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❈♦♠♣❧❡t❡ ❛①✐♦♠❛t✐③❛t✐♦♥s ♦❢ ❧❡①✐❝♦❣r❛♣❤✐❝ s✉♠s ❛♥❞ ♣r♦❞✉❝ts ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s

P❤✐❧✐♣♣❡ ❇❛❧❜✐❛♥✐✶ ■❧②❛ ❙❤❛♣✐r♦✈s❦②✷

✶ ■♥st✐t✉t ❞❡ ❘❡❝❤❡r❝❤❡ ❡♥ ■♥❢♦r♠❛t✐q✉❡ ❞❡ ❚♦✉❧♦✉s❡✱

❈◆❘❙ ✖ ❚♦✉❧♦✉s❡ ❯♥✐✈❡rs✐t②

✷ ■♥st✐t✉t❡ ❢♦r ■♥❢♦r♠❛t✐♦♥ ❚r❛♥s♠✐ss✐♦♥ Pr♦❜❧❡♠s✱

❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s

❚♦♣♦❧♦❣②✱ ❆❧❣❡❜r❛✱ ❛♥❞ ❈❛t❡❣♦r✐❡s ✐♥ ▲♦❣✐❝ ✭❚❆❈▲ ✷✵✶✺✮ ■s❝❤✐❛ ✭■t❛❧②✮✱ ❏✉♥❡ ✷✷✱ ✷✵✶✺

❲❡ ❝♦♥s✐❞❡r t✇♦ ♥❛t✉r❛❧ ♦♣❡r❛t✐♦♥s ♦♥ ♠♦❞❛❧ ❧♦❣✐❝s ✖ ❧❡①✐❝♦❣r❛♣❤✐❝ ✭♦r ♦r❞❡r❡❞✮ s✉♠s ❛♥❞ ♣r♦❞✉❝ts✳ ❋♦r s✉❝❤ s②st❡♠s ✇❡ ♣r❡s❡♥t ❣❡♥❡r❛❧ ❝♦♠♣❧❡t❡♥❡ss r❡s✉❧ts✳ ▲✐❦❡ ✏✉s✉❛❧✑ ♣r♦❞✉❝t ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ s✉♠ ❛♥❞ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s ❛r❡ ❞❡✜♥❡❞ s❡♠❛♥t✐❝❛❧❧② ✖ ✈✐❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♣❡r❛t✐♦♥ ♦♥ t❤❡✐r ❢r❛♠❡s✳

❲❡ ❝♦♥s✐❞❡r t✇♦ ♥❛t✉r❛❧ ♦♣❡r❛t✐♦♥s ♦♥ ♠♦❞❛❧ ❧♦❣✐❝s ✖ ❧❡①✐❝♦❣r❛♣❤✐❝ ✭♦r ♦r❞❡r❡❞✮ s✉♠s ❛♥❞ ♣r♦❞✉❝ts✳ ❋♦r s✉❝❤ s②st❡♠s ✇❡ ♣r❡s❡♥t ❣❡♥❡r❛❧ ❝♦♠♣❧❡t❡♥❡ss r❡s✉❧ts✳ ▲✐❦❡ ✏✉s✉❛❧✑ ♣r♦❞✉❝t ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s✱ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ s✉♠ ❛♥❞ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s ❛r❡ ❞❡✜♥❡❞ s❡♠❛♥t✐❝❛❧❧② ✖ ✈✐❛ ❝♦rr❡s♣♦♥❞✐♥❣ ♦♣❡r❛t✐♦♥ ♦♥ t❤❡✐r ❢r❛♠❡s✳

❙✉♠ ♦❢ ❢r❛♠❡s

❉❡✜♥✐t✐♦♥ ▲❡t ■ = (I, S) ❜❡ ❛ ❢r❛♠❡✱ {❋i = (Wi, Ri) | i ∈ I} ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❢r❛♠❡s✳ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ✭♦r ♦r❞❡r❡❞✮ s✉♠

