words 2011
play

WORDS 2011 Bounded Parikh Automata Cadilhac , Finkel &McKenzie - PowerPoint PPT Presentation

Mar 01, 2023 •370 likes •1.48k views

WORDS 2011 Bounded Parikh Automata Cadilhac , Finkel &McKenzie Introduction Result and definitions Bounded Parikh Automata BSL DetPA BSL DetAPA-X DetAPA-X DetpA M. Cadilhac 1 , A. Finkel 2 , and P. McKenzie 1 Corollaries and

  1. The result Bounded Parikh Automata Definition: semilinear Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definitions DetAPA-X ⊆ DetpA E = { ⃗ c 0 + ∑ m i = 1 ⃗ c i ⋅ k i ∣ k i ∈ N } ▸ Linear set: of the form Corollaries and 2 ) ⋅ k 1 } linear, { 2 n ∣ n ∈ N } not linear Further Work E.g., {( 0 0 ) + ( 1 Bibliography ▸ Semilinear set: finite union of linear sets (equiv. FO [ + ] ) Why are they natural? One of many reasons: Theorem ([Parikh, 1966]) With Σ = { a 1 ,..., a n } and w ∈ Σ ∗ , let Parikh ( w ) = (∣ w ∣ a 1 ,..., ∣ w ∣ a n ) ∈ N n be the Parikh image of w. L context-free ⇒ Parikh ( L ) semilinear 5 / 18

  2. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and Further Work Bibliography 6 / 18

  3. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work Bibliography 6 / 18

  4. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work Bibliography a ( t 4 ) a ( t 1 ) a ( t 3 ) b ( t 2 ) 6 / 18

  5. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) a ( t 3 ) b ( t 2 ) 6 / 18

  6. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎪ ⎪ ⎪ a ( t 3 ) ⎪ ⎪ C = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ b ( t 2 ) 6 / 18

  7. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎪ ⎛ ⎞ ⎪ ⎪ 1 a ( t 3 ) ⎪ ⎪ ⎜ ⎟ C = ⎨ ⎜ ⎟ 0 ⎜ ⎟ ⎪ ⋅ k 1 + ⎪ ⎪ ⎪ 0 ⎝ ⎠ ⎪ ⎩ 1 b ( t 2 ) 6 / 18

  8. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎪ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ 1 0 a ( t 3 ) ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ 0 1 ⎜ ⎟ ⎜ ⎟ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ 0 0 ⎝ ⎠ ⎝ ⎠ ⎪ ⎩ 1 1 b ( t 2 ) 6 / 18

  9. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) 6 / 18

  10. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Word: aabaaaa 6 / 18

  11. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa π = 6 / 18

  12. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa (0, 0, 0, 0) π = ▴ 6 / 18

  13. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa (1, 0, 0, 0) π = t 1 ▴ 6 / 18

  14. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa (2, 0, 0, 0) π = t 1 t 1 ▴ 6 / 18

  15. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa (2, 1, 0, 0) π = t 1 t 1 t 2 ▴ 6 / 18

  16. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa (2, 1, 1, 0) π = t 1 t 1 t 2 t 3 ▴ 6 / 18

  17. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa (2, 1, 1, 1) π = t 1 t 1 t 2 t 3 t 4 ▴ 6 / 18

  18. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa (2, 1, 1, 2) π = t 1 t 1 t 2 t 3 t 4 t 4 ▴ 6 / 18

  19. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : Word: aabaaaa (2, 1, 1, 3) π = t 1 t 1 t 2 t 3 t 4 t 4 t 4 ▴ 6 / 18

  20. The result Bounded Parikh Automata Example: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Example DetAPA-X ⊆ L = { w ⋅ a ∣ w ∣+ 1 ∣ w ∈ { a , b } ∗ } ∈ PA DetpA Corollaries and Further Work δ = { t 1 , t 2 , t 3 , t 4 } Bibliography a ( t 4 ) a ( t 1 ) ⎧ ⎫ ⎪ ⎪ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎪ ⎪ ⎪ 1 0 0 ⎪ a ( t 3 ) ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C = ⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ 0 1 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⋅ k 1 + ⋅ k 2 + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ 1 1 0 b ( t 2 ) Parikh ( traced run ) : (2, 1, 1, 3) ∈ C Word: aabaaaa π = t 1 t 1 t 2 t 3 t 4 t 4 t 4 ▴ 6 / 18

  21. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and Further Work Bibliography 7 / 18

  22. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition ([Klaedtke and Rueß, 2003]) DetAPA-X ⊆ DetpA ▸ Parikh automaton (PA): a pair ( A , C ) with: Corollaries and Further Work ▸ A a finite automaton of transition set δ Bibliography ▸ C ⊆ N ∣ δ ∣ semilinear 7 / 18

  23. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition ([Klaedtke and Rueß, 2003]) DetAPA-X ⊆ DetpA ▸ Parikh automaton (PA): a pair ( A , C ) with: Corollaries and Further Work ▸ A a finite automaton of transition set δ Bibliography ▸ C ⊆ N ∣ δ ∣ semilinear ▸ L ( A , C ) = { Label ( π ) ∣ π ∈ Runs ( A ) ∧ Parikh ( π ) ∈ C } 7 / 18

  24. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition ([Klaedtke and Rueß, 2003]) DetAPA-X ⊆ DetpA ▸ Parikh automaton (PA): a pair ( A , C ) with: Corollaries and Further Work ▸ A a finite automaton of transition set δ Bibliography ▸ C ⊆ N ∣ δ ∣ semilinear ▸ L ( A , C ) = { Label ( π ) ∣ π ∈ Runs ( A ) ∧ Parikh ( π ) ∈ C } ▸ Deterministic Parikh automaton (DetPA) if A is 7 / 18

