complexity 3: outline feasibility...complexity 3: feasibility dusko pavlovic introduction time and...

Complexity 3:Feasibility

Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

Complexity Theory — Part 3:Feasibility

Dusko Pavlovic

RHULSpring 2012


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

Outline

Introduction: Gödel’s letter

Time and space complexity

Abstract complexity

Reductions, hardness, completeness

Projections, provers and verifiers


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

Outline



Abstract complexity




Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

Gödel’s letter


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

Gödel’s letter

Context: Entscheidungsproblem

Hilbert (1928): Construct a machine Prov : Σ! → Σ! todecide valid formulas:

Prov(F ) ⇐⇒ ∃Π. Π % F

where F is a formula, and Π a proof.

Church (1935), Turing (1936): Entscheidungsproblem isundecidable: a machine Prove may searchforever.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

Gödel’s letter

Gödel’s idea

Bounded Entscheidungsproblem: Construct a machineProv : Σ! × N→ Σ!

Prov(F , n) ⇐⇒ ∃Π. Π % F ∧ |Π| ≤ n

where F is a formula, and Π a proof oflength |Π| ≤ n.

Decidable: Construct a TM to try all proofs of length ≤ n.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

Gödel’s letter

Gödel’s idea

Complexity of provability: Define

ψ(F , n) =∣∣∣ run of Prov(F , n)

∣∣∣

φ(n) = maxF{n,ψ(F , n)}! worst case

NP: If φ(n) = O(nc), then machines can feasiblyfind proofs.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

Gödel’s letter

Gödel’s idea

NP ={L(M) ⊆ Σ! | M ∈ NTM ∧

∣∣∣M(x)↓∣∣∣ ≤ O(|x |c)

}

(i.e., the languages decided by NTMs in polynomial time)


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

SAT ∈ NP

Task

Determine whether a propositional formula asSATisfiable.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

SAT ∈ NP


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

SAT ∈ NP Complexity 3:Feasibility

Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

SAT ∈ NP


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NP

SAT ∈ NP Complexity 3:Feasibility

Dusko Pavlovic

Introduction

Time and spaceDefinitions

Classes

Abstractcomplexity

Reducibility

P vs NP

Outline



Definitions

Classes

Abstract complexity




Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Time complexity

Idea

" The time requirement of a machine M for the input xis the length of its run⟨ι, inp (x)

⟩−−→$%

⟨q1, τ1

⟩−−→$%

⟨q2, τ2

⟩−−→$%· · · −−→

$%

⟨qk , τk

⟩

︸!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!︷︷!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!︸time=k


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Time complexity

Idea

" The time requirement of a machine M for the input xis the length of its run⟨ι, inp (x)

⟩−−→$%

⟨q1, τ1

⟩−−→$%

⟨q2, τ2

⟩−−→$%· · · −−→

$%

⟨qk , τk

⟩

︸!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!︷︷!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!︸time=k

" For nondeterministic M, then the time requirement ismaximized over the possible paths.


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Time complexity

Definition

For a program p

" the time requirement is ˜timep : Σ! ⇀ N

˜timep(x) =∨

y∈Σ!

{|r | | T (p, x , y , r)

}


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Time complexity

Definition

For a program p

" the time complexity measure is timep : N⇀ N

timep(n) =∨

|x |=n

{|r | | ∃y . T (p, x , y , r)

}


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Time complexity is a computable function

A Turing machine that measures time:

timep(n) = do for |x | = nrun U(m, x);count steps;if U(p, x) halts thenstore maximum;

od


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Space complexity

Idea

" The space requirement of a machine M for the inputx is the greatest span that the head reaches

0 nma b b b a a ab a a b baa

︸!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!︷︷!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!︸space=m+n


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Space complexity

Idea

" The space requirement of a machine M for the inputx is the greatest span that the head reaches

0 nma b b b a a ab a a b baa

︸!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!︷︷!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!︸space=m+n

" For nondeterministic M, the space requirement ismaximized over the possible paths.


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Space complexity

Definition

For a program p

" the space requirement is spacep : Σ! ⇀ N

˜spacep(x) =∨

y∈Σ!

{r | T (p, x , y , r)

}

where r is the span of the head in the run r .


