Computability Theory and Asymptotic Density Denis R. Hirschfeldt - PowerPoint PPT Presentation

Computability Theory and Asymptotic Density Denis R. Hirschfeldt — University of Chicago Groups and Computation: In Honor of Paul Schupp’s 80th Birthday

Cantor space and relative computability We work in the space 2 ω of infinite binary sequences, identifying A ∈ 2 ω with { n : A ( n ) = 1 } .

Cantor space and relative computability We work in the space 2 ω of infinite binary sequences, identifying A ∈ 2 ω with { n : A ( n ) = 1 } . For σ ∈ 2 <ω , let [ σ ] = { A ∈ 2 ω : σ ≺ A } . These are the basic open sets of the standard topology on 2 ω .

Cantor space and relative computability We work in the space 2 ω of infinite binary sequences, identifying A ∈ 2 ω with { n : A ( n ) = 1 } . For σ ∈ 2 <ω , let [ σ ] = { A ∈ 2 ω : σ ≺ A } . These are the basic open sets of the standard topology on 2 ω . Let µ the measure on 2 ω defined by µ ([ σ ]) = 2 −| σ | .

Cantor space and relative computability We work in the space 2 ω of infinite binary sequences, identifying A ∈ 2 ω with { n : A ( n ) = 1 } . For σ ∈ 2 <ω , let [ σ ] = { A ∈ 2 ω : σ ≺ A } . These are the basic open sets of the standard topology on 2 ω . Let µ the measure on 2 ω defined by µ ([ σ ]) = 2 −| σ | . We write A � T B to mean that A is computable relative to B . A ≡ T B if A � T B and B � T A . The ≡ T -equivalence classes are the (Turing) degrees.

Describing a set We consider partial descriptions of elements of 2 ω .

Describing a set We consider partial descriptions of elements of 2 ω . For an input n , such a description might: • give an answer (correctly or not); • never give an answer; • declare that it will not give an answer.

Describing a set We consider partial descriptions of elements of 2 ω . For an input n , such a description might: • give an answer (correctly or not); • never give an answer; • declare that it will not give an answer. A description of A ∈ 2 ω is a partial function ∆ : N → { 0 , 1 , � } . We write ∆( n ) ↓ if ∆( n ) is defined, and ∆( n ) ↑ otherwise.

Describing a set We consider partial descriptions of elements of 2 ω . For an input n , such a description might: • give an answer (correctly or not); • never give an answer; • declare that it will not give an answer. A description of A ∈ 2 ω is a partial function ∆ : N → { 0 , 1 , � } . We write ∆( n ) ↓ if ∆( n ) is defined, and ∆( n ) ↑ otherwise. When might we say that ∆ describes A “almost everywhere”?

Asymptotic density Let S ⊆ N . | S ∩ [ 0 , n ) | The upper (asymptotic) density of S is ρ ( S ) = lim sup n . n | S ∩ [ 0 , n ) | The lower (asymptotic) density of S is ρ ( S ) = lim inf n . n If ρ ( S ) = ρ ( S ) then S has (asymptotic) density ρ ( S ) = ρ ( S ) .

Asymptotic density Let S ⊆ N . | S ∩ [ 0 , n ) | The upper (asymptotic) density of S is ρ ( S ) = lim sup n . n | S ∩ [ 0 , n ) | The lower (asymptotic) density of S is ρ ( S ) = lim inf n . n If ρ ( S ) = ρ ( S ) then S has (asymptotic) density ρ ( S ) = ρ ( S ) . We think of sets of density 0 as negligible.

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n : • ∆( n ) ↑ • ∆( n ) ↓ = 1 − A ( n ) • ∆( n ) ↓ = �

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n : • ∆( n ) ↑ • ∆( n ) ↓ = 1 − A ( n ) • ∆( n ) ↓ = � Let D = { n : ∆( n ) ↑} , let M = { n : ∆( n ) ↓ = 1 − A ( n ) } , and let R = { n : ∆( n ) ↓ = � } .

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n : • ∆( n ) ↑ • ∆( n ) ↓ = 1 − A ( n ) • ∆( n ) ↓ = � Let D = { n : ∆( n ) ↑} , let M = { n : ∆( n ) ↓ = 1 − A ( n ) } , and let R = { n : ∆( n ) ↓ = � } . ∆ is a dense description of A if D ∪ M ∪ R has density 0.