  25. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition ([Klaedtke and Rueß, 2003]) DetAPA-X ⊆ DetpA ▸ Parikh automaton (PA): a pair ( A , C ) with: Corollaries and Further Work ▸ A a finite automaton of transition set δ Bibliography ▸ C ⊆ N ∣ δ ∣ semilinear ▸ L ( A , C ) = { Label ( π ) ∣ π ∈ Runs ( A ) ∧ Parikh ( π ) ∈ C } ▸ Deterministic Parikh automaton (DetPA) if A is ▸ L = { w ⋅ a ∣ w ∣+ 1 } ∈ PA but ∉ DetPA 7 / 18

  26. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition ([Klaedtke and Rueß, 2003]) DetAPA-X ⊆ DetpA ▸ Parikh automaton (PA): a pair ( A , C ) with: Corollaries and Further Work ▸ A a finite automaton of transition set δ Bibliography ▸ C ⊆ N ∣ δ ∣ semilinear ▸ L ( A , C ) = { Label ( π ) ∣ π ∈ Runs ( A ) ∧ Parikh ( π ) ∈ C } ▸ Deterministic Parikh automaton (DetPA) if A is ▸ L = { w ⋅ a ∣ w ∣+ 1 } ∈ PA but ∉ DetPA, same for PAL 7 / 18

  27. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition ([Klaedtke and Rueß, 2003]) DetAPA-X ⊆ DetpA ▸ Parikh automaton (PA): a pair ( A , C ) with: Corollaries and Further Work ▸ A a finite automaton of transition set δ Bibliography ▸ C ⊆ N ∣ δ ∣ semilinear ▸ L ( A , C ) = { Label ( π ) ∣ π ∈ Runs ( A ) ∧ Parikh ( π ) ∈ C } ▸ Deterministic Parikh automaton (DetPA) if A is ▸ L = { w ⋅ a ∣ w ∣+ 1 } ∈ PA but ∉ DetPA, same for PAL ▸ PAL ∉ PA 7 / 18

  28. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Why is PA a relevant model? DetpA Corollaries and Further Work Bibliography 8 / 18

  29. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Why is PA a relevant model? DetpA Corollaries and ▸ Nice decidability properties (emptiness) Further Work Bibliography 8 / 18

  30. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Why is PA a relevant model? DetpA Corollaries and ▸ Nice decidability properties (emptiness) Further Work ▸ Nice closure properties for DetPA Bibliography 8 / 18

  31. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Why is PA a relevant model? DetpA Corollaries and ▸ Nice decidability properties (emptiness) Further Work ▸ Nice closure properties for DetPA Bibliography ▸ Equivalent to reversal bounded counter machines (RBCM) of [Ibarra, 1978] 8 / 18

  32. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Why is PA a relevant model? DetpA Corollaries and ▸ Nice decidability properties (emptiness) Further Work ▸ Nice closure properties for DetPA Bibliography ▸ Equivalent to reversal bounded counter machines (RBCM) of [Ibarra, 1978] ▸ Equivalent to extended automata over ( Z k , + , 0 ) of [Mitrana and Stiebe, 2001] 8 / 18

  33. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Why is PA a relevant model? DetpA Corollaries and ▸ Nice decidability properties (emptiness) Further Work ▸ Nice closure properties for DetPA Bibliography ▸ Equivalent to reversal bounded counter machines (RBCM) of [Ibarra, 1978] ▸ Equivalent to extended automata over ( Z k , + , 0 ) of [Mitrana and Stiebe, 2001] ▸ Related deterministic models used for model checking 8 / 18

  34. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Why is PA a relevant model? DetpA Corollaries and ▸ Nice decidability properties (emptiness) Further Work ▸ Nice closure properties for DetPA Bibliography ▸ Equivalent to reversal bounded counter machines (RBCM) of [Ibarra, 1978] ▸ Equivalent to extended automata over ( Z k , + , 0 ) of [Mitrana and Stiebe, 2001] ▸ Related deterministic models used for model checking ▸ Logical characterization (WS1S + cardinalities) 8 / 18

  35. The result Bounded Parikh Automata Definition: Parikh automata Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Why is PA a relevant model? DetpA Corollaries and ▸ Nice decidability properties (emptiness) Further Work ▸ Nice closure properties for DetPA Bibliography ▸ Equivalent to reversal bounded counter machines (RBCM) of [Ibarra, 1978] ▸ Equivalent to extended automata over ( Z k , + , 0 ) of [Mitrana and Stiebe, 2001] ▸ Related deterministic models used for model checking ▸ Logical characterization (WS1S + cardinalities) ▸ Low complexity (PA ⊊ NL, DetPA ⊊ NC 1 ) 8 / 18

  36. The result Bounded Parikh Automata Definition: Bounded languages Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and Further Work Bibliography 9 / 18

  37. The result Bounded Parikh Automata Definition: Bounded languages Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition DetAPA-X ⊆ DetpA ▸ L bounded : L ⊆ w ∗ Corollaries and 1 ⋯ w ∗ n for some words w i ’s Further Work Bibliography 9 / 18

  38. The result Bounded Parikh Automata Definition: Bounded languages Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition DetAPA-X ⊆ DetpA ▸ L bounded : L ⊆ w ∗ Corollaries and 1 ⋯ w ∗ n for some words w i ’s Further Work w ( L ) = {( i 1 ,..., i n ) ∣ w i 1 n ∈ L } ⊆ N n 1 ⋯ w i n ▸ Define Iter ⃗ Bibliography 9 / 18

