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■♥tr♦❞✉❝t✐♦♥ t♦ Pr♦♣♦s✐t✐♦♥❛❧ ❉②♥❛♠✐❝ ▲♦❣✐❝ ❛♥❞

• ❛♠❡ ▲♦❣✐❝

❊r✐❝ P❛❝✉✐t ■▲▲❈✱ ❯♥✐✈❡rs✐t② ♦❢ ❆♠st❡r❞❛♠ st❛❢❢✳s❝✐❡♥❝❡✳✉✈❛✳♥❧✴∼❡♣❛❝✉✐t ❡♣❛❝✉✐t❅s❝✐❡♥❝❡✳✉✈❛✳♥❧ ◆♦✈❡♠❜❡r ✷✼✱ ✷✵✵✻ ■♥tr♦❞✉❝t✐♦♥ t♦ ▲♦❣✐❝ ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡

❖✈❡r✈✐❡✇

• Pr♦✈✐♥❣ ❈♦rr❡❝t♥❡ss ♦❢ Pr♦❣r❛♠s✿ ❋r♦♠ ❍♦❛r❡ ▲♦❣✐❝ t♦ PDL
• ■♥tr♦❞✉❝t✐♦♥ t♦ Pr♦♣♦s✐t✐♦♥❛❧ ❉②♥❛♠✐❝ ▲♦❣✐❝ ✭PDL✮
• ❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝
• ❙❡♠❛♥t✐❝s ❢♦r ●❛♠❡ ▲♦❣✐❝
• ❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ Pr♦❝❡❞✉r❡

❲❤❛t ✐s ❛ Pr♦❣r❛♠❄ ❆ ❝♦♠♣✉t❡r ♣r♦❣r❛♠ ✐s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✐♥str✉❝t✐♦♥s t❤❛t ❞❡s❝r✐❜❡ ❛ t❛s❦✱ ♦r s❡t ♦❢ t❛s❦s✱ t♦ ❜❡ ❝❛rr✐❡❞ ♦✉t ❜② ❛ ❝♦♠♣✉t❡r✳ ✭❲✐❦❡♣❡❞✐❛✮ ❆ ♣r♦❣r❛♠ ✐s ❛ r❡❝✐♣❡ ✇r✐tt❡♥ ✐♥ ❛ ❢♦r♠❛❧ ❧❛♥❣✉❛❣❡ ❢♦r ❝♦♠♣✉t✐♥❣ ❞❡s✐r❡❞ ♦✉t♣✉t ❞❛t❛ ❢r♦♠ ❣✐✈❡♥ ✐♥♣✉t ❞❛t❛✳ ✭❍❛r❡❧✱ ❑♦③❡♥ ❛♥❞ ❚✐✉r②♥✮

❊①❛♠♣❧❡✿ ❊✉❝❧✐❞✬s ❆❧❣♦r✐t❤♠ x := u; y := v; ✇❤✐❧❡ x = y ❞♦ ✐❢ x < y t❤❡♥ y := y − x; ❡❧s❡ x := x − y; ■♥♣✉t✿ x, y ∈ N ❖✉t♣✉t✿ gcd(x, y)

❲❤❡♥ ✐s ❛ Pr♦❣r❛♠ ❈♦rr❡❝t❄ ❋♦r♠❛❧ ❙♣❡❝✐✜❝❛t✐♦♥ ✉s❡ ♥♦t❛t✐♦♥s ❞❡r✐✈❡❞ ❢r♦♠ ❢♦r♠❛❧ ❧♦❣✐❝ t♦ ❞❡s❝r✐❜❡

• ❛ss✉♠♣t✐♦♥s ❛❜♦✉t t❤❡ ❡♥✈✐r♦♥♠❡♥t ✐♥ ✇❤✐❝❤ ❛ ♣r♦❣r❛♠ ✇✐❧❧

♦♣❡r❛t❡

• r❡q✉✐r❡♠❡♥ts ❛ ♣r♦❣r❛♠ ✐s t♦ ❛❝❤✐❡✈❡
• ❤♦✇ t♦ ❞❡s✐❣♥ t❤❡ ♣r♦❣r❛♠ t♦ ❛❝❤✐❡✈❡ t❤❡s❡ ❣♦❛❧s

❲❤❡♥ ✐s ❛ Pr♦❣r❛♠ ❈♦rr❡❝t❄ ❋♦r♠❛❧ ❱❡r✐✜❝❛t✐♦♥ ✉s❡ ♠❡t❤♦❞s ♦❢ ❢♦r♠❛❧ ❧♦❣✐❝ t♦

• ✈❛❧✐❞❛t❡ s♣❡❝✐✜❝❛t✐♦♥s ❜② ❝❤❡❝❦✐♥❣ ❝♦♥s✐st❡♥❝② ♦r ♣♦s✐♥❣

❝❤❛❧❧❡♥❣❡s

• ♣r♦✈❡ t❤❛t ❛ ♣r♦❣r❛♠ s❛t✐s✜❡s t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ✉♥❞❡r ❣✐✈❡♥

❛ss✉♠♣t✐♦♥s✱ ♦r ♣r♦✈❡ t❤❛t ❛ ♠♦r❡ ❞❡t❛✐❧❡❞ ♣r♦❣r❛♠ ✐♠♣❧❡♠❡♥ts ❛ ♠♦r❡ ❛❜str❛❝t ♦♥❡✳

❊①♦❣❡♥♦✉s ❛♥❞ ❊♥❞♦❣❡♥♦✉s Pr♦❣r❛♠ ▲♦❣✐❝s ❚✇♦ ♠❛✐♥ ❛♣♣r♦❛❝❤❡s t♦ t❤❡ ✭♠♦❞❛❧✮ ❧♦❣✐❝ ♦❢ ♣r♦❣r❛♠s✿ ❊①♦❣❡♥♦✉s✿ ♣r♦❣r❛♠s ❛r❡ ❡①♣❧✐❝✐t ✐♥ t❤❡ ❢♦r♠❛❧ ❧❛♥❣✉❛❣❡✳ ❊①❛♠♣❧❡s✿ ❍♦❛r❡ ▲♦❣✐❝✱ Pr♦♣♦s✐t✐♦♥❛❧ ❉②♥❛♠✐❝ ▲♦❣✐❝ ✭❞✐s❝✉ss❡❞ t♦❞❛②✮✳ ❊♥❞♦❣❡♥♦✉s✿ ❛ ♣r♦❣r❛♠ ✐s ✜①❡❞ ❛♥❞ ❝♦♥s✐❞❡r❡❞ ♣❛rt ♦❢ t❤❡ str✉❝t✉r❡ ♦✈❡r ✇❤✐❝❤ ❛ ♣r♦❣r❛♠ ✐s ✐♥t❡r♣r❡t❡❞✳ ❊①❛♠♣❧❡s✿ ▲✐♥❡❛r ❛♥❞ ❇r❛♥❝❤✐♥❣ ❚❡♠♣♦r❛❧ ▲♦❣✐❝s ✭♥♦t ❞✐s❝✉ss❡❞ t♦❞❛②✮✳

❈♦♠♣✉t❛t✐♦♥❛❧ ✈s✳ ❇❡❤❛✈✐♦r❛❧ ❙tr✉❝t✉r❡s

q0q0

x = 0

q0q1 x = 1

q0q0q0 q0q0q1 q0q1q0 q0q1q1 q0 x = 1 x = 0 x = 0 x = 1 x = 0

q0

q1

x = 0 x = 1

. . . . . .

❈♦♠♣✉t❛t✐♦♥❛❧ ✈s✳ ❇❡❤❛✈✐♦r❛❧ ❙tr✉❝t✉r❡s

q0q0

x = 0

q0q1 x = 1

q0q0q0 q0q0q1 q0q1q0 q0q1q1 q0 x = 1 x = 0 x = 0 x = 1 x = 0

q0

q1

x = 0 x = 1

. . . . . .

∃♦Px=1

❈♦♠♣✉t❛t✐♦♥❛❧ ✈s✳ ❇❡❤❛✈✐♦r❛❧ ❙tr✉❝t✉r❡s

q0q0

x = 0

q0q1 x = 1

q0q0q0 q0q0q1 q0q1q0 q0q1q1 q0 x = 1 x = 0 x = 0 x = 1 x = 0

q0

q1

x = 0 x = 1

. . . . . .

¬∀♦Px=1

▼♦r❡ ♦♥ ❚❡♠♣♦r❛❧ ▲♦❣✐❝

• ▲✐♥❡❛r ❚✐♠❡ ❚❡♠♣♦r❛❧ ▲♦❣✐❝✿ ❘❡❛s♦♥✐♥❣ ❛❜♦✉t ❝♦♠♣✉t❛t✐♦♥

♣❛t❤s✿ ♦φ✿ φ ✐s tr✉❡ s♦♠❡ t✐♠❡ ✐♥ t❤❡ ❢✉t✉r❡✳

❆✳ P♥✉❡❧❧✐✳ ❆ ❚❡♠♣♦r❛❧ ▲♦❣✐❝ ♦❢ Pr♦❣r❛♠s✳ ✐♥ Pr♦❝✳ ✶✽t❤ ■❊❊❊ ❙②♠♣♦s✐✉♠ ♦♥ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ✭✶✾✼✼✮✳

• ❇r❛♥❝❤✐♥❣ ❚✐♠❡ ❚❡♠♣♦r❛❧ ▲♦❣✐❝✿ ❆❧❧♦✇s q✉❛♥t✐✜❝❛t✐♦♥ ♦✈❡r

♣❛t❤s✿ ∃♦φ✿ t❤❡r❡ ✐s ❛ ♣❛t❤ ✐♥ ✇❤✐❝❤ φ ✐s ❡✈❡♥t✉❛❧❧② tr✉❡✳

❊✳ ▼✳ ❈❧❛r❦❡ ❛♥❞ ❊✳ ❆✳ ❊♠❡rs♦♥✳ ❉❡s✐❣♥ ❛♥❞ ❙②♥t❤❡s✐s ♦❢ ❙②♥❝❤r♦♥✐③❛t✐♦♥ ❙❦❡❧❡t♦♥s ✉s✐♥❣ ❇r❛♥❝❤✐♥❣✲t✐♠❡ ❚❡♠♣r♦❛❧✲❧♦❣✐❝ ❙♣❡❝✐✜❝❛t✐♦♥s✳ ■♥ Pr♦❝❡❡❞✐♥❣s ❲♦r❦s❤♦♣ ♦♥ ▲♦❣✐❝ ♦❢ Pr♦❣r❛♠s✱ ▲◆❈❙ ✭✶✾✽✶✮✳

