Real Parameterized and 2 nd nd Real Parameterized and 2 Order Complexity Theory: Order Complexity Theory: From Computability in Analysis From Computability in Analysis to Numerical Practice to Numerical Practice Martin Ziegler Martin Ziegler
TECHNISCHE �������� ���� ���������� �������� UNIVERSITÄT DARMSTADT, Martin Ziegler Real Parameterized and 2 nd -Order Complexity Theory: From Computability in Analysis to Numerical Practice 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 ( a n ) ⊆ � s.t. | x – a n /2 n +1 | ≤ 2 - n numerics / iRRAM d) There exist recursive sequences ( p n ),( q n ) ⊆ � s.t. sup n p n = x = inf n q n Folklore: Folklore: Folklore: Folklore: Every computable f :[0;1] → � with f (0)· f (1)<0 has a computable root. i) only ���� uniformly Obstacles to practice: ii) no running time bound
������������� ���������� TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler Real Parameterized and 2 nd -Order Complexity Theory: From Computability in Analysis to Numerical Practice Function f :[0,1] → � computable in time t computable in time ( n ) t ( n ) iRRAM if some TM can, on input of n ∈ � and of ( a m ) ⊆ � with | x-a m /2 m +1 | ≤ 2 - m ≡ ρ sd =: ρ .name p p output b ∈ � with | f ( x ) -b /2 n +1 | ≤ 2 - n . in time t in time ( n ) t ( n ) on [0;1] ! ��������� a) + + , , × × , , exp polytime exp n iff L ≡ ∑ ) ≡ ⊆ { ∑ n L ⊆ b) f 0 , 1 } decidable { 0 , 1 polytime. ( x - n * 4 - } * f ( x ) L 4 ∈ L n ∈ c) 1/ ) not polytime.computable c) sign Heaviside not computable 1/ln(e ln(e/ / x sign, , Heaviside x ) If ƒ ƒ computable ����������� i) i) If computable ⇒ ⇒ continuous. continuous. ����������� ii) ) If If f f computable computable in in time time t , then then ii ) , ( n t ( n ) ( t O ( ) is ) ) is a a modulus modulus of uniform of uniform continuity continuity of of f . O ( O( f . t ( ) ) O( n n )
TECHNISCHE �������� ���������� �������� UNIVERSITÄT DARMSTADT, Martin Ziegler Real Parameterized and 2 nd -Order Complexity Theory: From Computability in Analysis to Numerical Practice ⊆ {0,1}* L ⊆ TM � � decides ⊆ { ���� � L {0,1}* is is verifiable verifiable in in polyn L ⊆ polyn. time . time if if ��� 0 , 1 }* ����������� TM decides set set L { 0 , 1 }* if if ����������� ∈ L x ∈ • on on inputs inputs x L prints n | prints 1 1 and and terminates terminates, ) : , • ∈ {0,1} ∈ � ∃ y ∈ {0,1 〈 x 〉 ∈ V x ∈ n ∈ , ∃ y ∈ : 〈 y 〉 ∈ { x V } } = { � , ( n q ( n ) L = {0,1} n | n {0,1} } q , y x , L ∉ L x ∉ • on on inputs inputs x L prints prints 0 0 and and terminates terminates. . • � and ∈ � ∈ � V ∈ q ∈ for some some V and q . for ] . [ N � [ N ] Example: : L Example 10 , 11 , 101 , 111 , 1011 , 1101 , ={ 10 , 11 , 101 , 111 , 1011 , 1101 , … …} } L ={ ����� �� � ∃ p ∈ � ��� � runs if ∃ p ∈ runs in in polynom polynom. time . time /space space / if [ N ]: � [ N ]: n makes ∈ { � on x ∈ � 0 , 1 } on input input x { 0 , 1 makes at at most most p ) steps steps } n ( n p ( n ) / uses uses at at most most p ) bits bits of of memory memory. . / ( n p ( n ) � � � � = { � � � � ⊆ { L ⊆ 0 , 1 }* ����� ��� { 0 , 1 ����� ��� decidable in in polynomial polynomial time time } = { L }* decidable } ⊆ �� �� �� = { �� �� �� �� �� ⊆ 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 } = { L }
