Part 1 - University of Cambridge

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10 Finitary Elementary Equivalence For two structures A and B, we say A ≡ p B if for any sentence ϕ with qr(ϕ) ≤ p, A |= ϕ if, and only if, B |= ϕ. Key fact: a class of structures S is definable by a first order sentence if, and only if, S is closed under the relation ≡ p for some p. In a finite relational vocabulary, for any structure A there is a sentence θ p A such that B |= θ p A if, and only if, A ≡ p B Anuj Dawar February 2013
11 Ehrenfeucht-Fraïssé Game The p-round Ehrenfeucht game on structures A and B proceeds as follows: • There are two players called Spoiler and Duplicator. • At the ith round, Spoiler chooses one of the structures (say B) and one of the elements of that structure (say b i ). • Duplicator must respond with an element of the other structure (say a i ). • If, after p rounds, the map a i ↦→ b i is a partial isomorphism, then Duplicator has won the game, otherwise Spoiler has won. Theorem (Fraïssé 1954; Ehrenfeucht 1961) Duplicator has a strategy for winning the p-round Ehrenfeucht game on A and B if, and only if, A ≡ p B. Anuj Dawar February 2013
Page 1 and 2: 1 Games and Isomorphism in Finite M
Page 3 and 4: 3 Spoiler and Duplicator The two pl
Page 5 and 6: 5 Expressive Power of Logics We are
Page 7 and 8: 7 The Power of First-Order Logic Fo
Page 9: 9 Quantifier Rank The quantifier ra
Page 13 and 14: 13 Using Games To show that a class
Page 15 and 16: 15 Stratifying Isomorphism In order
Page 17 and 18: 17 IFP The logic IFP is formed by c
Page 19 and 20: 19 Finite Variable Logic We write L
Page 21 and 22: 21 Pebble Games The k-pebble game i
Page 23 and 24: 23 Evenness To show that Evenness i
Page 25 and 26: 25 Stratifications of Isomorphism I
Page 27 and 28: 27 Expressive Power of IFPC There a
Page 29 and 30: 29 Counting Game Immerman and Lande
Page 31 and 32: 31 Equivalence of Games It is easy
Page 33 and 34: 33 Cai-Fürer-Immerman Graphs There
Page 35 and 36: 35 Properties If G is connected and
Page 37 and 38: 37 TreeWidth Formal Definition: For
Page 39 and 40: 39 Strategies and Decompositions Th
Page 41 and 42: 41 Undefinability Results for IFPC
Page 43 and 44: 43 Systems of Linear Equations Cons
Page 45 and 46: 45 Satisfiability Lemma E G is sati

Part 1 - University of Cambridge

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