  39. The result Bounded Parikh Automata Definition: Bounded languages Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition DetAPA-X ⊆ DetpA ▸ L bounded : L ⊆ w ∗ Corollaries and 1 ⋯ w ∗ n for some words w i ’s Further Work w ( L ) = {( i 1 ,..., i n ) ∣ w i 1 n ∈ L } ⊆ N n 1 ⋯ w i n ▸ Define Iter ⃗ Bibliography ▸ BSL = { L ⊆ w ∗ n ∣ Iter ⃗ w ( L ) semilinear } 1 ⋯ w ∗ 9 / 18

  40. The result Bounded Parikh Automata Definition: Bounded languages Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition DetAPA-X ⊆ DetpA ▸ L bounded : L ⊆ w ∗ Corollaries and 1 ⋯ w ∗ n for some words w i ’s Further Work w ( L ) = {( i 1 ,..., i n ) ∣ w i 1 n ∈ L } ⊆ N n 1 ⋯ w i n ▸ Define Iter ⃗ Bibliography ▸ BSL = { L ⊆ w ∗ n ∣ Iter ⃗ w ( L ) semilinear } 1 ⋯ w ∗ ▸ { a i b 2 i } ∈ BSL 9 / 18

  41. The result Bounded Parikh Automata Definition: Bounded languages Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition DetAPA-X ⊆ DetpA ▸ L bounded : L ⊆ w ∗ Corollaries and 1 ⋯ w ∗ n for some words w i ’s Further Work w ( L ) = {( i 1 ,..., i n ) ∣ w i 1 n ∈ L } ⊆ N n 1 ⋯ w i n ▸ Define Iter ⃗ Bibliography ▸ BSL = { L ⊆ w ∗ n ∣ Iter ⃗ w ( L ) semilinear } 1 ⋯ w ∗ ▸ { a i b 2 i } ∈ BSL ▸ Σ ∗ not bounded 9 / 18

  42. The result Bounded Parikh Automata Definition: Bounded languages Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition DetAPA-X ⊆ DetpA ▸ L bounded : L ⊆ w ∗ Corollaries and 1 ⋯ w ∗ n for some words w i ’s Further Work w ( L ) = {( i 1 ,..., i n ) ∣ w i 1 n ∈ L } ⊆ N n 1 ⋯ w i n ▸ Define Iter ⃗ Bibliography ▸ BSL = { L ⊆ w ∗ n ∣ Iter ⃗ w ( L ) semilinear } 1 ⋯ w ∗ ▸ { a i b 2 i } ∈ BSL ▸ Σ ∗ not bounded ▸ { a 2 n } bounded ∉ BSL 9 / 18

  43. The result Bounded Parikh Automata Definition: Bounded languages Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X Definition DetAPA-X ⊆ DetpA ▸ L bounded : L ⊆ w ∗ Corollaries and 1 ⋯ w ∗ n for some words w i ’s Further Work w ( L ) = {( i 1 ,..., i n ) ∣ w i 1 n ∈ L } ⊆ N n 1 ⋯ w i n ▸ Define Iter ⃗ Bibliography ▸ BSL = { L ⊆ w ∗ n ∣ Iter ⃗ w ( L ) semilinear } 1 ⋯ w ∗ ▸ { a i b 2 i } ∈ BSL ▸ Σ ∗ not bounded ▸ { a 2 n } bounded ∉ BSL ▸▸ BSL intensively studied, e.g., Ginsburg & Spanier, 60’s 9 / 18

  44. The result Bounded Parikh Automata The big picture Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and Further Work Bibliography 10 / 18

  45. The result Bounded Parikh Automata The big picture Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Theorem, restated: DetpA Corollaries and Theorem Further Work PA ∩ BOUNDED ⊆ ⊆ DetPA ∩ BOUNDED Bibliography BSL 10 / 18

  46. The result Bounded Parikh Automata The big picture Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Theorem, restated: DetpA Corollaries and Theorem Further Work PA ∩ BOUNDED ⊆ ⊆ DetPA ∩ BOUNDED Bibliography BSL Parikh ( any L in PA ) semilinear PA closed under h − 1 , ∩ 10 / 18

  47. The result Bounded Parikh Automata The big picture Cadilhac , Finkel &McKenzie Introduction Theorem Result and Parikh automata and their deterministic variant recognize the definitions same bounded languages: those with a semilinear iteration set BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ Theorem, restated: DetpA Corollaries and Theorem Further Work PA ∩ BOUNDED ⊆ ⊆ DetPA ∩ BOUNDED Bibliography BSL Parikh ( any L in PA ) semilinear Rest of this talk PA closed under h − 1 , ∩ 10 / 18

  48. Outline Bounded Parikh Automata Cadilhac , Finkel &McKenzie Introduction Result and definitions BSL ⊆ DetPA Result and definitions BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA BSL ⊆ DetPA Corollaries and Further Work Bibliography Corollaries and Further Work 10 / 18

  49. Preliminary Bounded Parikh Automata Cadilhac , Finkel &McKenzie We make use of a related model: Introduction Result and definitions BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and Further Work Bibliography 11 / 18

  50. Preliminary Bounded Parikh Automata Cadilhac , Finkel &McKenzie We make use of a related model: Introduction Definition Result and definitions ▸ Affine Parikh automaton given by: BSL ⊆ DetPA ▸ A finite automaton BSL ⊆ DetAPA-X DetAPA-X ⊆ ▸ A labelling of the transitions by affine functions DetpA ▸ A semilinear set Corollaries and Further Work a [ ⃗ x ← M . ⃗ x + ⃗ v ] Bibliography 11 / 18