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Space complexity

Definition

For a program p

" the space complexity measure is spacep : N⇀ N

spacep(n) =∨

|x |=n

{r | ∃y . T (p, x , y , r)

}


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Space complexity is a computable function

A Turing machine that measures space:

spacep(n) = do for |x | = nrun U(m, x);record the offsets;if U(p, x) halts thenadd up the offsets;store maximum;

od


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Definition

For any f : N→ N define

DTIME(f ) ={L(M) ⊆ Σ! | M ∈ DTM . time#M$ ≤ f

}

NTIME(f ) ={L(M) ⊆ Σ! | M ∈ NTM . time#M$ ≤ f

}

DSPACE(f ) ={L(M) ⊆ Σ! | M ∈ DTM . space#M$ ≤ f

}

NSPACE(f ) ={L(M) ⊆ Σ! | M ∈ NTM . space#M$ ≤ f

}


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Preorder of functions

Notation

f ≤ g ⇐⇒ ∀x . f (x) ≤ g(x)


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Two views of time-bounded machines

1. TM that normally reach a decision in ≤ f (n) stepsapproach: try to construct a machine that solves

the problem in time

2. TM with a counter that clocks out after f (n) stepsapproach: • consider all machines that solve the

problem,• find out which ones reach the decisionin time


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Facts.

" obvious:

DTIME(f ) ⊆ NTIME(f ) DTIME(f ) ⊆ NTIME(f )

" "you cannot use more space than time"

DTIME(f ) ⊆ DSPACE(f ) NTIME(f ) ⊆ NSPACE(f )


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Proposition 1

NTIME(f ) ⊆ DTIME(2O(f ))


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Proposition 2

NTIME(f ) ⊆ DSPACE(f )


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Proposition 3

For f ≥ log

NSPACE(f ) ⊆ DTIME(2O(f ))


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Proposition 4 (Savich)

For f ≥ log

NSPACE(f ) ⊆ DSPACE(f 2)


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Proposition 5 (Immerman-Szelepcsényi)

For f ≥ log

NSPACE(f ) = co-NSPACE(f )


Dusko Pavlovic

Introduction


Classes

Abstractcomplexity

Reducibility

P vs NP

Complexity classes

Prop. 4

Prop. 5

DTIME(f ) DSPACE(f )

NTIME(f )

co-NSPACE(f )

Prop. 1

Prop. 2

Prop. 3

NSPACE(f )

=

2O2O

f>log f>log

f2


Dusko Pavlovic

Introduction

Time and space

AbstractcomplexityDefinition

Classes

The Hierarchy Theorem

Complexity bounds

Reducibility

P vs NP

OutlineIntroduction: Gödel’s letter


Abstract complexity

Definition

Classes


Complexity bounds




Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Task

" Compute computational complexity" Machines that measure complexity of machines


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Computer Complexity 3:Feasibility

Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Computer so far

Machines

Behaviors

Programs

−(−)

#−${−}Universal: U (#M$ , x) = M(x)

Notation: {n} (x) = U(n, x)

Extensional?: #{n}$ = n and {#M$} = M


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Computer so far

M

Σ∗⊥Σ!

Σ∗

−(−)

#−$

{−}"

U ′

U ∈M

U : Σ∗ × Σ! ⇀ Σ!

U ′ : Σ∗ → Σ∗Σ!


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Convention

For simplicity, henceforth fix the alphabet

Σ = {0, 1,%}

Notation

" N = {0, 1}∗

" N = N ∪ {∞}


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Computer so far

M

NN

N

−(−)

#−$

{−}"

U ′

U ∈M

u : N × N⇀ N

u : N→ NN


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Computer with complexity measure

M

NN

N

−(−)

#−$

{−}

NN

u c

U ,C ∈M

u, c : N × N⇀ N

u, c : N→ NN


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity measure

Definition

An (abstract) complexity requirement is a partial functionc : N × N⇀ N such that

" c(#M$ , x)↓ ⇐⇒ M(x)↓

" c(m, x) ≤ k is decidable

" c (#M ◦M ′$ , x) ≤ c (#M$ ,M ′(x)) + c (#M ′$ , x)

" c (#M ×M ′$ , 〈x ! y〉) ≤ c (#M$ , x) + c (#M ′$ , y)

" c (#read$ , x) = |x |


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity measure

Definition (continued)

The (abstract) complexity measure associated with c is afunction c : N→ N

N which maps the program m ∈ N tothe induced partial function

cm : N −→ N

n 7−→∨

|x |=nc(m, x)


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Time is a complexity measure

Lemma 1

˜time is a complexity measure with

˜time (#M ◦M ′$ , x) =˜time (#M$ ,M ′(x)) + ˜time (#M ′$ , x)

˜time (#M ×M ′$ , 〈x ! y〉) =

max{ ˜time (#M$ , x) , ˜time (#M ′$ , y)

}


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Space is a complexity measure

Lemma 2

˜space is a complexity requirement with

˜space (#M ◦M ′$ , x) =

max{space (#M$ ,M ′(x)) , space (#M ′$ , x)

}

˜space (#M ×M ′$ , 〈x ! y〉) =

space (#M$ , x) + space (#M ′$ , y)


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Nonexamples

" c(m, x) = u(m, x) is not a complexity measurebecause the predicate U(m, x) = k is not decidable.

