1 Finite Model Theory Lecture 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 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 L k 1,  ) formulas

5 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

6 Definition of B&F for J  For EF games: Forth: 8 f 2 I  8 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   ;

7 B&F v.s. Games EF games: Duplicator wins (A,B) game iff there exists a family J k 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

8 The Proofs EF Lemma 1. Let A, B agree on all sentences in FO[k]. Then there exists a family J k with the B&F property Proof in class Pebble games Lemma 1. Let A, B agree on all sentences in L k 1,  of qr < . Then there exists a family J  with the B&F property Proof in class

9 The Proofs EF Lemma 2. Let A, B have a family J k 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 J  with the B&F property. Then they agree on all sentences in L k 1,  of qr < . Proof in class