Finite Model Theory
Lecture 10
Second Order Logic
1
Outline
Chapter 7 in the textbook:
• SO, MSO, 9 SO, 9 MSO
• Games for SO
• Reachability
• Buchi’s theorem
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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
3
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
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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.
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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.
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EVEN  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 ?]
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EVEN  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 ? ]
8
EVEN 2 MSO + <
• Why ?
9
MSO Games
• Very hard to prove winning strategies for
duplicator
• I don’t know of any other application of
bare-bones MSO games !
10
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 ?
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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)
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9 MSO
• For an undirected graph G:
s, t are connected , G ² F
• Hence Undirected-Reachability 2 9 MSO
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9 MSO
• For an undirected graph G:
s, t are connected , G ² F
• But this fails for directed graphs:
s
t
• Which direction fails ?
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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 ?
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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)
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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 f 2 FO[k]
• Problem: the game is still hard to play
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Games for 9MSO
• The l, k – Ajtai-Fagin game on a property P
• Duplicator selects A 2 P
• Spoiler selects U1, …, Ul µ A
• Duplicator selects B  P,
then selects V1, …, Vl µ B
• Next they play EF on (A, U1, …, Ul) and
(B, V1, …, Vl)
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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 ? ]
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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 ?
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MSO on Strings
Theorem [Buchi]
On strings: MSO = regular languages.
• Proof [in class; next time ?]
Corollary.
On strings: MSO = 9MSO = 8MSO
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MSO and TrCl
Theorem
On 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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