Game Semantics

The Interactive Copier

Copycat on B → B. Opponent asks on the result (right) copy; Proponent mirrors the question into the argument (left) copy, waits for an answer, and copies it back. The constant-true strategy never inspects its argument.

Type B → B

OP?!₁P-viewΩ…₊ / ₋₁ ₂ ₃

Arena

Enabling structure of the arena (each move drawn once). Hover a move for its role. Enabling.

Strategies

copycat

Identity: copy every Opponent move to the other component.

Trace against ask-then-true. (truncated by bound)

q₊? 0₀
q₋? 1₁
tt₋! 2₂
tt₊! 3₁
q₊? 4₀
q₋? 5₅
tt₋! 6₆
tt₊! 7₅
q₊? 8₀
q₋? 9₉
tt₋! 10₁₀
tt₊! 11₉

P-view

Strategy tree

Unfolding against every legal Opponent continuation, up to the page bound. O branches fan out; P is a single reply. Click a reply count to expand folded Opponent options.

ε
Oq₊?
Pq₋?

const-true

Answer tt on the result without ever querying the argument.

Trace against ask-then-true. (truncated by bound)

q₊? 0₀
tt₊! 1₁
q₊? 2₀
tt₊! 3₃
q₊? 4₀
tt₊! 5₅
q₊? 6₀
tt₊! 7₇
q₊? 8₀
tt₊! 9₉
q₊? 10₀
tt₊! 11₁₁

P-view

Strategy tree

Unfolding against every legal Opponent continuation, up to the page bound. O branches fan out; P is a single reply. Click a reply count to expand folded Opponent options.

ε
Oq₊?
Ptt₊!
Oq₊?
Ptt₊!
Oq₊?
Ptt₊!
Oq₊?
Ptt₊!
… (bound)

bot

The empty strategy Ω: Proponent never moves.

Trace against ask-then-true. (stuck — Ω)

q₊? 0₀

P-view

Strategy tree

Unfolding against every legal Opponent continuation, up to the page bound. O branches fan out; P is a single reply. Click a reply count to expand folded Opponent options.

ε
Oq₊?
Ω