  51. Preliminary Bounded Parikh Automata Cadilhac , Finkel &McKenzie We make use of a related model: Introduction Definition Result and definitions ▸ Affine Parikh automaton given by: BSL ⊆ DetPA ▸ A finite automaton BSL ⊆ DetAPA-X DetAPA-X ⊆ ▸ A labelling of the transitions by affine functions DetpA ▸ A semilinear set Corollaries and Further Work a [ ⃗ x ← M . ⃗ x + ⃗ v ] Bibliography ▸ Its language: accepted words which take ⃗ 0 to some ⃗ x in the semilinear set 11 / 18

  52. Preliminary Bounded Parikh Automata Cadilhac , Finkel &McKenzie We make use of a related model: Introduction Definition Result and definitions ▸ Affine Parikh automaton given by: BSL ⊆ DetPA ▸ A finite automaton BSL ⊆ DetAPA-X DetAPA-X ⊆ ▸ A labelling of the transitions by affine functions DetpA ▸ A semilinear set Corollaries and Further Work a [ ⃗ x ← M . ⃗ x + ⃗ v ] Bibliography ▸ Its language: accepted words which take ⃗ 0 to some ⃗ x in the semilinear set ▸ PA: APA in which every M = Identity 11 / 18

  53. Preliminary Bounded Parikh Automata Cadilhac , Finkel &McKenzie We make use of a related model: Introduction Definition Result and definitions ▸ Affine Parikh automaton given by: BSL ⊆ DetPA ▸ A finite automaton BSL ⊆ DetAPA-X DetAPA-X ⊆ ▸ A labelling of the transitions by affine functions DetpA ▸ A semilinear set Corollaries and Further Work a [ ⃗ x ← M . ⃗ x + ⃗ v ] Bibliography ▸ Its language: accepted words which take ⃗ 0 to some ⃗ x in the semilinear set ▸ PA: APA in which every M = Identity ▸ Known facts: PAL, COPY, { a 2 n } ∈ APA ∖ PA ▸ Open: Dyck ∉ APA, PAL ∉ DetAPA 11 / 18

  54. Outline Bounded Parikh Automata Cadilhac , Finkel &McKenzie Introduction Result and definitions Result and definitions BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA BSL ⊆ DetPA Corollaries and Further Work BSL ⊆ DetAPA with some property X Bibliography DetAPA with this property ⊆ DetPA Corollaries and Further Work 11 / 18

  55. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel ▸ Let L ⊆ w ∗ 3 ∈ BSL. Then a PA describes L : &McKenzie 1 w ∗ 2 w ∗ Introduction w 3 w 1 w 2 Result and definitions BSL ⊆ DetPA ǫ ǫ q f A : BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA and C = Iter ⃗ w ( L ) Corollaries and Further Work Bibliography 12 / 18

  56. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel ▸ Let L ⊆ w ∗ 3 ∈ BSL. Then a PA describes L : &McKenzie 1 w ∗ 2 w ∗ Introduction w 3 w 1 w 2 Result and definitions BSL ⊆ DetPA ǫ ǫ q f A : BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA and C = Iter ⃗ w ( L ) Corollaries and Further Work Bibliography ▸ Let π 1 ,π 2 two accepting paths with same label: Parikh ( π 1 ) ∈ C ⇔ Parikh ( π 2 ) ∈ C 12 / 18

  57. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel ▸ Let L ⊆ w ∗ 3 ∈ BSL. Then a PA describes L : &McKenzie 1 w ∗ 2 w ∗ Introduction w 3 w 1 w 2 Result and definitions BSL ⊆ DetPA ǫ ǫ q f A : BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA and C = Iter ⃗ w ( L ) Corollaries and Further Work Bibliography ▸ Let π 1 ,π 2 two accepting paths with same label: Parikh ( π 1 ) ∈ C ⇔ Parikh ( π 2 ) ∈ C ▸▸ In SubsetDeterminize( A ), when reaching a final state { q f ,... } , we need only recall the Parikh image of one possible run to q f 12 / 18

  58. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel ▸ Let L ⊆ w ∗ 3 ∈ BSL. Then a PA describes L : &McKenzie 1 w ∗ 2 w ∗ Introduction w 3 w 1 w 2 Result and definitions BSL ⊆ DetPA ǫ ǫ q f A : BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA and C = Iter ⃗ w ( L ) Corollaries and Further Work Bibliography ▸ Let π 1 ,π 2 two accepting paths with same label: Parikh ( π 1 ) ∈ C ⇔ Parikh ( π 2 ) ∈ C ▸▸ In SubsetDeterminize( A ), when reaching a final state { q f ,... } , we need only recall the Parikh image of one possible run to q f ▸▸ In SubsetDeterminize( A ), when reaching some state { q 1 ,..., q k } , we need only recall the Parikh image of one possible run to each q i 12 / 18

  59. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and definitions BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and Further Work Bibliography 13 / 18

  60. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and Further Work Bibliography 13 / 18

  61. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and Further Work Bibliography 13 / 18

  62. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � Further Work Bibliography a ( t 1 ) 1 3 a ( t 2 ) 2 13 / 18

  63. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work Bibliography a ( t 1 ) 1 1, 2 3 a ( t 2 ) 2 3 13 / 18

  64. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work ⃗ Bibliography a ( t 1 ) 1 ⃗ x = ( j ) i ⃗ 1, 2 3 a ( t 2 ) 2 3 13 / 18

