Game Semantics

The Divergence Tester

Four sequential disjunctions on B → B → B. Left-or (lor) queries the first argument first and can return true without looking at the second; right-or is the symmetric sequential strategy. Parallel-or is not among them: no innocent strategy can return true when either argument is true and the other diverges.

Type 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

lor

Left-or: query x, return tt immediately on tt, else query y.

Trace against auto. (truncated by bound)

q? 0₀
q₁? 1₁
tt₁! 2₂
tt! 3₁
q? 4₀
q₁? 5₅
tt₁! 6₆
tt! 7₅
q? 8₀
q₁? 9₉
tt₁! 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?
Pq₁?

ror

Right-or: query y first.

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?

lror

Strict left-then-right or: always inspect both, left first.

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?

rlor

Strict right-then-left or.

Trace against auto. (truncated by bound)

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