Compmng Qanmes mn arov eward odes mcmae mmes German erospace - PowerPoint PPT Presentation

Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s Ťmcmae¯ Ùmme¯s German Πerospace Cenℓer michael.ummels@dlr.de (Şomnℓ Wor¯ wmℓm Cmrmsℓe¯ Ψamer, TÙ Dresden) ŘΘSSΠCS 2013 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 1 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 1 1 2 4 1 2 3 4 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 + 0 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 + 0 + 1 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 + 0 + 1 + 1 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 + 0 + 1 + 1 + 0 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 + 0 + 1 + 1 + 0 + 1 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 + 0 + 1 + 1 + 0 + 1 + 2 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 + 0 + 1 + 1 + 0 + 1 + 2 = 7 Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

Ťar¯ov Úeward Ťode¯s Ťode¯: Ťar¯ov decmsmon processes wmℓm nonnegaℓmve rewards on sℓaℓes. 2 1 0 1 0 1 1 2 4 0 0 1 2 3 4 Πcc№ m № ¯ aℓed r ewa r d: 0 + 1 + 1 + 0 + 1 + 1 + 0 + 1 + 2 = 7 Ţ oℓ e: Scmed№¯er r e so¯v e s non de ℓ e rmmnmsm. Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 2 θ 15

ÞÚCTL 2 1 0 1 0 1 1 2 4 s 0 a a a b b 0 0 1 2 3 4 c a Ŕxa mp¯ e prop e r ℓ m e s mn ÞÚCTL ( Π n d o va eℓ a ¯.) : Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 3 θ 15

ÞÚCTL 2 1 0 1 0 1 1 2 4 s 0 a a a b b 0 0 1 2 3 4 c a Ŕxa mp¯ e prop e r ℓ m e s mn ÞÚCTL ( Π n d o va eℓ a ¯.) : ▶ s 0 ⊧ P > 0 . 2 ( a U ≤ 3 b ) Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 3 θ 15

ÞÚCTL 2 1 0 1 0 1 1 2 4 s 0 a a a b b 0 0 1 2 3 4 c a Ŕxa mp¯ e prop e r ℓ m e s mn ÞÚCTL ( Π n d o va eℓ a ¯.) : ▶ s 0 ⊧ P > 0 . 2 ( a U ≤ 3 b ) ▶ s 0 ⊧ P = 0 ( a U ≤ 1 b ) Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 3 θ 15

ÞÚCTL 2 1 0 1 0 1 1 2 4 s 0 a a a b b 0 0 1 2 3 4 c a Ŕxa mp¯ e prop e r ℓ m e s mn ÞÚCTL ( Π n d o va eℓ a ¯.) : ▶ s 0 ⊧ P > 0 . 2 ( a U ≤ 3 b ) ▶ s 0 ⊧̹ P ≤ 0 . 2 ( a U ≤ 2 b ) ▶ s 0 ⊧ P = 0 ( a U ≤ 1 b ) Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 3 θ 15

ÞÚCTL 2 1 0 1 0 1 1 2 4 s 0 a a a b b 0 0 1 2 3 4 c a Ŕxa mp¯ e prop e r ℓ m e s mn ÞÚCTL ( Π n d o va eℓ a ¯.) : ▶ s 0 ⊧ P > 0 . 2 ( a U ≤ 3 b ) ▶ s 0 ⊧̹ P ≤ 0 . 2 ( a U ≤ 2 b ) ▶ s 0 ⊧ P = 0 ( a U ≤ 1 b ) ▶ s 0 ⊧̹ P > 0 ( a U ≤ 2 c ) Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 3 θ 15

ÞÚCTL 2 1 0 1 0 1 1 2 4 s 0 a a a b b 0 0 1 2 3 4 c a Ŕxa mp¯ e prop e r ℓ m e s mn ÞÚCTL ( Π n d o va eℓ a ¯.) : ▶ s 0 ⊧ ∀ P > 0 . 2 ( a U ≤ 3 b ) ▶ s 0 ⊧ ∃ P > 0 . 2 ( a U ≤ 2 b ) ▶ s 0 ⊧ ∀ P = 0 ( a U ≤ 1 b ) ▶ s 0 ⊧ ∃ P = 0 ( a U ≤ 2 b ) Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 3 θ 15