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n : • ∆( n ) ↑ • ∆( n ) ↓ = 1 − A ( n ) • ∆( n ) ↓ = � Let D = { n : ∆( n ) ↑} , let M = { n : ∆( n ) ↓ = 1 − A ( n ) } , and let R = { n : ∆( n ) ↓ = � } . ∆ is a dense description of A if D ∪ M ∪ R has density 0. If ∆ is a dense description then it is: • a generic description of A if M ∪ R = ∅ . • a coarse description of A if D ∪ R = ∅ .

Asymptotic descriptions There are three ways a description ∆ of A can be incorrect at n : • ∆( n ) ↑ • ∆( n ) ↓ = 1 − A ( n ) • ∆( n ) ↓ = � Let D = { n : ∆( n ) ↑} , let M = { n : ∆( n ) ↓ = 1 − A ( n ) } , and let R = { n : ∆( n ) ↓ = � } . ∆ is a dense description of A if D ∪ M ∪ R has density 0. If ∆ is a dense description then it is: • a generic description of A if M ∪ R = ∅ . • a coarse description of A if D ∪ R = ∅ . • an effective dense description of A if D ∪ M = ∅ .

Asymptotic computability A is densely computable if it has a computable dense description. A is generically computable if it has a computable generic description. A is coarsely computable if it has a computable coarse description. A is effectively densely computable if it has a computable effective dense description.

Asymptotic computability A is densely computable if it has a computable dense description. A is generically computable if it has a computable generic description. A is coarsely computable if it has a computable coarse description. A is effectively densely computable if it has a computable effective dense description. These notions can be relativized to define dense computability relative to X , etc.

Asymptotic computability: history and prehistory Generic computability was introduced by Kapovich, Myasnikov, Schupp, and Shpilrain (2003). It was studied by Jockusch and Schupp (2012), who also studied coarse computability.

Asymptotic computability: history and prehistory Generic computability was introduced by Kapovich, Myasnikov, Schupp, and Shpilrain (2003). It was studied by Jockusch and Schupp (2012), who also studied coarse computability. Dense and effective dense computability were studied by Astor, Hirschfeldt, and Jockusch (ta).

Asymptotic computability: history and prehistory Generic computability was introduced by Kapovich, Myasnikov, Schupp, and Shpilrain (2003). It was studied by Jockusch and Schupp (2012), who also studied coarse computability. Dense and effective dense computability were studied by Astor, Hirschfeldt, and Jockusch (ta). Meyer (1973) defined effective dense computability, and asked a question answered by Lynch (1974). Terwijn (1998) studied coarse computability.

Relationships between notions of asymptotic computability computable ↓ effectively densely computable ւ ց generically coarsely computable computable ց ւ densely computable No implications other than the ones shown hold in general.

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X , consider { 2 n : n ∈ X } .

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X , consider { 2 n : n ∈ X } . Every nontrivial Turing degree contains a set that is not densely n ∈ X [ 2 n , 2 n + 1 ) . computable: Given X , consider �

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X , consider { 2 n : n ∈ X } . Every nontrivial Turing degree contains a set that is not densely n ∈ X [ 2 n , 2 n + 1 ) . computable: Given X , consider � A set is simple if it is co-infinite and computably enumerable, and its complement does not contain an infinite c.e. set. A simple set of density 0 is coarsely computable but not generically computable.

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X , consider { 2 n : n ∈ X } . Every nontrivial Turing degree contains a set that is not densely n ∈ X [ 2 n , 2 n + 1 ) . computable: Given X , consider � A set is simple if it is co-infinite and computably enumerable, and its complement does not contain an infinite c.e. set. A simple set of density 0 is coarsely computable but not generically computable. There is a c.e. set that is generically computable but not coarsely computable.

Examples from Jockusch and Schupp Every Turing degree contains a set that is effectively densely computable: Given X , consider { 2 n : n ∈ X } . Every nontrivial Turing degree contains a set that is not densely n ∈ X [ 2 n , 2 n + 1 ) . computable: Given X , consider � A set is simple if it is co-infinite and computably enumerable, and its complement does not contain an infinite c.e. set. A simple set of density 0 is coarsely computable but not generically computable. There is a c.e. set that is generically computable but not coarsely computable. { A : A is densely computable } has measure 0 and is meager.

Asymptotic reducibilities A is coarsely reducible to B , written A � c B , if every coarse description of B computes a coarse description of A .

Asymptotic reducibilities A is coarsely reducible to B , written A � c B , if every coarse description of B computes a coarse description of A . There are nonuniform and uniform versions of this reducibility, but we will ignore the distinction.