TECHNISCHE !��������� ���������� ������������ UNIVERSITÄT DARMSTADT, Martin Ziegler Real Parameterized and 2 nd -Order Complexity Theory: ƒ :[0;1] → [0;1] polytime computable ( ⇒ continuous) From Computability in Analysis to Numerical Practice [Friedman&Ko'82] [Friedman&Ko'82] ƒ → → Max( ƒ ): x → → max ƒ ( ≤ x Max: ƒ Max( ƒ { ƒ t ≤ • Max: ): x max{ ( t ): t x } } t ): ƒ ) Max( ƒ ) computable in exponential time; Max( polytime.computable iff � � � � = �� �� �� �� even when when even x ƒ restricting restricting • ∫ ∫ : ƒ → → ∫ ƒ ∫ ƒ : ( x → → ∫ ∫ 0 ƒ ( : ƒ x : ( x ( t ) dt ) t ) dt ) ƒ∈ C to ƒ∈ ∞ C ∞ to 0 ∫ ƒ computable in exponential time; ∫ ƒ �� �� � � � � .complete" � another class between �� and ������ × [ ∋ ƒ ƒ → → z ƒ ( : C[0;1] × 1;1] ∋ )= ƒ ż ( : ż • dsolve dsolve: C[0;1] [- -1;1] ( t ( t , z ), z (0)=0. z : t )= t , z ), z (0)=0 � in general no computable solution z ( t ) z ( t ) 1 ������ � for ƒ∈ ƒ∈ C ������ ."complete" ������ ������ [Kawamura'10, C 1 [Kawamura'10, k �� � for ƒ∈ ƒ∈ C Kawamura et al] et al] Kawamura �� ."hard" �� �� C k
" Max �� �� �� �� �� #$���" TECHNISCHE UNIVERSITÄT DARMSTADT, Martin Ziegler Real Parameterized and 2 nd -Order Complexity Theory: From Computability in Analysis to Numerical Practice �� �� �� �� ∋ ⊆ � ∈ � � � � � �� �� �� �� ∋ L ∈ � ∃ M V ∈ � 〈 N,M � ⇔ f V polytime 〉 ∈ ∈ V V ∈ � � � N ∈ V ⊆ | ∃ : 〈 N,M 〉 � � polytime { N V } } , � � | = { L = , V <N : V M<N t → ∑ ϕ (2 tN ²-2 N )/ N ln N t → ∑ ϕ (3 tN ³-3 N ²- M )/ N ln N ∞ C ∞ g L : f V : C 1 〈 N , M 〉 ∈ V N 0.8 t ln(1/ t ) 0.6 0.4 0.2 0 -1 -0.5 0 0.5 1 φ ( φ ( t ) = exp( exp(- - t ²/1 /1- - t ²) ) t ) = t ² t ² pulse' function function ' pulse' ∞ ' C ∞ t = ⅓ t =½ C t =1 0 N =5 t =¼ N =3 N =2 N =4 N =1 polytime computable computable polytime M =0..3 M =0,1,2 M =1 M =0 M =0 �� there �� ∈ �� �� �� �� �� �� L ∈ To every every L there exists exists a a polytime polytime To ∞ function computable C function g → � � s.t.: s.t.: computable C ∞ :[0,1] → L :[0,1] g L ∋ t ∈ � � � � � � � � [0,1] ∋ L ∈ → max again polytime polytime iff iff L t → ] again max g | [0, [0,1] L | g L [0, t t ]
TECHNISCHE ������� ���������� �������������% UNIVERSITÄT DARMSTADT, Martin Ziegler Real Parameterized and 2 nd -Order Complexity Theory: ƒ :[0;1] → [0;1] polytime computable ( ⇒ continuous) From Computability in Analysis to Numerical Practice [Friedman&Ko [ ƒ → → Max( ƒ ): x → → max ƒ ( ≤ x Max: ƒ Max( ƒ { ƒ t ≤ Friedman&Ko] • Max: ): x max{ ( t ): t x } } t ): ƒ ) Max( ƒ ) computable in exponential time; Max( even when polytime.computable iff � � � � = �� �� �� �� restricting x ƒ non. ���# . # ��� non ƒ∈ C • ∫ ∫ : ƒ → → ∫ ƒ ∫ ƒ : ( x → → ∫ ∫ 0 ƒ ( : ƒ x to ƒ∈ ∞ C ∞ : ( x ( t ) dt ) t ) dt ) ������� uniform ������� uniform 0 ��� ��� ��� ��� ∫ ƒ computable in exponential time; ∫ ƒ ] �������� ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ �������� [N.M [ � � � �� � ."complete" � � N.Mü �������� �������� × [ ∋ ƒ ƒ → → z ƒ ( : C[0;1] × 1;1] ∋ )= ƒ ż ( : ż • dsolve dsolve: C[0;1] [- -1;1] ( t ( t , z ), z (0)=0. z : t )= t , z ), z (0)=0 üller ller] � in general no computable solution z ( t ) z ( t ) 1 ������ ] � for ƒ∈ ƒ∈ C ������ ."complete" ������ ������ [Kawamura'10, C 1 [Kawamura'10, k �� � for ƒ∈ ƒ∈ C Kawamura et al] et al] Kawamura �� �� �� ."hard" C k
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