•Logicians Tarski, Alonzo Church (PhD advisor)

Computability and Complexity

in Analysis

Martin Ziegler

IRTG 1529 IRTG 1529

Martin Ziegler

Contents

••inin--/computability,/computability,

•Halting problem, •Halting problem,

••Reduction, Reduction, enumerabilityenumerability

•computability of real numbers•computability of real numbers

•Specker•Specker sequence, sequence,

in-in-/effective convergence/effective convergence Real

Real functionfunction computabilitycomputability

•nonuniform•nonuniform vs. uniformvs. uniform

•Main Theorem•Main Theorem

•modulus•modulus of of continuitycontinuity

•Computable•Computable WeierstrassWeierstrass

•power•power seriesseries

•computable•computable joinjoin, max, max, , ∫∫

•uncomputable•uncomputable: : argmax

argmax, , roots,roots, ∂ ∂_xx

•Wave •Wave EquationEquation

Minicourse

Minicourse DiscreteDiscrete ComplexityComplexity

••bitbit--modelmodel of of computationcomputation

•asymptotic•asymptotic runtimeruntime/memory/memory

•example•example algorithms: algorithms: SieveSieve, , Euler

Euler CircuitCircuit, Edge Cover, Edge Cover

•SAT, 3SAT, •SAT, 3SAT, VertexVertex Cover, Cover, Hamilton

Hamilton CiruitCiruit, TSP, TSP

•• polynomialpolynomial reductionreduction

•• 4SAT 4SAT ≤≤ 3SAT 3SAT ≤≤ VertexVertex CoverCover

•• NPNP-completeness-completeness

Real function complexity Real function complexity

•polyn•polyn. continuity, . continuity, 1/ln(e/1/ln(e/xx))

•complexity of max•complexity of max

•Laplace•Laplace/Poisson /Poisson EqEq..

••iRRAMiRRAM

Martin Ziegler

decision decision problem problem

•Logicians Tarski, Alonzo Church (PhD advisor)

•Kurt Gödel (1931): There exist arithmetical

statements which are true but cannot be proven so.

Alan M. Turing 1936

• • first first scientific scientific calculations calculations on digital on digital computers computers

• • What What are are its its fundamental fundamental limitations limitations ? ?

• • Uncountably Uncountably many many P P ⊆ ⊆

• • but but countably countably many many ' ' algorithms algorithms ' '

• • Undecidable Undecidable Halting Problem H : : No No algorithm algorithm B B can can always always correctly correctly answer answer the the following following question question

Given

Given 〈 〈 A, A, x〉 x 〉 , , does does algorithm algorithm A A terminate terminate on on input input x x ? ? Proof (by contradiction): Consider Consider algor algor . . B B ' ' that that , , on on input input A _A , , executes executes B B on on 〈 〈 A,A A,A 〉 〉

1941

Halting

Halting Problem Problem H H

Proof

Proof ( ( by by contradiction contradiction ): ):

A A

x x B B + +

− −

A A

A A

B' B'

∞ ∞

How How does does B B ' ' behave behave on on B' B' ? ? answer

answer , , loops loops infinitely infinitely . .

and,

and, upon upon a positive a positive

simulator

simulator / / interpreter interpreter B B ? ? B' B' B' B'

B' B'

Martin Ziegler

Formalities & Tools

''Definition:Definition:' ' AlgorithmAlgorithm

A A

decidesdecides setset

L L ⊆ ⊆ { { 0 0 , , 1 1 }* }*

ifif

•• on on inputsinputs

x x ∈ ∈ L L

printsprints 11 and and terminatesterminates,,

•• on on inputsinputs

x x ∉ ∉ L L

printsprints 00 and and terminatesterminates..

A A

semisemi--decidesdecides ifif terminatesterminates on on

x x ∈ ∈ L L

,, elseelse divergediverge..

all finite all finite

binary binary sequences sequences e.ge.g. "Turing . "Turing

machine machine""

Consider

Consider algor algor . . B B ' ' that that , on , on input input A _A , , executes executes B B on on

〈 〈 A,A A,A 〉 〉 and, and, upon upon a positive a positive answer answer , , loops loops infinitely infinitely . .

countable countable!! Techniques

Techniques:: a) a) simulationsimulation c) c) dovetailingdovetailing

Theorem:

Theorem:

L L

decidabledecidable iffiff bothboth

L L , , L L

^CC semisemi-decidable-decidable Infinite

Infinite

L L ⊆ ⊆ { { 0 0 , , 1 1 }* }*

isis semisemi-decidable-decidable iffiff

L L =range( =range( f f ) )

forfor somesome computablecomputable injectiveinjective

f f : : N N → → { { 0 0 , , 1 1 }* }*

b) b) diagonalizationdiagonalization egeg. . UU={ algorithms={ algorithms } } ×× { { inputsinputs }} Universes

Universes UU otherother thanthan

{ { 0 0 , , 1 1 }* }*

((e.ge.g. . NN):): encodeencode..

