Computational Complexity Lecture 8 More of the Polynomial - - PowerPoint PPT Presentation

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Computational Complexity Lecture 8 More of the Polynomial - - PowerPoint PPT Presentation

Apr 07, 2024 148 likes •1.04k views

Computational Complexity Lecture 8 More of the Polynomial Hierarchy Oracle-based Definition 1 Recall PH { x | u 1 u 2 u 3 F( x ,u 1 ,u 2 ,u 3 )} { x | u 1 u 2 u 3 F( x ,u 1 ,u 2 ,u 3 )} 3P 3P { x | u 1 u 2

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Computational Complexity

Lecture 8 More of the Polynomial Hierarchy Oracle-based Definition

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Recall PH

P NP coNP

{x|F(x)} {x|∃u1 F(x,u1)}

Σ2P Π2P Σ3P Π3P

{x|∃u1∀u2∃u3 F(x,u1,u2,u3)} {x|∀u1∃u2∀u3 F(x,u1,u2,u3)} {x|∀u1 F(x,u1)} {x|∃u1∀u2 F(x,u1,u2)} {x|∀u1∃u2 F(x,u1,u2)}

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Oracle Machines

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Oracle Machines

Recall Oracle Machine

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Oracle Machines

Recall Oracle Machine Writes queries on query-tape, enters and leaves query state, and expects answer from oracle on the tape

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Oracle Machines

Recall Oracle Machine Writes queries on query-tape, enters and leaves query state, and expects answer from oracle on the tape Can run an oracle machine with any oracle

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Oracle Machines

Recall Oracle Machine Writes queries on query-tape, enters and leaves query state, and expects answer from oracle on the tape Can run an oracle machine with any oracle Oracle fully specified by the input-output behavior

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Oracle Machines

Recall Oracle Machine Writes queries on query-tape, enters and leaves query state, and expects answer from oracle on the tape Can run an oracle machine with any oracle Oracle fully specified by the input-output behavior Language oracle: answer is a single bit

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Oracle Machines

Recall Oracle Machine Writes queries on query-tape, enters and leaves query state, and expects answer from oracle on the tape Can run an oracle machine with any oracle Oracle fully specified by the input-output behavior Language oracle: answer is a single bit This is what we consider

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Oracle Machines (ctd.)

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Oracle Machines (ctd.)

Non-deterministic oracle machine

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Oracle Machines (ctd.)

Non-deterministic oracle machine Can make non-deterministic choices and make

  • racle queries. (Note: oracles are deterministic!)

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Oracle Machines (ctd.)

Non-deterministic oracle machine Can make non-deterministic choices and make

  • racle queries. (Note: oracles are deterministic!)

Said to accept if any thread reaches accept state

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Oracle Machines (ctd.)

Non-deterministic oracle machine Can make non-deterministic choices and make

  • racle queries. (Note: oracles are deterministic!)

Said to accept if any thread reaches accept state Equivalently, a deterministic oracle machine which takes a (read-once) certificate w (the list of non- deterministic choices)

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Oracle Machines (ctd.)

Non-deterministic oracle machine Can make non-deterministic choices and make

  • racle queries. (Note: oracles are deterministic!)

Said to accept if any thread reaches accept state Equivalently, a deterministic oracle machine which takes a (read-once) certificate w (the list of non- deterministic choices) Said to accept x if ∃w such that (x,w) takes it to accepting state

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Oracle Machines (ctd.)

Non-deterministic oracle machine Can make non-deterministic choices and make

  • racle queries. (Note: oracles are deterministic!)

Said to accept if any thread reaches accept state Equivalently, a deterministic oracle machine which takes a (read-once) certificate w (the list of non- deterministic choices) Said to accept x if ∃w such that (x,w) takes it to accepting state co- ^ all threads reach ∀w

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NPA

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NPA

NPA : class of languages accepted by oracle NTMs with

  • racle for A in poly time

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NPA

NPA : class of languages accepted by oracle NTMs with

  • racle for A in poly time

Certificate version: NPA has languages of the form

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NPA

NPA : class of languages accepted by oracle NTMs with

  • racle for A in poly time

Certificate version: NPA has languages of the form B = {x | ∃w MA(x,w) = 1}

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NPA

NPA : class of languages accepted by oracle NTMs with

  • racle for A in poly time

Certificate version: NPA has languages of the form B = {x | ∃w MA(x,w) = 1} where M deterministic oracle machine

