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Proof of the Pebble Games Theorem |
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Recall connection to complexity classes: |
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DTC + <
= LOGSPACE |
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TC + <
= NLOGSPACE |
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LFP + <
= PTIME |
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PFP + <
= PSPACE |
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Note: |
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DTC = TC ) LOGSPACE = NLOGSPACE |
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LFP = PFP ) PTIME = PSPACE |
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What about the converse ? |
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DTC ( TC
(Paper 1) |
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PTIME=PSPACE ) LFP = PFP (Paper 2) |
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Ehrenfeucht-Fraisse: |
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k
pebbles |
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k rounds |
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Main Theorem: |
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Duplicator wins (A,B) iff A, B agree on all
formulas in FO[k] |
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Pebble games |
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k
pebbles |
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n (or w)
rounds |
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Main Theorem |
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Duplicator wins for n (or w) rounds iff A, B
agree on all Lw1,w[n] (or Lk1,w)
formulas |
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For an ordinal a, will define Ja = {
Ib, b < a } to have the “back-and-forth” property |
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Ib = a set of partial isomorphisms
from A to B |
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Intuition: Ib contains set of
positions from which the duplicator can win if only b rounds remain |
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Intuition: duplicator has a winning strategy for
a rounds iff there exists a set Ja with b&f property |
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For EF games: |
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Forth: 8 f 2 Ib+1 8 a 2 A, 9 g 2 Ib
s.t. f µ g and a 2 dom(g) |
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Back: symmetric |
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Only need b < k |
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Pebble games |
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Forth: 8 f 2 Ib+1 |dom(f)| < k,
8 a 2 A, 9 g 2 Ib s.t. f µ g and a 2 dom(g) |
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Back: symmetric |
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Downwards closed: if f µ g, g 2 Ib,
then f 2 Ib |
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Antimonotone: b < g implies Ig µ Ib |
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Nonempty: Ib ą ; |
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EF games: |
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Duplicator wins (A,B) game iff there exists a
family Jk with the B&F property |
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Pebble games: |
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Duplicator wins (A,B) for a rounds iff there
exists a family Ja with the B&F property |
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B&F stronger than games |
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EF |
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Lemma 1.
Let A, B agree on all sentences in FO[k]. Then there exists a family Jk with the B&F
property |
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Proof in class |
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Pebble games |
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Lemma 1. Let A, B agree on all sentences in Lk1,w
of qr < a. Then there exists a
family Ja with the B&F property |
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Proof in class |
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EF |
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Lemma 2.
Let A, B have a family Jk with the B&F property. Then they agree on all formulas in FO[k] |
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Proof in class |
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Pebble games |
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Lemma 2. Let A, B have a family Ja
with the B&F property. Then they agree on all sentences in Lk1,w
of qr < a. |
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Proof in class |
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Notes