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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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outline
Outline

Chapter 7 in the textbook:

  • SO, MSO, 9 SO, 9 MSO
  • Games for SO
  • Reachability
  • Buchi’s theorem
second order logic
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
fragments
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
games for mso
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.
games for mso1
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.

even mso
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 ?]
even mso1
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 ? ]
mso games
MSO Games
  • Very hard to prove winning strategies for duplicator
  • I don’t know of any other application of bare-bones MSO games !
9 mso
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 ?
9 mso1
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)
9 mso2
9 MSO
  • For an undirected graph G:s, t are connected , G ²F
  • Hence Undirected-Reachability29 MSO
9 mso3
9 MSO
  • For an undirected graph G:s, t are connected , G ²F
  • But this fails for directed graphs:
  • Which direction fails ?

s

t

9 mso4
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 ?
games for 9 mso
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)
games for 9 mso1
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
games for 9 mso2
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)
games for 9 mso3
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 ? ]

mso and regular languages
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 ?
mso on strings
MSO on Strings

Theorem [Buchi] On strings: MSO = regular languages.

  • Proof [in class; next time ?]

Corollary. On strings: MSO = 9MSO = 8MSO

mso and trcl
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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