Using Taylor Models in TTE 1 1.5 1.5 f 1 f 2 R R R 1 1 0.5 0.5 �− → ̺ ̺ ̺ ̺ ̺ ̺ 0 0 -0.5 -0.5 I N I N I N I N I N -1 -1 Γ 1 Γ 1 Γ 2 Γ 2 -1.5 -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 F . Brauße M. Korovina N. Müller Universität Trier, IIS Novosibirsk CCC Kochel, 2015-09-15 1The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n ◦ PIRSES-GA-2011-294962-COMPUTAL and from the DFG/RFBR grant CAVER BE 1267/14-1 and 14-01-91334. F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 1 / 29
Computability on real numbers Computability on real numbers 1 Wrapping Sets 2 3 Taylor Models Examples 4 Closing remarks 5 F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 2 / 29
Computability on real numbers A real number x is usually represented as follows: use open intervals with dyadic endpoints �� m 1 2 k , m 2 � � input output I := | m 1 , m 2 ∈ Z , k ∈ N 2 k machine M interpret word functions as sequences queries answers and use OTM [Σ ∗ → Σ ∗ ] oracle I N ∼ [ N → I ] = define representation ̺ : ⊆ I N → R : x ∈ R is represented by ( I m ) m ∈ N iff m ∈ N I m ∈ I � m →∞ diam ( I m ) = 0 lim ∧ I m = { x } x m ∈ N F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 3 / 29
Computability on real numbers A real function f is computed using a machine M as follows: If n ∈ N J n ∈ I � � ( I m ) m ∈ N = x ̺ f ( x ) and machine M Γ M : ( I m ) m ∈ N ❀ ( J n ) n ∈ N m ∈ N I m ∈ I then x � � ( J n ) n ∈ N = f ( x ) ̺ F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 4 / 29
Computability on real numbers Computable analysis (via ‘representations’): f 1 f 2 R R R ̺ ̺ ̺ ̺ ̺ ̺ ̺ ̺ ̺ I N I N I N I N I N I N I N I N Γ 1 Γ 1 Γ 1 Γ 2 Γ 2 Γ 2 Remember: Computable functions are continuous! F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 5 / 29
Wrapping Sets Computability on real numbers 1 Wrapping Sets 2 3 Taylor Models Examples 4 Closing remarks 5 F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 6 / 29
Wrapping Sets Modified view on computations: Use vectors x i from R d as states during a ‘real’ computation. Each step is a function f i : ⊆ R d → R d computed by some Γ i . Use sequences O i . j of approximating vectors, i.e. x i = ̺ d � � ( O i . j ) j ∈ N f n − 1 f 0 f 1 f 2 x 0 �− → x 1 �− → x 2 �− → . . . �− → x n ↑ ↑ ↑ ↑ . . . . . . . . . . . . Γ n − 1 Γ 0 Γ 1 Γ 2 O 0 , 2 O 1 , 2 O 2 , 2 O n , 2 �− → �− → �− → · · · �− → O 0 , 1 O 1 , 1 O 2 , 1 O n , 1 O 0 , 0 O 1 , 0 O 2 , 0 O n , 0 Optimized for memory (like in iRRAM): Work line by line... Γ 0 ( j ) Γ 1 ( j ) Γ 2 ( j ) Γ n − 1 ( j ) O 0 , j �− → O 1 , j �− → O 2 , j �− → · · · �− → O n , j F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 7 / 29
Wrapping Sets Necessary condition: f i ( O i , j ) ⊆ O i + 1 , j ... ... but there is overestimation in interval vector computations: images of boxes O ∈ I d aren’t boxes again... example: function f : R 2 → R 2 with f ( x , y ) = ( x 2 − y 2 , x · y ) 1.5 1.5 1 1 0.5 0.5 �− → 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 ‘wrapping’ effects accumulate... so try to avoid or reduce them F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 8 / 29
Wrapping Sets Replacement for open boxes I d : Definition 2.1 A countable family A = { A n | n ∈ N } of sets A n ⊆ R d is wrapping ∀ x ∈ R d ∀ ε ∈ R > 0 iff ∃ A ∈ A , diam ( A ) ≤ ε ∧ x ∈ int ( A ) Corresponding representation τ A : ⊆A N → R d : τ A ( p ) := x iff the sequence p ∈ A N satisfies � n ∈ N diam ( p n ) = 0 lim ∧ p n = { x } n ∈ N Examples: I d , closed boxes (also with point intervals), unions of boxes... DAGs, symbolic representations, ... Taylor models F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 9 / 29