■

❋i ✐s t❤❡ ❢r❛♠❡ (W , R+, S+), ✇❤❡r❡ W ✐s t❤❡ ❞✐s❥♦✐♥ s✉♠

• I

Wi = {(w, i) | i ∈ I, w ∈ Wi}, ❛♥❞ (w, i)R+(u, j) ⇐ ⇒ i = j & wRiu, (w, i)S+(u, j) ⇐ ⇒ iSj. ■ ✐s ✏✈❡rt✐❝❛❧✑✱ ❋i ❛r❡ ✏❤♦r✐③♦♥t❛❧✑✳

❙✉♠ ♦❢ ❧♦❣✐❝s

❉❡✜♥✐t✐♦♥

• L✷

L✶ ✐s t❤❡ ❧♦❣✐❝ ♦❢ s✉♠s ✇❤❡r❡ ✏❤♦r✐③♦♥t❛❧✑ ❢r❛♠❡s ❛r❡ L✶✲❢r❛♠❡s✱ ❛♥❞ t❤❡ ✏✈❡rt✐❝❛❧✑ ❢r❛♠❡ ✐s ❛♥ L✷✲❢r❛♠❡✿

• L✷

L✶ = ▲♦❣({

• ■

❋i | ■ | = L✷, {❋i | i ✐♥ ■} | = L✶}). Pr♦❜❧❡♠ ❚♦ ❝♦♥str✉❝t t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢

L✷

L✶✱ ❦♥♦✇✐♥❣ t❤❡ ❧♦❣✐❝s L✶, L✷✳

❙♦♠❡ ❤✐st♦r②

■♥ ✷✵✵✼✱ ▲❡✈ ❇❡❦❧❡♠✐s❤❡✈ ❝♦♥str✉❝t❡❞ t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢

• GL

GL ❬▲✳ ❇❡❦❧❡♠✐s❤❡✈✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❢♦r ♣r♦✈❛❜✐❧✐t② ❧♦❣✐❝ ✳ ✷✵✶✵❪ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t②✱ t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ ❛ ❣♦♦❞ ♦♣❡r❛t✐♦♥✦ ■♥ ♠❛♥② ❝❛s❡s t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ ♣r❡s❡r✈❡s ❝♦♠♣❧❡①✐t② ♦❢ ❧♦❣✐❝s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛❧❧ t❤❡ ❛❜♦✈❡ ❧♦❣✐❝s ❛r❡ ✐♥ ✭❬❙❤✱ ✷✵✵✽❪✮❀ ✐t ❢♦❧❧♦✇s t❤❛t ✐s ✐♥ ✳ ❙✐♠✉❧t❛♥❡♦✉s❧②✱ ❙❡r❣❡② ❇❛❜❡♥②s❤❡✈ ❛♥❞ ❱❧❛❞✐♠✐r ❘②❜❛❦♦✈ ❞❡✈❡❧♦♣❡❞ ✜❧tr❛t✐♦♥s ❢♦r s✉♠s✱ ❛♥❞ ♣r♦✈❡❞ ❛ ♥✉♠❜❡r ♦❢ ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts✳ ❬❇❛❜❡♥②s❤❡✈✱ ❘②❜❛❦♦✈✳ ▲♦❣✐❝s ♦❢ ❑r✐♣❦❡ ♠❡t❛✲♠♦❞❡❧s✳ ✷✵✶✵❪

❙♦♠❡ ❤✐st♦r②

■♥ ✷✵✵✼✱ ▲❡✈ ❇❡❦❧❡♠✐s❤❡✈ ❝♦♥str✉❝t❡❞ t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢

