1 / 22

# Finite Model Theory Lecture 10 - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Finite Model Theory Lecture 10' - aimee

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Finite Model TheoryLecture 10

Second Order Logic

Chapter 7 in the textbook:

• SO, MSO, 9 SO, 9 MSO

• Games for SO

• Reachability

• Buchi’s 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.

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 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 ? ]

EVEN 2 MSO + <

• Why ?

• Very hard to prove winning strategies for duplicator

• I don’t know of any other application of bare-bones MSO games !

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 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 MSO

• For an undirected graph G:s, t are connected , G ²F

• Hence Undirected-Reachability29 MSO

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 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 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 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 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 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 ? ]

• 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

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