Computing with Compact Sets: The Gray Code Case Dieter Spreen - PowerPoint PPT Presentation

Computing with Compact Sets: The Gray Code Case Dieter Spreen University of Siegen joint work with Hideki Tsuiki Kyoto University Continuity, Computability, Constructivity: From Logic to Algorithms Kochel am See, Bavaria, 14-18 September 2015

1. Signed digit repesentation Let x = 0 . a 1 a 2 a 3 . . . be a real number in I = [ − 1 , +1]. Then writing a digit a ∈ SD := {− 1 , 0 , 1 } in front of a 1 a 2 a 3 . . . corresponds to applying the contraction d a defined by d a ( x ) = x + a 2 to x . In this way we can think of 0 . a 1 a 2 a 3 . . . as an iterated application of operations d a ∈ D := { d a | a ∈ SD } d a 1 ◦ d a 2 ◦ d a 3 ◦ · · ·

Lemma I is a complete bounded metric space and D a finite set of contractions that covers I : � I = { d a [ I ] | a ∈ SD } . In particular I is compact. Consequently, for α = α 0 α 1 . . . ∈ SD ω � d α 0 ◦ · · · ◦ d α n − 1 [ I ] n ≥ 0 contains exactly one element. Let [ [ α ] ] be the unique � x ∈ d α 0 ◦ · · · ◦ d α n − 1 [ I ] n ≥ 0

Lemma SD ω is a bounded complete and hence compact metric space with metric � 0 if α = β , δ ( α, β ) := 2 − min { n | α n � = β n } otherwise. Proposition ]: SD ω → I is onto and uniformly continuous. ◮ [ [ · ] ◮ The Euclidean topology on I is equivalent to the quotient topology induced by [ [ · ] ] . In particular we have that each α ∈ SD ω denotes a real number in I , and vice versa.

2. Pre-Gray code representation Let pG := { U , D , L , R , FinL , FinR } G := { U , L , R } H := { D , FinL , FinR } Now, not every sequence in pG ω is meaningful! The set co G of meaningful sequences is defined together with a set co H by mutual coninduction: ( co G , co H ) is the largest subset of pG ω × pG ω such that α ∈ co G → ( ∃ β ∈ co G ) α ∈ { L , R } β ∨ ( ∃ β ∈ co H ) α = U β α ∈ co H → ( ∃ β ∈ co G ) α ∈ { FinL , FinR } β ∨ ( ∃ β ∈ co H ) α = D β

Proposition 1. ( co G , δ ) is complete. 2. ( co H , δ ) is complete. Recall that δ is bounded. Corollary Both, ( co G , δ ) and ( co H , δ ) are compact.

Let [ [ L ] ] := LR − 1 [ [ R ] ] := LR 1 [ [ U ] ] := U [ [ FinL ] ] := Fin − 1 [ [ FinR ] ] := Fin 1 [ [ D ] ] := D where for a ∈ {− 1 , 1 } LR a ( x ) := − ax − 1 U ( x ) := x 2 2 Fin a ( x ) := ax + 1 D ( x ) := x 2 2

Lemma ◮ d 1 ( x ) = Fin 1 ( x ) = LR 1 ( − x ) ◮ d − 1 ( x ) = Fin − 1 ( − x ) = LR − 1 ( x ) ◮ d 0 ( x ) = U ( x ) = D ( x ) Corollary ◮ range ( d 1 ) = range ( Fin 1 ) = − − → range ( LR 1 ) = ← − − [0 , 1] , [0 , 1] ◮ range ( d − 1 ) = range ( LR − 1 ) = − [ − 1 , 0] , range ( Fin − 1 ) = ← − − − → − − − − [ − 1 , 0] ◮ range ( d 0 ) = range ( U ) = range ( D ) = [ − 1 / 2 , +1 / 2]

Consequently, both { U , LR a | a ∈ {− 1 , 1 } } and { D , Fin a | a ∈ {− 1 , 1 } } are finite covering sets of contractions on I . Hence, for α = α 0 α 1 . . . ∈ co G and β = β 0 β 1 . . . ∈ co H � � [ [ α 0 ] ] ◦ · · · ◦ [ [ α n ] ][ I ] and [ [ β 0 ] ] ◦ · · · ◦ [ [ β n ] ][ I ] n ≥ 0 n ≥ 0 contain exactly one element. Let � � { [ [ α ] ] G } := [ [ α 0 ] ] ◦· · ·◦ [ [ α n ] ][ I ] and { [ [ β ] ] H } := [ [ β 0 ] ] ◦· · ·◦ [ [ β n ] ][ I ] n ≥ 0 n ≥ 0 Proposition ◮ [ ] G : co G → I , [ ] H : co H → I are onto and uniformly [ · ] [ · ] continuous. ◮ The Euclidean topology on I is equivalent to the quotient topology induced by [ [ · ] ] G ( [ [ · ] ] H ).

