Computable Real Functions Parameterized Uniform Parameterized Uniform From NP -hard to polytime from smooth to analytic Akitoshi Akitoshi Complexity in Numerics: Complexity in Numerics: Kawamura, , Kawamura A computable A computable f Norbert Müller Norbert Müller from Smooth to Analytic, from Smooth to Analytic, function must must function Carsten Carsten Rösnick Rösnick NP - NP NP NP from NP NP NP NP -hard to Polytime hard to Polytime from be continuous continuous be Martin Ziegler Martin Ziegler x x' n for x ∈ ∈ � computable ⇔ ⇔ | ) ⊆ ⊆ � | ≤ ≤ 2 � computable n +1 +1 | - n � | x x - - a a n /2 n 2 - for recursive recursive ( ( a a n x n /2 n ) Real Function Complexity 3 Effects in Real Complexity From NP -hard to polytime From NP -hard to polytime iRRAM (GMP/MPFR) Function f :[0,1] → � computable from smooth to analytic from smooth to analytic � Consider multivalued 'functions' computable in time in time t t ( ( n n ) ) with if some TM can, on input of n ∈ � and of � additional discrete data ('enrichment'). ( a m ) ⊆ � with | x-a m /2 m +1 |<2 - m ≡ ρ sd =: ρ dy -name p p not Example c1): exp exp not computable on entire � , output b ∈ � with | f ( x ) -b /2 n +1 |<2 - n . in time t in time t ( ( n n ) ) c2) Evaluation ( ƒ , x ) →ƒ ( x ) is not computable on [0;1] ! Examples: a) + + , , × × , , exp polytime exp in time depending only on output precision n . n iff L ≡ ∑ ) ≡ ∑ n ⊆ { b) f L ⊆ 0 , 1 } decidable { 0 , 1 polytime- f ( ( x x ) 4 - - n } * * L 4 Example b): Given real symmetric d × d n ∈ ∈ L c) 1/ c) sign Heaviside not computable ) not polytime-computable 1/ln(e ln(e/ / x x ) sign, , Heaviside matrix A , find an eigenvector: incomputable; but computable when knowing Card σ ( A ) [Z+B'04] If ƒ ƒ computable Observation i) i) If computable ⇒ ⇒ continuous continuous. . Observation ii) If ii) If f f computable computable in in time time t ) , , then then � parameterized real complexity t ( ( n n ) sign( x ) +2) is is a a modulus modulus of uniform of uniform continuity continuity of of f f . . � � t ( ( n n +2) t Example a): Tests for in- canonical C++ canonical cos(1 /ε ) sin(1 /ε ) C++ x ε · � n := { k /2 n : k ∈ � }, � = n � n dyadic rationals /equality are undecidable declaration/ /interface interface declaration sin(1 /ε ) − cos(1 /ε )
' Max is NP NP NP -hard' NP Nonuniform Complexity of Operators From NP -hard to polytime From NP -hard to polytime ƒ :[0;1] → [0;1] polytime computable ( ⇒ continuous) NP ∋ ∋ L ∈ {0,1} n | ∃ y ∈ {0,1} 〈 x 〉 ∈ V NP x ∈ | ∃ y ∈ : 〈 y 〉 ∈ from smooth to analytic from smooth to analytic { x V } } = { p ( ( n n ) ) : L = {0,1} n {0,1} p x , , y [Friedman&Ko'80ies] [Friedman&Ko'80ies] ƒ → → Max( ƒ ): x → → max ƒ ( ≤ x Max: ƒ Max( ƒ { ƒ t ≤ • Max: ): x max{ ( t t ): ): t x } } 1 1 C ∞ ' pulse' function 〈 x 〉 ∈ V 1 〈 y 〉 ∈ x , , y V ƒ ) Max( ƒ ) computable in exponential time; φ ( φ Max( ( t t ) = ) = exp( exp(- - t t ² ²/1 /1- - t t ² ²), | ), | t t |<1 |<1 0.8 even when polytime-computable iff P = NP polytime computable 〈 x 〈 y 〉 ∉ 〉 ∉ V x , , y V restricting 0.6 ½ x ƒ ½ non- non non non- - - n φ n n ≤ ? ≤ • ∫ ∫ : ƒ → → ∫ ƒ ∫ ƒ : x → → ∫ ∫ 0 x ƒ ( ƒ∈ C