Ťoℓmvaℓmon Ŕxamp¯e: Úandommsed Ť№ℓ№a¯ exc¯№smon. nn wn nw 1 c 1 cn ww nc 5 c 1 cw wc Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 4 θ 15

Ťoℓmvaℓmon Ŕxamp¯e: Úandommsed Ť№ℓ№a¯ exc¯№smon. nn wn nw 1 c 1 cn ww nc 5 c 1 cw wc Q№esℓmon: Śow many sℓeps may process 1 wamℓ №nℓm¯ wmℓm 90% cmance mn crmℓmca¯ secℓmon? Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 4 θ 15

Ťoℓmvaℓmon Ŕxamp¯e: Úandommsed Ť№ℓ№a¯ exc¯№smon. nn wn nw 1 c 1 cn ww nc 5 c 1 cw wc Q№esℓmon: Śow many sℓeps may process 1 wamℓ №nℓm¯ wmℓm 90% cmance mn crmℓmca¯ secℓmon? Comp№ℓe: ¯easℓ r s№cm ℓmaℓ wn ⊧ ∀ P ≥ 0.9 ( ℓr№e U ≤ r c 1 ) . Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 4 θ 15

Ťore Ťoℓmvaℓmon Ŕxamp¯e: Úeso№rce Cons№mpℓmon. 1,000,000 ⋯ 1,000 prod№cℓ s 1 ⋯ fam¯№re Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 5 θ 15

Ťore Ťoℓmvaℓmon Ŕxamp¯e: Úeso№rce Cons№mpℓmon. 1,000,000 ⋯ 1,000 prod№cℓ s 1 ⋯ fam¯№re Q№esℓmon: Śow m№cm ℓo mnvesℓ ℓo s№ccessf№¯¯y prod№ce wmℓm 99%? Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 5 θ 15

Ťore Ťoℓmvaℓmon Ŕxamp¯e: Úeso№rce Cons№mpℓmon. 1,000,000 ⋯ 1,000 prod№cℓ s 1 ⋯ fam¯№re Q№esℓmon: Śow m№cm ℓo mnvesℓ ℓo s№ccessf№¯¯y prod№ce wmℓm 99%? Comp№ℓe: ¯easℓ r s№cm ℓmaℓ s ⊧ ∃ P ≥ 0.99 ( ℓr№e U ≤ r prod№cℓ ) . Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 5 θ 15

Q№anℓm¯e Q№ermes Q№anℓm¯e Q№ery φ = ∀ P ⋈ p ( a U ≤ ? b ) or φ = ∃ P ⋈ p ( a U ≤ ? b ) wmere ▶ a , b ∈ AP , ▶ p ∈ [ 0, 1 ] , and ▶ ⋈ ∈ { < , ≤ , ≥ , > } . Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 6 θ 15

Q№anℓm¯e Q№ermes Q№anℓm¯e Q№ery φ = ∀ P ⋈ p ( a U ≤ ? b ) or φ = ∃ P ⋈ p ( a U ≤ ? b ) wmere ▶ a , b ∈ AP , ▶ p ∈ [ 0, 1 ] , and ▶ ⋈ ∈ { < , ≤ , ≥ , > } . Wrmℓe φ [ r ] for ℓme ÞÚCTL form№¯a ℓmaℓ res№¯ℓs from rep¯acmng ? by r . Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 6 θ 15

Q№anℓm¯e Q№ermes Q№anℓm¯e Q№ery φ = ∀ P ⋈ p ( a U ≤ ? b ) or φ = ∃ P ⋈ p ( a U ≤ ? b ) wmere ▶ a , b ∈ AP , ▶ p ∈ [ 0, 1 ] , and ▶ ⋈ ∈ { < , ≤ , ≥ , > } . Wrmℓe φ [ r ] for ℓme ÞÚCTL form№¯a ℓmaℓ res№¯ℓs from rep¯acmng ? by r . Defjne ℓme va¯№e of s wrℓ. φ ℓo be ℓme ¯eas ℓθ ¯arges ℓ r s № cm ℓ ma ℓ s ⊧ φ [ r ] : Ťmcmae¯ Ùmme¯s – Comp№ℓmng Q№anℓm¯es mn Ťar¯ov Úeward Ťode¯s 6 θ 15