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝ ▼♦t✐✈❛t✐♦♥✿ ❋♦r♠❛❧❧② ✈❡r✐❢② t❤❡ ✏❝♦rr❡❝t♥❡ss✑ ♦❢ ❛ ♣r♦❣r❛♠ ✈✐❛ ♣❛rt✐❛❧ ❝♦rr❡❝t♥❡ss ❛ss❡rt✐♦♥s✿

{φ}α{ψ}

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝ ▼♦t✐✈❛t✐♦♥✿ ❋♦r♠❛❧❧② ✈❡r✐❢② t❤❡ ✏❝♦rr❡❝t♥❡ss✑ ♦❢ ❛ ♣r♦❣r❛♠ ✈✐❛ ♣❛rt✐❛❧ ❝♦rr❡❝t♥❡ss ❛ss❡rt✐♦♥s✿

{φ}α{ψ}

■♥t❡♥❞❡❞ ■♥t❡r♣r❡t❛t✐♦♥✿ ■❢ t❤❡ ♣r♦❣r❛♠ α ❜❡❣✐♥s ✐♥ ❛ st❛t❡ ✐♥ ✇❤✐❝❤ φ ✐s tr✉❡✱ t❤❡♥ ❛❢t❡r α t❡r♠✐♥❛t❡s ✭✦✮✱ ψ ✇✐❧❧ ❜❡ tr✉❡✳

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝ ▼♦t✐✈❛t✐♦♥✿ ❋♦r♠❛❧❧② ✈❡r✐❢② t❤❡ ✏❝♦rr❡❝t♥❡ss✑ ♦❢ ❛ ♣r♦❣r❛♠ ✈✐❛ ♣❛rt✐❛❧ ❝♦rr❡❝t♥❡ss ❛ss❡rt✐♦♥s✿

{φ}α{ψ}

■♥t❡♥❞❡❞ ■♥t❡r♣r❡t❛t✐♦♥✿ ■❢ t❤❡ ♣r♦❣r❛♠ α ❜❡❣✐♥s ✐♥ ❛ st❛t❡ ✐♥ ✇❤✐❝❤ φ ✐s tr✉❡✱ t❤❡♥ ❛❢t❡r α t❡r♠✐♥❛t❡s ✭✦✮✱ ψ ✇✐❧❧ ❜❡ tr✉❡✳

❈✳ ❆✳ ❘✳ ❍♦❛r❡✳ ❆♥ ❆①✐♦♠❛t✐❝ ❇❛s✐s ❢♦r ❈♦♠♣✉t❡r Pr♦❣r❛♠♠✐♥❣✳✳ ❈♦♠♠✳ ❆ss♦❝✳ ❈♦♠♣✉t✳ ▼❛❝❤✳ ✶✾✻✾✳

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝ ▼❛✐♥ ❘✉❧❡s✿

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝ ▼❛✐♥ ❘✉❧❡s✿ ❆ss✐❣♥♠❡♥t ❘✉❧❡✿ {φ[x/e]} x := e {φ}

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝ ▼❛✐♥ ❘✉❧❡s✿ ❆ss✐❣♥♠❡♥t ❘✉❧❡✿ {φ[x/e]} x := e {φ} ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡✿ {φ} α {σ} {σ} β {ψ} {φ} α; β {ψ}

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝ ▼❛✐♥ ❘✉❧❡s✿ ❆ss✐❣♥♠❡♥t ❘✉❧❡✿ {φ[x/e]} x := e {φ} ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡✿ {φ} α {σ} {σ} β {ψ} {φ} α; β {ψ} ❈♦♥❞✐t✐♦♥❛❧ ❘✉❧❡✿ {φ ∧ σ} α {ψ} {φ ∧ ¬σ} β {ψ} {φ} if σ then α else β {ψ}

❇❛❝❦❣r♦✉♥❞✿ ❍♦❛r❡ ▲♦❣✐❝ ▼❛✐♥ ❘✉❧❡s✿ ❆ss✐❣♥♠❡♥t ❘✉❧❡✿ {φ[x/e]} x := e {φ} ❈♦♠♣♦s✐t✐♦♥ ❘✉❧❡✿ {φ} α {σ} {σ} β {ψ} {φ} α; β {ψ} ❈♦♥❞✐t✐♦♥❛❧ ❘✉❧❡✿ {φ ∧ σ} α {ψ} {φ ∧ ¬σ} β {ψ} {φ} if σ then α else β {ψ} ❲❤✐❧❡ ❘✉❧❡✿ {φ ∧ σ} α {φ} {φ} while σ do α {φ ∧ ¬σ}

❊①❛♠♣❧❡✿ ❊✉❝❧✐❞✬s ❆❧❣♦r✐t❤♠ x := u; y := v; ✇❤✐❧❡ x = y ❞♦ ✐❢ x < y t❤❡♥ y := y − x; ❡❧s❡ x := x − y; ▲❡t φ := gcd(x, y) = gcd(u, v)

❊①❛♠♣❧❡✿ ❊✉❝❧✐❞✬s ❆❧❣♦r✐t❤♠ x := u; y := v; ✇❤✐❧❡ x = y ❞♦ ✐❢ x < y t❤❡♥ y := y − x; ❡❧s❡ x := x − y; ▲❡t α ❜❡ t❤❡ ✐♥♥❡r ✐❢ st❛t❡♠❡♥t✳

❊①❛♠♣❧❡✿ ❊✉❝❧✐❞✬s ❆❧❣♦r✐t❤♠ x := u; y := v; ✇❤✐❧❡ x = y ❞♦ ✐❢ x < y t❤❡♥ y := y − x; ❡❧s❡ x := x − y; ▲❡t α ❜❡ t❤❡ ✐♥♥❡r ✐❢ st❛t❡♠❡♥t✳ ❚❤❡♥ {gcd(x, y) = gcd(u, v)} α {gcd(x, y) = gcd(u, v)}

❊①❛♠♣❧❡✿ ❊✉❝❧✐❞✬s ❆❧❣♦r✐t❤♠ x := u; y := v; ✇❤✐❧❡ x = y ❞♦ ✐❢ x < y t❤❡♥ y := y − x; ❡❧s❡ x := x − y; ❍❡♥❝❡ ❜② t❤❡ ✇❤✐❧❡✲r✉❧❡ ✭✉s✐♥❣ ❛ ✏✇❡❛❦❡♥✐♥❣ r✉❧❡✑✮ {(gcd(x, y) = gcd(u, v)) ∧ (x = y)} α {gcd(x, y) = gcd(u, v))} {gcd(x, y) = gcd(u, v)} while σ do α {(gcd(x, y) = gcd(u, v)) ∧ ¬(x = y)}

▼♦r❡ ♦♥ ❍♦❛r❡ ▲♦❣✐❝

❑✳ ❆♣t✳ ❚❡♥ ❨❡❛rs ♦❢ ❍♦❛r❡✬s ▲♦❣✐❝✿ ❆ ❙✉r✈❡② ✖ P❛rt ■✳ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ Pr♦❣r❛♠♠✐♥❣ ▲❛♥❣✉❛❣❡s ❛♥❞ ❙②st❡♠s✱ ✶✾✽✶✳

Pr♦❣r❛♠s ❛s ❙t❛t❡ ❚r❛♥s❢♦r♠❡rs ❆ st❛t❡ ✐s ✭✐♥❢♦r♠❛❧❧②✮ ❛♥ ✐♥st❛♥t❛♥❡♦✉s ❞❡s❝r✐♣t✐♦♥ ♦❢ r❡❛❧✐t②✳ ❋♦r♠❛❧❧②✱ ❛ st❛t❡ ♦❢ ❛ ♣r♦❣r❛♠ ✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t ❛ss✐❣♥s t♦ ❡✈❡r② ✈❛r✐❛❜❧❡ ❛ ✈❛❧✉❡ ❢r♦♠ ✐ts ❞♦♠❛✐♥ ♦❢ ✐♥t❡r♣r❡t❛t✐♦♥✳ ❊①❛♠♣❧❡✿ (x, y, u, v) = (15, 27, 15, 27) ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ ❛❜♦✈❡ ♣r♦❣r❛♠✳

Pr♦❣r❛♠s ❛s ❙t❛t❡ ❚r❛♥s❢♦r♠❡rs ❆ st❛t❡ ✐s ✭✐♥❢♦r♠❛❧❧②✮ ❛♥ ✐♥st❛♥t❛♥❡♦✉s ❞❡s❝r✐♣t✐♦♥ ♦❢ r❡❛❧✐t②✳ ❋♦r♠❛❧❧②✱ ❛ st❛t❡ ♦❢ ❛ ♣r♦❣r❛♠ ✐s ❛ ❢✉♥❝t✐♦♥ t❤❛t ❛ss✐❣♥s t♦ ❡✈❡r② ✈❛r✐❛❜❧❡ ❛ ✈❛❧✉❡ ❢r♦♠ ✐ts ❞♦♠❛✐♥ ♦❢ ✐♥t❡r♣r❡t❛t✐♦♥✳ ❊①❛♠♣❧❡✿ (x, y, u, v) = (15, 27, 15, 27) ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡ ♦❢ t❤❡ ❛❜♦✈❡ ♣r♦❣r❛♠✳ ❆ ♣r♦❣r❛♠ ✐s ❛ s❡t ♦❢ ✐♥str✉❝t✐♦♥s t❤❛t tr❛♥s❢♦r♠s ❛ st❛t❡✿ (15, 27, 15, 27), (15, 12, 15, 27), (3, 12, 15, 27), (3, 9, 15, 27), (3, 6, 15, 27), (3, 3, 15, 27)

Pr♦♣♦s✐t✐♦♥❛❧ ❉②♥❛♠✐❝ ▲♦❣✐❝ ▲❡t P ❜❡ ❛ s❡t ♦❢ ❛t♦♠✐❝ ♣r♦❣r❛♠s ❛♥❞ At ❛ s❡t ♦❢ ❛t♦♠✐❝ ♣r♦♣♦s✐t✐♦♥s✳ ❋♦r♠✉❧❛s ♦❢ PDL ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②♥t❛❝t✐❝ ❢♦r♠✿ φ := p | ⊥ | ¬φ | φ ∨ ψ | [α]φ α := a | α ∪ β | α; β | α∗ | φ? ✇❤❡r❡ p ∈ At ❛♥❞ a ∈ P✳

Pr♦♣♦s✐t✐♦♥❛❧ ❉②♥❛♠✐❝ ▲♦❣✐❝ ▲❡t P ❜❡ ❛ s❡t ♦❢ ❛t♦♠✐❝ ♣r♦❣r❛♠s ❛♥❞ At ❛ s❡t ♦❢ ❛t♦♠✐❝ ♣r♦♣♦s✐t✐♦♥s✳ ❋♦r♠✉❧❛s ♦❢ PDL ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s②♥t❛❝t✐❝ ❢♦r♠✿ φ := p | ⊥ | ¬φ | φ ∨ ψ | [α]φ α := a | α ∪ β | α; β | α∗ | φ? ✇❤❡r❡ p ∈ At ❛♥❞ a ∈ P✳ {φ} α {ψ} ✐s r❡♣❧❛❝❡❞ ✇✐t❤ φ → [α]ψ

P❉▲✿ ■♥t❡♥❞❡❞ ▼❡❛♥✐♥❣s [α]φ✿ ✏■t ✐s ♥❡❝❡ss❛r② t❤❛t ❛❢t❡r ❡①❡❝✉t✐♥❣ α✱ φ ✐s tr✉❡✑ α; β✿ ✏❊①❡❝✉t❡ α t❤❡♥ β α ∪ β✿ ✏❈❤♦♦s❡ ❡✐t❤❡r α ♦r β α∗✿ ✏❊①❡❝✉t❡ α ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝❛❧❧② ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t✐♠❡s ✭③❡r♦ ♦r ♠♦r❡✮✑ φ?✿ ✏❚❡st φ✱ ♣r♦❝❡❡❞ ✐❢ tr✉❡✱ ❢❛✐❧ ✐❢ ❢❛❧s❡✑

❋❛♠✐❧✐❛r Pr♦❣r❛♠s skip := ⊤? fail := ⊥? if φ then α else β := φ?; α ∪ ¬φ?; β while φ do α := (φ?; α)∗; ¬φ? do φ1 → α1 | · · · | φn → αn od := (φ1?; α1 ∪ · · · ∪ φn?; αn)∗; (¬φ1 ∧ · · · ∧ ¬φn)?

