Real Parameterized and 2 Order Complexity Theory: Order Complexity - - PowerPoint PPT Presentation

SLIDE 1

Real Parameterized and 2 Real Parameterized and 2nd

nd

Order Complexity Theory: Order Complexity Theory: From Computability in Analysis From Computability in Analysis to Numerical Practice to Numerical Practice

Martin Ziegler Martin Ziegler

SLIDE 2

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

Folklore: Folklore: Folklore: Folklore: For x∈ the following are equivalent: a) x has a decidable binary expansion b) x has a recursive signed-digit expansion c) There exists a recursive sequence (an) ⊆ s.t. | x – an/2n+1 | ≤ 2-n d) There exist recursive sequences (pn),(qn)⊆ s.t. supn pn = x = infn qn

i) only uniformly ii) no running time bound

Folklore: Folklore: Folklore: Folklore: Every computable f:[0;1]→ with f(0)·f(1)<0 has a computable root.

numerics / iRRAM

Obstacles to practice:

SLIDE 3

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

• n [0;1]!

Function f:[0,1]→ computable computable if some TM can, on input of n∈ and of

(am)⊆ with |x-am/2m+1|≤2-m

• utput b∈ with |f(x)-b/2n+1|≤2-n.

=:ρ.name ≡ ρsd

p p

in time in time t

t( (n n) )

iRRAM

a) + +, , × ×, , exp

exp

b) f

f( (x x) )≡ ≡∑

∑n

n∈ ∈L L 4

4-

• n

n iff L

L⊆ ⊆{ {0 0, ,1 1} }*

*

decidable in time in time t

t( (n n) )

• i)

i) If If ƒ

ƒ computable

computable ⇒ ⇒ continuous. continuous. ii ii) ) If If f

f computable

computable in in time time t

t( (n n) ),

, then then O O(

(t

t(

(O(

O(n n) ))

))

) is

is a a modulus modulus of uniform

• f uniform continuity

continuity of

• f f

f.

.

polytime polytime. c) 1/

1/ln(e ln(e/ /x x) ) not polytime.computable

c) sign

sign, , Heaviside Heaviside not computable

SLIDE 4

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

• L

L⊆ ⊆{0,1}* {0,1}* is

is verifiable verifiable in in polyn

• polyn. time

. time if if

L L = = {

{ x

x∈ ∈{0,1} {0,1}n

n |

| n n∈ ∈ , , ∃ ∃y y∈ ∈{0,1 {0,1} }q

q( (n n) ) :

: 〈 〈x x, ,y y〉∈ 〉∈V V }

}

for for some some V

V∈ ∈ and

and q

q∈ ∈

[

[N N] ].

.

• TM

TM

decides

decides set set L

L⊆ ⊆{ {0 0, ,1 1}* }* if

if

• on
• n inputs

inputs x

x∈ ∈L L prints

prints 1 1 and and terminates terminates, ,

• on
• n inputs

inputs x

x∉ ∉L L prints

prints 0 0 and and terminates terminates. .

• runs

runs in in polynom

• polynom. time

. time if if ∃

∃p p∈ ∈

[

[N N]: ]:

• n
• n input

input x

x∈ ∈{ {0 0, ,1 1} }n

n makes

makes at at most most p

p( (n n) ) steps

steps / / uses uses at at most most p

p( (n n) ) bits

bits of

• f memory

memory. . / /space space

Example Example: : L L={ ={ 10 10, , 11 11, , 101 101, , 111 111, , 1011 1011, , 1101 1101, , … …} }

• = {

= { L

L⊆ ⊆{ {0 0, ,1 1}* }* decidable

decidable in in polynomial polynomial time time }

} ⊆ ⊆

• = {

= { L

L

in in polynomial polynomial time time }

} ⊆ ⊆

• := {

:= { L

L decidable

decidable in in polyn polyn. . space space }

} ⊆ ⊆

• = {

= { L

L decidable

decidable in exponential time in exponential time }

}

SLIDE 5

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

!