• GL

GL,

• GL
• GL

GL

• ,
• GL
• GL
• GL

GL

• ,

. . . ❬▲✳ ❇❡❦❧❡♠✐s❤❡✈✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❢♦r ♣r♦✈❛❜✐❧✐t② ❧♦❣✐❝ GLP✳ ✷✵✶✵❪ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t②✱ t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ ❛ ❣♦♦❞ ♦♣❡r❛t✐♦♥✦ ■♥ ♠❛♥② ❝❛s❡s t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ ♣r❡s❡r✈❡s ❝♦♠♣❧❡①✐t② ♦❢ ❧♦❣✐❝s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛❧❧ t❤❡ ❛❜♦✈❡ ❧♦❣✐❝s ❛r❡ ✐♥ ✭❬❙❤✱ ✷✵✵✽❪✮❀ ✐t ❢♦❧❧♦✇s t❤❛t ✐s ✐♥ ✳ ❙✐♠✉❧t❛♥❡♦✉s❧②✱ ❙❡r❣❡② ❇❛❜❡♥②s❤❡✈ ❛♥❞ ❱❧❛❞✐♠✐r ❘②❜❛❦♦✈ ❞❡✈❡❧♦♣❡❞ ✜❧tr❛t✐♦♥s ❢♦r s✉♠s✱ ❛♥❞ ♣r♦✈❡❞ ❛ ♥✉♠❜❡r ♦❢ ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts✳ ❬❇❛❜❡♥②s❤❡✈✱ ❘②❜❛❦♦✈✳ ▲♦❣✐❝s ♦❢ ❑r✐♣❦❡ ♠❡t❛✲♠♦❞❡❧s✳ ✷✵✶✵❪

❙♦♠❡ ❤✐st♦r②

■♥ ✷✵✵✼✱ ▲❡✈ ❇❡❦❧❡♠✐s❤❡✈ ❝♦♥str✉❝t❡❞ t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢

• GL

GL,

• GL
• GL

GL

• ,
• GL
• GL
• GL

GL

• ,

. . . ❬▲✳ ❇❡❦❧❡♠✐s❤❡✈✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❢♦r ♣r♦✈❛❜✐❧✐t② ❧♦❣✐❝ GLP✳ ✷✵✶✵❪ ■♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t②✱ t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ t✉r♥s ♦✉t t♦ ❜❡ ❛ ❣♦♦❞ ♦♣❡r❛t✐♦♥✦ ■♥ ♠❛♥② ❝❛s❡s t❤❡ s✉♠ ♦♣❡r❛t✐♦♥ ♣r❡s❡r✈❡s ❝♦♠♣❧❡①✐t② ♦❢ ❧♦❣✐❝s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❛❧❧ t❤❡ ❛❜♦✈❡ ❧♦❣✐❝s ❛r❡ ✐♥ PSPACE ✭❬❙❤✱ ✷✵✵✽❪✮❀ ✐t ❢♦❧❧♦✇s t❤❛t GLP ✐s ✐♥ PSPACE✳ ❙✐♠✉❧t❛♥❡♦✉s❧②✱ ❙❡r❣❡② ❇❛❜❡♥②s❤❡✈ ❛♥❞ ❱❧❛❞✐♠✐r ❘②❜❛❦♦✈ ❞❡✈❡❧♦♣❡❞ ✜❧tr❛t✐♦♥s ❢♦r s✉♠s✱ ❛♥❞ ♣r♦✈❡❞ ❛ ♥✉♠❜❡r ♦❢ ❞❡❝✐❞❛❜✐❧✐t② r❡s✉❧ts✳ ❬❇❛❜❡♥②s❤❡✈✱ ❘②❜❛❦♦✈✳ ▲♦❣✐❝s ♦❢ ❑r✐♣❦❡ ♠❡t❛✲♠♦❞❡❧s✳ ✷✵✶✵❪

α = ✷p → ✶✷p, β = ✷p → ✷✶p, γ = ♦✷p → ✶♦✷p

• GL

GL = GL ∗ GL + {α, β, γ} ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛s ❄ ❛r❡ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛s✳ ❋♦r ❋

