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Finite Model Theory Lecture 10

Finite Model Theory Lecture 10. Second Order Logic. Outline. Chapter 7 in the textbook: SO, MSO, 9 SO, 9 MSO Games for SO Reachability Buchi’s theorem. Second Order Logic. Add second order quantifiers: 9 X. f or 8 X. f

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Finite Model Theory Lecture 10

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  1. Finite Model TheoryLecture 10 Second Order Logic

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

  3. Second Order Logic • Add second order quantifiers:9 X.f or 8 X.f • All 2nd order quantifiers can be done before the 1st order quantifiers [ why ?] • Hence: Q1 X1. … Qm Xm. Q1 x1 … Qn xn. f, where f is quantifier free

  4. Fragments • MSO = X1, … Xm are all unary relations • 9 SO = Q1, …, Qm are all existential quantifiers • 9 MSO = [ what is that ? ] • 9 MSO is also called monadic NP

  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. 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. EVEN Ï MSO Proposition EVEN is not expressible in MSO Proof: • Will show that if s = ; and |A|, |B| ¸ 2k 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. EVEN Ï MSO • So let spoiler choose U µ A. • |U| · 2k-1. Pick any V µ B s.t. |V| = |U| • |A-U| · 2k-1. Pick any V µ B s.t. |V| = |U| • |U| > 2k-1 and |A-U| > 2k-1. We pick a V s.t. |V| > 2k-1 and |A-V| > 2k-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. EVEN 2 MSO + < • Why ?

  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. 9MSO 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 8MSO [ how ??? ] • But are they expressible in 9MSO ?

  12. 9 MSO Reachability: • Try this:F = 9 X. f • Where f 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. 9 MSO • For an undirected graph G:s, t are connected , G ²F • Hence Undirected-Reachability29 MSO

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

  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. Games for 9MSO The l,k-Fagin game on two structures A, B: • Spoiler selects l subsets U1, …, Ul of A • Duplicator replies with L subsets V1, …, Vl of B • Then they play an Ehrenfeucht-Fraisse game on (A, U1, …, Ul) and (B, Vl, …, Vl)

  17. Games for 9MSO 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 f2 FO[k] • Problem: the game is still hard to play

  18. Games for 9MSO • The l, k – Ajtai-Fagin game on a property P • Duplicator selects A 2 P • Spoiler selects U1, …, Ulµ A • Duplicator selects B Ï P,then selects V1, …, Vlµ B • Next they play EF on (A, U1, …, Ul) and (B, V1, …, Vl)

  19. Games for 9MSO 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 9MSO [ in class ? ]

  20. MSO and Regular Languages • Let S = {a, b} and s = (<, Pa, Pb) • Then S*' STRUCT[s] • What can we express in FO over strings ? • What can we express in MSO over strings ?

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

  22. MSO and TrCl TheoremOn strings, MSO = TrCl1 However, TrCl2 can express an.bn [ how ? ] Question: what is the relationship between these languages: • MSO on arbitrary graphs and TrCl1 • MSO on arbitrary graphs and TrCl

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