Part 1 - University of Cambridge

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26 Fixed-point Logic with Counting Immerman proposed IFPC—the extension of IFP with a mechanism for counting Two sorts of variables: • x 1 , x 2 , . . . range over |A|—the domain of the structure; • ν 1 , ν 2 , . . . which range over numbers in the range 0, . . .,|A| If ϕ(x) is a formula with free variable x, then ν = #xϕ denotes that ν is the number of elements of A that satisfy the formula ϕ. We also have the order ν 1 < ν 2 , which allows us (using recursion) to define arithmetic operations. Anuj Dawar February 2013
27 Expressive Power of IFPC There are an even number of elements satisfying ϕ(x): ∃ν 1 ∃ν 2 (ν 1 = [#xϕ] ∧ (ν 2 + ν 2 = ν 1 )) Many “obviously” polynomial-time algorithms can be expressed in IFPC. IFPC captures all of PTime over many interesting classes of structures, such as any proper minor-closed class of graphs (Grohe 2010) Matching on graphs can be defined in IFPC. • bipartite graphs (Blass, Gurevich, Shelah 2005) • general graphs (Anderson, D., Holm 2013) 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 and 10: 9 Quantifier Rank The quantifier ra
Page 11 and 12: 11 Ehrenfeucht-Fraïssé Game The p
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: 25 Stratifications of Isomorphism I
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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