" c(m, x) = |m| · |x | is not a complexity measurebecause it is defined even when {m} (x) is not.


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

The additivity induces the polynomial bounds

" If p[q] denotes a program p that calls a procedure qthen

cp[q](n) ≤ cp(n) · cq(n)

follows from the additivity of c, because p can call qat most cp(n) times.

" The exponential growth of complexity occurs wheneach procedure call makes further procedure calls,like in the exhaustive search algorithms.


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Abstract complexity classes

Definition

For any complexity measure c define

DCLASSc(f ) ={L(M) ⊆ Σ! | M ∈ DTM . c#M$ ≤ f

}

NCLASSc(f ) ={L(M) ⊆ Σ! | M ∈ NTM . c#M$ ≤ f

}

Pc =∞⋃

k=0DCLASSc(nk )

NPc =∞⋃

k=0NCLASSc(nk )


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Concrete complexity classes

Definition

P = Ptime =∞⋃

k=0DTIME(nk )

NP = NPtime =∞⋃

k=0NTIME(nk )

PSPACE = Pspace =∞⋃

k=0DSPACE(nk )

NPSPACE = NPspace =∞⋃

k=0NSPACE(nk )


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Concrete complexity classes

Remarks

" Props. 4 and 5 imply

PSPACE = NPSPACE = co-NPSPACE

" The quest forP # NP

is ongoing.


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

The hierarchy of complexity classes

Recall for RE and REC

For

K(x) =

1 if U(x , x)↓0 otherwise

we provedK ∈ RE \ REC


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

The hierarchy of complexity classes

Now for CLASSc(f )

We show that for

Kf (x) =

1 if c(x , x) ≤ f (|x |)0 otherwise

holds

Kf ∈ CLASSc(O(f · logk f )

)\ CLASSc (O(f ))


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Proof of Kf $ CLASSc(f )

Define

Kf (x) =

1 if Kf (x) = 0↑ if Kf (x) = 1

and note

Kf(#Kf $

)⇐⇒ c

(#Kf $ , #Kf $

)≤ f(∣∣∣∣#Kf $

∣∣∣∣)

=⇒ c(#Kf $ , #Kf $

)↓

⇐⇒ Kf(#Kf $

)↓

⇐⇒ ¬Kf(#Kf $

)


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Proof of Kf $ CLASSc(f )

HenceKf(#Kf $

)=⇒ ¬Kf

(#Kf $

)

¬Kf(#Kf $

)

c(#Kf $ , #Kf $

)> f(∣∣∣∣#Kf $

∣∣∣∣)

Kf $ CLASSc(f )


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP


Theorem

For every f : N→ N and every complexity measure cthere is k ≥ 1 and a language

Kf ∈ CLASSc(O(f · logk f )

)\ CLASSc (O(f ))


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP


Corollary

For every f and c there is some k ≥ 1

" CLASSc (O(f )) " CLASSc(O(f k+1)

)

" CLASSc(O(logk )

)" Pc " CLASSc (O(exp))


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP


Corollary

For every f and c there is some k ≥ 1

" CLASSc (O(f )) " CLASSc(O(f k+1)

)

" CLASSc(O(logk )

)" Pc " CLASSc (O(exp))

" NCLASSc (O(f )) " NCLASSc(O(f k+1)

)

" NCLASSc(O(logk )

)" NPc " NCLASSc (O(exp))


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity bounds

Question

Which other functions besides polynomials, logarithmsand exponentials are meaningful as complexity bounds?

Answer

" Pathology: Gap theorem

" Health: c-costructible functions = functions for whichc is the universal machine.


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity bounds

Gap Theorem

For every complexity measure c there is a recursivefunction r : N→ N such that

c#M$(x) > r (|x |) =⇒ c#M$(x) > 2r(|x |)

for all M ∈ TM and all but finitely many x .

Hence

CLASSc (r (|x |)) ⊇ CLASSc(2r(|x |)

)


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity bounds

Definition

A function f : N→ N is c-constructible if there is there is1F ∈ DTM such that f (|x |) = c (#F$ , x), for all x ∈ Σ!.