✶ ✷ ✱ ✇❡ ❤❛✈❡✿

❋

✶ ✷ ✷

❋

✷ ✶ ✷

❋

✶ ✶ ✷ ✷✳

▲❡♠♠❛ ✭✷✵✶✹✮ ❈♦♥s✐❞❡r ❛ r♦♦t❡❞ ❢r❛♠❡ ❋

✶ ✷ ✳

❋ ✐✛ ❋ ✐s ❛ ✲♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉♠✳ ❈♦r♦❧❧❛r②

α = ✷p → ✶✷p, β = ✷p → ✷✶p, γ = ♦✷p → ✶♦✷p

• GL

GL = GL ∗ GL + {α, β, γ} ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛s α, β, γ❄ ❛r❡ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛s✳ ❋♦r ❋

✶ ✷ ✱ ✇❡ ❤❛✈❡✿

❋

✶ ✷ ✷

❋

✷ ✶ ✷

❋

✶ ✶ ✷ ✷✳

▲❡♠♠❛ ✭✷✵✶✹✮ ❈♦♥s✐❞❡r ❛ r♦♦t❡❞ ❢r❛♠❡ ❋

✶ ✷ ✳

❋ ✐✛ ❋ ✐s ❛ ✲♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉♠✳ ❈♦r♦❧❧❛r②

α = ✷p → ✶✷p, β = ✷p → ✷✶p, γ = ♦✷p → ✶♦✷p

• GL

GL = GL ∗ GL + {α, β, γ} ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛s α, β, γ❄ α, β, γ ❛r❡ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛s✳ ❋♦r ❋ = (W , R✶, R✷)✱ ✇❡ ❤❛✈❡✿ ❋ | = α ⇐ ⇒ R✶ ◦ R✷ ⊆ R✷; ❋ | = β ⇐ ⇒ R✷ ◦ R✶ ⊆ R✷; ❋ | = γ ⇐ ⇒ R−✶

✶

• R✷ ⊆ R✷✳

▲❡♠♠❛ ✭✷✵✶✹✮ ❈♦♥s✐❞❡r ❛ r♦♦t❡❞ ❢r❛♠❡ ❋

✶ ✷ ✳

❋ ✐✛ ❋ ✐s ❛ ✲♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉♠✳ ❈♦r♦❧❧❛r②

α = ✷p → ✶✷p, β = ✷p → ✷✶p, γ = ♦✷p → ✶♦✷p

• GL

GL = GL ∗ GL + {α, β, γ} ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛s α, β, γ❄ α, β, γ ❛r❡ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛s✳ ❋♦r ❋ = (W , R✶, R✷)✱ ✇❡ ❤❛✈❡✿ ❋ | = α ⇐ ⇒ R✶ ◦ R✷ ⊆ R✷; ❋ | = β ⇐ ⇒ R✷ ◦ R✶ ⊆ R✷; ❋ | = γ ⇐ ⇒ R−✶

✶

• R✷ ⊆ R✷✳

▲❡♠♠❛ ✭✷✵✶✹✮ ❈♦♥s✐❞❡r ❛ r♦♦t❡❞ ❢r❛♠❡ ❋ = (W , R✶, R✷)✳ ❋ | = α ∧ β ∧ γ ✐✛ ❋ ✐s ❛ p✲♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉♠✳ ❈♦r♦❧❧❛r②

α = ✷p → ✶✷p, β = ✷p → ✷✶p, γ = ♦✷p → ✶♦✷p

• GL

GL = GL ∗ GL + {α, β, γ} ❲❤❛t ✐s t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❢♦r♠✉❧❛s α, β, γ❄ α, β, γ ❛r❡ ❙❛❤❧q✈✐st ❢♦r♠✉❧❛s✳ ❋♦r ❋ = (W , R✶, R✷)✱ ✇❡ ❤❛✈❡✿ ❋ | = α ⇐ ⇒ R✶ ◦ R✷ ⊆ R✷; ❋ | = β ⇐ ⇒ R✷ ◦ R✶ ⊆ R✷; ❋ | = γ ⇐ ⇒ R−✶

