Approximating incomputable sets Timothy H. McNicholl Department of - - PowerPoint PPT Presentation

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density

Approximating incomputable sets

Timothy H. McNicholl

Department of Mathematics Iowa State University mcnichol@iastate.edu

Groups and Computation, June 2017

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density

Outline

1

Computability Theory Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

2

Approximating the incomputable- the Ershov hierarchy

3

Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Outline

1

Computability Theory Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

2

Approximating the incomputable- the Ershov hierarchy

3

Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Computability theory/ theory of computation Limitations of computers (i.e. discrete computing devices) When a problem can be solved by a computer, how difficult is it to solve? (in terms of time/memory) When a problem can not be solved by any computer, how impossible is it (relative to other impossible problems)? (Complexity)

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Fundamental notion: algorithm i.e. a procedure that can be carried out without thinking. Viewpoint: computers are just devices for implementing algorithms, so algorithms are the real focus of study.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Fundamental definition

Let N = {0, 1, 2, . . .}. Definition A function f :⊆ Nm → N is computable if there is an algorithm that given any n1, . . . , nm ∈ N as input, halts with output f(n1, . . . , nm) if (n1, . . . , nm) ∈ dom(f) and does not halt if (n1, . . . , nm) ∈ dom(f). Examples: Addition, multiplication, division gcd’s Probably any function you can think of.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Formalizing ‘algorithm’

Turing machines (A. Turing) Partial recursive functions (A. Church) Unlimited register machine Flowchart computability (Wang) These are all equivalent!

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Notation When A ⊆ N, χA is the characteristic function of A. Definition A ⊆ N is computable if χA is computable.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Definition A ⊆ N is computably enumerable if either A = ∅ or if there is a sequence {an}n∈N such that A = {a0, a1, a2, . . .} and such that n → an is a computable function. i.e. A is the range of a computable function f :⊆ N → N. Proposition Every computable set is computably enumerable. Question Does the converse hold?

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Theorem (Church, Turing, Kleene, Gödel, 1936) There is a computably enumerable set that is not computable. Proposition A ⊆ N is computable if and only if A and N − A are c.e..

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Definition A coding of a set X is a function c :⊆ N → X that is onto. Definition Suppose cj is a coding of Xj for j = 1, 2 and that f :⊆ X1 → X2. With respect to these codings, we say that f is computable if there is a computable function F :⊆ N → N such that dom(F) = c1[dom(f)] and F(c1(x)) = c2(f(x)) for all x ∈ dom(f).

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

! !

F X1 X2 c1 c2 f

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

By means of codings, can extend computability to Z Q set of all finite graphs, etc. Examples of incomputable sets and functions have been found throughout mathematics. e.g. Algebra: Hilbert’s tenth problem, word problem for groups Analysis: differentiation is not a computable operator Computer science: the halting set (set of all computer programs that halt on at least one input).

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Relative computability

In computability theory, when it is known that something can’t be computed, we often seek to describe how impossible it is to compute. Definition (Informal) An oracle algorithm (procedure) is an algorithm that is allowed to ask set-membership questions of an oracle (i.e. can ‘call a friend’). Definition (Informal) Suppose A, B ⊆ N. We say that B computes A (or that A is Turing reducible to B) if there is an oracle algorithm that can compute A when it uses B as an oracle. In this case we write A ≤T B. Simple examples: A ≤T A, A ≤T N − A, ∅ ≤T A. A computable if and only if A .

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Turing degrees

≤T is a preorder on P(N) The resulting equivalence classes are called Turing degrees. Study of the partial order of Turing degrees and some substructures has directed a lot of computability theory.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Theorem A ⊆ N is c.e. if and only if there is a computable R ⊆ N × N such that A = {n ∈ N : ∃m ∈ N (n, m) ∈ R}

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Definition Suppose A ⊆ N.

1

A is Σ0

1 if there is a computable R ⊆ N × N such that

A = {x ∈ N | ∃y ∈ N(x, y) ∈ R}.