ƒ:[0;1]→[0;1] polytime computable (⇒ continuous)

• Max:

Max: ƒ ƒ → → Max( Max(ƒ ƒ): ): x x → → max max{ { ƒ ƒ( (t t): ): t t≤ ≤x x } } Max( Max(ƒ ƒ) ) computable in exponential time;

polytime.computable iff

• =
• ∫

∫: : ƒ ƒ → → ∫ƒ ∫ƒ: ( : (x x → → ∫ ∫0

x x ƒ

ƒ( (t t) ) dt dt) ) ∫ƒ ∫ƒ computable in exponential time;

• .complete"
• dsolve

dsolve: C[0;1] : C[0;1]× ×[ [-

• 1;1]

1;1] ∋ ∋ ƒ ƒ → → z z: : ż ż( (t t)= )=ƒ ƒ( (t t, ,z z), ), z z(0)=0 (0)=0.

in general no computable solution z

z( (t t) )

for ƒ∈

ƒ∈C C1

1

• ."complete"

for ƒ∈

ƒ∈C Ck

k

• ."hard"

even even when when restricting restricting to to ƒ∈

ƒ∈C C∞

∞

[Friedman&Ko'82] [Friedman&Ko'82] [Kawamura'10, [Kawamura'10, Kawamura Kawamura et al] et al]

another class between and

SLIDE 6

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

To To every every L

L∈ ∈

• there

there exists exists a a polytime polytime computable computable C

C∞

∞ function

function g

gL

L:[0,1]

:[0,1]→ → s.t.:

s.t.:

[0,1] [0,1]∋ ∋t t→ →max max g gL

L|

|[0,

[0,t t] ] again

again polytime polytime iff iff L

L∈ ∈

• ∋

∋ L L = = {

{ N

N∈ ∈ | | ∃ ∃M M<N <N: : 〈 〈N,M N,M〉 〉 ∈ ∈V V }

},

, V V ∈ ∈

• "Max
• #$"

0.2 0.4 0.6 0.8 1

• 1
• 0.5

0.5 1

t → ∑ ϕ(3tN³-3N²-M)/Nln N

〈N,M〉

N=1

N=2

N=3

N=4

N=5

M=0

M=0

M=0,1,2

M=0..3

t=1 t=½

t=⅓

t=¼

M=1

tln(1/t)

t → ∑ ϕ(2tN²-2N)/Nln N

N

V∈

• ⇔ fV polytime

polytime

V V⊆ ⊆

• fV:

∈V

gL:

φ φ( (t t) = ) = exp( exp(-

• t

t² ²/1 /1-

• t

t² ²) )

C C∞

∞ ' 'pulse'

pulse' function function

polytime polytime computable computable

C C∞

∞

SLIDE 7

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

%

ƒ:[0;1]→[0;1] polytime computable (⇒ continuous)

• Max:

Max: ƒ ƒ → → Max( Max(ƒ ƒ): ): x x → → max max{ { ƒ ƒ( (t t): ): t t≤ ≤x x } } Max( Max(ƒ ƒ) ) computable in exponential time;

polytime.computable iff

• =
• ∫

∫: : ƒ ƒ → → ∫ƒ ∫ƒ: ( : (x x → → ∫ ∫0

x x ƒ

ƒ( (t t) ) dt dt) ) ∫ƒ ∫ƒ computable in exponential time;

• ."complete"
• dsolve

dsolve: C[0;1] : C[0;1]× ×[ [-

• 1;1]

1;1] ∋ ∋ ƒ ƒ → → z z: : ż ż( (t t)= )=ƒ ƒ( (t t, ,z z), ), z z(0)=0 (0)=0.