✶

• R✷ ⊆ R✷✳

▲❡♠♠❛ ✭✷✵✶✹✮ ❈♦♥s✐❞❡r ❛ r♦♦t❡❞ ❢r❛♠❡ ❋ = (W , R✶, R✷)✳ ❋ | = α ∧ β ∧ γ ✐✛ ❋ ✐s ❛ p✲♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉♠✳ ❈♦r♦❧❧❛r②

• K

K = K ∗ K + {α, β, γ}

❇② ❛ ❝❧♦s❡❞ s❡♥t❡♥❝❡ ✇❡ ♠❡❛♥ t❤❡ st❛♥❞❛r❞ tr❛♥s❧❛t✐♦♥ ♦❢ ❛ ❝❧♦s❡❞ ♠♦❞❛❧ ❢♦r♠✉❧❛✳ ❍♦r♥ s❡♥t❡♥❝❡s✿ ∀x✶ . . . xn(ψ✶ ∧ . . . ∧ ψk → ψ✵), ✇❤❡r❡ ψi ❛r❡ ❛t♦♠s✳ ❆ ❧♦❣✐❝ L ✐s ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡✱ ✐❢ ❋r❛♠❡s(L) ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❝❧❛ss t❤❛t ✐s ❞❡✜♥❡❞ ❜② ❍♦r♥ s❡♥t❡♥❝❡s ❛♥❞ ❝❧♦s❡❞ s❡♥t❡♥❝❡s✳ ❚❤❡ st❛♥❞❛r❞ s②st❡♠s ❑, ❚, ❇, ❑✹, ❙✹, ❙✺, . . . ❛r❡ ❡①❛♠♣❧❡s ♦❢ ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡ ❧♦❣✐❝s✳ ❚❤❡♦r❡♠ ✶ ▲❡t L✶ ∗ L✷ + {α, β, γ} ❜❡ ❑r✐♣❦❡ ❝♦♠♣❧❡t❡✱ L✷ ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡✳ ❚❤❡♥

L✷

L✶ = L✶ ∗ L✷ + {α, β, γ}. ❈♦r♦❧❧❛r② ▲❡t L✶ ❛♥❞ L✷ ❜❡ ❝❛♥♦♥✐❝❛❧ ✉♥✐♠♦❞❛❧ ❧♦❣✐❝s✱ L✷ ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡✳ ❚❤❡♥

L✷

L✶ = L✶ ∗ L✷ + {α, β, γ}.

▲❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts ♦❢ ❢r❛♠❡s

❉❡✜♥✐t✐♦♥ ▲❡t ■ = (I, S) ❜❡ ❛ ❢r❛♠❡✱ {❋i = (Wi, Ri) | i ∈ I} ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❢r❛♠❡s✳ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ✭♦r ♦r❞❡r❡❞✮ s✉♠

■

❋i ✐s t❤❡ ❢r❛♠❡ (W , R+, S+), ✇❤❡r❡ W ✐s t❤❡ ❞✐s❥♦✐♥ s✉♠

• I

Wi = {(w, i) | i ∈ I, w ∈ Wi}, ❛♥❞ (w, i)R+(u, j) ⇐ ⇒ i = j & wRiu, (w, i)S+(u, j) ⇐ ⇒ iSj. ■❢ ❢♦r ❛❧❧ ❋ ❋✱ ✇❡ ✇r✐t❡ ❋ ■ ❢♦r

■

❋ ❀ t❤❡ ❢r❛♠❡ ❋ ■ ✐s ❝❛❧❧❡❞ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❢r❛♠❡s ❋ ❛♥❞ ■✳

▲❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts ♦❢ ❢r❛♠❡s

❉❡✜♥✐t✐♦♥ ▲❡t ■ = (I, S) ❜❡ ❛ ❢r❛♠❡✱ {❋i = (Wi, Ri) | i ∈ I} ❜❡ ❛ ❢❛♠✐❧② ♦❢ ❢r❛♠❡s✳ ❚❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ✭♦r ♦r❞❡r❡❞✮ s✉♠

