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The Duplicator-Spoiler (Ehrenfeucht-Fra ss e) Game for an Ordinal - - PowerPoint PPT Presentation
The Duplicator-Spoiler (Ehrenfeucht-Fra ss e) Game for an Ordinal - - PowerPoint PPT Presentation
The Duplicator-Spoiler (Ehrenfeucht-Fra ss e) Game for an Ordinal Number of Turns Gasarch, Pinkerton Lets Play a Game! Lets play G ( Q ; Z ). Q Rationals Z Integers Lets Play a Game! Lets play G ( Q ;
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Let’s Play a Game!
Let’s play G(Q; Z). Q
Rationals
Z
Integers
· · · · · ·
1
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Let’s Play a Game!
Let’s play G(Q; Z). Q
Rationals
Z
Integers
· · · · · ·
1 1
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Let’s Play a Game!
Let’s play G(Q; Z). Q
Rationals
Z
Integers
· · · · · ·
1 1 2
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Let’s Play a Game!
Let’s play G(Q; Z). Q
Rationals
Z
Integers
· · · · · ·
1 1 2 2
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Let’s Play a Game!
Let’s play G(Q; Z). Q
Rationals
Z
Integers
· · · · · ·
1 1 2 2 1 1
2
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Let’s Play a Game!
Let’s play G(Q; Z). Q
Rationals
Z
Integers
· · · · · ·
1 1 2 2 1 1
2
4
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And Now with Logic
This sentence is true of Q but not of Z. (∀x)(∀y)[x < y = ⇒ (∃z)(x < z < y)] 3 2 1
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Connection to Finite Model Theory
Theorem
Let S1 and S2 be sets. Let R be a relation on S1 and S2. The following are equivalent:
◮ Duplicator wins the m-turn game G(S1; S2) with relation R. ◮ For all first-order m-quantifier-depth sentences φ in the
language of standard logical symbols and R [S1 | = φ ⇐ ⇒ S2 | = φ].
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Applications of Duplicator-Spoiler Games
◮ Proving limits of expressibility ◮ Separating complexity classes
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Finite Linear Orderings
Theorem
The spoiler wins the game G(Fm; Fn) (m > n) in ⌊log2(n + 1)⌋ + 1 turns.
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The Problem. . .
Can the spoiler win G(N + Z; N)? Suppose there are 1000 turns. N + Z · · · + · · · · · · Z N N
21000
· · ·
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Ordinal Numbers of Turns
Definition
Let β and γ be ordinals. If a game has γ turns then after one turn, the spoiler chooses a β < γ and there are β turns remaining.
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G(N + Z; N)
Can the spoiler win G(N + Z; N)? N + Z · · · + · · · · · · Z N N
x
· · · This game takes ω turns.
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G(Zk + Zk; Zk)
Definition
- 1. Z2 = Z ∗ Z = . . . + Z + Z + Z + Z + . . .
- 2. Z3 = Z2 ∗ Z = . . . + Z2 + Z2 + Z2 + . . .
etc.
Theorem
G(Zk + Zk; Zk) takes ω ∗ k + 1 turns.
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Higher Ordinal Games
Definition
- 1. Z0 = F1
- 2. Zγ ∗ Z = Zγ+1
- 3. Zλ = ( (Zγ ∗ ω|γ < λ))∗ + (Zγ ∗ ω|γ < λ)
Theorem
For any ordinal λ, G(Zλ + Zλ; Zλ) takes ω ∗ λ + 1 turns.
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The Addition Game
Let’s play G(Q; Z) with addition. Q
Rationals
Z
Integers
· · · · · ·
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The Addition Game
Let’s play G(Q; Z) with addition. Q
Rationals
Z
Integers
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1
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The Addition Game
Let’s play G(Q; Z) with addition. Q
Rationals
Z
Integers
· · · · · ·
1 x
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The Addition Game
Let’s play G(Q; Z) with addition. Q
Rationals
Z
Integers
· · · · · ·
1 x
x 2
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Let’s Play another Plus Game
G(Q; R) with addition takes ω + 1 turns. Q R
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Let’s Play another Plus Game
G(Q; R) with addition takes ω + 1 turns. Q R
3 3 π 3.1
ω + 1 ω
LCM LCM
finite
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Vectors of Rationals
G(Q × Q; Q) Q × Q Q
(1,0) 1.1 (0,1) 1.3 LCM LCM
Theorem: Let m, n ∈ N, m > n. G(Qm; Qn) takes ω + n turns.
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Bounds on Operation Games
Theorem
Unary operation games take fewer than ω ∗ 2 turns.
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Back to Logic
G(Z + Z; Z) takes ω + 1 turns. ω + 1 corresponds to: two moves of delay then declaring the number of remaining turns. (∀x)(∀y) (∃S)(∀z)[x < z < y = ⇒ z ∈ S]
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