1can always assume multitape


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity bounds

Lemma

If f : N→ N is c-constructible then for any g : N→ N holds

CLASSc(f ) ⊆ CLASSc(f ) ⇐⇒ f ≤ g


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity bounds

Corollary

The recursive function r from Gap Theorem is notc-constructible for any c.


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity bounds

Corollary

The recursive function r from Gap Theorem is notc-constructible for any c.

(because CLASSc(2r ) ⊆ CLASSc(r) but not 2r ≤ r )


Dusko Pavlovic

Introduction

Time and space


Classes


Complexity bounds

Reducibility

P vs NP

Complexity bounds

Comments

" c-constructible functions are just those for which cplays the role of universal TM.

" c-complexity bounds should always bec-constructible

" to avoid Gap Theorem


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

ReducibilityReductions

Cook-Levin

P vs NP

Outline



Abstract complexity


Reductions

Cook-Levin: NP-completeness



Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Reductions

Definition

Let A,B,R ∈ TM. We say that A is reducible to B by Rand write

AR−→ B

if for all inputs x ∈ Σ! holds

" A(x)↓ ⇐⇒ B(R(x))↓

" A(x)↓ =⇒ A(x) = B(R(x))


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Reductions

Definition

Let C ⊆ TM. We say that A is C-reducible to B when

A −→CB ⇐⇒ ∃R ∈ C. A

R−→ B


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Reductions

Definition

A and B are C-equivalent when

A←→CB ⇐⇒ ∃R ∈ C. A

R−→ B ∧ B

R−→ A


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Reductions

Definition

Let A,B ∈ ℘Σ! and C ⊆ ℘Σ!. The notions of reducibilityand equivalence extend by

A −→CB ⇐⇒ ∃A,B ∈ TM ∃C ⊆ TM .

A = L(A) ∧ B = L(B) ∧C = L[C] ∧A −→CB

A←→C

B ⇐⇒ ∃A,B ∈ TM ∃C ⊆ TM .

A = L(A) ∧ B = L(B) ∧C = L[C] ∧A←→CB


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Reductions

Examples

Let D ⊆ TM be any class containing constants andidentities and C a complexity measure. Then

" M −→DU for allM ∈ TM

" F −→DC iff F is a C-constructible function.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Properties of −→C

Proposition 6

(a) preordering

A −→CA and A −→

CB ∧ B −→

CD =⇒ A −→

CD

(b) lower closed in C

A −→CB ∧ B ∈ C =⇒ A ∈ C


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Equivalence

Proposition 7

Any two complexity measures C and D are recursivelyequivalent:

C ←→REC

D


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Hardness and completeness

Definition

Let B ∈ ℘Σ! and L,C ⊆ ℘Σ!. We say that B is L-hardover C

X ∈ L =⇒ X −→CB

We write L −→CB.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP


Definition

Let B ∈ ℘Σ! and L,C ⊆ ℘Σ!. We say that B isL-complete over C if

" L −→CB

" B ∈ L


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP


Examples

Let D be any class of languages containing constantsand identities (i.e. the diagonal languages {〈x , x〉}) and Ca complexity class.

" The halting language H = {〈p, x〉 | U(p, x)↓} isRE -complete over D.

" The language consisting of programs in someC-bounded class is complete over D for the family ofC-constructible functions.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP


Proposition 8

If C ⊆ ℘Σ! is closed under composition, i.e.

M ,N ∈ C =⇒ M ◦ N ∈ C

then(∃B ∈ C. L −→

CB)

=⇒ L ⊆ C


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP


Corollary

(∃B ∈ P. NP −→

PB)

=⇒ NP = P


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP


Trouble

If C ⊆ REC is closed under composition and conditionals,i.e.

M ,N ∈ C =⇒ M ◦ N ∈ C∧ (if M then N) ∈ C

then every nonempty B " Σ! is C-hard over C

C −→CB


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP


Trouble

If C ⊆ REC is closed under composition and conditionals,i.e.

M ,N ∈ C =⇒ M ◦ N ∈ C∧ (if M then N) ∈ C

then every nonempty B " Σ! is C-hard over C

C −→CB

(All main classes are closed under composition and conditionals.)


