The Duplicator-Spoiler (Ehrenfeucht-Fra ss e) Game for an Ordinal - PowerPoint PPT Presentation

The Duplicator-Spoiler (Ehrenfeucht-Fra¨ ıss´ e) Game for an Ordinal Number of Turns Gasarch, Pinkerton

Let’s Play a Game! Let’s play G ( Q ; Z ). Q Rationals · · · · · · Z Integers

Let’s Play a Game! Let’s play G ( Q ; Z ). Q Rationals · · · · · · Z 1 Integers

Let’s Play a Game! Let’s play G ( Q ; Z ). 1 Q Rationals · · · · · · Z 1 Integers

Let’s Play a Game! Let’s play G ( Q ; Z ). 1 Q Rationals · · · · · · Z 1 2 Integers

Let’s Play a Game! Let’s play G ( Q ; Z ). 1 2 Q Rationals · · · · · · Z 1 2 Integers

Let’s Play a Game! Let’s play G ( Q ; Z ). 1 1 1 2 2 Q Rationals · · · · · · Z 1 2 Integers

Let’s Play a Game! Let’s play G ( Q ; Z ). 1 1 1 2 2 Q Rationals · · · · · · Z 1 2 4 Integers

And Now with Logic This sentence is true of Q but not of Z . ( ∀ x )( ∀ y )[ x < y = ⇒ ( ∃ z )( x < z < y )] 1 2 3

Connection to Finite Model Theory Theorem Let S 1 and S 2 be sets. Let R be a relation on S 1 and S 2 . The following are equivalent: ◮ Duplicator wins the m-turn game G ( S 1 ; S 2 ) with relation R . ◮ For all first-order m-quantifier-depth sentences φ in the language of standard logical symbols and R [ S 1 | = φ ⇐ ⇒ S 2 | = φ ] .

Applications of Duplicator-Spoiler Games ◮ Proving limits of expressibility ◮ Separating complexity classes

Finite Linear Orderings Theorem The spoiler wins the game G ( F m ; F n ) ( m > n ) in ⌊ log 2 ( n + 1) ⌋ + 1 turns.

The Problem . . . Can the spoiler win G ( N + Z ; N )? Suppose there are 1000 turns. N Z · · · + · · · · · · N + Z · · · N 2 1000

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.

G ( N + Z ; N ) Can the spoiler win G ( N + Z ; N )? N Z · · · + · · · · · · N + Z · · · N x This game takes ω turns.

G ( Z k + Z k ; Z k ) Definition 1. Z 2 = Z ∗ Z = . . . + Z + Z + Z + Z + . . . 2. Z 3 = Z 2 ∗ Z = . . . + Z 2 + Z 2 + Z 2 + . . . etc. Theorem G ( Z k + Z k ; Z k ) takes ω ∗ k + 1 turns.

Higher Ordinal Games Definition 1. Z 0 = F 1 2. Z γ ∗ Z = Z γ +1 3. Z λ = ( � ( Z γ ∗ ω | γ < λ )) ∗ + � ( Z γ ∗ ω | γ < λ ) Theorem For any ordinal λ , G ( Z λ + Z λ ; Z λ ) takes ω ∗ λ + 1 turns.

The Addition Game Let’s play G ( Q ; Z ) with addition . Q Rationals · · · · · · Z Integers

The Addition Game Let’s play G ( Q ; Z ) with addition . Q Rationals · · · · · · Z 1 Integers

The Addition Game Let’s play G ( Q ; Z ) with addition . x Q Rationals · · · · · · Z 1 Integers

The Addition Game Let’s play G ( Q ; Z ) with addition . x x 2 Q Rationals · · · · · · Z 1 Integers

Let’s Play another Plus Game G ( Q ; R ) with addition takes ω + 1 turns. Q R

Let’s Play another Plus Game G ( Q ; R ) with addition takes ω + 1 turns. 3 3 . 1 LCM Q 3 LCM π R ω + 1 ω finite

Vectors of Rationals G ( Q × Q ; Q ) LCM LCM (0,1) 1.3 1.1 (1,0) Q × Q Q Theorem: Let m , n ∈ N , m > n . G ( Q m ; Q n ) takes ω + n turns.

Bounds on Operation Games Theorem Unary operation games take fewer than ω ∗ 2 turns.

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 ]

Future Work ◮ Formalize matching higher-order system. ◮ Find natural operation game which take ω ∗ 2 turns. ◮ Explore generalized graphs.