Presentation-invariant definability Steven Lindell, Haverford - - PowerPoint PPT Presentation
Presentation-invariant definability Steven Lindell, Haverford College Scott Weinstein, University of Pennsylvania 1 Elementary definability Simple graph: V x , y V ( x x ) ( x y y x ) Total
Steven Lindell, Haverford College Scott Weinstein, University of Pennsylvania
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Simple graph: – ⊆ V² ∀x, y ∈ V
Total ordering: < ⊆ D² ∀x, y ,z ∈ D
(x < y < z → x < z)
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even: The number of vertices is even. connected: The graph is connected. acyclic: The graph is acyclic. None of these are elementary over finite graphs. Because first-order logic is local (compactness).
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Augment each graph with an arbitrary ordering: (G, <) Elementary definability invariant of particular <: (G, <) ⊧ θ ⇔ (G, <') ⊧ θ But: even, connected, acyclic ∉ FO(<) ≠ FO
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Expand each graph by a definable relation R: (∃R) (G, R) ⊧ σ Special case: σ depends only on |G| and R. Using R, define a graph query Q, invariant of R: (∀R (G, R) ⊧ σ) [(G, R) ⊧ θ ⇔ G ∈ Q]
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Degree: zero
two isolated barbell chain even: barbells with at most one isolated point. parity: barbells where both ends are in P. majority: barbells with ends in P and ¬P. Fact: Distance is not bounded-degree invariant.
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An ordering of its components, each with the property that every initial segment is connected: [ .. ] .. [x, y] .. [ .. ] (∀v)(∃x)(∃y)[x ≤ v ≤ y](∀z)(x ≤ z ≤ y) {(∀w – z)[x ≤ w ≤ y]} ∧ {z ≠ x → (∃w – z)[w < z]}
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Connected: consists of one component interval Acyclic: no node with two prior neighbors Reachable: both nodes are in same component
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