The class NP Isabel Oitavem CMAF-UL and FCT-UNL - - PowerPoint PPT Presentation

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the class np isabel oitavem cmaf ul and fct unl recursion
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The class NP Isabel Oitavem CMAF-UL and FCT-UNL - - PowerPoint PPT Presentation

Dec 27, 2023 47 likes •193 views

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

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

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Recursion-theoretic approach

Theorem

FPtime ≃ [ B ; SC , SRW ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TRW ] (O 2008) P ⊆ NP ⊆ Pspace

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Recursion-theoretic approach

Theorem

FPtime ≃ [ B ; SC , SRW ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TRW ] (O 2008) P ⊆ NP ⊆ Pspace FPtime ⊆ · · · ⊆ FPspace

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Recursion-theoretic approach

Theorem

FPtime ≃ [ B ; SC , SRW ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TRW ] (O 2008) P ⊆ NP ⊆ Pspace FPtime ⊆ FPtime ∪ NP ⊆ FPspace

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Recursion-theoretic approach

Theorem

FPtime ≃ [ B ; SC , SRW ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TRW ] (O 2008) P ⊆ NP ⊆ Pspace FPtime determ. ⊆ FPtime ∪ NP

  • non-determ.
  • ⊆

FPspace

  • alternating
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Recursion-theoretic approach

Theorem

FPtime ≃ [ B ; SC , SRW ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TRW ] (O 2008) P ⊆ NP ⊆ Pspace FPtime determ.

recursion

⊆ FPtime ∪ NP

  • non-determ.
  • ?

⊆ FPspace

  • alternating
  • tree-recursion
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Recursion-theoretic approach

FPtime ≃ [ B ; SC , SRW ] (Bellantoni-Cook 1992) FPspace ≃ [ B ; SC , TRW ] (O 2008)

◮ f = SC(g,¯

r,¯ s) f (¯ x; ¯ y) = g(¯ r(¯ x; );¯ s(¯ x; ¯ y))

◮ f = SRW(g, h)

f (ǫ, ¯ x; ¯ y) = g(¯ x; ¯ y) f (z0, ¯ x; ¯ y) = h(z0, ¯ x; ¯ y, f (z, ¯ x; ¯ y)) f (z1, ¯ x; ¯ y) = h(z1, ¯ x; ¯ y, f (z, ¯ x; ¯ y))

◮ f = TRW(g, h)

f (p, ǫ, ¯ x; ¯ y) = g(p, ¯ x; ¯ y) f (p, z0, ¯ x; ¯ y) = h(p, z0, ¯ x; ¯ y, f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y)) f (p, z1, ¯ x; ¯ y) = h(p, z1, ¯ x; ¯ y, f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y))

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Recursion-theoretic approach

FPtime ≃ [ B ; SC , SRW ] (Bellantoni-Cook 1992) NP ≃ · · · FPspace ≃ [ B ; SC , TRW ] (O 2008)

◮ f = SC(g,¯

r,¯ s) f (¯ x; ¯ y) = g(¯ r(¯ x; );¯ s(¯ x; ¯ y))

◮ f = SRW(g, h)

f (ǫ, ¯ x; ¯ y) = g(¯ x; ¯ y) f (z0, ¯ x; ¯ y) = h(z0, ¯ x; ¯ y, f (z, ¯ x; ¯ y)) f (z1, ¯ x; ¯ y) = h(z1, ¯ x; ¯ y, f (z, ¯ x; ¯ y))

◮ f = TRW(g, h)

f (p, ǫ, ¯ x; ¯ y) = g(p, ¯ x; ¯ y) f (p, z0, ¯ x; ¯ y) = h(p, z0, ¯ x; ¯ y, f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y)) f (p, z1, ¯ x; ¯ y) = h(p, z1, ¯ x; ¯ y, f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y))

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Recursion-theoretic approach

FPtime ≃ [ B ; SC , SRW ] = ST0 (Bellantoni-Cook 1992) NP ≃ [ ST0 ; SC0 , · · · ] FPspace ≃ [ B ; SC , TRW ] (O 2008)

◮ f = SC0(g,¯

r,¯ s) f (¯ x; ¯ y) = g(¯ r(¯ x; );¯ s(¯ x; ¯ y)), ¯ r,¯ s ∈ ST0

◮ f = SRW(g, h)

f (ǫ, ¯ x; ¯ y) = g(¯ x; ¯ y) f (z0, ¯ x; ¯ y) = h(z0, ¯ x; ¯ y, f (z, ¯ x; ¯ y)) f (z1, ¯ x; ¯ y) = h(z1, ¯ x; ¯ y, f (z, ¯ x; ¯ y))