3. Compact sets Let K ( I ) be the collection of all nonempty compact subsets of I . Proposition K ( I ) endowed with the Hausdorff metric µ H is a bounded complete, hence compact metric space. Berger/Spreen: There is no finite set of contractions on K ( I ) that covers K ( I ). So, it is impossible to represent nonempty compact subsets of I by infinite streams. As we will see, however, they can be represented by infinite trees.

Definition A digital tree is a nonempty set T ⊆ pG ∗ of finite sequences of elements of pG that is closed under initial seqments and has no maximal elements, that is, the empty sequence [] is in T and whenever R 0 . . . R n ∈ T , then R 0 . . . R n − 1 ∈ T and R 0 . . . R n R ∈ T , for some R ∈ pG . Note that each such tree is finitely branching. Moreover, each element R 0 . . . R n ∈ T can be continued to an infinite path α in T . In the following we write α ∈ T to mean that α is an infinite path of T and identify T with the set of its infinite paths. Let T be the set of all digital trees and T G := { T ∩ co G | T ∈ T } T H := { T ∩ co H | T ∈ T }

Lemma ( T G , T H ) is the largest subset of T × T such that T ∈ T G → ( ∃ T 1 , . . . , T n ∈ T )( ∃ R 1 , . . . , R n ∈ G )( ∀ 1 ≤ i ≤ n ) [ R i ∈ { L , R } → T i ∈ T G ] ∧ [ R i = U → T i ∈ T H ] n � ∧ T = R i T i i =1 T ∈ T H → ( ∃ T 1 , . . . , T n ∈ T )( ∃ R 1 , . . . , R n ∈ H )( ∀ 1 ≤ i ≤ n ) [ R i ∈ { FinL , FinR } → T i ∈ T G ] ∧ [ R i = D → T i ∈ T H ] n � ∧ T = R i T i . i =1

Proposition ◮ T G endowed with the Hausdorff metric δ H induced by δ is a bounded complete, hence compact metric space. ◮ T H endowed with the Hausdorff metric δ H induced by δ is a bounded complete, hence compact metric space. Lemma Every tree in T G is compact and every nonempty compact subset of co G is the set of infinite paths of a digital tree in T G . Similarly, for T H . Definition For T ∈ T G set [ [ T ] ] T G := { [ [ α ] ] G | α ∈ T } .

Since [ [ · ] ] G is continuous, [ [ T ] ] T G is compact. Conversely, for any ] − 1 K ∈ K ( I ), [ [ · ] G [ K ] is closed and hence compact, as T D is compact. Lemma The nonempty compact subsets of I are exactly the values of trees in T G . Proposition ◮ [ [ · ] ] T G : T G → I is onto and uniformly continuous. ◮ The topology on K ( I ) induced by the Hausdorff metric is equivalent to the quotient topology induced by [ [ · ] ] T G .

Let D := { d a | a ∈ {− 1 , 0 , +1 } } , G := { U , LR − 1 , LR 1 } , H := { D , Fin − 1 , Fin 1 } . In Berger/Spreen a representation-free coinductive approach to computations with compact subsets of I was presented. Let C K be the largest subset of K ( I ) such that C K ( A ) → ( ∃ B ⊆ D )( ∃ ( A b ) b ∈ B ∈ K ( I ) B )( ∀ b ∈ B ) C K ( A b ) � ∧ A = b [ A b ] . b ∈ B Lemma (Berger/Spreen) K ( I ) = C K .

Let ( G K , H K ) be the largest subset of K ( I ) × K ( I ) such that G K ( A ) → ( ∃ E ⊆ G )( ∃ ( A e ) e ∈ E ∈ K ( I ) E ) A = � e [ A e ] ∧ ( ∀ e ∈ E ) e ∈ E [ e ∈ { LR − 1 , LR 1 } → G K ( A e )] ∧ [ e = U → H K ( A e )] , � H K ( A ) → ( ∃ F ⊆ H )( ∃ ( A f ) f ∈ F ∈ K ( I ) F ) A = f [ A f ] ∧ ( ∀ f ∈ F ) f ∈ F [ f ∈ { Fin − 1 , Fin 1 } → G K ( A f )] ∧ [ f = D → H K ( A f )] . Theorem G K = C K .

◮ From the proof a computable translator between realizers for both sides can be derived, that is, between trees in T G and trees over SD . ◮ In Berger/Spreen the equivalence between the latter representation and the Cauchy representation for nonempty compact sets was shown. ◮ Consequently, pre-Gray computability of nonempty compact subsets of I is equivalent to TTE computability.