s, t are connected, G ²  Hence Undirected-Reachability 2 9 MSO

14. 14 9 MSO For an undirected graph G: s, t are connected, G ²  But this fails for directed graphs: Which direction fails ? s t

15. 15 9 MSO Theorem Reachability on directed graphs is not expressible in 9 MSO What if G is a DAG ? What if G has degree · k ?

16. 16 Games for 9 MSO The l,k-Fagin game on two structures A, B : Spoiler selects l subsets U 1, …, U l of A Duplicator replies with L subsets V 1, …, V l of B Then they play an Ehrenfeucht-Fraisse game on ( A, U 1, …, U l ) and ( B, V l, …, V l )

17. 17 Games for 9 MSO Theorem If duplicator has a winning strategy for the l,k-Fagin game, then A, B are indistinguishable in MSO[l, k] MSO[l,k] = has l second order 9 quantifiers, followed by  2 FO[k] Problem: the game is still hard to play

18. 18 Games for 9 MSO The l, k – Ajtai-Fagin game on a property P Duplicator selects A 2 P Spoiler selects U 1, …, U l µ A Duplicator selects B  P, then selects V 1, …, V l µ B Next they play EF on (A, U 1, …, U l ) and (B, V 1, …, V l )

19. 19 Games for 9 MSO Theorem If spoiler has winning strategy, then P cannot be expressed by a formula in MSO[l, k] Application: prove that reachability is not in 9 MSO [ in class ? ]

20. 20 MSO and Regular Languages Let  = {a, b} and  = (<, P a, P b ) Then  * ' STRUCT[  ] What can we express in FO over strings ? What can we express in MSO over strings ?

21. 21 MSO on Strings Theorem [Buchi] On strings: MSO = regular languages. Proof [in class; next time ?] Corollary. On strings: MSO = 9 MSO = 8 MSO

22. 22 MSO and TrCl Theorem On strings, MSO = TrCl 1 However, TrCl 2 can express a n.b n [ how ? ] Question: what is the relationship between these languages: MSO on arbitrary graphs and TrCl 1 MSO on arbitrary graphs and TrCl