❙❡♠❛♥t✐❝s ❙❡♠❛♥t✐❝s✿ M = W, {Ra | a ∈ P}, V ✇❤❡r❡ ❢♦r ❡❛❝❤ a ∈ P✱ Ra ⊆ W × W ❛♥❞ V : At → 2W

• Rα∪β := Rα ∪ Rβ
• Rα;β := Rα ◦ Rβ
• Rα∗ := ∪n≥0Rn

α

• Rφ? = {(w, w) | M, w |

= φ} M, w | = [α]φ ✐✛ ❢♦r ❡❛❝❤ v✱ ✐❢ wRαv t❤❡♥ M, v | = φ

❙❡❣❡r❜❡r❣ ❆①✐♦♠s ✶✳ ❆①✐♦♠s ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ✷✳ [α](φ → ψ) → ([α]φ → [α]ψ) ✸✳ [α ∪ β]φ ↔ [α]φ ∧ [β]φ ✹✳ [α; β]φ ↔ [α][β]φ ✺✳ [ψ?]φ ↔ (ψ → φ) ✻✳ φ ∧ [α][α∗]φ ↔ [α∗]φ ✼✳ φ ∧ [α∗](φ → [α]φ) → [α∗]φ ✽✳ ▼♦❞✉s P♦♥❡♥s ❛♥❞ ◆❡❝❡ss✐t❛t✐♦♥ ✭❢♦r ❡❛❝❤ ♣r♦❣r❛♠ α✮

❙❡❣❡r❜❡r❣ ❆①✐♦♠s ✶✳ ❆①✐♦♠s ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ✷✳ [α](φ → ψ) → ([α]φ → [α]ψ) ✸✳ [α ∪ β]φ ↔ [α]φ ∧ [β]φ ✹✳ [α; β]φ ↔ [α][β]φ ✺✳ [ψ?]φ ↔ (ψ → φ) ✻✳ φ ∧ [α][α∗]φ ↔ [α∗]φ ✭❋✐①❡❞✲P♦✐♥t ❆①✐♦♠✮ ✼✳ φ ∧ [α∗](φ → [α]φ) → [α∗]φ ✭■♥❞✉❝t✐♦♥ ❆①✐♦♠✮ ✽✳ ▼♦❞✉s P♦♥❡♥s ❛♥❞ ◆❡❝❡ss✐t❛t✐♦♥ ✭❢♦r ❡❛❝❤ ♣r♦❣r❛♠ α✮

❊①t❡♥❞✐♥❣ P❉▲

• ■♥t❡rs❡❝t✐♦♥ ✭α ∩ β✮✿ wRα∩βv ✐✛ wRαv ❛♥❞ wRβv
• ❈♦♠♣❧❡♠❡♥t❛t✐♦♥ ✭α✮✿ wRαv ✐✛ ✐t ✐s ♥♦t t❤❡ ❝❛s❡ t❤❛t wRαv
• ❈♦♥✈❡rs❡ ✭α−1✮✿ wRα−1v ✐✛ vRαw

❊①t❡♥❞✐♥❣ P❉▲

• ■♥t❡rs❡❝t✐♦♥ ✭α ∩ β✮✿ wRα∩βv ✐✛ wRαv ❛♥❞ wRβv
• ❈♦♠♣❧❡♠❡♥t❛t✐♦♥ ✭α✮✿ wRαv ✐✛ ✐t ✐s ♥♦t t❤❡ ❝❛s❡ t❤❛t wRαv
• ❈♦♥✈❡rs❡ ✭α−1✮✿ wRα−1v ✐✛ vRαw

❙❡❡ ✇♦r❦ ♦♥ ❇♦♦❧❡❛♥ ♠♦❞❛❧ ❧♦❣✐❝✳

P❛ss② ❛♥❞ ❚✐♥❝❤❡✈✳ ❆♥ ❡ss❛② ✐♥ ❝♦♠❜✐♥❛t♦r② ❞②♥❛♠✐❝ ❧♦❣✐❝✳ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✾✸ ✭✶✾✾✶✮✳

❊①t❡♥❞✐♥❣ P❉▲

• ■♥t❡rs❡❝t✐♦♥ ✭α ∩ β✮✿ wRα∩βv ✐✛ wRαv ❛♥❞ wRβv
• ❈♦♠♣❧❡♠❡♥t❛t✐♦♥ ✭α✮✿ wRαv ✐✛ ✐t ✐s ♥♦t t❤❡ ❝❛s❡ t❤❛t wRαv
• ❈♦♥✈❡rs❡ ✭α−1✮✿ wRα−1v ✐✛ vRαw

❙❡❡ ✇♦r❦ ♦♥ ❇♦♦❧❡❛♥ ♠♦❞❛❧ ❧♦❣✐❝✳

P❛ss② ❛♥❞ ❚✐♥❝❤❡✈✳ ❆♥ ❡ss❛② ✐♥ ❝♦♠❜✐♥❛t♦r② ❞②♥❛♠✐❝ ❧♦❣✐❝✳ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✾✸ ✭✶✾✾✶✮✳

❉②♥❛♠✐❝ ▲♦❣✐❝ ✭✜rst✲♦r❞❡r✮✱ ♥♦♥r❡❣✉❧❛r P❉▲✱ ❡t❝✳

❉✳ ❍❛r❡❧✱ ❉✳ ❑♦③❡♥ ❛♥❞ ❚✐✉r②♥✳ ❉②♥❛♠✐❝ ▲♦❣✐❝✳ ✷✵✵✵✳

❆ ❙✉r✈❡② ♦❢ ❘❡s✉❧ts ❚❤❡♦r❡♠ ✭P❛r✐❦❤✱ ❑♦③❡♥ ❛♥❞ P❛r✐❦❤✮ PDL ✐s s♦✉♥❞ ❛♥❞ ✇❡❛❦❧② ❝♦♠♣❧❡t❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❙❡❣❡r❜❡r❣ ❆①✐♦♠s✳ ❚❤❡♦r❡♠ ❚❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r PDL ✐s ❞❡❝✐❞❛❜❧❡ ✭EXPTIME✲❈♦♠♣❧❡t❡✮✳

❉✳ ❑♦③❡♥ ❛♥❞ ❘✳ P❛r✐❦❤✳ ❆♥ ❊❧❡♠❡♥t❛r② Pr♦♦❢ ♦❢ t❤❡ ❈♦♠♣❧❡t❡♥❡ss ♦❢ P❉▲✳ ✶✾✽✶✳ ❉✳ ❍❛r❡❧✱ ❉✳ ❑♦③❡♥ ❛♥❞ ❚✐✉r②♥✳ ❉②♥❛♠✐❝ ▲♦❣✐❝✳ ✷✵✵✵✳

❆ ❙✉r✈❡② ♦❢ ❘❡s✉❧ts R ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐❢ ❢♦r ❡❛❝❤ w ∈ W t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ v ∈ W s✉❝❤ t❤❛t wRv✳ ❚❤❡♦r❡♠ ❆ss✉♠✐♥❣ t❤❛t ❛❧❧ ❛t♦♠✐❝ ♣r♦❣r❛♠s ❛r❡ ❞❡t❡r♠✐♥✐st✐❝✱ t❤❡♥ P❉▲ ✇✐t❤ ✐♥t❡rs❡❝t✐♦♥ ✐s ❤✐❣❤❧② ✉♥❞❡❝✐❞❛❜❧❡✱ ✐✳❡✳✱ Π1

1✲❝♦♠♣❧❡t❡✳

❆ ❙✉r✈❡② ♦❢ ❘❡s✉❧ts R ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐❢ ❢♦r ❡❛❝❤ w ∈ W t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ v ∈ W s✉❝❤ t❤❛t wRv✳ ❚❤❡♦r❡♠ ❆ss✉♠✐♥❣ t❤❛t ❛❧❧ ❛t♦♠✐❝ ♣r♦❣r❛♠s ❛r❡ ❞❡t❡r♠✐♥✐st✐❝✱ t❤❡♥ P❉▲ ✇✐t❤ ✐♥t❡rs❡❝t✐♦♥ ✐s ❤✐❣❤❧② ✉♥❞❡❝✐❞❛❜❧❡✱ ✐✳❡✳✱ Π1

1✲❝♦♠♣❧❡t❡✳

❚❤❡♦r❡♠ ❚❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r P❉▲ ✇✐t❤ ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ✐s ✉♥❞❡❝✐❞❛❜❧❡✳

❆ ❙✉r✈❡② ♦❢ ❘❡s✉❧ts R ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐❢ ❢♦r ❡❛❝❤ w ∈ W t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ v ∈ W s✉❝❤ t❤❛t wRv✳ ❚❤❡♦r❡♠ ❆ss✉♠✐♥❣ t❤❛t ❛❧❧ ❛t♦♠✐❝ ♣r♦❣r❛♠s ❛r❡ ❞❡t❡r♠✐♥✐st✐❝✱ t❤❡♥ P❉▲ ✇✐t❤ ✐♥t❡rs❡❝t✐♦♥ ✐s ❤✐❣❤❧② ✉♥❞❡❝✐❞❛❜❧❡✱ ✐✳❡✳✱ Π1