  65. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work Parikh ( a path to 1 ) ⃗ Bibliography a ( t 1 ) 1 ⃗ x = ( j ) i ⃗ 1, 2 3 a ( t 2 ) 2 3 13 / 18

  66. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work Parikh ( a path to 1 ) ⃗ Bibliography a ( t 1 ) 1 ⃗ x = ( j ) i ⃗ 1, 2 3 Parikh ( a path to 2 ) a ( t 2 ) 2 3 13 / 18

  67. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work Parikh ( a path to 1 ) ⃗ Bibliography a ( t 1 ) 1 ⃗ x = ( j ) i ⃗ 1, 2 3 Parikh ( a path to 2 ) a ( t 2 ) 2 ⃗ Parikh ( a path to 3 ) ⃗ x = ( 0 ) i ⃗ 3 13 / 18

  68. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work Parikh ( a path to 1 ) ⃗ Bibliography a ( t 1 ) 1 ⃗ x = ( j ) i ⃗ 1, 2 3 Parikh ( a path to 2 ) ⃗ ⃗ i + Parikh ( t 1 ) a [( j ) ← ( )] a ( t 2 ) i ⃗ ⃗ 2 0 ⃗ Parikh ( a path to 3 ) ⃗ x = ( 0 ) i ⃗ 3 13 / 18

  69. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work Parikh ( a path to 1 ) ⃗ Bibliography a ( t 1 ) 1 ⃗ x = ( j ) i ⃗ 1, 2 3 Parikh ( a path to 2 ) ⃗ ⃗ ⃗ ⃗ j ) + ( Parikh ( t 1 ) a [( j ) ← ( 0 ) ⋅ ( )] a ( t 2 ) i 1 0 i ⃗ ⃗ ⃗ ⃗ ⃗ 2 0 0 ⃗ Parikh ( a path to 3 ) ⃗ x = ( 0 ) i ⃗ 3 13 / 18

  70. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work Parikh ( a path to 1 ) ⃗ Bibliography a ( t 1 ) 1 ⃗ x = ( j ) i ⃗ 1, 2 3 Parikh ( a path to 2 ) ⃗ ⃗ i + Parikh ( t 1 ) a [( j ) ← ( )] a ( t 2 ) i ⃗ ⃗ 2 0 ⃗ Parikh ( a path to 3 ) ⃗ x = ( 0 ) i ⃗ 3 13 / 18

  71. BSL ⊆ DetAPA with some property X Bounded Parikh Automata Cadilhac , Finkel &McKenzie Thus we do the following: 1. Let ( A , C ) be the PA for L ∈ BSL Introduction Result and 2. Determinize A by the subset construction definitions BSL ⊆ DetPA 3. Associate functions to compute the Parikh image: BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA Corollaries and A � SubsetDeterminize( A ) Further Work Parikh ( a path to 1 ) ⃗ Bibliography a ( t 1 ) 1 ⃗ x = ( j ) i ⃗ 1, 2 3 Parikh ( a path to 2 ) ⃗ ⃗ j + Parikh ( t 2 ) a [( j ) ← ( )] a ( t 2 ) i ⃗ ⃗ 2 0 ⃗ Parikh ( a path to 3 ) ⃗ x = ( 0 ) i ⃗ 3 13 / 18

  72. Outline Bounded Parikh Automata Cadilhac , Finkel &McKenzie Introduction Result and definitions Result and definitions BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ DetpA BSL ⊆ DetPA Corollaries and Further Work BSL ⊆ DetAPA with some property X Bibliography DetAPA with this property ⊆ DetPA Corollaries and Further Work 13 / 18

  73. DetAPA with the property X ⊆ DetPA Bounded Parikh Automata Cadilhac , Finkel &McKenzie ▸ We can assume the DetAPA, of language L , is of the form: Introduction Result and definitions u 3 [ ⃗ x ← M 3 . ⃗ x + ⃗ u 1 [ ⃗ x ← M 1 . ⃗ x + ⃗ u 2 [ ⃗ x ← M 2 . ⃗ x + ⃗ v 1 ] v 2 ] v 3 ] BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ u ′ u ′ DetpA 1 2 Corollaries and Further Work Bibliography 14 / 18

  74. DetAPA with the property X ⊆ DetPA Bounded Parikh Automata Cadilhac , Finkel &McKenzie ▸ We can assume the DetAPA, of language L , is of the form: Introduction Result and definitions u 3 [ ⃗ x ← M 3 . ⃗ x + ⃗ u 1 [ ⃗ x ← M 1 . ⃗ x + ⃗ u 2 [ ⃗ x ← M 2 . ⃗ x + ⃗ v 1 ] v 2 ] v 3 ] BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ u ′ u ′ DetpA 1 2 Corollaries and Further Work = M p i + k i Bibliography ▸ Property X: For all i , there are p i , k i s.t. M p i i i 14 / 18

  75. DetAPA with the property X ⊆ DetPA Bounded Parikh Automata Cadilhac , Finkel &McKenzie ▸ We can assume the DetAPA, of language L , is of the form: Introduction Result and definitions u 3 [ ⃗ x ← M 3 . ⃗ x + ⃗ u 1 [ ⃗ x ← M 1 . ⃗ x + ⃗ u 2 [ ⃗ x ← M 2 . ⃗ x + ⃗ v 1 ] v 2 ] v 3 ] BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ u ′ u ′ DetpA 1 2 Corollaries and Further Work = M p i + k i Bibliography ▸ Property X: For all i , there are p i , k i s.t. M p i i i ▸ For any ⃗ a ∈ { p i ,..., p i + k i } { 1 , 2 , 3 } , we give a DetPA for L ∩ ( u a 1 1 )( u k 1 1 ) ∗ ⋅ u ′ 1 ⋅ ( u a 2 2 )( u k 2 2 ) ∗ ⋅ u ′ 2 ⋅ ( u a 3 3 )( u k 3 3 ) ∗ 14 / 18

