Computer Science Laboratory, SRI International ❋♦r♠❛❧ ❚❡❝❤♥✐q✉❡s ❢♦r ❆♥❛❧②③✐♥❣ ❍②❜r✐❞ ❙②st❡♠s ❆s❤✐s❤ ❚✐✇❛r✐
❘❡♣r❡s❡♥t✐♥❣ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❍♦✇ t♦ s♦❧✈❡ t❤❡ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ♣r♦❜❧❡♠ S | = φ ❄ ❈r✉❝✐❛❧❧② ❞❡♣❡♥❞s ♦♥ ❤♦✇ ✇❡ r❡♣r❡s❡♥t S ❘❡❝❛❧❧ t❤❛t S = ( X , F , Init) ❲❡ ♥❡❡❞ t♦ r❡♣r❡s❡♥t t❤❡ st❛t❡ s♣❛❝❡✱ t❤❡ ✐♥✐t✐❛❧ s❡t Init ✱ ❛♥❞ t❤❡ s❡t ♦❢ tr❛❥❡❝t♦r✐❡s ✶
❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ❙t❛t❡ ❙♣❛❝❡ ❙t❛t❡ s♣❛❝❡ X • ❙t❛t❡ s♣❛❝❡ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ s❡t ♦❢ ❛❧❧ ✈❛❧✉❛t✐♦♥s ♦❢ ❛ s❡t ♦❢ ✈❛r✐❛❜❧❡s • ❲❡ ❥✉st ♥❡❡❞ t♦ ❞❡❝❧❛r❡ ✈❛r✐❛❜❧❡s X ❛♥❞ t❤❡✐r t②♣❡s ❚❤❡ ✐♥✐t✐❛❧ st❛t❡s Init • ❚❤✐s ✐s ❥✉st ❛ s✉❜s❡t ♦❢ t❤❡ st❛t❡ s♣❛❝❡ • ❈❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ❢♦r♠✉❧❛ ♦✈❡r X ✷
❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ❚r❛❥❡❝t♦r✐❡s ❚❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s ✐s ✉s✉❛❧❧② r❡♣r❡s❡♥t❡❞ ❜② ❧♦❝❛❧ r✉❧❡s ❚❤❛t s♣❡❝✐❢② t❤❡ ♥❡①t st❛t❡✭s✮ ❣✐✈❡♥ t❤❡ ❝✉rr❡♥t st❛t❡ ✭❛♥❞ ✐♥♣✉t✱ ✐❢ ❛♥②✮ ❚❤❡ ♥♦t✐♦♥ ♦❢ ♥❡①t ✐s ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ T ❲❤❡♥ T = N ✱ ♥❡①t ✐s ❥✉st ✭❝✉rr❡♥t t✐♠❡ ✰ ✶ ✮ ✸
❋r♦♠ ❝♦♥t✐♥✉♦✉s t✐♠❡ t♦ ❞✐s❝r❡t❡✲t✐♠❡ ❲❤❡♥ T = R ≥ 0 ✱ t❤❡r❡ ✐s ❛ ♣♦♣✉❧❛r ✇❛② t♦ ♠❛♣ ✐t t♦ T = N s → s ′ ✐❢ t❤❡r❡ ❡①✐sts ❛ t ′ > t s✳t✳ s ′ = F ( t ′ ) ❛♥❞ s = F ( t ) ❋r♦♠ ♠② ❝✉rr❡♥t st❛t❡ s ✱ t❤❡ s②st❡♠ ❝❛♥ ♥♦♥❞❡t❡r♠✐♥✐st✐❝❛❧❧② tr❛♥s✐t✐♦♥ ❛♥② st❛t❡ r❡❛❝❤❛❜❧❡ ✐♥ ❛♥② ❢✉t✉r❡ t✐♠❡ ✐♥st❛♥❝❡ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ dx dt = 1 ✐s ❛ ❝❧♦❝❦✱ ✐ts ❞✐s❝r❡t❡ t✐♠❡ s❡♠❛♥t✐❝s ✐s✿ x → x ′ ✐❢ x ′ > x ❚❤✐s ✐s t✐♠❡ ♦❜❧✐✈✐♦✉s ■❢ ✇❡ ✇❛♥t t♦ ❦❡❡♣ tr❛❝❦ ♦❢ t✐♠❡ ❡❧❛♣s❡❞✱ ✇❡ ❝❛♥ ❛❞❞ ❛ ♥❡✇ st❛t❡ ✈❛r✐❛❜❧❡ t ❛♥❞ ❤❛✈❡ ( x, t ) → ( x ′ , t ′ ) ✐❢ x ′ > x ❛♥❞ x ′ − x = t ′ − t ✹
❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ❚r❛❥❡❝t♦r✐❡s ▲❡t ✉s ❢♦❝✉s ♦♥ t✐♠❡ ❞♦♠❛✐♥ N ✱ ❛♥❞ ❝♦♥s✐❞❡r ✇❛②s t♦ s♣❡❝✐❢② ❛ s✐♥❣❧❡ st❡♣ s → s ′ st❛t❡ tr❛♥s✐t✐♦♥ ❚❤✐s ❝❛♥ ❜❡ ❣✐✈❡♥ ❛s ❛ ❢♦r♠✉❧❛ ♦✈❡r ❛ ♣❛✐r ♦❢ st❛t❡ ✈❛r✐❛❜❧❡s✿ φ ( x, x ′ ) ❋♦r ♦♣❡♥ s②st❡♠s✱ t❤✐s ❢♦r♠✉❧❛ ❝❛♥ ❞❡♣❡♥❞ ♦♥ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡s ❋♦r ❝♦♥t✐♥✉♦✉s t✐♠❡ s②st❡♠s✱ t❤❡ r✉❧❡ ✐s ❛ ❢♦r♠✉❧❛ ♦✈❡r ( x, ˙ x ) ✺
❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ▼♦❞✉❧❛r❧② ❲❡ ❝❛♥ r❡♣r❡s❡♥t ❞②♥❛♠✐❝❛❧ s②st❡♠s s✉❝❝✐♥❝t❧② ❜② ❞❡s❝r✐❜✐♥❣ t❤❡♠ ❛s ❛ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ♦r ♠♦r❡ s♠❛❧❧❡r ❞②♥❛♠✐❝❛❧ s②st❡♠s✿ state local output input local input output ❲❡ ♥❡❡❞ t♦ t❤✐♥❦ ❛❜♦✉t ♦♣❡♥ s②st❡♠s ✜rst ✻
❖♣❡♥ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s ❙②st❡♠s ✇✐t❤ ✐♥♣✉ts ■♥♣✉ts ❞♦ ♥♦t ❝❤❛♥❣❡ t❤❡ st❛t❡ s♣❛❝❡ ♦❢ t❤❡ s②st❡♠ st❛t❡ s♣❛❝❡ st✐❧❧ ❞❡✜♥❡❞ ❜② t❤❡ ❧♦❝❛❧ ❛♥❞ ♦✉t♣✉t ✈❛r✐❛❜❧❡s ■♥♣✉ts ❝❛♥ ✐♥✢✉❡♥❝❡ t❤❡ ♣♦ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s ❍❡♥❝❡✱ r✉❧❡ φ ( s, i, s ′ ) ❝❛♥ ❞❡♣❡♥❞ ♦♥ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡s i s → s ′ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♥♣✉t i ∈ I : φ ( s, i , s ′ ) ✼
❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ▼♦❞✉❧❛r❧② ❙t❛t❡ s♣❛❝❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❂ ❝r♦ss ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ st❛t❡ s♣❛❝❡s ❂ ✈❛❧✉❛t✐♦♥s ♦❢ L 1 ∪ O 1 ∪ L 2 ∪ O 2 ❚r❛❥❡❝t♦r✐❡s ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❂ ❄❄ ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥❛❧ ❢r❛♠❡✇♦r❦ ✽
❚✇♦ ❈♦♠♣♦s✐t✐♦♥❛❧ ❋r❛♠❡✇♦r❦s ❙②♥❝❤r♦♥♦✉s✿ • ( s 1 , s 2 ) → ( s ′ 1 , s ′ 2 ) ✐❢ s 1 → s ′ 1 ❆◆❉ s 2 → s ′ 2 • ❇♦t❤ s✉❜s②st❡♠s ♠❛❦❡ ❛ tr❛♥s✐t✐♦♥ s✐♠✉❧t❛♥❡♦✉s❧② ❆s②♥❝❤r♦♥♦✉s✿ • ( s 1 , s 2 ) → ( s ′ 1 , s ′ 2 ) ✐❢ s 1 → s ′ 1 , s ′ 2 = s 2 ❖❘ s 2 → s ′ 2 , s ′ 1 = s 1 • ❖♥❡ ♦❢ t❤❡ t✇♦ s✉❜s②st❡♠s ♠❛❦❡s ❛ tr❛♥s✐t✐♦♥ ■❢ t❤❡ t✇♦ s✉❜s②st❡♠s ❛r❡ ✇✐r❡❞ t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡ tr❛♥s✐t✐♦♥s ♦♥ t❤❡ s✉❜s②st❡♠s s❤♦✉❧❞ ❜❡ ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ✐♥♣✉ts ✭❣❡♥❡r❛t❡❞ ❜② t❤❡ ♦t❤❡r s✉❜s②st❡♠✮ ✾