◮ f = TRW(g, h)

f (p, ǫ, ¯ x; ¯ y) = g(p, ¯ x; ¯ y) f (p, z0, ¯ x; ¯ y) = h(p, z0, ¯ x; ¯ y, f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y)) f (p, z1, ¯ x; ¯ y) = h(p, z1, ¯ x; ¯ y, f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y))

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Recursion-theoretic approach

FPtime ≃ [ B ; SC , SRW ] = ST0 (Bellantoni-Cook 1992) NP ≃ [ ST0 ; SC0 , · · · ] FPspace ≃ [ B ; SC , TRW ] (O 2008)

◮ f = SC0(g,¯

r,¯ s) f (¯ x; ¯ y) = g(¯ r(¯ x; );¯ s(¯ x; ¯ y)), ¯ r,¯ s ∈ ST0

◮ f = SRW(g, h)

f (ǫ, ¯ x; ¯ y) = g(¯ x; ¯ y) f (z0, ¯ x; ¯ y) = h(z0, ¯ x; ¯ y, f (z, ¯ x; ¯ y)) f (z1, ¯ x; ¯ y) = h(z1, ¯ x; ¯ y, f (z, ¯ x; ¯ y))

◮ f = TRW(g, ∨)

f (p, ǫ, ¯ x; ¯ y) = g(p, ¯ x; ¯ y) f (p, z0, ¯ x; ¯ y) = ∨( ; f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y)) f (p, z1, ¯ x; ¯ y) = ∨( ; f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y))

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Recursion-theoretic approach

FPtime ≃ [ B ; SC , SRW ] = ST0 (Bellantoni-Cook 1992) NP ≃ [ ST0 ; SC0 , ∨-TRL

W ] ◮ f = SC0(g,¯

r,¯ s) f (¯ x; ¯ y) = g(¯ r(¯ x; );¯ s(¯ x; ¯ y)), ¯ r,¯ s ∈ ST0

◮ f = TRW(g, ∨) = ∨-TRL W(g)

f (p, ǫ, ¯ x; ¯ y) = g(p, ¯ x; ¯ y) f (p, z0, ¯ x; ¯ y) = ∨( ; f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y)) f (p, z1, ¯ x; ¯ y) = ∨( ; f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y))

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Recursion-theoretic approach

FPtime ≃ [ B ; SC , SRW ] = ST0 (Bellantoni-Cook 1992) NP ≃ [ ST0 ; SC0 , ∨-TRL

W ] ◮ f = SC0(g,¯

r,¯ s) f (¯ x; ¯ y) = g(¯ r(¯ x; );¯ s(¯ x; ¯ y)), ¯ r,¯ s ∈ ST0

◮ f = TRW(g, ∨) = ∨-TRL W(g)

f (p, ǫ, ¯ x; ¯ y) = g(p, ¯ x; ¯ y) f (p, z0, ¯ x; ¯ y) = ∨( ; f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y)) f (p, z1, ¯ x; ¯ y) = ∨( ; f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y))

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Recursion-theoretic approach

FPtime ≃ [ B ; SC , SRW ] = ST0 (Bellantoni-Cook 1992) NP ≃ [ ST0 ; SC0 , ∨-TRL

W ]

≃ [ ST0 ; SC0 , ∨-TRR

W ] ◮ f = SC0(g,¯

r,¯ s) f (¯ x; ¯ y) = g(¯ r(¯ x; );¯ s(¯ x; ¯ y)), ¯ r,¯ s ∈ ST0

◮ f = TRW(g, ∨) = ∨-TRL W(g)

f (p, ǫ, ¯ x; ¯ y) = g(p, ¯ x; ¯ y) f (p, z0, ¯ x; ¯ y) = ∨( ; f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y)) f (p, z1, ¯ x; ¯ y) = ∨( ; f (p0, z, ¯ x; ¯ y), f (p1, z, ¯ x; ¯ y))

◮ f = ∨-TRR W(g)

f (ǫ, ¯ x; ¯ y, p) = g(¯ x; ¯ y, p) f (z0, ¯ x; ¯ y, p) = ∨( ; f (z, ¯ x; ¯ y, p0), f (z, ¯ x; ¯ y, p1)) f (z1, ¯ x; ¯ y, p) = ∨( ; f (z, ¯ x; ¯ y, p0), f (z, ¯ x; ¯ y, p1))

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Recursion-theoretic approach

Theorem

NP ≃ [ ST0 ; SC0 , ∨-TRL

W ]

≃ [ ST0 ; SC0 , ∨-TRR

W ]

Lemma

For all f ∈ [ ST0 ; SC0 , ∨-TRL

W ] there exists

F ∈ [ ST0 ; SC0 , ∨-TRR

W ] such that

∀¯ x∀¯ y f (¯ x; ¯ y) = F(¯ x, ¯ y; ).