1✲❝♦♠♣❧❡t❡✳

❚❤❡♦r❡♠ ❚❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r P❉▲ ✇✐t❤ ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ✐s ✉♥❞❡❝✐❞❛❜❧❡✳ ▲❡t Sφ ❜❡ ❛❧❧ t❤❡ s✉❜st✐t✉t✐♦♥ ✐♥st❛♥❝❡s ♦❢ φ✳ ❚❤❡♦r❡♠ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❞❡❝✐❞✐♥❣ ✇❤❡t❤❡r Sφ | = ψ ✐s ❤✐❣❤❧② ✉♥❞❡❝✐❞❛❜❧❡ ✭Π1

1✲❝♦♠♣❧❡t❡✮✳

❉✳ ❍❛r❡❧✱ ❉✳ ❑♦③❡♥ ❛♥❞ ❚✐✉r②♥✳ ❉②♥❛♠✐❝ ▲♦❣✐❝✳ ✷✵✵✵✳

❆ ❙✉r✈❡② ♦❢ ✭❘❡❝❡♥t✮ ❘❡s✉❧ts ❚❤❡♦r❡♠ P❉▲ ✇✐t❤ ✐♥t❡rs❡❝t✐♦♥ ❜✉t ✇✐t❤♦✉t ✐t❡r❛t✐♦♥ ✐s ❛①✐♦♠❛t✐③❛❜❧❡✳

❇❛❧❜✐❛♥✐ ❛♥❞ ❱❛❦❛r❡❧♦✈✳ ■t❡r❛t✐♦♥✲❢r❡❡ ♣❞❧ ✇✐t❤ ✐♥t❡rs❡❝t✐♦♥✿ ❛ ❝♦♠♣❧❡t❡ ❛①✐♦♠✲ ❛t✐③❛t✐♦♥✳ ❋✉♥❞❛♠❡♥t❛ ■♥❢♦r♠❛t✐❝❛❡ ✹✺ ✭✷✵✵✶✮ ✳

❚❤❡♦r❡♠ ❚❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r P❉▲ ✇✐t❤ ❝♦♠♣❧❡♠❡♥t ❛♣♣❧✐❡❞ ♦♥❧② t♦ ❛t♦♠✐❝ ♣r♦❣r❛♠s ✐s ❞❡❝✐❞❛❜❧❡✳

▲✉t③ ❛♥❞ ❲❛❧t❤❡r✳ P❉▲ ✇✐t❤ ♥❡❣❛t✐♦♥ ♦❢ ❛t♦♠✐❝ ♣r♦❣r❛♠s✳ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ ◆♦♥✲❈❧❛ss✐❝❛❧ ▲♦❣✐❝s ✶✹✭✷✮ ✭✷✵✵✹✮ ✳

❆ ❙✉r✈❡② ♦❢ ✭❘❡❝❡♥t✮ ❘❡s✉❧ts ❚❤❡♦r❡♠ ❚❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r P❉▲ ✇✐t❤ ✐♥t❡rs❡❝t✐♦♥ ✭❛♥❞ ✇✐t❤♦✉t ❝♦♠♣❧❡♠❡♥t✮ ✐s ✷✲❊❳P❙P❆❈❊✲❝♦♠♣❧❡t❡✳

▲❛♥❣❡ ❛♥❞ ▲✉t③✳ ✷✲❡①♣t✐♠❡ ❧♦✇❡r ❜♦✉♥❞s ❢♦r ♣r♦♣♦s✐t✐♦♥❛❧ ❞②♥❛♠✐❝ ❧♦❣✐❝s ✇✐t❤ ✐♥t❡rs❡❝t✐♦♥✳ ❏♦✉r♥❛❧ ♦❢ ❙②♠❜♦❧✐❝ ▲♦❣✐❝ ✼✵✭✹✮ ✭✷✵✵✺✮✳

❚❤❡♦r❡♠ ❚❤❡ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ♣r♦❜❧❡♠ ❢♦r P❉▲ ♠♦❞❡❧s ✐s P❚■▼❊✳

▲❛♥❣❡✳ ▼♦❞❡❧ ❝❤❡❝❦✐♥❣ ♣r♦♣♦s✐t✐♦♥❛❧ ❞②♥❛♠✐❝ ❧♦❣✐❝ ✇✐t❤ ❛❧❧ ❡①tr❛s✳ ❏♦✉r♥❛❧ ♦❢ ❆♣♣❧✐❡❞ ▲♦❣✐❝ ✹ ✭✷✵✵✻✮✳

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝

• ❛♠❡ ▲♦❣✐❝ ✭GL✮ ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❘♦❤✐t P❛r✐❦❤ ✐♥

❘✳ P❛r✐❦❤✳ ❚❤❡ ▲♦❣✐❝ ♦❢ ●❛♠❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳✳ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✳ ✭✶✾✽✺✮ ✳

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝

• ❛♠❡ ▲♦❣✐❝ ✭GL✮ ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❘♦❤✐t P❛r✐❦❤ ✐♥

❘✳ P❛r✐❦❤✳ ❚❤❡ ▲♦❣✐❝ ♦❢ ●❛♠❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳✳ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✳ ✭✶✾✽✺✮ ✳

▼❛✐♥ ■❞❡❛✿ ■♥ PDL✿ w | = πφ✿ t❤❡r❡ ✐s ❛ r✉♥ ♦❢ t❤❡ ♣r♦❣r❛♠ π st❛rt✐♥❣ ✐♥ st❛t❡ w t❤❛t ❡♥❞s ✐♥ ❛ st❛t❡ ✇❤❡r❡ φ ✐s tr✉❡✳ ❚❤❡ ♣r♦❣r❛♠s ✐♥ PDL ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s s✐♥❣❧❡ ♣❧❛②❡r ❣❛♠❡s✳

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝

• ❛♠❡ ▲♦❣✐❝ ✭GL✮ ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❘♦❤✐t P❛r✐❦❤ ✐♥

❘✳ P❛r✐❦❤✳ ❚❤❡ ▲♦❣✐❝ ♦❢ ●❛♠❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳✳ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✳ ✭✶✾✽✺✮ ✳

▼❛✐♥ ■❞❡❛✿ ■♥ PDL✿ w | = πφ✿ t❤❡r❡ ✐s ❛ r✉♥ ♦❢ t❤❡ ♣r♦❣r❛♠ π st❛rt✐♥❣ ✐♥ st❛t❡ w t❤❛t ❡♥❞s ✐♥ ❛ st❛t❡ ✇❤❡r❡ φ ✐s tr✉❡✳ ❚❤❡ ♣r♦❣r❛♠s ✐♥ PDL ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s s✐♥❣❧❡ ♣❧❛②❡r ❣❛♠❡s✳

• ❛♠❡ ▲♦❣✐❝ ❣❡♥❡r❛❧✐③❡❞ PDL ❜② ❝♦♥s✐❞❡r✐♥❣ t✇♦ ♣❧❛②❡rs✿

■♥ GL✿ w | = γφ✿ ❆♥❣❡❧ ❤❛s ❛ str❛t❡❣② ✐♥ t❤❡ ❣❛♠❡ γ t♦ ❡♥s✉r❡ t❤❛t t❤❡ ❣❛♠❡ ❡♥❞s ✐♥ ❛ st❛t❡ ✇❤❡r❡ φ ✐s tr✉❡✳

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❈♦♥s❡q✉❡♥❝❡s ♦❢ t✇♦ ♣❧❛②❡rs✿

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❈♦♥s❡q✉❡♥❝❡s ♦❢ t✇♦ ♣❧❛②❡rs✿ γφ✿ ❆♥❣❡❧ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ [γ]φ✿ ❉❡♠♦♥ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❈♦♥s❡q✉❡♥❝❡s ♦❢ t✇♦ ♣❧❛②❡rs✿ γφ✿ ❆♥❣❡❧ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ [γ]φ✿ ❉❡♠♦♥ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ ❊✐t❤❡r ❆♥❣❡❧ ♦r ❉❡♠♦♥ ❝❛♥ ✇✐♥✿ γφ ∨ [γ]¬φ

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❈♦♥s❡q✉❡♥❝❡s ♦❢ t✇♦ ♣❧❛②❡rs✿ γφ✿ ❆♥❣❡❧ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ [γ]φ✿ ❉❡♠♦♥ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ ❊✐t❤❡r ❆♥❣❡❧ ♦r ❉❡♠♦♥ ❝❛♥ ✇✐♥✿ γφ ∨ [γ]¬φ ❇✉t ♥♦t ❜♦t❤✿ ¬(γφ ∧ [γ]¬φ)

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❈♦♥s❡q✉❡♥❝❡s ♦❢ t✇♦ ♣❧❛②❡rs✿ γφ✿ ❆♥❣❡❧ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ [γ]φ✿ ❉❡♠♦♥ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ ❊✐t❤❡r ❆♥❣❡❧ ♦r ❉❡♠♦♥ ❝❛♥ ✇✐♥✿ γφ ∨ [γ]¬φ ❇✉t ♥♦t ❜♦t❤✿ ¬(γφ ∧ [γ]¬φ) ❚❤✉s✱ [γ]φ ↔ ¬γ¬φ ✐s ❛ ✈❛❧✐❞ ♣r✐♥❝✐♣❧❡

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❈♦♥s❡q✉❡♥❝❡s ♦❢ t✇♦ ♣❧❛②❡rs✿ γφ✿ ❆♥❣❡❧ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ [γ]φ✿ ❉❡♠♦♥ ❤❛s ❛ str❛t❡❣② ✐♥ γ t♦ ❡♥s✉r❡ φ ✐s tr✉❡ ❊✐t❤❡r ❆♥❣❡❧ ♦r ❉❡♠♦♥ ❝❛♥ ✇✐♥✿ γφ ∨ [γ]¬φ ❇✉t ♥♦t ❜♦t❤✿ ¬(γφ ∧ [γ]¬φ) ❚❤✉s✱ [γ]φ ↔ ¬γ¬φ ✐s ❛ ✈❛❧✐❞ ♣r✐♥❝✐♣❧❡ ❍♦✇❡✈❡r✱ [γ]φ ∧ [γ]ψ → [γ](φ ∧ ψ) ✐s ♥♦t ❛ ✈❛❧✐❞ ♣r✐♥❝✐♣❧❡

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❘❡✐♥t❡r♣r❡t ♦♣❡r❛t✐♦♥s ❛♥❞ ✐♥✈❡♥t ♥❡✇ ♦♥❡s✿

• ?φ✿ ❈❤❡❝❦ ✇❤❡t❤❡r φ ❝✉rr❡♥t❧② ❤♦❧❞s

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❘❡✐♥t❡r♣r❡t ♦♣❡r❛t✐♦♥s ❛♥❞ ✐♥✈❡♥t ♥❡✇ ♦♥❡s✿