Wrapping Sets Lemma 2.2 If A and B are wrapping, then τ A and τ B are topologically equivalent. Basic idea of topological reduction τ A ≤ τ B : W.l.o.g. R d ∈ B For all A ∈ A , n ∈ N there is B ∈ B with A ⊆ B ∧ diam ( B ) ≤ inf { diam ( B ′ ) | A ⊆ B ′ ∈ B} + 2 − n Use arbitrary such function w B A : ( A , n ) �→ B . Define W : A N → B N : q n := w B W ( p ) := q where A ( p n , n ) W is continuous, transformation stays strictly local! W is realizer for id R d . F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 10 / 29
Wrapping Sets Special case: Lemma 2.3 A , B wrapping families: Suppose there is a (( ν A , ν N ) , ν B ) -computable multivalued function w B A : A × N ⇒ B such that for all A ∈ A , n ∈ N , A ⊆ w B A ( A , n ) ∧ A ( A , n ) ) ≤ 2 − n + inf { diam ( B ′ ) | A ⊆ B ′ ∈ B} . diam ( w B Then τ A is computably reducible to τ B . F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 11 / 29
Wrapping Sets Remarks: We always(!) have topological equivalence to ̺ d . In interesting cases we have computational equivalence to ̺ d . ❀ wrapping is not important for computability ... but for efficiency! Computations will usually change only single component of a state in one step. Instead of general f : ⊆ R d → R d we aim at f with x 1 . . . f ( x 1 , x 2 , . . . , x n ) = g ( x 1 , x 2 , . . . , x n ) . . . x n for a corresponding computable function g : ⊆ R d → R F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 12 / 29
Taylor Models Computability on real numbers 1 Wrapping Sets 2 3 Taylor Models Examples 4 Closing remarks 5 F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 13 / 29
Taylor Models Taylor models [Makino/Berz]: aim at R d with dynamical changeable d use hypercube U k with U = [ − 1 , 1 ] , k independent from d use vectors λ = ( λ 1 , . . . , λ k ) of ‘error symbols’ λ i ∈ U n of multivariate polynomials in λ � consider T ( λ ) = c n · λ n coefficients c n are vectors from R d example with d = k = 2 : � 0 � � 0 � � 1 � � − 1 � · λ 2 · λ 2 T ( λ ) = + · λ 1 · λ 2 + 1 + 2 0 1 0 0 1.5 1.5 1 1 0.5 0.5 U 2 ⊆ R 2 �− → 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 ❀ Taylor models lead to wrapping families: A = T ( U k ) F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 14 / 29
Taylor Models c n , 1 . . Coefficient space K for components c n = ? . c n , d in practice: coefficients c n , i are in double precision generalize to: ◮ dyadic numbers D ◮ rational numbers Q ◮ computable real numbers R c ◮ computable complex numbers C c further generalizations: ◮ each/some c n , i might be an interval itself in practice: coefficient c 0 is usually vector of intervals F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 15 / 29
Taylor Models Variants of Taylor models: Affine arithmetic: order 1 , only c 0 , i as non-point intervals Generalized interval arithmetic: order 1 , all c n , i are arbitrary intervals Classical Taylor models: arbitrary order, only c 0 , i as non-point intervals Interval Taylor models: arbitrary order, all c n , i are arbitrary intervals ❀ all versions of Taylor models yield same notion of computability! F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 16 / 29
Taylor Models Functions on R d are implemented as transformations of polynomials: Addition/subtraction on R d ❀ polynomial addition/subtraction Example computation: x=...; y=...; y=x+y; x=y-x; use d = 2 , here with linear Taylor model and k = 2 : interval Taylor model 4 � 0 � � 0 � � 2 � x = ... ; [ − 2 , 2 ] 2 + · λ 1 + · λ 2 0 y = ... ; [ − 1 , 1 ] 0 1 0 -2 -4 -6 -4 -2 0 2 4 6 4 � 0 � � 0 � � 2 � [ − 2 , 2 ] 2 y = x + y ; + · λ 1 + · λ 2 0 [ − 3 , 3 ] 0 1 2 -2 -4 -6 -4 -2 0 2 4 6 4 � 0 � � 1 � � 0 � [ − 5 , 5 ] 2 x = y − x ; + · λ 1 + · λ 2 0 [ − 3 , 3 ] 0 1 2 -2 -4 -6 -4 -2 0 2 4 6 here there is no overestimation functional dependencies are completely retained... F. Brauße, M. Korovina, N. Müller Taylor models in TTE CCC Kochel, 2015-09-15 17 / 29
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