in general no computable solution z

z( (t t) )

for ƒ∈

ƒ∈C C1

1

• ."complete"

for ƒ∈

ƒ∈C Ck

k

• ."hard"

even when restricting to ƒ∈

ƒ∈C C∞

∞

• ƒ

ƒ ƒ ƒ ƒ ƒ ƒ ƒ

• non

non. . uniform uniform [ [Friedman&Ko Friedman&Ko] ]

• #

#

• [Kawamura'10,

[Kawamura'10, Kawamura Kawamura et al] et al] [ [N.M N.Mü üller ller] ]

SLIDE 8

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

&$

a) natural emergence of multivaluedness (aka non.extensionality) b) Uniform computation may require

• r otherwise 'enriched' representations (TTE)

― which yield canonical C++ declarations c) Parameterized uniform upper complexity bounds (as well.established in Discrete Complexity)

that numerical scientists might be interested in / should be aware of

(Brattka&Z, ) Finding an eigenvector (basis) to a given real symmetric d×d matrix A is computable; becomes computable when knowing Card σ(A).

REAL **diagonalize(int d, int nDistinctEValues, REAL **matrix);

→ ε.semantics of "<"

but +, exp computable in time polynomial

in n on [0;1];

n+k,

independent of independent of x

x

• n
• n

dom dom

• n [0;2k]: + in time polynomial in

exp in time polynomial in n+2k.

• cmp. "feasible real.RAM"

SLIDE 9

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

'( &&

a) A partial F:⊆{0,1}ω→{0,1}ω is computable computable in in time time t:→

τ=F(σ)

b) For spaces X,Y equipped with representations α,β, (multivalued partial) f:⊆X ⇒Y is (

(α α, ,β β) )–

–computable computable in time in time t

t

if it admits an (α,β)–realizer F computable in time t. c) A parameter parameter to a space X with representation α is a mapping k:dom(α)→. d) For X,Y spaces with representations α,β and parameters

k,ℓ,

it admits an (α,β)–realizer F and a polynomial p such that a Type.2 machine can compute F on inputs σ within

p(n+k(σ)) steps

) Some computably reasonable (e.g. admissible) representations induce trivial notions of complexity. [Weihrauch'03] and [Schröder'04] have devised (meta.) conditions on representations to avoid such degeneracies. call f as above fully fully polytime polytime (

(α α, ,k k, ,β β, ,ℓ ℓ) )–

–computable computable if if a Type.2 machine can convert σ∈dom(F) to s.t. the n.th symbol of τ appears within t(n) steps. call and ℓ◦F≤p◦k holds.

SLIDE 10

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

• access

access

&$

Evaluation Eval:(f,x)→f(x) a) requires ≥µ(n) steps, µ:→ mod. of continuity to f. "Parameter" "Parameter" µ

µ( (f f) ) is

is not not . .valued valued but but

• .

.valued valued! ! b) Even restricted to the compact domain 1 := { f:[0;1]→[0;1] 1.Lipschitz } there exists no representation δ:⊆{0,1}ω→1 rendering Eval computable in exponential time.

Parameter to representation α: a mapping k:dom(α)→. Representations α,β and parameters k,ℓ: (α,β)–realizer F required computable on inputs σ within poly(n+k(σ)) steps.

2-n

≈2n-1 'hats'

≥22.1 functions pairwise differing when evaluating up to error 2. but only 2() different initial segments of δ.names that can be read within t(n) steps. q.e.d.

• ?

?

(Arzela.Ascoli)

SLIDE 11

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

• access

access

&$

Evaluation Eval:(f,x)→f(x) a) requires ≥µ(n) steps, µ:→ mod. of continuity to f. Parameter Parameter µ

µ( (f f) ) is

is not not . .valued valued but but

• .

