|
|
|
|
Chapter 7 in the textbook: |
|
SO, MSO, 9 SO, 9 MSO |
|
Games for SO |
|
Reachability |
|
Buchis theorem |
|
|
|
|
|
|
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 |
|
|
|
|
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 |
|
|
|
|
|
|
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. |
|
|
|
|
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. |
|
|
|
|
Proposition EVEN is not expressible in MSO |
|
|
|
Proof: |
|
Will show that if s = ; and |A|, |B| Έ 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 ?] |
|
|
|
|
|
So let spoiler choose U ΅ A. |
|
|U| · 2k-1. Pick any V ΅ B s.t. |V| = |U| |
|
|A-U| · 2k-1. Pick any V ΅ 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 ? ] |
|
|
|
|
|
Very hard to prove winning strategies for
duplicator |
|
|
|
I dont know of any other application of
bare-bones MSO games ! |
|
|
|
|
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 ? |
|
|
|
|
|
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) |
|
|
|
|
For an undirected graph G:
s, t are connected ,
G ² F |
|
|
|
Hence Undirected-Reachability 2 9 MSO |
|
|
|
|
For an undirected graph G:
s, t are connected ,
G ² F |
|
|
|
But this fails for directed graphs: |
|
|
|
|
|
Which direction fails ? |
|
|
|
|
Theorem Reachability on directed graphs is not
expressible in 9 MSO |
|
|
|
What if G is a DAG ? |
|
|
|
What if G has degree · k ? |
|
|
|
|
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) |
|
|
|
|
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 |
|
|
|
|
The l, k Ajtai-Fagin game on a property P |
|
|
|
Duplicator selects A 2 P |
|
Spoiler selects U1,
, Ul ΅
A |
|
Duplicator selects B Ο P,
then selects V1,
, Vl ΅ B |
|
Next they play EF on (A, U1,
, Ul)
and (B, V1,
, Vl) |
|
|
|
|
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 ? ] |
|
|
|
|
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 ? |
|
|
|
|
Theorem [Buchi]
On strings: MSO = regular
languages. |
|
|
|
Proof [in class; next time ?] |
|
|
|
Corollary.
On strings: MSO = 9MSO = 8MSO |
|
|
|
|
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 |
|