■

❋i ✐s t❤❡ ❢r❛♠❡ (W , R+, S+), ✇❤❡r❡ W ✐s t❤❡ ❞✐s❥♦✐♥ s✉♠

• I

Wi = {(w, i) | i ∈ I, w ∈ Wi}, ❛♥❞ (w, i)R+(u, j) ⇐ ⇒ i = j & wRiu, (w, i)S+(u, j) ⇐ ⇒ iSj. ■❢ ❢♦r ❛❧❧ i ❋i = ❋✱ ✇❡ ✇r✐t❡ ❋ ⋋ ■ ❢♦r

■

❋i❀ t❤❡ ❢r❛♠❡ ❋ ⋋ ■ ✐s ❝❛❧❧❡❞ t❤❡ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝t ♦❢ ❢r❛♠❡s ❋ ❛♥❞ ■✳

▲❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts ♦❢ ❧♦❣✐❝s

❉❡✜♥✐t✐♦♥ ❋♦r ❧♦❣✐❝s L✶✱ L✷✱ ♣✉t L✶ ⋋ L✷ = ▲♦❣({❋ ⋋ ■ | ❋ | = L✶, ■ | = L✷}). Pr♦❜❧❡♠ ❚♦ ❝♦♥str✉❝t t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢ L✶ ⋋ L✷✱ ❦♥♦✇✐♥❣ t❤❡ ❧♦❣✐❝s L✶, L✷✳ ❬P❤✳ ❇❛❧❜✐❛♥✐✱ ❆①✐♦♠❛t✐③❛t✐♦♥ ❛♥❞ ❝♦♠♣❧❡t❡♥❡ss ♦❢ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s✳ ✷✵✵✾✳❪

▲❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts ♦❢ ❧♦❣✐❝s

❉❡✜♥✐t✐♦♥ ❋♦r ❧♦❣✐❝s L✶✱ L✷✱ ♣✉t L✶ ⋋ L✷ = ▲♦❣({❋ ⋋ ■ | ❋ | = L✶, ■ | = L✷}). Pr♦❜❧❡♠ ❚♦ ❝♦♥str✉❝t t❤❡ ❛①✐♦♠❛t✐③❛t✐♦♥ ♦❢ L✶ ⋋ L✷✱ ❦♥♦✇✐♥❣ t❤❡ ❧♦❣✐❝s L✶, L✷✳ ❬P❤✳ ❇❛❧❜✐❛♥✐✱ ❆①✐♦♠❛t✐③❛t✐♦♥ ❛♥❞ ❝♦♠♣❧❡t❡♥❡ss ♦❢ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s✳ ✷✵✵✾✳❪

❚❤❡♦r❡♠ ✷ ✭✷✵✵✾❀ ✷✵✶✹✮ ■❢ L✶ ❛♥❞ L✷ ❛r❡ ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡ ❑r✐♣❦❡ ❝♦♠♣❧❡t❡ ❧♦❣✐❝s✱ ♦⊤ ∈ L✶✱ t❤❡♥ L✶ ⋋ L✷ = L✶ ∗ L✷ + {α, β, γ}, ❛♥❞ ❤❡♥❝❡✱

✶ ✷

✷

✶

◗✉❡st✐♦♥ ✭✷✵✵✾✮ ❑ ❑

❚❤❡♦r❡♠ ✷ ✭✷✵✵✾❀ ✷✵✶✹✮ ■❢ L✶ ❛♥❞ L✷ ❛r❡ ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡ ❑r✐♣❦❡ ❝♦♠♣❧❡t❡ ❧♦❣✐❝s✱ ♦⊤ ∈ L✶✱ t❤❡♥ L✶ ⋋ L✷ = L✶ ∗ L✷ + {α, β, γ}, ❛♥❞ ❤❡♥❝❡✱ L✶ ⋋ L✷ =

• L✷

L✶. ◗✉❡st✐♦♥ ✭✷✵✵✾✮ ❑ ❑

❚❤❡♦r❡♠ ✷ ✭✷✵✵✾❀ ✷✵✶✹✮ ■❢ L✶ ❛♥❞ L✷ ❛r❡ ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡ ❑r✐♣❦❡ ❝♦♠♣❧❡t❡ ❧♦❣✐❝s✱ ♦⊤ ∈ L✶✱ t❤❡♥ L✶ ⋋ L✷ = L✶ ∗ L✷ + {α, β, γ}, ❛♥❞ ❤❡♥❝❡✱ L✶ ⋋ L✷ =

• L✷

L✶. ◗✉❡st✐♦♥ ✭✷✵✵✾✮ ❑ ⋋ ❑ =?

