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Finite Model TheoryLecture 14

Proof of the Pebble Games Theorem

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More Motivation

Recall connection to complexity classes:

• DTC + < = LOGSPACE

• TC + < = NLOGSPACE

• LFP + < = PTIME

• PFP + < = PSPACE

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More Motivation

Note:

• DTC = TC ) LOGSPACE = NLOGSPACE

• LFP = PFP ) PTIME = PSPACE

What about the converse ?

• DTC ( TC (Paper 1)

• PTIME=PSPACE ) LFP = PFP (Paper 2)

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EF v.s. Pebble Games

Ehrenfeucht-Fraisse:• k pebbles• k rounds

Main Theorem:• Duplicator wins (A,B)

iff A, B agree on all formulas in FO[k]

Pebble games• k pebbles• n (or ) rounds

Main Theorem• Duplicator wins for n

(or ) rounds iff A, B agree on all L

1,[n] (or Lk

1,) formulas

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Back-and-forth

• For an ordinal , will define J = { I, < } to have the “back-and-forth” property

• I = a set of partial isomorphisms from A to B

• Intuition: I contains set of positions from which the duplicator can win if only rounds remain

• Intuition: duplicator has a winning strategy for rounds iff there exists a set J with b&f property

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Definition of B&F for J

For EF games:

• Forth: 8 f 2 I8 a 2 A, 9 g 2 I s.t. f µ g and a 2 dom(g)

• Back: symmetric

• Only need < k

Pebble games

• Forth: 8 f 2 I|dom(f)| < k,8 a 2 A, 9 g 2 I s.t. f µ g and a 2 dom(g)

• Back: symmetric

• Downwards closed: if f µ g, g 2 I, then f 2 I

• Antimonotone: < implies I µ I

• Nonempty: I ;

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B&F v.s. Games

• EF games:

• Duplicator wins (A,B) game iff there exists a family Jk with the B&F property

• Pebble games:

• Duplicator wins (A,B) for rounds iff there exists a family J with the B&F property

• B&F stronger than games

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The Proofs

EF

Lemma 1. Let A, B agree on all sentences in FO[k]. Then there exists a family Jk with the B&F property

Proof in class

Pebble gamesLemma 1. Let A, B

agree on all sentences in Lk

1, of qr < . Then there exists a family J with the B&F property

Proof in class

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The Proofs

EF

Lemma 2. Let A, B have a family Jk with the B&F property. Then they agree on all formulas in FO[k]

Proof in class

Pebble gamesLemma 2. Let A, B have

a family J with the B&F property. Then they agree on all sentences in Lk

1, of qr < .

Proof in class