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NPA

NPA : class of languages accepted by oracle NTMs with

  • racle for A in poly time

Certificate version: NPA has languages of the form B = {x | ∃w MA(x,w) = 1} where M deterministic oracle machine MA runs in poly(|x|) time and |w|=poly(|x|)

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NPA

NPA : class of languages accepted by oracle NTMs with

  • racle for A in poly time

Certificate version: NPA has languages of the form B = {x | ∃w MA(x,w) = 1} where M deterministic oracle machine MA runs in poly(|x|) time and |w|=poly(|x|) i.e., B = {x | ∃w (x,w) ∈ L}, where L in PA

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NPA

NPA : class of languages accepted by oracle NTMs with

  • racle for A in poly time

Certificate version: NPA has languages of the form B = {x | ∃w MA(x,w) = 1} where M deterministic oracle machine MA runs in poly(|x|) time and |w|=poly(|x|) i.e., B = {x | ∃w (x,w) ∈ L}, where L in PA co-(NPA) = (co-NP)A

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NPA

NPA : class of languages accepted by oracle NTMs with

  • racle for A in poly time

Certificate version: NPA has languages of the form B = {x | ∃w MA(x,w) = 1} where M deterministic oracle machine MA runs in poly(|x|) time and |w|=poly(|x|) i.e., B = {x | ∃w (x,w) ∈ L}, where L in PA co-(NPA) = (co-NP)A languages of the form {x | ∀w (x,w) ∈ L}, where L in PA

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NPA

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NPA

If A in P, NPA = NP

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NPA

If A in P, NPA = NP Can “implement” the oracle as a subroutine

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NPA

If A in P, NPA = NP Can “implement” the oracle as a subroutine If A in NP?

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NPA

If A in P, NPA = NP Can “implement” the oracle as a subroutine If A in NP? Oracle for A is an oracle for Ac too! NPA = NPAc

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NPA

If A in P, NPA = NP Can “implement” the oracle as a subroutine If A in NP? Oracle for A is an oracle for Ac too! NPA = NPAc NP ∪ co-NP ⊆ NPSAT

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NPA

If A in P, NPA = NP Can “implement” the oracle as a subroutine If A in NP? Oracle for A is an oracle for Ac too! NPA = NPAc NP ∪ co-NP ⊆ NPSAT Can we better characterize NPSAT?

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NPNP and relatives

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NPNP and relatives

NPSAT = ∪A∈NP NPA

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NPNP and relatives

NPSAT = ∪A∈NP NPA Oracle for A can be implemented using oracle for SAT in polynomial time (deterministically)

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NPNP and relatives

NPSAT = ∪A∈NP NPA Oracle for A can be implemented using oracle for SAT in polynomial time (deterministically) NPSAT also called NPNP

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NPNP and relatives

NPSAT = ∪A∈NP NPA Oracle for A can be implemented using oracle for SAT in polynomial time (deterministically) NPSAT also called NPNP NPΣk = ∪A∈Σk NPA = NPΣkSAT

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NPNP and relatives

NPSAT = ∪A∈NP NPA Oracle for A can be implemented using oracle for SAT in polynomial time (deterministically) NPSAT also called NPNP NPΣk = ∪A∈Σk NPA = NPΣkSAT Will show NPΣk = Σk+1P (alt. definition for Σk+1P)

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NPNP and relatives

NPSAT = ∪A∈NP NPA Oracle for A can be implemented using oracle for SAT in polynomial time (deterministically) NPSAT also called NPNP NPΣk = ∪A∈Σk NPA = NPΣkSAT Will show NPΣk = Σk+1P (alt. definition for Σk+1P) In particular, NPNP = Σ2P