Halting

Halting Problem Problem H H only only semi semi - - decidable decidable

Hilbert Hotel Hilbert Hotel

d) d) reductionreduction (in/(in/outputoutput translationtranslation))

Martin Ziegler

Some Undecidable Problems

''Definition:Definition:' ' AlgorithmAlgorithm

A A

decidesdecides setset

L L ⊆ ⊆ { { 0 0 , , 1 1 }* }*

ifif

•• on on inputsinputs

x x ∈ ∈ L L

printsprints 11 and and terminatesterminates,,

•• on on inputsinputs

x x ∉ ∉ L L

printsprints 00 and and terminatesterminates..

Techniques

Techniques:: a) a) simulationsimulation c) c) dovetailingdovetailing

b) b) diagonalizationdiagonalization

For For

L,L' L,L' ⊆ ⊆ { { 0 0 , , 1 1 }* }*

writewrite

L L ≼ ≼ L' L'

ifif therethere isis a computablea computable

f f : : { { 0 0 , , 1 1 }* }* → → { { 0 0 , , 1 1 }* }*

such such thatthat

∀ ∀ x x : : x x ∈ ∈ L L ⇔ ⇔ f f ( ( x x ) ) ∈ ∈ L L '. '.

a) a)

L' L'

decidabledecidable ⇒⇒ so so

L L

. . b) b)

L L ≼ ≼ L' L' ≼ ≼ L'' L''

⇒⇒

L L ≼ ≼ L'' L''

Universes

Universes UU otherother thanthan

{ { 0 0 , , 1 1 }* }*

((e.ge.g. . NN):): encodeencode.. d) d) reductionreduction (in/(in/outputoutput translationtranslation)) Halting

Halting problemproblem: :

H H = { = { 〈 〈 A A , , x x 〉 〉 : : A A

terminatesterminates onon

x x } }

Hilbert's

Hilbert's 10th:10th: TheThe followingfollowing setset isis undecidableundecidable::

{ { 〈 〈 p p 〉 〉 | | p p ∈ ∈

N_N

[ [ X X

₁₁

, , … … X X

_nn

], ], n n ∈ ∈

N_N

, , ∃ ∃ x x

₁₁

… … x x

_nn

∈ ∈ N N p p ( ( x x

₁₁

, , … … x x

_nn

)=0 } )=0 }

Word Problem

Word Problem forfor finitelyfinitely presentedpresented groupsgroups Mortality

Mortality Problem Problem forfor twotwo 2121××21 21 matricesmatrices Homeomorphy

Homeomorphy of 2 finite of 2 finite simplicialsimplicial complexescomplexes

Martin Ziegler

integer integer

Exercise Questions

Which

Which of of thethe followingfollowing areare unun--/semi/semi--//decidabledecidable?? a) a) GivenGiven an integer, an integer, isis itit a prime a prime numbernumber??

b) b) GivenGiven a finite a finite stringstring overover ++,,××,(,(,,)),,00,1,1,,

X X

₁₁

, , … … X X

_{n n}

isis itit syntacticallysyntactically correctcorrect??

c) c) GivenGiven a a BooleanBoolean formulaformula

ϕ ϕ ( ( X X

₁₁

, , … … X X

_nn

) )

,, doesdoes itit havehave a a satisfyingsatisfying assignmentassignment?? d) d) GivenGiven

M M ∈ ∈

^nn××nn and and

b b ∈ ∈

ⁿⁿ, ,

doesdoes therethere existsexists a a vectorvector

x x

s.t. s.t.

M M · · x x ≤ ≤ b b

?? e) e) GivenGiven an an algorithmalgorithm AA, , inputinput xx, and integer , and integer NN,,

doesdoes AA terminateterminate on on inputinput xx withinwithin NN stepssteps ??

f) Doesf) Does a givena given algorithmalgorithm terminateterminate on all on all inputsinputs?? g) g) DoesDoes givengiven algorithmalgorithm terminateterminate on on somesome inputinput??

realreal