  76. DetAPA with the property X ⊆ DetPA Bounded Parikh Automata Cadilhac , Finkel &McKenzie ▸ We can assume the DetAPA, of language L , is of the form: Introduction Result and definitions u 3 [ ⃗ x ← M 3 . ⃗ x + ⃗ u 1 [ ⃗ x ← M 1 . ⃗ x + ⃗ u 2 [ ⃗ x ← M 2 . ⃗ x + ⃗ v 1 ] v 2 ] v 3 ] BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ u ′ u ′ DetpA 1 2 Corollaries and Further Work = M p i + k i Bibliography ▸ Property X: For all i , there are p i , k i s.t. M p i i i ▸ For any ⃗ a ∈ { p i ,..., p i + k i } { 1 , 2 , 3 } , we give a DetPA for L ∩ ( u a 1 1 )( u k 1 1 ) ∗ ⋅ u ′ 1 ⋅ ( u a 2 2 )( u k 2 2 ) ∗ ⋅ u ′ 2 ⋅ ( u a 3 3 )( u k 3 3 ) ∗ ▸▸ Construction main idea: suppose a word contains t 2 . Final value of ⃗ times u k 2 x contains: 14 / 18

  77. DetAPA with the property X ⊆ DetPA Bounded Parikh Automata Cadilhac , Finkel &McKenzie ▸ We can assume the DetAPA, of language L , is of the form: Introduction Result and definitions u 3 [ ⃗ x ← M 3 . ⃗ x + ⃗ u 1 [ ⃗ x ← M 1 . ⃗ x + ⃗ u 2 [ ⃗ x ← M 2 . ⃗ x + ⃗ v 1 ] v 2 ] v 3 ] BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ u ′ u ′ DetpA 1 2 Corollaries and Further Work = M p i + k i Bibliography ▸ Property X: For all i , there are p i , k i s.t. M p i i i ▸ For any ⃗ a ∈ { p i ,..., p i + k i } { 1 , 2 , 3 } , we give a DetPA for L ∩ ( u a 1 1 )( u k 1 1 ) ∗ ⋅ u ′ 1 ⋅ ( u a 2 2 )( u k 2 2 ) ∗ ⋅ u ′ 2 ⋅ ( u a 3 3 )( u k 3 3 ) ∗ ▸▸ Construction main idea: suppose a word contains t 2 . Final value of ⃗ times u k 2 x contains: ∑ t i = 1 ⋯ M i × k 2 M a 2 2 ⋯ 2 14 / 18

  78. DetAPA with the property X ⊆ DetPA Bounded Parikh Automata Cadilhac , Finkel &McKenzie ▸ We can assume the DetAPA, of language L , is of the form: Introduction Result and definitions u 3 [ ⃗ x ← M 3 . ⃗ x + ⃗ u 1 [ ⃗ x ← M 1 . ⃗ x + ⃗ u 2 [ ⃗ x ← M 2 . ⃗ x + ⃗ v 1 ] v 2 ] v 3 ] BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ u ′ u ′ DetpA 1 2 Corollaries and Further Work = M p i + k i Bibliography ▸ Property X: For all i , there are p i , k i s.t. M p i i i ▸ For any ⃗ a ∈ { p i ,..., p i + k i } { 1 , 2 , 3 } , we give a DetPA for L ∩ ( u a 1 1 )( u k 1 1 ) ∗ ⋅ u ′ 1 ⋅ ( u a 2 2 )( u k 2 2 ) ∗ ⋅ u ′ 2 ⋅ ( u a 3 3 )( u k 3 3 ) ∗ ▸▸ Construction main idea: suppose a word contains t 2 . Final value of ⃗ times u k 2 x contains: 2 ⋯ = t × ⋯ M a 2 ∑ t i = 1 ⋯ M i × k 2 M a 2 2 ⋯ 2 14 / 18

  79. DetAPA with the property X ⊆ DetPA Bounded Parikh Automata Cadilhac , Finkel &McKenzie ▸ We can assume the DetAPA, of language L , is of the form: Introduction Result and definitions u 3 [ ⃗ x ← M 3 . ⃗ x + ⃗ u 1 [ ⃗ x ← M 1 . ⃗ x + ⃗ u 2 [ ⃗ x ← M 2 . ⃗ x + ⃗ v 1 ] v 2 ] v 3 ] BSL ⊆ DetPA BSL ⊆ DetAPA-X DetAPA-X ⊆ u ′ u ′ DetpA 1 2 Corollaries and Further Work = M p i + k i Bibliography ▸ Property X: For all i , there are p i , k i s.t. M p i i i ▸ For any ⃗ a ∈ { p i ,..., p i + k i } { 1 , 2 , 3 } , we give a DetPA for L ∩ ( u a 1 1 )( u k 1 1 ) ∗ ⋅ u ′ 1 ⋅ ( u a 2 2 )( u k 2 2 ) ∗ ⋅ u ′ 2 ⋅ ( u a 3 3 )( u k 3 3 ) ∗ ▸▸ Construction main idea: suppose a word contains t 2 . Final value of ⃗ times u k 2 x contains: 2 ⋯ = t × ⋯ M a 2 ∑ t i = 1 ⋯ M i × k 2 M a 2 2 ⋯ 2 ▸▸ Contribution of ( u k 2 2 ) constant 14 / 18