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Σk+1 = NPΣk

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Σk+1 = NPΣk

Consider L ∈ Σk+1P

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Σk+1 = NPΣk

Consider L ∈ Σk+1P L = { x| ∃w (x,w) ∈ L ’}, where L ’ in ΠkP

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Σk+1 = NPΣk

Consider L ∈ Σk+1P L = { x| ∃w (x,w) ∈ L ’}, where L ’ in ΠkP So L in NPL

’ where L

’ in ΠkP

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Σk+1 = NPΣk

Consider L ∈ Σk+1P L = { x| ∃w (x,w) ∈ L ’}, where L ’ in ΠkP So L in NPL

’ where L

’ in ΠkP So NPL

’ ⊆ NPΠk = NPΣk

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Σk+1 = NPΣk

Consider L ∈ Σk+1P L = { x| ∃w (x,w) ∈ L ’}, where L ’ in ΠkP So L in NPL

’ where L

’ in ΠkP So NPL

’ ⊆ NPΠk = NPΣk

So Σk+1P ⊆ NPΣk

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Σk+1 = NPΣk

Consider L ∈ Σk+1P L = { x| ∃w (x,w) ∈ L ’}, where L ’ in ΠkP So L in NPL

’ where L

’ in ΠkP So NPL

’ ⊆ NPΠk = NPΣk

So Σk+1P ⊆ NPΣk Now to show NPΣk ⊆ Σk+1P

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NPΣk ⊆ Σk+1

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NPΣk ⊆ Σk+1

To show NPA ⊆ Σk+1P if A in ΣkP

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NPΣk ⊆ Σk+1

To show NPA ⊆ Σk+1P if A in ΣkP For B ∈ NPA poly-time TM M s.t. B = { x| ∃w MA(x,w)=1}

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NPΣk ⊆ Σk+1

To show NPA ⊆ Σk+1P if A in ΣkP For B ∈ NPA poly-time TM M s.t. B = { x| ∃w MA(x,w)=1} i.e., B = { x| ∃w ∃ans M<ans>(x,w)=1 and “ans correct”}

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NPΣk ⊆ Σk+1

To show NPA ⊆ Σk+1P if A in ΣkP For B ∈ NPA poly-time TM M s.t. B = { x| ∃w MA(x,w)=1} i.e., B = { x| ∃w ∃ans M<ans>(x,w)=1 and “ans correct”} To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P

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NPΣk ⊆ Σk+1

To show NPA ⊆ Σk+1P if A in ΣkP For B ∈ NPA poly-time TM M s.t. B = { x| ∃w MA(x,w)=1} i.e., B = { x| ∃w ∃ans M<ans>(x,w)=1 and “ans correct”} To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P Then B also in Σk+1P

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NPΣk ⊆ Σk+1

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NPΣk ⊆ Σk+1

To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P

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NPΣk ⊆ Σk+1

To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P Suppose M makes only one query z=Z(x,w). ans is a single bit saying if z in A or not

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NPΣk ⊆ Σk+1

To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P Suppose M makes only one query z=Z(x,w). ans is a single bit saying if z in A or not “ans correct”: (ans=1 ∧ z ∈ A) or (ans=0 ∧ z ∉ A)

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NPΣk ⊆ Σk+1

To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P Suppose M makes only one query z=Z(x,w). ans is a single bit saying if z in A or not “ans correct”: (ans=1 ∧ z ∈ A) or (ans=0 ∧ z ∉ A) C={(x,w,ans) | M<ans>(x,w)=1 ∧ [(ans=1 ∧ ∃u1∀u2...Qkuk F(z,u1,...)=1) or (ans=0 ∧ ∀v1∃v2...Q’kvk F(z,v1,...)=0)] }

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NPΣk ⊆ Σk+1

To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P Suppose M makes only one query z=Z(x,w). ans is a single bit saying if z in A or not “ans correct”: (ans=1 ∧ z ∈ A) or (ans=0 ∧ z ∉ A) C={(x,w,ans) | M<ans>(x,w)=1 ∧ [(ans=1 ∧ ∃u1∀u2...Qkuk F(z,u1,...)=1) or (ans=0 ∧ ∀v1∃v2...Q’kvk F(z,v1,...)=0)] } C={(x,w,ans) | ∃u1∀u2v1∃u3v2...Qkukvk-1Q’kvk M<ans>(x,w)=1 ∧ [(ans=1 ∧ F(z,u1,...)=1) or (ans=0 ∧ F(z,v1,...)=0)] }

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NPΣk ⊆ Σk+1

To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P Suppose M makes only one query z=Z(x,w). ans is a single bit saying if z in A or not “ans correct”: (ans=1 ∧ z ∈ A) or (ans=0 ∧ z ∉ A) C={(x,w,ans) | M<ans>(x,w)=1 ∧ [(ans=1 ∧ ∃u1∀u2...Qkuk F(z,u1,...)=1) or (ans=0 ∧ ∀v1∃v2...Q’kvk F(z,v1,...)=0)] } C={(x,w,ans) | ∃u1∀u2v1∃u3v2...Qkukvk-1Q’kvk M<ans>(x,w)=1 ∧ [(ans=1 ∧ F(z,u1,...)=1) or (ans=0 ∧ F(z,v1,...)=0)] } i n Σk+1P