Φ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❝❧♦s❡❞ ❢♦r♠✉❧❛s ✐♥ t❤❡ ♠♦❞❛❧ ❧❛♥❣✉❛❣❡ ML(✶)✳ ❚❤❡♦r❡♠ ✸ ■❢ L✶ ❛♥❞ L✷ ❛r❡ ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡ ❑r✐♣❦❡ ❝♦♠♣❧❡t❡ ❧♦❣✐❝s✱ t❤❡♥ L✶ ⋋ L✷ = L✶ ∗ L✷ + {α, β, γ} ∪ Ξ✶ ∪ Ξ✷ ∪ Ξ✸, ✇❤❡r❡ Ξ✶ = {♦✷♦✷p ∧ ♦✷ϕ → ♦✷(♦✷p ∧ ϕ) | ϕ ∈ Φ}, Ξ✷ = {♦✷✷⊥ ∧ ♦✷ϕ → ♦✷(✷⊥ ∧ ϕ) | ϕ ∈ Φ}, Ξ✸ = {♦i

✷ϕ → j ✷(♦✷⊤ → ♦✷ϕ) | i, j ≥ ✵, ϕ ∈ Φ}.

◆♦t❡ t❤❛t ✐❢ ♦⊤ ∈ L✶✱ t❤❡♥ L✶ ∗ L✷ + {α, β, γ} ∪ Ξ✶ ∪ Ξ✷ ∪ Ξ✸ = L✶ ∗ L✷ + {α, β, γ}

❇② t❤❡ ✇❛②✳✳✳

❙✐♠✐❧❛r s✐t✉❛t✐♦♥ ❛♣♣❡❛rs ✐♥ t♦♣♦❧♦❣✐❝❛❧ ✭♥❡✐❣❤❜♦r❤♦♦❞✮ ♣r♦❞✉❝ts ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s✿ ❬❏✳ ✈❛♥ ❇❡♥t❤❡♠✱ ● ❇❡③❤❛♥✐s❤✈✐❧✐✱ ❇✳ t❡♥ ❈❛t❡✱ ❉✳ ❙❛r❡♥❛❝✱ ✷✵✵✻❪✱ ❬❑✉❞✐♥♦✈✱ ✷✵✶✷❪ ❙✹ ×N ❙✹ = ❙✹ ∗ ❙✹✱ (❑ + ♦⊤) ×N ❙✹ = (❑ + ♦⊤) ∗ ❙✹✱ (❑ + ♦⊤) ×N (❑ + ♦⊤) = (❑ + ♦⊤) ∗ (❑ + ♦⊤), . . . ❬❑✉❞✐♥♦✈✱ ✷✵✶✹❪ ❑ ❑ ❑ ❑ ✇❤❡r❡