• ?φ✿ ❈❤❡❝❦ ✇❤❡t❤❡r φ ❝✉rr❡♥t❧② ❤♦❧❞s
• γ1; γ2✿ ❋✐rst ♣❧❛② γ1 t❤❡♥ γ2

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❘❡✐♥t❡r♣r❡t ♦♣❡r❛t✐♦♥s ❛♥❞ ✐♥✈❡♥t ♥❡✇ ♦♥❡s✿

• ?φ✿ ❈❤❡❝❦ ✇❤❡t❤❡r φ ❝✉rr❡♥t❧② ❤♦❧❞s
• γ1; γ2✿ ❋✐rst ♣❧❛② γ1 t❤❡♥ γ2
• γ1 ∪ γ2✿ ❆♥❣❡❧ ❝❤♦♦s❡ ❜❡t✇❡❡♥ γ1 ❛♥❞ γ2

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❘❡✐♥t❡r♣r❡t ♦♣❡r❛t✐♦♥s ❛♥❞ ✐♥✈❡♥t ♥❡✇ ♦♥❡s✿

• ?φ✿ ❈❤❡❝❦ ✇❤❡t❤❡r φ ❝✉rr❡♥t❧② ❤♦❧❞s
• γ1; γ2✿ ❋✐rst ♣❧❛② γ1 t❤❡♥ γ2
• γ1 ∪ γ2✿ ❆♥❣❡❧ ❝❤♦♦s❡ ❜❡t✇❡❡♥ γ1 ❛♥❞ γ2
• γ∗✿ ❆♥❣❡❧ ❝❛♥ ❝❤♦♦s❡ ❤♦✇ ♦❢t❡♥ t♦ ♣❧❛② γ ✭♣♦ss✐❜❧② ♥♦t ❛t ❛❧❧✮❀

❡❛❝❤ t✐♠❡ s❤❡ ❤❛s ♣❧❛②❡❞ γ✱ s❤❡ ❝❛♥ ❞❡❝✐❞❡ ✇❤❡t❤❡r t♦ ♣❧❛② ✐t ❛❣❛✐♥ ♦r ♥♦t✳

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❘❡✐♥t❡r♣r❡t ♦♣❡r❛t✐♦♥s ❛♥❞ ✐♥✈❡♥t ♥❡✇ ♦♥❡s✿

• ?φ✿ ❈❤❡❝❦ ✇❤❡t❤❡r φ ❝✉rr❡♥t❧② ❤♦❧❞s
• γ1; γ2✿ ❋✐rst ♣❧❛② γ1 t❤❡♥ γ2
• γ1 ∪ γ2✿ ❆♥❣❡❧ ❝❤♦♦s❡ ❜❡t✇❡❡♥ γ1 ❛♥❞ γ2
• γ∗✿ ❆♥❣❡❧ ❝❛♥ ❝❤♦♦s❡ ❤♦✇ ♦❢t❡♥ t♦ ♣❧❛② γ ✭♣♦ss✐❜❧② ♥♦t ❛t ❛❧❧✮❀

❡❛❝❤ t✐♠❡ s❤❡ ❤❛s ♣❧❛②❡❞ γ✱ s❤❡ ❝❛♥ ❞❡❝✐❞❡ ✇❤❡t❤❡r t♦ ♣❧❛② ✐t ❛❣❛✐♥ ♦r ♥♦t✳

• γd✿ ❙✇✐t❝❤ r♦❧❡s✱ t❤❡♥ ♣❧❛② γ

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❘❡✐♥t❡r♣r❡t ♦♣❡r❛t✐♦♥s ❛♥❞ ✐♥✈❡♥t ♥❡✇ ♦♥❡s✿

• ?φ✿ ❈❤❡❝❦ ✇❤❡t❤❡r φ ❝✉rr❡♥t❧② ❤♦❧❞s
• γ1; γ2✿ ❋✐rst ♣❧❛② γ1 t❤❡♥ γ2
• γ1 ∪ γ2✿ ❆♥❣❡❧ ❝❤♦♦s❡ ❜❡t✇❡❡♥ γ1 ❛♥❞ γ2
• γ∗✿ ❆♥❣❡❧ ❝❛♥ ❝❤♦♦s❡ ❤♦✇ ♦❢t❡♥ t♦ ♣❧❛② γ ✭♣♦ss✐❜❧② ♥♦t ❛t ❛❧❧✮❀

❡❛❝❤ t✐♠❡ s❤❡ ❤❛s ♣❧❛②❡❞ γ✱ s❤❡ ❝❛♥ ❞❡❝✐❞❡ ✇❤❡t❤❡r t♦ ♣❧❛② ✐t ❛❣❛✐♥ ♦r ♥♦t✳

• γd✿ ❙✇✐t❝❤ r♦❧❡s✱ t❤❡♥ ♣❧❛② γ
• γ1 ∩ γ2 := (γd

1 ∪ γd 2)d✿ ❉❡♠♦♥ ❝❤♦♦s❡s ❜❡t✇❡❡♥ γ1 ❛♥❞ γ2

❋r♦♠ PDL t♦ ●❛♠❡ ▲♦❣✐❝ ❘❡✐♥t❡r♣r❡t ♦♣❡r❛t✐♦♥s ❛♥❞ ✐♥✈❡♥t ♥❡✇ ♦♥❡s✿

• ?φ✿ ❈❤❡❝❦ ✇❤❡t❤❡r φ ❝✉rr❡♥t❧② ❤♦❧❞s
• γ1; γ2✿ ❋✐rst ♣❧❛② γ1 t❤❡♥ γ2
• γ1 ∪ γ2✿ ❆♥❣❡❧ ❝❤♦♦s❡ ❜❡t✇❡❡♥ γ1 ❛♥❞ γ2
• γ∗✿ ❆♥❣❡❧ ❝❛♥ ❝❤♦♦s❡ ❤♦✇ ♦❢t❡♥ t♦ ♣❧❛② γ ✭♣♦ss✐❜❧② ♥♦t ❛t ❛❧❧✮❀

❡❛❝❤ t✐♠❡ s❤❡ ❤❛s ♣❧❛②❡❞ γ✱ s❤❡ ❝❛♥ ❞❡❝✐❞❡ ✇❤❡t❤❡r t♦ ♣❧❛② ✐t ❛❣❛✐♥ ♦r ♥♦t✳

• γd✿ ❙✇✐t❝❤ r♦❧❡s✱ t❤❡♥ ♣❧❛② γ
• γ1 ∩ γ2 := (γd

1 ∪ γd 2)d✿ ❉❡♠♦♥ ❝❤♦♦s❡s ❜❡t✇❡❡♥ γ1 ❛♥❞ γ2

• γx := ((γd)∗)d✿ ❉❡♠♦♥ ❝❛♥ ❝❤♦♦s❡ ❤♦✇ ♦❢t❡♥ t♦ ♣❧❛② γ

✭♣♦ss✐❜❧② ♥♦t ❛t ❛❧❧✮❀ ❡❛❝❤ t✐♠❡ ❤❡ ❤❛s ♣❧❛②❡❞ γ✱ ❤❡ ❝❛♥ ❞❡❝✐❞❡ ✇❤❡t❤❡r t♦ ♣❧❛② ✐t ❛❣❛✐♥ ♦r ♥♦t✳

• ❛♠❡ ▲♦❣✐❝✿ ❙②♥t❛①

❙②♥t❛①

▲❡t Γ0 ❜❡ ❛ s❡t ♦❢ ❛t♦♠✐❝ ❣❛♠❡s ❛♥❞ At ❛ s❡t ♦❢ ❛t♦♠✐❝ ♣r♦♣♦s✐t✐♦♥s✳ ❚❤❡♥ ❢♦r♠✉❧❛s ♦❢ ●❛♠❡ ▲♦❣✐❝ ❛r❡ ❞❡✜♥❡❞ ✐♥❞✉❝t✐✈❡❧② ❛s ❢♦❧❧♦✇s✿ γ := g | φ? | γ; γ | γ ∪ γ | γ∗ | γd φ := ⊥ | p | ¬φ | φ ∨ φ | γφ | [γ]φ ✇❤❡r❡ p ∈ At, g ∈ Γ0✳

• ❛♠❡ ▲♦❣✐❝✿ ❙❡♠❛♥t✐❝s ■

❆ ♥❡✐❣❤❜♦r❤♦♦❞ ❣❛♠❡ ♠♦❞❡❧ ✐s ❛ t✉♣❧❡ M = W, {Eg | g ∈ Γ0}, V ✇❤❡r❡ W ✐s ❛ ♥♦♥❡♠♣t② s❡t ♦❢ st❛t❡s ❋♦r ❡❛❝❤ g ∈ Γ0✱ Eg : W → 22W ✐s ❛♥ ❡✛❡❝t✐✈✐t② ❢✉♥❝t✐♦♥ s✉❝❤ t❤❛t ✐❢ X ⊆ X′ ❛♥❞ X ∈ Eg(w) t❤❡♥ X′ ∈ Eg(w)✳ X ∈ Eg(w) ♠❡❛♥s ✐♥ st❛t❡ s✱ ❆♥❣❡❧ ❤❛s ❛ str❛t❡❣② t♦ ❢♦r❝❡ t❤❡ ❣❛♠❡ t♦ ❡♥❞ ✐♥ s♦♠❡ st❛t❡ ✐♥ X ✭✇❡ ♠❛② ✇r✐t❡ wEgX✮ V : At → 2W ✐s ❛ ✈❛❧✉❛t✐♦♥ ❢✉♥❝t✐♦♥✳

• ❛♠❡ ▲♦❣✐❝✿ ❙❡♠❛♥t✐❝s ■

❆ ♥❡✐❣❤❜♦r❤♦♦❞ ❣❛♠❡ ♠♦❞❡❧ ✐s ❛ t✉♣❧❡ M = W, {Eg | g ∈ Γ0}, V ✇❤❡r❡ Pr♦♣♦s✐t✐♦♥❛❧ ❧❡tt❡rs ❛♥❞ ❜♦♦❧❡❛♥ ❝♦♥♥❡❝t✐✈❡s ❛r❡ ❛s ✉s✉❛❧✳ M, w | = γφ ✐✛ (φ)M ∈ Eγ(w)

• ❛♠❡ ▲♦❣✐❝✿ ❙❡♠❛♥t✐❝s ■

❆ ♥❡✐❣❤❜♦r❤♦♦❞ ❣❛♠❡ ♠♦❞❡❧ ✐s ❛ t✉♣❧❡ M = W, {Eg | g ∈ Γ0}, V ✇❤❡r❡ Pr♦♣♦s✐t✐♦♥❛❧ ❧❡tt❡rs ❛♥❞ ❜♦♦❧❡❛♥ ❝♦♥♥❡❝t✐✈❡s ❛r❡ ❛s ✉s✉❛❧✳ M, w | = γφ ✐✛ (φ)M ∈ Eγ(w) ❙✉♣♣♦s❡ Eγ(Y ) = {s | Y ∈ Eg(s)}