.valued valued! ! b) Even restricted to the compact domain 1 := { f:[0;1]→[0;1] 1.Lipschitz } (Arzela.Ascoli) there exists no representation δ:⊆{0,1}ω→1 rendering Eval computable in exponential time. Kawamura&Cook'10 (based on Cook&Kapron'96): Kawamura&Cook'10 (based on Cook&Kapron'96): Remedy Remedy to a): to a): . .order

• rder complexity

complexity theory theory Remedy Remedy to b): to b): . .order

• rder representations

representations

SLIDE 12

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

*# &$

Evaluation Eval:(f,x)→f(x) a) requires ≥µ(n) steps, µ:→ mod. of continuity to f. Parameter Parameter µ

µ( (f f) ) is

is not not . .valued valued but but

• .

.valued valued! !

• A

A second second. .order

• rder polynomial

polynomial P

P( (n n, ,λ λ) ) is

is a a term term built built from from

+ +,

, ×

×, integer constants and (first

, integer constants and (first. .order) variable

• rder) variable n

n (rang

(rang. . ing ing over

• ver

) ) and second and second. .order variable

• rder variable λ

λ (ranging

(ranging over

• ver
• ).

).

a) Second.order polynomials are closed under kinds of composition

(Q◦P)(n,λ) := Q(P(n,λ),λ)

and (Q▫P)(n,λ) := Q(n,P(·,λ)) b) For λ∈[n], P(n,λ) is an ordinary polynomial. λ³(λ(n²)·n+λ²(n))+n17

SLIDE 13

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

| |ψ ψ| |(

(n n) )

:=| :=|ψ ψ( (1 1n

n)|

)|

*#

Recall that ({0,1}*){0,1}* denotes the set { ψ:{0,1}* →{0,1}* } For F:⊆({0,1}*){0,1}*→({0,1}*){0,1}*, oracle Turing machine ? computes computes F if ψ on input v∈{0,1}* outputs w=F(ψ)(v). Call ψ∈({0,1}*){0,1}* length length. .monotone monotone if |ψ(v)|≤|ψ(w)| ∀|v|≤|w|. ? runs in 2 2nd

nd.

.order

• rder polytime

polytime if, for some 2nd.order poly. nomial P, ψ on input v∈{0,1}* makes ≤P(|v|,|ψ|) steps.

• A

A second second. .order

• rder polynomial

polynomial P

P( (n n, ,λ λ) ) is

is a a term term built built from from

+ +,

, ×

×, integer constants and (first

, integer constants and (first. .order) variable

• rder) variable n

n (rang

(rang. . ing ing over

• ver

) ) and second and second. .order variable

• rder variable λ

λ (ranging

(ranging over

• ver
• ).

). {0,1}{0,1}* ∋ Q can be computed in 2nd.order polytime by a non.determin.

• racle machine

→ ( {0,1}* ∋ v → ∃u∈{0,1}|v|: Q〈v,u〉 )

but by a deterministic one.

provably

SLIDE 14

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

{ length.monotone ψ }

• access

access

*#

A second second. .order

• rder representation

representation of X is a surjective partial mapping ∆:⊆LM→X

A ( (R R, ,Γ Γ) ). .realizer realizer of f:⊆X⇒Y is a mapping F:LM→LM s.t.…

Even on compact 1 = { f:[0;1]→[0;1] 1.Lipschitz } there is no representation δ:⊆{0,1}ω→1 rendering Eval computable in subexponential time.

For F:⊆({0,1}*){0,1}*→({0,1}*){0,1}*, oracle Turing machine ? computes computes F if ψ on input v∈{0,1}* outputs w=F(ψ)(v). Call ψ∈({0,1}*){0,1}* length length. .monotone monotone if |ψ(v)|≤|ψ(w)| ∀|v|≤|w|. ? runs in 2 2nd

nd.

.order

• rder polytime

polytime if, for some 2nd.order poly. nomial P, ψ on input v∈{0,1}* makes ≤P(|v|,|ψ|) steps.

SLIDE 15

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

+#

a) An ordinary representation δ:⊆{0,1}ω→X induces induces a 2nd.order representation ∆ where ψ:{0,1}*→{0,1} is a ∆.name of x∈X iff

(ψ(1n))n is a δ.name of x.

b) a ρ ρ

.

