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Understanding Pebble Games Theorem in Finite Model Theory

Explore the Proof of the Pebble Games Theorem in Finite Model Theory with a focus on Ehrenfeucht-Fraisse games, Duplicator strategies, and Back-and-Forth techniques. Understand the connections to complexity classes like LOGSPACE, NLOGSPACE, PTIME, and PSPACE. Learn how Duplicator wins in these games and the significance of the B&F property.

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Understanding Pebble Games Theorem in Finite Model Theory

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  1. Finite Model TheoryLecture 14 Proof of the Pebble Games Theorem

  2. More Motivation Recall connection to complexity classes: • DTC + < = LOGSPACE • TC + < = NLOGSPACE • LFP + < = PTIME • PFP + < = PSPACE

  3. 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)

  4. 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 w) rounds Main Theorem Duplicator wins for n (or w) rounds iff A, B agree on all Lw1,w[n] (or Lk1,w) formulas EF v.s. Pebble Games

  5. Back-and-forth • For an ordinal a, will define Ja = { Ib, b < a } to have the “back-and-forth” property • Ib = a set of partial isomorphisms from A to B • Intuition: Ib contains set of positions from which the duplicator can win if only b rounds remain • Intuition: duplicator has a winning strategy for a rounds iff there exists a set Ja with b&f property

  6. For EF games: Forth: 8 f 2 Ib+1 8 a 2 A, 9 g 2 Ib s.t. f µ g and a 2 dom(g) Back: symmetric Only need b < k Pebble games 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) Back: symmetric Downwards closed: if f µ g, g 2 Ib, then f 2 Ib Antimonotone: b < g implies Igµ Ib Nonempty: Ib¹; Definition of B&F for Ja

  7. 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 a rounds iff there exists a family Ja with the B&F property B&F stronger than games B&F v.s. Games

  8. 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 games 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 Proof in class The Proofs

  9. 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 games 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. Proof in class The Proofs

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