• Eγ1;γ2(Y ) := Eγ1(Eγ2(Y ))
• Eγ1∪γ2(Y ) := Eγ1(Y ) ∪ Eγ2(Y )
• Eφ?(Y ) := (φ)M ∩ Y
• Eγd(Y ) := Eγ(Y )
• Eγ∗(Y ) := µX.Y ∪ Eγ(X)

• ❛♠❡ ▲♦❣✐❝✿ ❆①✐♦♠s

✶✳ ❆❧❧ ♣r♦♣♦s✐t✐♦♥❛❧ t❛✉t♦❧♦❣✐❡s ✷✳ α; βφ ↔ αβφ ❈♦♠♣♦s✐t✐♦♥ ✸✳ α ∪ βφ ↔ αφ ∨ βφ ❯♥✐♦♥ ✹✳ ψ?φ ↔ (ψ ∧ φ) ❚❡st ✺✳ αdφ ↔ ¬α¬φ ❉✉❛❧ ✻✳ (φ ∨ αα∗φ) → α∗φ ▼✐① ❛♥❞ t❤❡ r✉❧❡s✱ φ φ → ψ ψ φ → ψ αφ → αψ (φ ∨ αψ) → ψ α∗φ → ψ

❙♦♠❡ ❘❡s✉❧ts

• ●❛♠❡ ▲♦❣✐❝ ✐s ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ PDL

❙♦♠❡ ❘❡s✉❧ts

• ●❛♠❡ ▲♦❣✐❝ ✐s ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ PDL

(gd)∗⊥

❙♦♠❡ ❘❡s✉❧ts

• ●❛♠❡ ▲♦❣✐❝ ✐s ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ PDL

(gd)∗⊥

• ❚❤❡ ✐♥❞✉❝t✐♦♥ ❛①✐♦♠ ✐s ♥♦t ✈❛❧✐❞ ✐♥ ●▲✳

❘✳ P❛r✐❦❤✳ ❚❤❡ ▲♦❣✐❝ ♦❢ ●❛♠❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳✳ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✳ ✭✶✾✽✺✮ ✳

❙♦♠❡ ❘❡s✉❧ts

• ●❛♠❡ ▲♦❣✐❝ ✐s ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ PDL

(gd)∗⊥

• ❚❤❡ ✐♥❞✉❝t✐♦♥ ❛①✐♦♠ ✐s ♥♦t ✈❛❧✐❞ ✐♥ ●▲✳

❘✳ P❛r✐❦❤✳ ❚❤❡ ▲♦❣✐❝ ♦❢ ●❛♠❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳✳ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✳ ✭✶✾✽✺✮ ✳

• ❆❧❧ ●▲ ❣❛♠❡s ❛r❡ ❞❡t❡r♠✐♥❡❞✳ ❚❤✐s ✐s ♥♦t ❛ tr✐✈✐❛❧ r❡s✉❧t s✐♥❝❡

♥❡✐t❤❡r ❩❡r♠❡❧♦✬s ❚❤❡♦r❡♠ ♥♦r t❤❡ ●❛❧❡✲❙t❡✇❛rt ❚❤❡♦r❡♠ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞✳

▼✳ P❛✉❧②✳

• ❛♠❡

▲♦❣✐❝ ❢♦r

• ❛♠❡

❚❤❡♦r✐sts✳ ❆✈❛✐❧❛❜❧❡ ❛t ❤tt♣✿✴✴✇✇✇✳st❛♥❢♦r❞✳❡❞✉✴ ♣✐❛♥♦♠❛♥✴✳

❙♦♠❡ ❘❡s✉❧ts ❚❤❡♦r❡♠ ❬✶❪ ❉✉❛❧✲❢r❡❡ ❣❛♠❡ ❧♦❣✐❝ ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ❣❛♠❡ ♠♦❞❡❧s✳ ❚❤❡♦r❡♠ ❬✷❪ ■t❡r❛t✐♦♥✲❢r❡❡ ❣❛♠❡ ❧♦❣✐❝ ✐s s♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ❣❛♠❡ ♠♦❞❡❧s✳ ❖♣❡♥ ◗✉❡st✐♦♥ ■s ✭❢✉❧❧✮ ❣❛♠❡ ❧♦❣✐❝ ❝♦♠♣❧❡t❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ ❣❛♠❡ ♠♦❞❡❧s❄

❬✶❪ ❘✳ P❛r✐❦❤✳ ❚❤❡ ▲♦❣✐❝ ♦❢ ●❛♠❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳✳ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✳ ✭✶✾✽✺✮ ✳ ❬✷❪ ▼✳ P❛✉❧②✳ ▲♦❣✐❝ ❢♦r ❙♦❝✐❛❧ ❙♦❢t✇❛r❡✳ P❤✳❉✳ ❚❤❡s✐s✱ ❯♥✐✈❡rs✐t② ♦❢ ❆♠st❡r✲ ❞❛♠ ✭✷✵✵✶✮✳✳

❙♦♠❡ ❘❡s✉❧ts ❚❤❡♦r❡♠ ❬✷❪ ●✐✈❡♥ ❛ ❣❛♠❡ ❧♦❣✐❝ ❢♦r♠✉❧❛ φ ❛♥❞ ❛ ✜♥✐t❡ ❣❛♠❡ ♠♦❞❡❧ M✱ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ t✐♠❡ O(|M|ad(φ)+1 × |φ|) ❚❤❡♦r❡♠ ❬✶✱✷❪ ❚❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r ❣❛♠❡ ❧♦❣✐❝ ✐s ✐♥ ❊❳P❚■▼❊✳ ❚❤❡♦r❡♠ ❬✶❪ ●❛♠❡ ❧♦❣✐❝ ❝❛♥ ❜❡ tr❛♥s❧❛t❡❞ ✐♥t♦ t❤❡ ♠♦❞❛❧ µ✲❝❛❧❝✉❧✉s

❬✶❪ ❘✳ P❛r✐❦❤✳ ❚❤❡ ▲♦❣✐❝ ♦❢ ●❛♠❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳✳ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✳ ✭✶✾✽✺✮ ✳ ❬✷❪ ▼✳ P❛✉❧②✳ ▲♦❣✐❝ ❢♦r ❙♦❝✐❛❧ ❙♦❢t✇❛r❡✳ P❤✳❉✳ ❚❤❡s✐s✱ ❯♥✐✈❡rs✐t② ♦❢ ❆♠st❡r✲ ❞❛♠ ✭✷✵✵✶✮✳✳

❙♦♠❡ ❘❡s✉❧ts ❙❛② t✇♦ ❣❛♠❡s γ1 ❛♥❞ γ2 ❛r❡ ❡q✉✐✈❛❧❡♥t ♣r♦✈✐❞❡❞ Eγ1 = Eγ2 ✐✛ γ1p ↔ γ2p ✐s ✈❛❧✐❞ ❢♦r ❛ p ✇❤✐❝❤ ♦❝❝✉rs ♥❡✐t❤❡r ✐♥ γ1 ♥♦r ✐♥ γ2✳ ❚❤❡♦r❡♠ ❬✶✱✷❪❪ ❙♦✉♥❞ ❛♥❞ ❝♦♠♣❧❡t❡ ❛①✐♦♠❛t✐③❛t✐♦♥s ♦❢ ✭✐t❡r❛t✐♦♥ ❢r❡❡✮ ❣❛♠❡ ❧♦❣✐❝ ❚❤❡♦r❡♠ ❬✸❪ ◆♦ ✜♥✐t❡ ❧❡✈❡❧ ♦❢ t❤❡ ♠♦❞❛❧ µ✲❝❛❧❝✉❧✉s ❤✐❡r❛r❝❤② ❝❛♣t✉r❡s t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ♦❢ ❣❛♠❡ ❧♦❣✐❝✳

❬✶❪ ❨✳ ❱❡♥❡♠❛✳ ❘❡♣r❡s❡♥t✐♥❣ ●❛♠❡ ❆❧❣❡❜r❛s✳ ❙t✉❞✐❛ ▲♦❣✐❝❛ ✼✺ ✭✷✵✵✸✮✳✳ ❬✷❪ ❱✳ ●♦r❛♥❦♦✳ ❚❤❡ ❇❛s✐❝ ❆❧❣❡❜r❛ ♦❢ ●❛♠❡ ❊q✉✐✈❛❧❡♥❝❡s✳ ❙t✉❞✐❛ ▲♦❣✐❝❛ ✼✺ ✭✷✵✵✸✮✳✳ ❬✸❪ ❉✳ ❇❡r✇❛♥❣❡r✳ ●❛♠❡ ▲♦❣✐❝ ✐s ❙tr♦♥❣ ❊♥♦✉❣❤ ❢♦r P❛r✐t② ●❛♠❡s✳ ❙t✉❞✐❛ ▲♦❣✐❝❛ ✼✺ ✭✷✵✵✸✮✳✳

▼♦r❡ ■♥❢♦r♠❛t✐♦♥

❊❞✐t♦rs✿ ▼✳ P❛✉❧② ❛♥❞ ❘✳ P❛r✐❦❤✳ ❙♣❡❝✐❛❧ ■ss✉❡ ♦♥ ●❛♠❡ ▲♦❣✐❝✳ ❙t✉❞✐❛ ▲♦❣✐❝❛ ✼✺✱ ✷✵✵✸✳ ▼✳ P❛✉❧② ❛♥❞ ❘✳ P❛r✐❦❤✳ ●❛♠❡ ▲♦❣✐❝ ✖ ❆♥ ❖✈❡r✈✐❡✇✳ ❙t✉❞✐❛ ▲♦❣✐❝❛ ✼✺✱ ✷✵✵✸✳ ❘✳ P❛r✐❦❤✳ ❚❤❡ ▲♦❣✐❝ ♦❢ ●❛♠❡s ❛♥❞ ✐ts ❆♣♣❧✐❝❛t✐♦♥s✳✳ ❆♥♥❛❧s ♦❢ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✳ ✭✶✾✽✺✮ ✳ ▼✳ P❛✉❧②✳

• ❛♠❡

▲♦❣✐❝ ❢♦r

• ❛♠❡

❚❤❡♦r✐sts✳ ❆✈❛✐❧❛❜❧❡ ❛t ❤tt♣✿✴✴✇✇✇✳st❛♥❢♦r❞✳❡❞✉✴ ♣✐❛♥♦♠❛♥✴✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ❆❧❣♦r✐t❤♠✿