.name name of f∈C[0;1] as a ψ∈LM s.t.

| f( bin(v)/2|v|+1 ) – bin(ψ(v))/2|v|+1 | ≤ 2-|v|

A 2 2nd

nd.

.order

• rder representation

representation of X is a surjective ∆:⊆LM→X Call ψ∈({0,1}*){0,1}* length length. .monotone monotone if |ψ(v)|≤|ψ(w)| ∀|v|≤|w|. ? runs in 2 2nd

nd.

.order

• rder polytime

polytime if, for some 2nd.order poly. nomial P, ψ on input v∈{0,1}* makes ≤P(|v|,|ψ|) steps.

, , a) a) Polytime Polytime δ

δ.

.computability computability is is uniformly uniformly equivalent equivalent to 2 to 2nd

nd.

.order

• rder polytime

polytime ∆

∆.

.computability computability b) Evaluation b) Evaluation (

(f f, ,x x) )→ →f f( (x x) ) is

is (

(ρ ρ

×

×Ρ Ρ, ,Ρ Ρ) ).

.computable computable. .

SLIDE 16

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

b) Evaluation b) Evaluation (

(f f, ,x x) )→ →f f( (x x) ) is

is (

(ρ ρ

×

×Ρ Ρ, ,Ρ Ρ) ).

.computable computable. .

and µ a modulus

• f continuity of f

+#

b) a ρ ρ

.

.name name of f∈C[0;1] as a ψ∈LM s.t.

| f( bin(v)/2|v|+1 ) – bin(ψ(v))/2|v|+1 | ≤ 2-|v|

c) a ρ ρ

⊓

⊓Lip Lip. .name name of f∈Lip2ℓ[0;1] as

{0,1}* ∋ v → 1ℓ 0 ψ(v)

for a ρ.name ψ of f. d) A [ [ρ→ρ ρ→ρ] ]. .name name of f∈C[0;1]

{0,1}* ∋ v → 1µ(|v|) 0 ψ(v)

for ρ.name ψ of f.

? runs in 2 2nd

nd.

.order

• rder polytime

polytime if, for some 2nd.order poly. nomial P, ψ on input v∈{0,1}* makes ≤P(|v|,|ψ|) steps.

, , c) Evaluation on c) Evaluation on Lip[0;1]

Lip[0;1]

2 2nd

nd.

.order

• rder

polytime polytime ( (ρ ρ

⊓

⊓Lip Lip×Ρ ×Ρ, ,Ρ Ρ) ). .computable computable d) and 2 d) and 2nd

nd.

.order

• rder polytime

polytime ([ ([ρ→ρ ρ→ρ] ]×Ρ ×Ρ, ,Ρ Ρ) ). . computable computable on

• n C[0;1]

C[0;1].

.

SLIDE 17

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

*-.

• P.Oliva: "Polynomial.time Algorithms from Ineffective Proofs", pp.128.137

in (LiCS'03)

• K.Weihrauch: "Computational Complexity on Computable Metric Spaces",

pp.3.21 in ! " vol./0 (2003).

• M.Schröder: "Spaces Allowing Type.2 Complexity Theory Revisited",

pp.443.459 in ! " vol.1 (2004).

• M.Braverman: "On the Complexity of Real Functions", pp.155.164 in

#$%&' (FOCS'05).

• Z.Du, C.K.Yap: "Uniform Complexity of Approximating Hypergeometric

Functions with Absolute Error", pp.246.249 in (%! (ASCM 2005)

• N.Müller, X.Zhao: "Complexity of Operators on Compact Sets", pp.101.119 in

)% (CCA'07), vol.++ (2008)

• K..I.Ko, F.Yu: "On the Complexity of Convex Hulls of Subsets of the Two.