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NPΣk ⊆ Σk+1

To show C = {(x,w,ans) | M<ans>(x,w)=1 and “ans correct”} in Σk+1P Suppose M makes only one query z=Z(x,w). ans is a single bit saying if z in A or not “ans correct”: (ans=1 ∧ z ∈ A) or (ans=0 ∧ z ∉ A) C={(x,w,ans) | M<ans>(x,w)=1 ∧ [(ans=1 ∧ ∃u1∀u2...Qkuk F(z,u1,...)=1) or (ans=0 ∧ ∀v1∃v2...Q’kvk F(z,v1,...)=0)] } C={(x,w,ans) | ∃u1∀u2v1∃u3v2...Qkukvk-1Q’kvk M<ans>(x,w)=1 ∧ [(ans=1 ∧ F(z,u1,...)=1) or (ans=0 ∧ F(z,v1,...)=0)] } Changes for t queries: z=Z(x,w) ! (z(1)...z(t)) = Z(x,w,ans), ui ! ui(1)...ui(t), vi ! vi(1)...vi(t), and use conjunction of t checks (for j=1,...,t) of the form [ (ans(j)=1 ∧ F(z(j),u1(j),...)=1) or (ans(j)=0 ∧ F(z(j),v1(j),...)=0) ] i n Σk+1P

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Oracle Version

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Oracle Version

Σk+1P = NPΣk (with Σ0P = P)

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Oracle Version

Σk+1P = NPΣk (with Σ0P = P) Πk+1P = co-NPΠk (with Π0P = P)

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Oracle Version

Σk+1P = NPΣk (with Σ0P = P) Πk+1P = co-NPΠk (with Π0P = P) Πk+1P = co-(NPΣk) = co-NPΣk = co-NPΠk

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ΔkP

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ΔkP

Δk+1P = PΣk = PΠk

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ΔkP

Δk+1P = PΣk = PΠk Δ1P = P

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ΔkP

Δk+1P = PΣk = PΠk Δ1P = P Δ2P = PNP

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ΔkP

Δk+1P = PΣk = PΠk Δ1P = P Δ2P = PNP Note that Δ2P = co-Δ2P

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ΔkP

Δk+1P = PΣk = PΠk Δ1P = P Δ2P = PNP Note that Δ2P = co-Δ2P Δk+1P ⊇ ΣkP ∪ ΠkP

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ΔkP

Δk+1P = PΣk = PΠk Δ1P = P Δ2P = PNP Note that Δ2P = co-Δ2P Δk+1P ⊇ ΣkP ∪ ΠkP Δk+1P ⊆ Σk+1P ∩ Πk+1P (why?)

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ΔkP

Δk+1P = PΣk = PΠk Δ1P = P Δ2P = PNP Note that Δ2P = co-Δ2P Δk+1P ⊇ ΣkP ∪ ΠkP Δk+1P ⊆ Σk+1P ∩ Πk+1P (why?) PΣk ⊆ NPΣk ∩ coNPΣk

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

NPNP

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

NPNP coNPNP

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

NPNP coNPNP NPNPNP

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

NPNP coNPNP NPNPNP coNPNPNP

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

NPNP coNPNP NPNPNP coNPNPNP

Δ2P

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

NPNP coNPNP NPNPNP coNPNPNP

Δ3P Δ2P

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

NPNP coNPNP NPNPNP coNPNPNP PNP

Δ3P Δ2P

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PH

P NP coNP Σ2P Π2P Σ3P Π3P

NPNP coNPNP NPNPNP coNPNPNP PNP PNPNP

Δ3P Δ2P

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Today

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Today

Today, more PH

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Today

Today, more PH Oracle-based definitions (in particular NPNP = Σ2P)

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Today

Today, more PH Oracle-based definitions (in particular NPNP = Σ2P) Next lecture, more PH

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Today

Today, more PH Oracle-based definitions (in particular NPNP = Σ2P) Next lecture, more PH Alternating TM-based definitions

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Today

Today, more PH Oracle-based definitions (in particular NPNP = Σ2P) Next lecture, more PH Alternating TM-based definitions Time-Space tradeoffs

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