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ❆❧❣♦r✐t❤♠✿

• ❚❤❡ ✜rst ♣❡rs♦♥ ❝✉ts ♦✉t ❛ ♣✐❡❝❡ ✇❤✐❝❤ ❤❡ ❝❧❛✐♠s ✐s ❤✐s ❢❛✐r

s❤❛r❡✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ❆❧❣♦r✐t❤♠✿

• ❚❤❡ ✜rst ♣❡rs♦♥ ❝✉ts ♦✉t ❛ ♣✐❡❝❡ ✇❤✐❝❤ ❤❡ ❝❧❛✐♠s ✐s ❤✐s ❢❛✐r

s❤❛r❡✳

• ❚❤❡ ♣✐❡❝❡ ❣♦❡s ❛r♦✉♥❞ ❜❡✐♥❣ ✐♥s♣❡❝t❡❞ ❜② ❡❛❝❤ ❛❣❡♥t✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ❆❧❣♦r✐t❤♠✿

• ❚❤❡ ✜rst ♣❡rs♦♥ ❝✉ts ♦✉t ❛ ♣✐❡❝❡ ✇❤✐❝❤ ❤❡ ❝❧❛✐♠s ✐s ❤✐s ❢❛✐r

s❤❛r❡✳

• ❚❤❡ ♣✐❡❝❡ ❣♦❡s ❛r♦✉♥❞ ❜❡✐♥❣ ✐♥s♣❡❝t❡❞ ❜② ❡❛❝❤ ❛❣❡♥t✳
• ❊❛❝❤ ❛❣❡♥t✱ ✐♥ t✉r♥✱ ❝❛♥ ❡✐t❤❡r r❡❞✉❝❡ t❤❡ ♣✐❡❝❡✱ ♣✉tt✐♥❣ s♦♠❡

❜❛❝❦ t♦ t❤❡ ♠❛✐♥ ♣❛rt✱ ♦r ❥✉st ♣❛ss ✐t✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ❆❧❣♦r✐t❤♠✿

• ❚❤❡ ✜rst ♣❡rs♦♥ ❝✉ts ♦✉t ❛ ♣✐❡❝❡ ✇❤✐❝❤ ❤❡ ❝❧❛✐♠s ✐s ❤✐s ❢❛✐r

s❤❛r❡✳

• ❚❤❡ ♣✐❡❝❡ ❣♦❡s ❛r♦✉♥❞ ❜❡✐♥❣ ✐♥s♣❡❝t❡❞ ❜② ❡❛❝❤ ❛❣❡♥t✳
• ❊❛❝❤ ❛❣❡♥t✱ ✐♥ t✉r♥✱ ❝❛♥ ❡✐t❤❡r r❡❞✉❝❡ t❤❡ ♣✐❡❝❡✱ ♣✉tt✐♥❣ s♦♠❡

❜❛❝❦ t♦ t❤❡ ♠❛✐♥ ♣❛rt✱ ♦r ❥✉st ♣❛ss ✐t✳

• ❆❢t❡r t❤❡ ♣✐❡❝❡ ❤❛s ❜❡❡♥ ✐♥s♣❡❝t❡❞ ❜② pn✱ t❤❡ ❧❛st ♣❡rs♦♥ ✇❤♦

r❡❞✉❝❡❞ t❤❡ ♣✐❡❝❡✱ t❛❦❡s ✐t✳ ■❢ t❤❡r❡ ✐s ♥♦ s✉❝❤ ♣❡rs♦♥✱ t❤❡♥ t❤❡ ♣✐❡❝❡ ✐s t❛❦❡♥ ❜② p1✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ❆❧❣♦r✐t❤♠✿

• ❚❤❡ ✜rst ♣❡rs♦♥ ❝✉ts ♦✉t ❛ ♣✐❡❝❡ ✇❤✐❝❤ ❤❡ ❝❧❛✐♠s ✐s ❤✐s ❢❛✐r

s❤❛r❡✳

• ❚❤❡ ♣✐❡❝❡ ❣♦❡s ❛r♦✉♥❞ ❜❡✐♥❣ ✐♥s♣❡❝t❡❞ ❜② ❡❛❝❤ ❛❣❡♥t✳
• ❊❛❝❤ ❛❣❡♥t✱ ✐♥ t✉r♥✱ ❝❛♥ ❡✐t❤❡r r❡❞✉❝❡ t❤❡ ♣✐❡❝❡✱ ♣✉tt✐♥❣ s♦♠❡

❜❛❝❦ t♦ t❤❡ ♠❛✐♥ ♣❛rt✱ ♦r ❥✉st ♣❛ss ✐t✳

• ❆❢t❡r t❤❡ ♣✐❡❝❡ ❤❛s ❜❡❡♥ ✐♥s♣❡❝t❡❞ ❜② pn✱ t❤❡ ❧❛st ♣❡rs♦♥ ✇❤♦

r❡❞✉❝❡❞ t❤❡ ♣✐❡❝❡✱ t❛❦❡s ✐t✳ ■❢ t❤❡r❡ ✐s ♥♦ s✉❝❤ ♣❡rs♦♥✱ t❤❡♥ t❤❡ ♣✐❡❝❡ ✐s t❛❦❡♥ ❜② p1✳

• ❚❤❡ ❛❧❣♦r✐t❤♠ ❝♦♥t✐♥✉❡s ✇✐t❤ n − 1 ♣❛rt✐❝✐♣❛♥ts✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❈♦rr❡❝t♥❡ss✿ ❚❤❡ ❛❧❣♦r✐t❤♠ ✐s ✏❝♦rr❡❝t✑ ✐✛ ❡❛❝❤ ♣❧❛②❡r ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② ❢♦r ❛❝❤✐❡✈✐♥❣ ❛ ❢❛✐r ♦✉t❝♦♠❡ ✭1/n ♦❢ t❤❡ ♣✐❡ ❛❝❝♦r❞✐♥❣ t♦ pi✬s ♦✇♥ ✈❛❧✉❛t✐♦♥✮✳ ❚♦✇❛r❞s ❛ ❋♦r♠❛❧ Pr♦♦❢✿ ❆ st❛t❡ ✇✐❧❧ ❝♦♥s✐st ♦❢ t❤❡ ✈❛❧✉❡s ♦❢ n + 2 ✈❛r✐❛❜❧❡s✳

• ❚❤❡ ✈❛r✐❛❜❧❡ m ❤❛s ❛s ✐ts ✈❛❧✉❡ t❤❡ ♠❛✐♥ ♣❛rt ♦❢ t❤❡ ❝❛❦❡✳
• ❚❤❡ ✈❛r✐❛❜❧❡ x ✐s t❤❡ ♣✐❡❝❡ ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥✳
• ❋♦r i = 1, . . . , n✱ t❤❡ ✈❛r✐❛❜❧❡ xi ❤❛s ❛s ✐ts ✈❛❧✉❡ t❤❡ ♣✐❡❝❡✱ ✐❢

❛♥②✱ ❛ss✐❣♥❡❞ t♦ t❤❡ ♣❡rs♦♥ pi✳ ❱❛r✐❛❜❧❡s m, x, x1, . . . , xn r❛♥❣❡ ♦✈❡r s✉❜s❡ts ♦❢ t❤❡ ❝❛❦❡✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ❛❧❣♦r✐t❤♠ ✉s❡s t❤r❡❡ ❜❛s✐❝ ❛❝t✐♦♥s✳

• c ❝✉ts ❛ ♣✐❡❝❡ ❢r♦♠ m ❛♥❞ ❛ss✐❣♥s ✐t t♦ x✳ c ✇♦r❦s ♦♥❧② ✐❢ x ✐s ✵✳
• r ✭r❡❞✉❝❡✮ tr❛♥s❢❡rs s♦♠❡ ✭♥♦♥✲③❡r♦✮ ♣♦rt✐♦♥ ❢r♦♠ x ❜❛❝❦ t♦ m✳
• ai ✭❛ss✐❣♥✮ ❛ss✐❣♥s t❤❡ ♣✐❡❝❡ x t♦ ♣❡rs♦♥ pi✳ ❚❤✉s ai ✐s s✐♠♣❧②✱

(xi, x) := (x, 0)✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ❛❧❣♦r✐t❤♠ ✉s❡s t❤r❡❡ ❜❛s✐❝ ❛❝t✐♦♥s✳

• c ❝✉ts ❛ ♣✐❡❝❡ ❢r♦♠ m ❛♥❞ ❛ss✐❣♥s ✐t t♦ x✳ c ✇♦r❦s ♦♥❧② ✐❢ x ✐s ✵✳
• r ✭r❡❞✉❝❡✮ tr❛♥s❢❡rs s♦♠❡ ✭♥♦♥✲③❡r♦✮ ♣♦rt✐♦♥ ❢r♦♠ x ❜❛❝❦ t♦ m✳
• ai ✭❛ss✐❣♥✮ ❛ss✐❣♥s t❤❡ ♣✐❡❝❡ x t♦ ♣❡rs♦♥ pi✳ ❚❤✉s ai ✐s s✐♠♣❧②✱

(xi, x) := (x, 0)✳ ❆♥❞ ♣r❡❞✐❝❛t❡s✿

• F(u, k)✿ t❤❡ ♣✐❡❝❡ u ✐s ❜✐❣ ❡♥♦✉❣❤ ❢♦r k ♣❡♦♣❧❡✳
• F(u) ❛❜❜r❡✈✐❛t❡s F(u, 1) ❛♥❞ Fi ❛❜❜r❡✈✐❛t❡s F(xi)✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❆ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❆ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s ✶✳ F(m, k) → c(F(m, k − 1) ∧ F(x))

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❆ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s ✶✳ F(m, k) → c(F(m, k − 1) ∧ F(x)) 1′. F(m, k) → (c, i)(F(m, k − 1) ∧ F(x))

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❆ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s ✶✳ F(m, k) → c(F(m, k − 1) ∧ F(x)) 1′. F(m, k) → (c, i)(F(m, k − 1) ∧ F(x)) ✷✳ F(m, k) → [r∗]F(m, k)

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❆ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s ✶✳ F(m, k) → c(F(m, k − 1) ∧ F(x)) 1′. F(m, k) → (c, i)(F(m, k − 1) ∧ F(x)) ✷✳ F(m, k) → [r∗]F(m, k) ✸✳ F(m, k) → [c][r∗](F(m, k − 1) ∨ r(F(m, k − 1) ∧ F(x)))

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❆ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s ✶✳ F(m, k) → c(F(m, k − 1) ∧ F(x)) 1′. F(m, k) → (c, i)(F(m, k − 1) ∧ F(x)) ✷✳ F(m, k) → [r∗]F(m, k) ✸✳ F(m, k) → [c][r∗](F(m, k − 1) ∨ r(F(m, k − 1) ∧ F(x))) ✹✳ F(x) → [ai]Fi

