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The class NP Isabel Oitavem CMAF-UL and FCT-UNL Recursion-theoretic approach Theorem FPtime [ B ; SC , SR W ] (Bellantoni-Cook 1992) FPspace [ B ; SC , TR W ] (O 2008) P NP Pspace Recursion-theoretic approach Theorem FPtime

  1. The class NP Isabel Oitavem CMAF-UL and FCT-UNL

  2. Recursion-theoretic approach Theorem FPtime ≃ [ B ; SC , SR W ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TR W ] (O 2008) P ⊆ NP ⊆ Pspace

  3. Recursion-theoretic approach Theorem FPtime ≃ [ B ; SC , SR W ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TR W ] (O 2008) P ⊆ NP ⊆ Pspace FPtime ⊆ · · · ⊆ FPspace

  4. Recursion-theoretic approach Theorem FPtime ≃ [ B ; SC , SR W ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TR W ] (O 2008) P ⊆ NP ⊆ Pspace FPtime ⊆ FPtime ∪ NP ⊆ FPspace

  5. Recursion-theoretic approach Theorem FPtime ≃ [ B ; SC , SR W ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TR W ] (O 2008) P ⊆ NP ⊆ Pspace FPtime ⊆ FPtime ∪ NP ⊆ FPspace � �� � � �� � � �� � determ. non-determ. alternating � �� � � �� � � �� �

  6. Recursion-theoretic approach Theorem FPtime ≃ [ B ; SC , SR W ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TR W ] (O 2008) P ⊆ NP ⊆ Pspace FPtime ⊆ FPtime ∪ NP ⊆ FPspace � �� � � �� � � �� � determ. non-determ. alternating � �� � � �� � � �� � recursion ? tree-recursion

  7. Recursion-theoretic approach FPtime ≃ [ B ; SC , SR W ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TR W ] (O 2008) ◮ f = SC( g , ¯ r , ¯ s ) f (¯ x ; ¯ y ) = g (¯ r (¯ x ; );¯ s (¯ x ; ¯ y )) ◮ f = SR W ( g , h ) f ( ǫ, ¯ x ; ¯ y ) = g (¯ x ; ¯ y ) f ( z 0 , ¯ x ; ¯ y ) = h ( z 0 , ¯ x ; ¯ y , f ( z , ¯ x ; ¯ y )) f ( z 1 , ¯ x ; ¯ y ) = h ( z 1 , ¯ x ; ¯ y , f ( z , ¯ x ; ¯ y )) ◮ f = TR W ( g , h ) f ( p , ǫ, ¯ x ; ¯ y ) = g ( p , ¯ x ; ¯ y ) f ( p , z 0 , ¯ x ; ¯ y ) = h ( p , z 0 , ¯ x ; ¯ y , f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y )) f ( p , z 1 , ¯ x ; ¯ y ) = h ( p , z 1 , ¯ x ; ¯ y , f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y ))

  8. Recursion-theoretic approach FPtime ≃ [ B ; SC , SR W ] (Bellantoni-Cook 1992) NP ≃ · · · FPspace ≃ [ B ; SC , TR W ] (O 2008) ◮ f = SC( g , ¯ s ) r , ¯ f (¯ x ; ¯ y ) = g (¯ r (¯ x ; );¯ s (¯ x ; ¯ y )) ◮ f = SR W ( g , h ) f ( ǫ, ¯ x ; ¯ y ) = g (¯ x ; ¯ y ) f ( z 0 , ¯ x ; ¯ y ) = h ( z 0 , ¯ x ; ¯ y , f ( z , ¯ x ; ¯ y )) f ( z 1 , ¯ x ; ¯ y ) = h ( z 1 , ¯ x ; ¯ y , f ( z , ¯ x ; ¯ y )) ◮ f = TR W ( g , h ) f ( p , ǫ, ¯ x ; ¯ y ) = g ( p , ¯ x ; ¯ y ) f ( p , z 0 , ¯ x ; ¯ y ) = h ( p , z 0 , ¯ x ; ¯ y , f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y )) f ( p , z 1 , ¯ x ; ¯ y ) = h ( p , z 1 , ¯ x ; ¯ y , f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y ))

  9. Recursion-theoretic approach FPtime ≃ [ B ; SC , SR W ] = ST 0 (Bellantoni-Cook 1992) NP ≃ [ ST 0 ; SC 0 , · · · ] FPspace ≃ [ B ; SC , TR W ] (O 2008) ◮ f = SC 0 ( g , ¯ s ) r , ¯ f (¯ x ; ¯ y ) = g (¯ r (¯ x ; );¯ s (¯ x ; ¯ y )) , ¯ r , ¯ s ∈ ST 0 ◮ f = SR W ( g , h ) f ( ǫ, ¯ x ; ¯ y ) = g (¯ x ; ¯ y ) f ( z 0 , ¯ x ; ¯ y ) = h ( z 0 , ¯ x ; ¯ y , f ( z , ¯ x ; ¯ y )) f ( z 1 , ¯ x ; ¯ y ) = h ( z 1 , ¯ x ; ¯ y , f ( z , ¯ x ; ¯ y )) ◮ f = TR W ( g , h ) f ( p , ǫ, ¯ x ; ¯ y ) = g ( p , ¯ x ; ¯ y ) f ( p , z 0 , ¯ x ; ¯ y ) = h ( p , z 0 , ¯ x ; ¯ y , f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y )) f ( p , z 1 , ¯ x ; ¯ y ) = h ( p , z 1 , ¯ x ; ¯ y , f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y ))

