�������� ����� � � � � � � � � �� ���� ��� � �� ���� ���� ��� ����������� � �������������������������� ������������������������ ������ ���� ����� � Symbolic complexity Hardness CoPolish spaces Spaces of open sets Descriptive complexity on non-Polish spaces Mathieu Hoyrup joint work with Antonin Callard Loria - Inria, Nancy (France) 1 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets DST outside Polish spaces Descriptive Set Theory (DST): ‚ Mainly on Polish spaces (completely metrizable spaces). Theoretical Computer Science induces other spaces: ‚ Partial functions, ‚ Higher-order functionals, e.g. p N Ñ N q Ñ N , ‚ Computation with advice, ‚ etc. 2 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets DST outside Polish spaces Descriptive Set Theory (DST): ‚ Mainly on Polish spaces (completely metrizable spaces). Theoretical Computer Science induces other spaces: ‚ Partial functions, ‚ Higher-order functionals, e.g. p N Ñ N q Ñ N , ‚ Computation with advice, ‚ etc. Need to develop DST outside Polish spaces: ‚ Domains [Selivanov] , quasi-Polish spaces [de Brecht] ‚ Represented spaces [Brattka, de Brecht, Pauly, Schröder, Selivanov] 2 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Two measures of complexity In an admissibly represented space X , two measures of complexity of a set A Ď X . Topological complexity Complexity of describing A from open sets. Symbolic complexity Complexity of testing whether a point belongs to A . 3 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Two measures of complexity In an admissibly represented space X , two measures of complexity of a set A Ď X . Topological complexity Complexity of describing A from open sets. Symbolic complexity Complexity of testing whether a point belongs to A . Theorem (de Brecht, 2013) They coincide on countably-based spaces. What about other spaces? 3 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Motivating example " 1 * Let A “ n : n P N Ď R . How complicated is A ? Two approaches: ‚ How to test x P A it with an algorithm ? ‚ How to describe A in terms of simpler sets? 4 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Motivating example " 1 * Let A “ n : n P N Ď R . Algorithm 1 1 x ‰ 1 D n, n ` 1 ă x ă n ´ 1 ? n ? No Yes No Description n ` 1 , 1 1 ´ ¯ ď A “ p 0 , `8qz . n n 4 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Motivating example These approaches are equivalent: for any A Ď R , A is decidable with ď 2 mind changes No-Yes-No ð ñ A is a difference of two effective open sets ( A P D 2 p R q ). 5 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets More generally Are these two approaches always equivalent? ‚ Algorithms make sense on represented spaces, ‚ Descriptions using open sets make sense on topological spaces. So let’s work on topological spaces with an admissible representation. 6 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Polynomials Representation A polynomial P P R r X s is represented by: ‚ Some n ě deg p P q , ‚ The coefficients of P “ p 0 ` p 1 X ` . . . ` p n X n . 7 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Polynomials Representation A polynomial P P R r X s is represented by: ‚ Some n ě deg p P q , ‚ The coefficients of P “ p 0 ` p 1 X ` . . . ` p n X n . How complicated is " * 1 A “ P P R r X s : p 0 “ 0 or p 0 ą ? deg p P q 7 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Polynomials Representation A polynomial P P R r X s is represented by: ‚ Some n ě deg p P q , ‚ The coefficients of P “ p 0 ` p 1 X ` . . . ` p n X n . How complicated is " * 1 A “ P P R r X s : p 0 “ 0 or p 0 ą ? deg p P q ‚ Decidable with 2 mind-changes, ‚ But not a difference of two open sets! 7 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Polynomials " * 1 A “ P P R r X s : p 0 “ 0 or p 0 ą . deg p P q Algorithm Given P and n ě deg p P q , p 0 ‰ 0 ? Yes No p 0 » 0 1 p 0 ą deg p P q ? p 0 fi 0 No Yes 8 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Polynomials " * 1 A “ P P R r X s : p 0 “ 0 or p 0 ą . deg p P q Descriptive complexity A is not a difference of 2 open sets: n ` X n ` 1 1 1 Ý Ý Ý Ñ Ý Ý Ý Ñ 0 p n p Ñ8 n Ñ8 P A R A P A 8 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets The problem Algorithms and topology induce the same complexity on R but not on R r X s . ‚ Why? ‚ What about other spaces? ‚ What about other complexity levels ( Σ 0 α , etc.) � ‚ What do algorithms measure? 