▼♦❞❡❧✐♥❣ ✐♥ ❙❆▲ ❉✐s❝r❡t❡✲t✐♠❡ ❤②❜r✐❞✲s♣❛❝❡ ❞②♥❛♠✐❝❛❧ s②st❡♠s ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✐♥ ❙❆▲ ❊✈❡r② ❙❆▲ ▼❖❉❯▲❊ ❞❡s❝r✐❜❡s ❛ ❞②♥❛♠✐❝❛❧ ✭s✉❜✮s②st❡♠ ❝♦❧❧❛t③✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ❖❯❚P❯❚ ①✿ ◆❆❚❯❘❆▲ ■◆■❚■❆▲■❩❆❚■❖◆ ① ■◆ ④✈✿ ◆❆❚❯❘❆▲ ⑤ ❚r✉❡⑥❀ ❚❘❆◆❙■❚■❖◆ ❬ ① ▼❖❉ ✷ ❂ ✵ ✲✲❃ ①✬ ❂ ①✴✷❀ ❬❪ ❡❧s❡ ✲✲❃ ①✬ ❂ ✸✯①✰✶❀ ❪ ❊◆❉❀ ✶✵
❆s ❚✇♦ ❙✉❜s②st❡♠s ❝♦❧❧❛t③❊✈❡♥✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ■◆P❯❚ ①✿ ◆❆❚❯❘❆▲ ❖❯❚P❯❚ ②✿ ◆❆❚❯❘❆▲ ❚❘❆◆❙■❚■❖◆ ❬ ① ▼❖❉ ✷ ❂ ✵ ✲✲❃ ②✬ ❂ ①✴✷❀ ❬❪ ② ▼❖❉ ✷ ❂ ✵ ✲✲❃ ②✬ ❂ ②✴✷❀ ❪ ❊◆❉❀ ❝♦❧❧❛t③❖❞❞✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ■◆P❯❚ ②✿ ◆❆❚❯❘❆▲ ❖❯❚P❯❚ ①✿ ◆❆❚❯❘❆▲ ❚❘❆◆❙■❚■❖◆ ❬ ② ▼❖❉ ✷ ❂ ✶ ✲✲❃ ①✬ ❂ ✸✯②✰✶❀ ❪ ❊◆❉❀ ✶✶
✳✳✳ ❈♦♠♣♦s❡❞ ❚♦❣❡t❤❡r ❝♦❧❧❛t③✿ ▼❖❉❯▲❊ ❂ ❝♦❧❧❛t③❊✈❡♥ ❬❪ ❝♦❧❧❛t③❖❞❞ ❀ ■♥♣✉t ❛♥❞ ❖✉t♣✉t ♠❛t❝❤ ✉♣ ❜② ♥❛♠❡ ❤❡r❡❀ ❝♦✉❧❞ ❛♣♣❧② ♦✉t❡r r❡♥❛♠❡ ♦♣❡r❛t♦r t♦ ❛ ♠♦❞✉❧❡ ♦✳✇✳ ❆❧❧ ♠♦❞✉❧❡s ❝❛♥ ❜❡ ♣✉t ✇✐t❤✐♥ ❛ ❝♦♥t❡①t ✭✜❧❡✮ ❝♦❧❧❛t③✳s❛❧ ✿ ❝♦❧❧❛t③✿ ❈❖◆❚❊❳❚ ❂ ❇❊●■◆ ❝♦❧❧❛t③❊✈❡♥✿ ▼❖❉❯▲❊ ❂ ✳✳✳ ❝♦❧❧❛t③❖❞❞✿ ▼❖❉❯▲❊ ❂ ✳✳✳ ❝♦❧❧❛t③✿ ▼❖❉❯▲❊ ❂ ✳✳✳ ❊◆❉ ✶✷
❚❡♠♣♦r❛❧ Pr♦♣❡rt✐❡s ✐♥ ❙❆▲ ❚❤❡ ♦♣❡♥ ♣r♦❜❧❡♠✿ ❛♠❴■❴tr✉❡✿ ❚❍❊❖❘❊▼ ❝♦❧❧❛t③ ⑤✲ ❋✭ ① ❂ ✶ ✮❀ ✶✸
▼♦❞❡❧✐♥❣ ❊①❡r❝✐s❡ ❉✐s❝r❡t❡✲t✐♠❡ ❞✐s❝r❡t❡✲s♣❛❝❡ ❞②♥❛♠✐❝❛❧ s②st❡♠ ♠♦❞❡❧✐♥❣ ❛ ✈❡♥❞✐♥❣ ♠❛❝❤✐♥❡ • ❯s❡r ❝❛♥ st❛rt ✇✐t❤ ❞♦❧❧❛r ❛♠♦✉♥t ✐♥ t❤❡ r❛♥❣❡ [0 , . . . , 5] • ❯s❡r ❝❛♥ ❜✉② ❛ ❝❛❦❡ ❢♦r $1 • ❯s❡r ❝❛♥ ❜✉② ❛♥ ❛♣♣❧❡ ❢♦r $0 . 75 ❚❤❡ st❛t❡ s♣❛❝❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ✭✶✮ ❛♠♦✉♥t ♦❢ ♠♦♥❡② r❡♠❛✐♥✐♥❣✱ ✭✷✮ ♥✉♠❜❡r ♦❢ ❝❛❦❡s ❜♦✉❣❤t✱ ❛♥❞ ✭✸✮ ♥✉♠❜❡r ♦❢ ❛♣♣❧❡s ❜♦✉❣❤t ✶✹
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