  10. Recursion-theoretic approach FPtime ≃ [ B ; SC , SR W ] = ST 0 (Bellantoni-Cook 1992) NP ≃ [ ST 0 ; SC 0 , · · · ] FPspace ≃ [ B ; SC , TR W ] (O 2008) ◮ f = SC 0 ( g , ¯ s ) r , ¯ f (¯ x ; ¯ y ) = g (¯ r (¯ x ; );¯ s (¯ x ; ¯ y )) , ¯ r , ¯ s ∈ ST 0 ◮ f = SR W ( g , h ) f ( ǫ, ¯ x ; ¯ y ) = g (¯ x ; ¯ y ) f ( z 0 , ¯ x ; ¯ y ) = h ( z 0 , ¯ x ; ¯ y , f ( z , ¯ x ; ¯ y )) f ( z 1 , ¯ x ; ¯ y ) = h ( z 1 , ¯ x ; ¯ y , f ( z , ¯ x ; ¯ y )) ◮ f = TR W ( g , ∨ ) f ( p , ǫ, ¯ x ; ¯ y ) = g ( p , ¯ x ; ¯ y ) f ( p , z 0 , ¯ x ; ¯ y ) = ∨ ( ; f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y )) f ( p , z 1 , ¯ x ; ¯ y ) = ∨ ( ; f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y ))

  11. Recursion-theoretic approach FPtime ≃ [ B ; SC , SR W ] = ST 0 (Bellantoni-Cook 1992) NP ≃ [ ST 0 ; SC 0 , ∨ -TR L W ] ◮ f = SC 0 ( g , ¯ r , ¯ s ) f (¯ x ; ¯ y ) = g (¯ r (¯ x ; );¯ s (¯ x ; ¯ y )) , ¯ r , ¯ s ∈ ST 0 ◮ f = TR W ( g , ∨ ) = ∨ -TR L W ( g ) f ( p , ǫ, ¯ x ; ¯ y ) = g ( p , ¯ x ; ¯ y ) f ( p , z 0 , ¯ x ; ¯ y ) = ∨ ( ; f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y )) f ( p , z 1 , ¯ x ; ¯ y ) = ∨ ( ; f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y ))

  12. Recursion-theoretic approach FPtime ≃ [ B ; SC , SR W ] = ST 0 (Bellantoni-Cook 1992) NP ≃ [ ST 0 ; SC 0 , ∨ -TR L W ] ◮ f = SC 0 ( g , ¯ r , ¯ s ) f (¯ x ; ¯ y ) = g (¯ r (¯ x ; );¯ s (¯ x ; ¯ y )) , ¯ r , ¯ s ∈ ST 0 ◮ f = TR W ( g , ∨ ) = ∨ -TR L W ( g ) f ( p , ǫ, ¯ x ; ¯ y ) = g ( p , ¯ x ; ¯ y ) f ( p , z 0 , ¯ x ; ¯ y ) = ∨ ( ; f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y )) f ( p , z 1 , ¯ x ; ¯ y ) = ∨ ( ; f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y ))

  13. Recursion-theoretic approach FPtime ≃ [ B ; SC , SR W ] = ST 0 (Bellantoni-Cook 1992) NP ≃ [ ST 0 ; SC 0 , ∨ -TR L W ] ≃ [ ST 0 ; SC 0 , ∨ -TR R W ] ◮ f = SC 0 ( g , ¯ s ) r , ¯ f (¯ x ; ¯ y ) = g (¯ r (¯ x ; );¯ s (¯ x ; ¯ y )) , ¯ r , ¯ s ∈ ST 0 ◮ f = TR W ( g , ∨ ) = ∨ -TR L W ( g ) f ( p , ǫ, ¯ x ; ¯ y ) = g ( p , ¯ x ; ¯ y ) f ( p , z 0 , ¯ x ; ¯ y ) = ∨ ( ; f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y )) f ( p , z 1 , ¯ x ; ¯ y ) = ∨ ( ; f ( p 0 , z , ¯ x ; ¯ y ) , f ( p 1 , z , ¯ x ; ¯ y )) ◮ f = ∨ -TR R W ( g ) f ( ǫ, ¯ x ; ¯ y , p ) = g (¯ x ; ¯ y , p ) f ( z 0 , ¯ x ; ¯ y , p ) = ∨ ( ; f ( z , ¯ x ; ¯ y , p 0) , f ( z , ¯ x ; ¯ y , p 1)) f ( z 1 , ¯ x ; ¯ y , p ) = ∨ ( ; f ( z , ¯ x ; ¯ y , p 0) , f ( z , ¯ x ; ¯ y , p 1))

  14. Recursion-theoretic approach Theorem NP ≃ [ ST 0 ; SC 0 , ∨ -TR L W ] ≃ [ ST 0 ; SC 0 , ∨ -TR R W ] Lemma For all f ∈ [ ST 0 ; SC 0 , ∨ -TR L W ] there exists F ∈ [ ST 0 ; SC 0 , ∨ -TR R W ] such that ∀ ¯ x ∀ ¯ y f (¯ x ; ¯ y ) = F (¯ x , ¯ y ; ) .

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