Dimensional Plane", pp.121.135 in *vol.++ (2008)

• R.Rettinger: "Lower Bounds on the Continuation of Holomorphic Functions",

pp.207.217 in +') %(CCA 2008), ENTCS vol.++

• R.Rettinger: "Towards the Complexity of Riemann Mappings", in

$')% (CCA 2009)

SLIDE 18

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

*-.

• G.Hotz: "On in Polynomial Time Approximable Real Numbers and Analytic

Functions", pp.155.164 in ',- ./ %/ , Vieweg+Teubner (2009).

• A.Kawamura: "Lipschitz Continuous Ordinary Differential Equations are

Polynomial.Space Complete", pp.305.332 in ) vol.0+ (2010)

• O.Bournez, D.S.Graça, A.Pouly: "Solving Analytic Differential Equations in

Polynomial Time over Unbounded Domains", pp.170.181 in 0$ !&' (MFCS'2011), Springer LNCS vol.203

• A.Kawamura, S.A.Cook: "Complexity Theory for Operators in Analysis", %!

vol./+ (2012), article 5.

• A.Kawamura, H.Ota, C.Rösnick, M.Z.: "Computational Complexity of

Smooth Differential Equations", pp.578.589 in 0( !&'(MFCS'2012), Springer LNCS vol.3/2/

• C.Spandl: "Computational Complexity of Iterated Maps on the Interval",

pp.1459.1477 in ! vol.4+4 (2012).

• H.Férée, W.Gomaa, H.Hoyrup: "Query Complexity of Real Functionals",
• Proc. 28th ACM/IEEE Symp. on Logic in Computer Science (LiCS'2013).
• U.Brandt, K.Ambos.Spies, M.Z.: "Real Benefit of Promises and Advice",

pp.1.11 in 1' (CiE'2013).

SLIDE 19

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

!55

+ hardware support / large data / high.dim matrices − − − − heuristics, ad.hoc approaches, unspecified class of permitted inputs, non.guaranteed behavior, vague/ inconsistent semantics, various notions of error, not closed under composition, empirical "proofs" of correctness & performance, const..factor acceleration

nag_opt_one_var_deriv nag_opt_one_var_deriv (e04bbc) (e04bbc) normally normally computes computes a a sequence sequence of x

• f x values

values which which tend tend in in the the limit limit to a to a minimum minimum of F x

• f F x subject

subject to to the the given given bounds bounds

"The iterative methods used to solve problems of nonlinear "The iterative methods used to solve problems of nonlinear programming differ according to whether they evaluate programming differ according to whether they evaluate Hessians, gradients, or only function values. While evaluating Hessians, gradients, or only function values. While evaluating Hessians and gradients improves the rate of convergence, Hessians and gradients improves the rate of convergence, such evaluations increase the such evaluations increase the computational complexity computational complexity (or computational cost) of each iteration. In some cases, (or computational cost) of each iteration. In some cases, the the computational complexity computational complexity may be excessively high." may be excessively high."

• Math. Optimization
• Math. Optimization

Wikipedia Wikipedia

SLIDE 20

Real Parameterized and 2nd-Order Complexity Theory: From Computability in Analysis to Numerical Practice

TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler

• $

$ 6$ 6$ ! !

guaranteed error bounds

high accuracy test of conjectures in classical analysis

fully specified algorithms with runtime bounds consistent semantics closed under composition

modular software development of certified libraries

concepts such as multivaluedness and enrichment/information theory (TTE)

canonical interface declaration of implementation

Alan Turing Alan Turing / / also a also a Numerical Numerical Scientist! Scientist! Let's collaborate with, and approach, e.g. the Let's collaborate with, and approach, e.g. the

Interval Interval and

and Computer

Computer-

• Assisted Proof

Assisted Proof community

community

' '# # # #6 6

Bloch, Feigenbaum, Bremble Bloch, Feigenbaum, Bremble. .Hilbert Hilbert

Kreinovich,Yap Kreinovich,Yap Revol, Plum, Revol, Plum, v.Gudenberg? v.Gudenberg?