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Sturmian words, Lecture 3 Standard words Dominique Perrin 1 er d ecembre 2011 Dominique

Sturmian words, Lecture 3 Standard words Dominique Perrin 1 er d ecembre 2011 Dominique

Sturmian words, Lecture 3 Standard words Dominique Perrin 1 er d ecembre 2011 Dominique Perrin Sturmian words, Lecture 3 Standard words Consider two functions and from { 0 , 1 } { 0 , 1 } into itself defined by ( u , v )

306 views • 29 slides
Synonyms Antonyms Are words Are words that mean the that mean the same opposite

Synonyms Antonyms Are words Are words that mean the that mean the same opposite

Synonyms Antonyms Are words Are words that mean the that mean the same opposite Okay.Practice Time! On your computer open word, make a Table, (3 columns, 20 rows) label them (left) synonym and (right) antonym, and decide which words

572 views • 13 slides
Solving The Words Search Problem Ivan Kazmenko St. Petersburg State University Tuesday, July 5,

Solving The Words Search Problem Ivan Kazmenko St. Petersburg State University Tuesday, July 5,

Solving The Words Search Problem Ivan Kazmenko St. Petersburg State University Tuesday, July 5, 2011 Ivan Kazmenko (SPbSU) Words Search 05.07.2011 1 / 30 Outline The Problem 1 Al Zimmermanns Programming Contests The Words Search

804 views • 52 slides
MORPHOLOGY A Study of the internal structure of words and the relationships among words

MORPHOLOGY A Study of the internal structure of words and the relationships among words

MORPHOLOGY A Study of the internal structure of words and the relationships among words Definition of Words: Word is an arbitrary pairing of sound and meaning No way to distinguish between words and phrases or sentences It in It is raining has

179 views • 13 slides
Retail Presentation Retail Presentation Looking Back at 2011 Looking Back at 2011 and Moving

Retail Presentation Retail Presentation Looking Back at 2011 Looking Back at 2011 and Moving

Retail Presentation Retail Presentation Looking Back at 2011 Looking Back at 2011 and Moving Forward in 2012 and Moving Forward in 2012 Presented by Karen Rich Buzz Words Buzz Words Consolidation Upside Down Economic Slowdown Underwater

562 views • 25 slides
Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani Dipartimento di Sistemi e

Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani Dipartimento di Sistemi e

Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani Dipartimento di Sistemi e Informatica Universit` a degli Studi di Firenze WORDS 2011 Pattern 1 j 0 i avoiding binary words S.Bilotta E.Pergola R.Pinzani (dsi.unifi.it) WORDS

1.44k views • 109 slides
Simplicity in Practice https://xkcd.com/1349/ Words, words, words. Hamlet, Act 2 Scene

Simplicity in Practice https://xkcd.com/1349/ Words, words, words. Hamlet, Act 2 Scene

Simplicity in Practice https://xkcd.com/1349/ Words, words, words. Hamlet, Act 2 Scene 2 simple, adj. one fold/braid one role, task, concept, dimension antonym: complex, interleaved, of many parts objective measure

481 views • 44 slides
Words, Words, Words AND WHY THEY MATTER IN ADVERTISING AND MARKETING Steve Kaplan Becky

Words, Words, Words AND WHY THEY MATTER IN ADVERTISING AND MARKETING Steve Kaplan Becky

Words, Words, Words AND WHY THEY MATTER IN ADVERTISING AND MARKETING Steve Kaplan Becky Lobenherz June 28, 2016 Partner Associate +1 202 263 1043 +1 202 263 3436 skaplan@mayerbrown.com rlobenherz@mayerbrown.com Mayer Brown is a global

455 views • 31 slides
Words & Pictures Clustering and Bag of Words Many

Words & Pictures Clustering and Bag of Words Many

Words & Pictures Clustering and Bag of Words Many slides adapted from Svetlana Lazebnik, Fei-Fei Li, Rob Fergus, and Antonio Torralba Document

1.03k views • 55 slides
Proverbs Words: The Power of Life and Death Words: The Power of 3. Words: They Can Be

Proverbs Words: The Power of Life and Death Words: The Power of 3. Words: They Can Be

Proverbs Words: The Power of Life and Death Words: The Power of 3. Words: They Can Be Destructive- Life and Death THEY CAN RUIN RELATIONSHIPS Prov. 16:28 A perverse person stirs up conflict, and a gossip separates Part 13 in Proverbs:

355 views • 17 slides
Stories, Not Words: Abstract Datatype Instruction Sets Martha Kim Columbia University Workshop

Stories, Not Words: Abstract Datatype Instruction Sets Martha Kim Columbia University Workshop

Stories, Not Words: Abstract Datatype Instruction Sets Martha Kim Columbia University Workshop on New Directions in Computer Architecture 6/5/2011 Sunday, June 5, 2011 The Utilization Wall Exponential decrease in percentage of

603 views • 36 slides
Pattern Avoidability with Involution Bastian Bischoff Dirk Nowotka WORDS 2011 unavoidable over

Pattern Avoidability with Involution Bastian Bischoff Dirk Nowotka WORDS 2011 unavoidable over

Pattern Avoidability with Involution Bastian Bischoff Dirk Nowotka WORDS 2011 unavoidable over (actually occurs in every binary word longer than four) avoidable over witness: (square-free Thue-word) avoidability index: Generalization:

777 views • 49 slides
This week, we are going to look at words that have an /ear/ sound spelt using ere. This

This week, we are going to look at words that have an /ear/ sound spelt using ere. This

This week, we are going to look at words that have an /ear/ sound spelt using ere. This letter pattern usually appears at the end of words like: or sphere atmosphere Well start by practising these challenge words by trying to help out

612 views • 20 slides
A Few Words about Knowledge Compilation Pierre Marquis Universit e dArtois CRIL UMR CNRS

A Few Words about Knowledge Compilation Pierre Marquis Universit e dArtois CRIL UMR CNRS

A Few Words about Knowledge Compilation Pierre Marquis Universit e dArtois CRIL UMR CNRS 8188 France Third Workshop on Kernelization (WorKer 2011), Vienna, September 2 nd , 2011 What is Knowledge Compilation? A family of

787 views • 55 slides
Restatement Some unknown words are probably restated by using another word which is more familiar

Restatement Some unknown words are probably restated by using another word which is more familiar

Guessing Meanings of Words from Context sari_hid@yahoo.com/ sari_hidayati@uny.ac.id Context is a word, a group of words, or sentences that surround unknown words. A Context is a hint from the rest of the sentence or Clue paragraph that

147 views • 4 slides
Sixteen words Shakespeare did not invent gloomy http://www.oed.com/view/Entry/79096 David

Sixteen words Shakespeare did not invent gloomy http://www.oed.com/view/Entry/79096 David

Sixteen words Shakespeare did not invent gloomy http://www.oed.com/view/Entry/79096 David Crystal, 2011, Citing Scripture for my purpose, Around the Globe 48, 2011, 2021 available from

469 views • 28 slides
A Contraction Method to Decide MSO Theories of Trees Gabriele Puppis Departement of Mathematics

A Contraction Method to Decide MSO Theories of Trees Gabriele Puppis Departement of Mathematics

Introduction The contraction method Reducible trees Conclusions A Contraction Method to Decide MSO Theories of Trees Gabriele Puppis Departement of Mathematics and Computer Science University of Udine, Italy gabriele.puppis@dimi.uniud.it

788 views • 67 slides
Forks, finitely related clones, and finitely generated varieties Erhard Aichinger Department of

Forks, finitely related clones, and finitely generated varieties Erhard Aichinger Department of

Forks, finitely related clones, and finitely generated varieties Erhard Aichinger Department of Algebra Johannes Kepler University Linz, Austria 53 rd SSAOS, Srn, September 2015 Supported by the Austrian Science Fund (FWF) in P24077 Results

1.23k views • 100 slides
Computational Linguistics II: Parsing Formal Languages: Regular Languages II Frank Richter &

Computational Linguistics II: Parsing Formal Languages: Regular Languages II Frank Richter &

Computational Linguistics II: Parsing Formal Languages: Regular Languages II Frank Richter & Jan-Philipp S ohn fr@sfs.uni-tuebingen.de, jp.soehn@uni-tuebingen.de Computational Linguistics II: Parsing p.1 Reminder: The Big Picture

535 views • 29 slides
Finite State Transducers fmies:N 1 2 3 4 5 time:N fmies:V example 2: POS tagging / shallow

Finite State Transducers fmies:N 1 2 3 4 5 time:N fmies:V example 2: POS tagging / shallow

Finite State Transducers fmies:N 1 2 3 4 5 time:N fmies:V example 2: POS tagging / shallow parsing like:ADP like:V an:D arrow:N 0 1 2 0 Where do we use FSTs? PROPN:NP In this lecture, we treat an FSA as a simple FST that outputs

222 views • 3 slides
In the Maze of Data Data Languages Languages In the Maze of Loris D'Antoni Loris D'Antoni WPE

In the Maze of Data Data Languages Languages In the Maze of Loris D'Antoni Loris D'Antoni WPE

In the Maze of Data Data Languages Languages In the Maze of Loris D'Antoni Loris D'Antoni WPE II WPE II 05/08/2012 05/08/2012 Data Languages Data Languages Motivation Motivation Data Model Data Strings Automata and Logics

755 views • 72 slides
B uchi Automata and their Application to Software Verification Finite Automata Theory and

B uchi Automata and their Application to Software Verification Finite Automata Theory and

B uchi Automata and their Application to Software Verification Finite Automata Theory and Formal Languages Wolfgang Ahrendt 22nd April 2013 B uchi Automata: TMV027/DIT321 / GU 130423 1 / 25 Motivating Temporal Logic? But How to

832 views • 64 slides
Recall: Regular Languages The language recognized by a finite automaton M is denoted by L(M). A

Recall: Regular Languages The language recognized by a finite automaton M is denoted by L(M). A

CSE 2001: Introduction to Theory of Computation Summer 2013 Yves Lesperance lesperan@cse.yorku.ca Office: CSEB 3052A Phone: 416-736-2100 ext 70146 Course page: http://www.cse.yorku.ca/course/2001 Slides are mostly taken from Suprakash

395 views • 20 slides
CSC 473 Automata, Grammars & Languages 9/29/10 Automata, Grammars and Languages Discourse 03

CSC 473 Automata, Grammars & Languages 9/29/10 Automata, Grammars and Languages Discourse 03

CSC 473 Automata, Grammars & Languages 9/29/10 Automata, Grammars and Languages Discourse 03 Finite Automata C SC 473 Automata, Grammars & Languages Finite Automata / Switching Theory (CS) / (CE) Boolean operators / Gates

519 views • 32 slides

More recommend