φ ( : ƒ to ƒ∈ ? C ∞ ∞ n / ( C C· ·x x- -z z )/ )/ C C d n /dx dx n d : x ( t t ) ) dt dt uniform uniform uniform uniform 0.4 0 too, uniformly in C,z but for for but ∫ ƒ computable in exponential time; ∫ ƒ ¼ ¼ x = x = 0.2 analytic ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ analytic x =1 =1 x "#P -complete" x =0 x =0 00 01 01 10 10 11 11 00 Negativistic !! Negativistic !! polytime polytime 0 -1 -0.5 0 0.5 1 n =2 =2 y =000,001, y =000,001,… … n y =000,001, y =000,001,… … n =1 =1 n × [ ∋ ƒ ƒ → → z ƒ ( : C[0;1] × 1;1] ∋ )= ƒ : ż ż ( • dsolve dsolve: C[0;1] [- -1;1] z : ( t t )= ( t t , , z z ), ), z z (0)=0 (0)=0. ∈ NP NP there L ∈ To every To every L there exists exists a a polytime polytime � in general no computable solution z 2 - - n n , n - -th th large large interval interval: : size size 2 , n z ( ( t t ) ) ∞ function computable C function f → � � s.t.: s.t.: computable n subintervals C ∞ :[0,1] → ∈ {0,1} x ∈ 1 " PSPACE -complete" f L L :[0,1] n , � for ƒ∈ ƒ∈ C containing 2 2 n subintervals: : one one for for each each x {0,1} n , containing Positivistic !! Positivistic !! C 1 ) subsubintervals ∋ y ∈ P P [0,1] ∋ L ∈ → max 2 p ] polytime p ( polytime iff ( n n ) iff L k " CH -hard" in turn subdivided in turn subdivided into y → into 2 subsubintervals for for y y 's 's [0,1] max f f L L | | [0, � for ƒ∈ ƒ∈ C C ω → → C C ∞ Phase transition Phase transition C ω ω ω ω → ω → → → → → ∞ ∞ ∞ ∞ ∞ ω ω ∞ ∞ C k [0, y y ] Representing Power Series Parameterized Real Complexity From NP -hard to polytime From NP -hard to polytime incomputable [ZhWe'01] [ZhWe'01] incomputable • Classical complexity theory: from smooth to analytic from smooth to analytic ∑ j c j z j • radius of convergence R =1/limsup j | c j | 1/ j worst-case over all inputs of length n as parameter • to 0< r < R exist C ∈ � : | c j | ≤ C/r j (Cauchy-Hadamard) • parametrized complexity (FPT etc): 2 param.s ( n , k ) binary binary • � ∋ K : ≥ 1/log( r ) = Θ ( 1/( r -1) ) 1 1 • Complexity of a single real: n = output precision unary unary • tail bound | ∑ j ≥ N c j z j | ≤ C ·(| z |/ r ) N /(1-| z |/ r ) R R r r • of a real function ƒ : n = output precision Complexity uniform in | z | ≤ 1: ∈ [0;1] ∈ (i.e. R >1 ) in worst-case over all arguments x [0;1] compact compact! ! Convergence degrades es as r → 1 ; quantitatively? • or parameterized – e.g. in k = | x | or k = log | x | parametrized running running time time parametrized • TTE: encode x as in finite binary sequence, length=∞ Theorem 1: Represent series ∑ j c j z j with R>1 • x =( x j ) real sequence: access access time polynom. in n + j as [a ( ρ dy ) ω - name of] ( c j ) and K,C ∈ � as above. Must 'skip' over 2 n entries to access access ƒ (2 - n ) The following are uniformly computable in time • Real operator/functional Λ : encode input ƒ∈ Lip[0;1] i) eval eval, , polyn. in n+K+ log( C ) : i) ii) sum ii) sum, , iii) product iii) product, , • as values on dense sequence 0,1,½,¼,¾, ⅛ , ⅜ , ⅝ ,...=: � iv) iv ) derivative derivative, , v) anti v) anti- -derivative derivative, , vi) vi ) Max Max • and Lipschitz constant ℓ ∈ � as discrete data & advice
Recommend
More recommend