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❆ss✉♠❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥s ✶✳ F(m, k) → c(F(m, k − 1) ∧ F(x)) 1′. F(m, k) → (c, i)(F(m, k − 1) ∧ F(x)) ✷✳ F(m, k) → [r∗]F(m, k) ✸✳ F(m, k) → [c][r∗](F(m, k − 1) ∨ r(F(m, k − 1) ∧ F(x))) ✹✳ F(x) → [ai]Fi ❚❤❡r❡ ❛r❡ t❛❝✐t ❛ss✉♠♣t✐♦♥s ♦❢ r❡❧❡✈❛♥❝❡✱ ❡✳❣✳ t❤❛t r ❛♥❞ c ❝❛♥ ♦♥❧② ❛✛❡❝t st❛t❡♠❡♥ts ✐♥ ✇❤✐❝❤ m ♦r x ♦❝❝✉rs✳ ❲❡ ❛ss✉♠❡ ♠♦r❡♦✈❡r t❤❛t F(m, n) ✐s tr✉❡ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ✭✐♥✮❢♦r♠❛❧ ♣r♦♦❢✿ ✶✳ ❲❡ s❤♦✇ ♥♦✇ t❤❛t ❡❛❝❤ ♣❡rs♦♥ pi ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② s♦ t❤❛t ✐❢✱ ❛❢t❡r t❤❡ kt❤ ❝②❝❧❡✱ ✭s✮❤❡ ✐s st✐❧❧ ✐♥ t❤❡ ❣❛♠❡ t❤❡♥ F(m, n − k) ❛♥❞ ✐❢ ✭s✮❤❡ ✐s ❛ss✐❣♥❡❞ ❛ ♣✐❡❝❡✱ t❤❡♥ Fi ✐s tr✉❡✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ✭✐♥✮❢♦r♠❛❧ ♣r♦♦❢✿ ✶✳ ❲❡ s❤♦✇ ♥♦✇ t❤❛t ❡❛❝❤ ♣❡rs♦♥ pi ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② s♦ t❤❛t ✐❢✱ ❛❢t❡r t❤❡ kt❤ ❝②❝❧❡✱ ✭s✮❤❡ ✐s st✐❧❧ ✐♥ t❤❡ ❣❛♠❡ t❤❡♥ F(m, n − k) ❛♥❞ ✐❢ ✭s✮❤❡ ✐s ❛ss✐❣♥❡❞ ❛ ♣✐❡❝❡✱ t❤❡♥ Fi ✐s tr✉❡✳ ✷✳ ❚❤✐s ✐s tr✉❡ ❛t st❛rt s✐♥❝❡ k = 0✱ F(m, n) ❤♦❧❞s ❛♥❞ ♥♦ ♦♥❡ ②❡t ❤❛s ❛ ♣✐❡❝❡✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ✭✐♥✮❢♦r♠❛❧ ♣r♦♦❢✿ ✶✳ ❲❡ s❤♦✇ ♥♦✇ t❤❛t ❡❛❝❤ ♣❡rs♦♥ pi ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② s♦ t❤❛t ✐❢✱ ❛❢t❡r t❤❡ kt❤ ❝②❝❧❡✱ ✭s✮❤❡ ✐s st✐❧❧ ✐♥ t❤❡ ❣❛♠❡ t❤❡♥ F(m, n − k) ❛♥❞ ✐❢ ✭s✮❤❡ ✐s ❛ss✐❣♥❡❞ ❛ ♣✐❡❝❡✱ t❤❡♥ Fi ✐s tr✉❡✳ ✷✳ ❚❤✐s ✐s tr✉❡ ❛t st❛rt s✐♥❝❡ k = 0✱ F(m, n) ❤♦❧❞s ❛♥❞ ♥♦ ♦♥❡ ②❡t ❤❛s ❛ ♣✐❡❝❡✳ ✸✳ ❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ✐♥❞✉❝t✐✈❡ st❡♣ ❢r♦♠ k t♦ k + 1✳ ❲❡ ❛ss✉♠❡ ❜② ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s t❤❛t F(m, n − k) ❤♦❧❞s ❛t t❤✐s st❛❣❡✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ❚❤❡ ✭✐♥✮❢♦r♠❛❧ ♣r♦♦❢✿ ✶✳ ❲❡ s❤♦✇ ♥♦✇ t❤❛t ❡❛❝❤ ♣❡rs♦♥ pi ❤❛s ❛ ✇✐♥♥✐♥❣ str❛t❡❣② s♦ t❤❛t ✐❢✱ ❛❢t❡r t❤❡ kt❤ ❝②❝❧❡✱ ✭s✮❤❡ ✐s st✐❧❧ ✐♥ t❤❡ ❣❛♠❡ t❤❡♥ F(m, n − k) ❛♥❞ ✐❢ ✭s✮❤❡ ✐s ❛ss✐❣♥❡❞ ❛ ♣✐❡❝❡✱ t❤❡♥ Fi ✐s tr✉❡✳ ✷✳ ❚❤✐s ✐s tr✉❡ ❛t st❛rt s✐♥❝❡ k = 0✱ F(m, n) ❤♦❧❞s ❛♥❞ ♥♦ ♦♥❡ ②❡t ❤❛s ❛ ♣✐❡❝❡✳ ✸✳ ❲❡ ♥♦✇ ❝♦♥s✐❞❡r t❤❡ ✐♥❞✉❝t✐✈❡ st❡♣ ❢r♦♠ k t♦ k + 1✳ ❲❡ ❛ss✉♠❡ ❜② ✐♥❞✉❝t✐♦♥ ❤②♣♦t❤❡s✐s t❤❛t F(m, n − k) ❤♦❧❞s ❛t t❤✐s st❛❣❡✳ ✹✳ ■❢ i = 1 t❤❡♥ s✐♥❝❡ p1 ✭♦r ✇❤♦❡✈❡r ❞♦❡s t❤❡ ❝✉tt✐♥❣✮ ❞♦❡s t❤❡ ❝✉tt✐♥❣✱ ❜② ✭✶✮ ❛♥❞ ✭✶✬✮ s❤❡ ❝❛♥ ❛❝❤✐❡✈❡ F(m, n − k − 1) ∧ F(x)✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ✺✳ ■❢ ♥♦ ♦♥❡ ❞♦❡s ❛♥ r✱ s❤❡ ❣❡ts x ❛♥❞ F1 ✇✐❧❧ ❤♦❧❞ s✐♥❝❡ x ❞✐❞ ♥♦t ❝❤❛♥❣❡✳ ■❢ s♦♠❡♦♥❡ ❞♦❡s ❞♦ ❛♥ r✱ t❤❡♥ ❜② ✭✷✮✱ F(m, n − k − 1) ✇✐❧❧ st✐❧❧ ❤♦❧❞ ❛♥❞ t❤✐s ✐s ❖❑ s✐♥❝❡ s❤❡ ✇✐❧❧ t❤❡♥ ❜❡ ♣❛rt✐❝✐♣❛t✐♥❣ ❛t t❤❡ ♥❡①t st❛❣❡✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ✺✳ ■❢ ♥♦ ♦♥❡ ❞♦❡s ❛♥ r✱ s❤❡ ❣❡ts x ❛♥❞ F1 ✇✐❧❧ ❤♦❧❞ s✐♥❝❡ x ❞✐❞ ♥♦t ❝❤❛♥❣❡✳ ■❢ s♦♠❡♦♥❡ ❞♦❡s ❞♦ ❛♥ r✱ t❤❡♥ ❜② ✭✷✮✱ F(m, n − k − 1) ✇✐❧❧ st✐❧❧ ❤♦❧❞ ❛♥❞ t❤✐s ✐s ❖❑ s✐♥❝❡ s❤❡ ✇✐❧❧ t❤❡♥ ❜❡ ♣❛rt✐❝✐♣❛t✐♥❣ ❛t t❤❡ ♥❡①t st❛❣❡✳ ✻✳ ▲❡t ✉s ♥♦✇ ❝♦♥s✐❞❡r ❥✉st ♦♥❡ ♦❢ t❤❡ ♦t❤❡r ♣❡♦♣❧❡✳ ❚❤❡ ❧❛st ♣❡rs♦♥ pi t♦ ❞♦ r ✭✐❢ t❤❡r❡ ✐s s♦♠❡♦♥❡ ✇❤♦ ❞♦❡s r✮ ❝♦✉❧❞ ✭❜② ✭✸✮✮ ❛❝❤✐❡✈❡ F(x) ❛♥❞ t❤❡r❡❢♦r❡ ✇❤❡♥ x ✐s ❛ss✐❣♥❡❞ t♦ ❤✐♠✱ F1 ✇✐❧❧ ❤♦❧❞✳

❊①❛♠♣❧❡✿ ❇❛♥❛❝❤✲❑♥❛st❡r ❈❛❦❡ ❈✉tt✐♥❣ ❆❧❣♦r✐t❤♠ ✺✳ ■❢ ♥♦ ♦♥❡ ❞♦❡s ❛♥ r✱ s❤❡ ❣❡ts x ❛♥❞ F1 ✇✐❧❧ ❤♦❧❞ s✐♥❝❡ x ❞✐❞ ♥♦t ❝❤❛♥❣❡✳ ■❢ s♦♠❡♦♥❡ ❞♦❡s ❞♦ ❛♥ r✱ t❤❡♥ ❜② ✭✷✮✱ F(m, n − k − 1) ✇✐❧❧ st✐❧❧ ❤♦❧❞ ❛♥❞ t❤✐s ✐s ❖❑ s✐♥❝❡ s❤❡ ✇✐❧❧ t❤❡♥ ❜❡ ♣❛rt✐❝✐♣❛t✐♥❣ ❛t t❤❡ ♥❡①t st❛❣❡✳ ✻✳ ▲❡t ✉s ♥♦✇ ❝♦♥s✐❞❡r ❥✉st ♦♥❡ ♦❢ t❤❡ ♦t❤❡r ♣❡♦♣❧❡✳ ❚❤❡ ❧❛st ♣❡rs♦♥ pi t♦ ❞♦ r ✭✐❢ t❤❡r❡ ✐s s♦♠❡♦♥❡ ✇❤♦ ❞♦❡s r✮ ❝♦✉❧❞ ✭❜② ✭✸✮✮ ❛❝❤✐❡✈❡ F(x) ❛♥❞ t❤❡r❡❢♦r❡ ✇❤❡♥ x ✐s ❛ss✐❣♥❡❞ t♦ ❤✐♠✱ F1 ✇✐❧❧ ❤♦❧❞✳ ✼✳ ❆❧❧ t❤❡ ♦t❤❡r ❝❛s❡s ❛r❡ q✉✐t❡ ❛♥❛❧♦❣♦✉s✱ ❛♥❞ t❤❡ ✐♥❞✉❝t✐♦♥ st❡♣ ❣♦❡s t❤r♦✉❣❤✳ ❇② t❛❦✐♥❣ k = n ✇❡ s❡❡ t❤❛t ❡✈❡r② pi ❤❛s t❤❡ ❛❜✐❧✐t② t♦ ❛❝❤✐❡✈❡ Fi✳

❚❤❛♥❦ ②♦✉✳