✷

✐s ❝❧♦s❡❞

✶✲❢♦r♠✉❧❛ ✶

✐s ❝❧♦s❡❞

✷✲❢♦r♠✉❧❛ ✳

❇② t❤❡ ✇❛②✳✳✳

❙✐♠✐❧❛r s✐t✉❛t✐♦♥ ❛♣♣❡❛rs ✐♥ t♦♣♦❧♦❣✐❝❛❧ ✭♥❡✐❣❤❜♦r❤♦♦❞✮ ♣r♦❞✉❝ts ♦❢ ♠♦❞❛❧ ❧♦❣✐❝s✿ ❬❏✳ ✈❛♥ ❇❡♥t❤❡♠✱ ● ❇❡③❤❛♥✐s❤✈✐❧✐✱ ❇✳ t❡♥ ❈❛t❡✱ ❉✳ ❙❛r❡♥❛❝✱ ✷✵✵✻❪✱ ❬❑✉❞✐♥♦✈✱ ✷✵✶✷❪ ❙✹ ×N ❙✹ = ❙✹ ∗ ❙✹✱ (❑ + ♦⊤) ×N ❙✹ = (❑ + ♦⊤) ∗ ❙✹✱ (❑ + ♦⊤) ×N (❑ + ♦⊤) = (❑ + ♦⊤) ∗ (❑ + ♦⊤), . . . ❬❑✉❞✐♥♦✈✱ ✷✵✶✹❪ ❑ ×N ❑ = ❑ ∗ ❑ + ∆, ✇❤❡r❡ ∆ = {φ → ✷φ | φ ✐s ❝❧♦s❡❞ ✶✲❢♦r♠✉❧❛}∪ {ψ → ✶ψ | ψ ✐s ❝❧♦s❡❞ ✷✲❢♦r♠✉❧❛}✳

Φ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❝❧♦s❡❞ ❢♦r♠✉❧❛s ✐♥ t❤❡ ♠♦❞❛❧ ❧❛♥❣✉❛❣❡ ML(✶)✳ ❚❤❡♦r❡♠ ✸ ■❢ L✶ ❛♥❞ L✷ ❛r❡ ❍♦r♥ ❛①✐♦♠❛t✐③❛❜❧❡ ❑r✐♣❦❡ ❝♦♠♣❧❡t❡ ❧♦❣✐❝s✱ t❤❡♥ L✶ ⋋ L✷ = L✶ ∗ L✷ + {α, β, γ} ∪ Ξ✶ ∪ Ξ✷ ∪ Ξ✸, ✇❤❡r❡ Ξ✶ = {♦✷♦✷p ∧ ♦✷ϕ → ♦✷(♦✷p ∧ ϕ) | ϕ ∈ Φ}, Ξ✷ = {♦✷✷⊥ ∧ ♦✷ϕ → ♦✷(✷⊥ ∧ ϕ) | ϕ ∈ Φ}, Ξ✸ = {♦i

✷ϕ → j ✷(♦✷⊤ → ♦✷ϕ) | i, j ≥ ✵, ϕ ∈ Φ}.

◆♦t❡ t❤❛t ✐❢ ♦⊤ ∈ L✶✱ t❤❡♥ L✶ ∗ L✷ + {α, β, γ} ∪ Ξ✶ ∪ Ξ✷ ∪ Ξ✸ = L✶ ∗ L✷ + {α, β, γ}

❉❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t② ♦❢ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts

❋r♦♠ t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ❧❡①✐❝♦❣r❛♣❤✐❝ ♣r♦❞✉❝ts ❛r❡ s❛❢❡r t❤❛♥ ✏✉s✉❛❧✑ ♠♦❞❛❧ ♣r♦❞✉❝ts✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r ❙✹ ⋋ ❙✹ ✐s ✐♥ PSPACE✳ ❚❤❡♦r❡♠ ▲❡t L✶, L✷ ❜❡ ❑r✐♣❦❡ ❝♦♠♣❧❡t❡ ✉♥✐♠♦❞❛❧ ❧♦❣✐❝s✱ ❛♥❞ ❜♦t❤ L✶ ❛♥❞ L✷ ❛❞♠✐t ✜❧tr❛t✐♦♥✳ ❚❤❡♥ L✶ ❛♥❞ L✷ ❤❛✈❡ t❤❡ ⋋✲❢♠♣✱ ✐✳❡✳✱ L✶ ⋋ L✷ = ▲♦❣({❋✶ ⋋ ❋✷ | ❋i | = Li, ❋i ❛r❡ ✜♥✐t❡}).

❚❤❛♥❦ ②♦✉✦