Notes

Outline

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 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 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 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 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 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 ?

MSO Games 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 ² F Hence Undirected-Reachability 2 9 MSO

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

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 f 2 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, …, Ul µ A Duplicator selects B Ï P, then selects V1, …, Vl µ 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 ? ]

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

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