Game Semantics

If-Then-Else

Sequential conditional of type B → B → B → B. The canonical strategy queries the condition and then exactly one of the two branches. The unused branch is never interrogated — that is definable, unlike por.

Type B → B → 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

ite

Query the condition, then copy the chosen branch.

Trace against auto. (truncated by bound)

q? 0₀
q₁? 1₁
tt₁! 2₂
q₂? 3₁
tt₂! 4₄
tt! 5₁
q? 6₀
q₁? 7₇
tt₁! 8₈
q₂? 9₇
tt₂! 10₁₀
tt! 11₇

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?
Pq₁?

const-then

Ignore the condition and copy the then-branch.

Trace against auto. (truncated by bound)

q? 0₀
q₂? 1₁
tt₂! 2₂
tt! 3₁
q? 4₀
q₂? 5₅
tt₂! 6₆
tt! 7₅
q? 8₀
q₂? 9₉
tt₂! 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?
Pq₂?

bot

The empty strategy Ω: Proponent never moves.

Trace against auto. (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