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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

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