9 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets The problem Algorithms and topology induce the same complexity on R but not on R r X s . ‚ Why? ‚ What about other spaces? ‚ What about other complexity levels ( Σ 0 α , etc.) � ‚ What do algorithms measure? Guess Algorithms reflect the sequential rather than topological aspects of the space. 9 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Algorithms prefer sequences Only sequential spaces can be handled by representations (Schröder). Franklin 65 Sequential spaces ” quotients of metric spaces, Schröder 02 Adm. rep. spaces ” quotients of countably-based metric spaces. ‚ A subspace of a represented space is not a topological subspace but its sequentialization, ‚ A product of represented spaces is not the topological product but its sequentialization. 10 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Topology vs sequences Seq. continuity “ continuity Sequential Fréchet-Urysohn Seq. closure “ closure First-countable Countably-based 11 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Symbolic complexity Hardness CoPolish spaces Spaces of open sets 12 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Topological complexity ∆ 0 Borel 2 � ˇ Σ 0 Π 0 D α D α α α � � � � . . . . . . ˇ Σ 0 Π 0 D 3 D 3 3 3 � � � � D 3 X ˇ ∆ 0 D 3 3 � � � Σ 0 Π 0 ˇ D 2 D 2 2 2 � � � � ∆ 0 D 2 X ˇ D 2 2 � � � Σ 0 Π 0 Σ 0 Π 0 1 1 1 1 � � � � (a) Borel hierarchy (b) Hausdorff difference hierarchy 13 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Symbolic complexity We work on represented spaces with a total admissible representation p X, δ X q . Σ 0 α , Σ 0 Let Γ be some complexity class ( α , etc.). � Definition (Symbolic complexity) A set A Ď X belongs to r Γ s if δ ´ 1 X p A q P Γ p N q . One always has Γ Ď r Γ s , ( δ X is continuous) Σ 0 Σ 0 1 “ r 1 s (final topology) . � � 14 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Countably-based spaces Symbolic and topological complexity coincide on countably-based spaces. Theorem (De Brecht, 2013) If X is countably-based, then r Γ s “ Γ . Already in (Brattka 2005), (Saint-Raymond 2007) for Polish spaces. Theorem The following are equivalent: ‚ X is countably-based, ‚ r D 2 s “ D 2 in a uniform way. � � 15 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Symbolic complexity We have seen that on R r X s , r D 2 s Ę D 2 , � � and even r D 2 s Ę D 2 , � witnessed by " 1 * A “ P P R r X s : p 0 “ 0 or p 0 ą . deg p P q 16 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets We mainly study two classes of spaces: ‚ CoPolish spaces ” inductive limits of compact metric spaces, ‚ Spaces of open subsets of Polish spaces. 17 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Symbolic complexity Hardness CoPolish spaces Spaces of open sets 18 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Hardness Σ 0 ‚ In a Polish space, how to show that a set A is not 2 ? � 19 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Hardness Σ 0 ‚ In a Polish space, how to show that a set A is not 2 ? � 2 -subset of N N is Π 0 Π 0 § Prove that it is 2 -hard: every � � continuously reducible to A . 19 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Hardness Σ 0 ‚ In a Polish space, how to show that a set A is not 2 ? � 2 -subset of N N is Π 0 Π 0 § Prove that it is 2 -hard: every � � continuously reducible to A . Theorem (Wadge) Let Γ ‰ ˇ Γ . For any Borel subset A of a Polish space, ñ A is ˇ A R Γ ð Γ -hard. Σ 0 Π 0 ∆ 0 Applies to Γ “ D α , α , α , but not α . � � � � 19 / 40
Symbolic complexity Hardness CoPolish spaces Spaces of open sets Hardness outside Polish spaces ‚ What about non-Polish spaces? 20 / 40
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