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Moral

Complexity classes

" cannot be meaningfully studied in terms ofreducibility over themselves, but

" over a smaller class, say

L = DSPACE(log)


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

Convention

By default

A −→ B means A −→LB


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

NP-completeness

Cook-Levin Theorem

NP −→ SAT


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

NP-completeness


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity


Cook-Levin

P vs NP

NP-completeness Complexity 3:Feasibility

Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

OutlineIntroduction: Gödel’s letter


Abstract complexity



Projections

coNP

Summary


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Recall: Projections

Definition

For any family of languages FAM ⊆ ℘(Σ! × Σ!) theprojection family is defined

∃FAM = {L∃ ⊆ Σ! | L ∈ FAM}

where L∃ = {y ∈ Σ! | ∃x . 〈x , y〉 ∈ L}.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Recall: RE = ∃REC

Proof of ∃REC ⊆ RE

M∃(y) = µx . 1 −M(x , y) = 0


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Recall: RE = ∃REC

Proof of ∃REC ⊆ RE

M∃(y) = µx . 1 −M(x , y) = 0


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Recall: RE = ∃REC

Proof of RE ⊆ ∃REC

Kleene’s recursive predicate T : Σ!4 → {0, 1}

U(m, x) = y ⇐⇒ ∃r . T (m, x , y , r)


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Polynomial projections

Definition

For any family of languages FAM ⊆ ℘(Σ! × Σ!) theprojection family is defined

∃pFAM = {L∃q ⊆ Σ! | L ∈ FAM ∧ q ∈ Z[n]}

where L∃q = {y ∈ Σ! | ∃x . |x | ≤ q(|y |) ∧ 〈x , y〉 ∈ L}.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

NP = ∃pP

Proof of ∃pP ⊆ NP


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

NP = ∃pP

Proof of NP ⊆ ∃pP


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Interpretation


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Interpretation Complexity 3:Feasibility

Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Why is P " NP harder than REC " RE?

RE # REC;

some X ∈ ∃REC \ REC

NP # P;

some X ∈ ∃P \ P


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary


K ∈ ∃REC \ REC" does not show thatfinding witnesses ishard

" shows the decider ofhalting may not halt

" ∃ says that there isa halting run

X ∈ ∃P \ P" must show thatfinding witnesses ishard

" all NP deciders halteventually

" ∃ says that theyhalt in poly time


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary


K ∈ ∃REC \ REC" does not show thatfinding witnesses ishard

" shows the decider ofhalting may not halt

" ∃ says that there isa halting run

X ∈ ∃P \ P" must show thatfinding witnesses ishard

" all NP deciders arestopped in poly time

" ∃ says that thereis a deciding run


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

RE = ∃REC

B ∈ RE;

there is A ∈ REC.(y ∈ B ⇐⇒ ∃x ∈ Σ∗. 〈x , y〉 ∈ A

)


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

coRE = ∀REC

B ∈ coRE;

there is A ∈ REC.(y ∈ B ⇐⇒ ∀x ∈ Σ∗. 〈x , y〉 ∈ A

)


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

NP = ∃pP

B ∈ coNP;

there are A, q ∈ P.

y ∈ B ⇐⇒ ∃x ∈ Σ∗.

|x | ≤ q(|y |)⇓

〈x , y〉 ∈ A


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

coNP = ∀pP

B ∈ coNP;

there are A, q ∈ P.

y ∈ B ⇐⇒ ∀x ∈ Σ∗.

|x | ≤ q(|y |)⇓

〈x , y〉 ∈ A


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Interpretation Complexity 3:Feasibility

Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

Interpretation

" NP = B = set of satisfiable properties" if 〈x , y〉 ∈ A means "program x satisfies y"" then for every property y ∈ B = ∃pA there issome program x in A that satisfies y

" NP # P because it is hard to find it.

" coNP = B = set of satisfied properties" if 〈x , y〉 ∈ A means "program x satisfies y"" then for every property y ∈ B = ∃pAall programs x in A satisfy y

" coNP # P because it is hard to prove that


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

NP ∩ coNP

" although RE ∩ coRE = REC" if y ∈ L ∈ RE then ML(y) will halt and prove y ∈ L" if y $ L ∈ coRE then ML(y) will halt and prove y $ L.

" it is believed that NP ∩ coNP & P, because having" a family of programs A where y ∈ B ⇐⇒ some xsatisfies it with 〈x , y〉 ∈ A, and

" a family of programs C where y ∈ B ⇐⇒ every zsatisfies it with 〈z , y〉 ∈ C

does not yield feasible way to eliminate search forbuilding x and eliminating z.


Dusko Pavlovic

Introduction

Time and space

Abstractcomplexity

Reducibility

P vs NPProjections

Hierarchy

Summary

The Inner Eggs

complexity 3: outline feasibility...complexity 3: feasibility dusko pavlovic introduction time and...

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