2

A is Π0

1 if there is a computable R ⊆ N × N such that

A = {x ∈ N | ∀y ∈ N(x, y) ∈ R}.

3

A is ∆0

1 if it is Σ0 1 and Π0 1.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Definition Suppose A ⊆ N.

1

A is Σ0

2 if there is a computable R ⊆ N3 such that

A = {x ∈ N | ∃y1 ∈ N∀y2 ∈ N (x, y1, y2) ∈ R}.

2

A is Π0

2 if there is a computable R ⊆ N3 such that

A = {x ∈ N | ∀y1 ∈ N∃y2 ∈ N (x, y1, y2) ∈ R}.

3

A is ∆0

2 if it is Σ0 2 and Π0 2.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Definition Suppose A ⊆ N.

1

A is Σ0

3 if there is a computable R ⊆ N4 such that

A = {x ∈ N | ∃y1 ∈ N∀y2 ∈ N∃y3 ∈ N (x, y1, y2, y3) ∈ R}.

2

A is Π0

3 if there is a computable R ⊆ N3 such that

A = {x ∈ N | ∀y1 ∈ N∃y2 ∈ N∀y3 ∈ N (x, y1, y2, y3) ∈ R}.

3

A is ∆0

3 if it is Σ0 3 and Π0 3.

Etc.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

Theorem A set is ∆0

2 if and only if it is Turing reducible to the halting set.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density

Outline

1

Computability Theory Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

2

Approximating the incomputable- the Ershov hierarchy

3

Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density

Definition Suppose A ⊆ N and that ψ : N × N → {0, 1}. We say that ψ uniformly approximates A if:

1

ψ(n, 0) = 0 for all n, and if

2

χA(n) = limt→∞ ψ(n, t) for all n. Theorem A is ∆2

0 if and only if A has a computable uniform

approximation.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density

Notation Suppose ψ : N × N → {0, 1}. For each n ∈ N, let Mψ(n) = #{t : ψ(n, t) = ψ(n, t + 1)}. Idea behind Ershov hierarchy: rank each ∆0

2 set by the growth

rate of Mψ.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density

Definition Suppose A ⊆ N and that f : N → N. We say that A is f-c.e. if A has a computable uniform approximation ψ such that Mψ ≤ f. Definition A is n-c.e. if A is f-c.e. where f(n) = n for all n.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density

Definition A is ω-c.e. if A is f-c.e. for some computable f. Theorem (Epstein, Hass, Kramer 1981 )

1

For each n ∈ N, there is an (n + 1)-c.e. set that is not n-c.e..

2

There is an ω-c.e. set that is not n-c.e. for any n.

3

There is a ∆0

2-set that is not ω-c.e..

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density

Proposition Suppose A is an n-c.e. set where n is a positive integer.

1

If n = 2k where k ∈ N, then A can be written in the form (A1 − A2) ∪ . . . ∪ (A2k−1 − A2k) where A1, . . . , A2k are c.e. and A1 ⊇ . . . ⊇ A2k.

2

If n = 2k +1 where k ∈ N, then A can be written in the form (A1 − A2) ∪ . . . ∪ (A2k−1 − A2k) ∪ A2k+1 where A1, . . . , A2k+1 are c.e. and A1 ⊇ . . . ⊇ A2k+1.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Outline

1

Computability Theory Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy

2

Approximating the incomputable- the Ershov hierarchy

3

Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Notation When A ⊆ N and n ∈ N, let ρn(A) = #(A ∩ {0, . . . , n}) n + 1 . Definition Suppose A ⊆ N.

1

The upper density of A is ρ(A) := lim supn→∞ ρn(A).

2

The lower density of A is ρ(A) := lim infn→∞ ρn(A).

3

If ρ(A) = ρ(A), then the density of A is defined to be ρ(A) := ρ(A) = ρ(A).

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Motivation

Classify complexity of sets according to the complexity of their asymptotic densities (when defined). Use asymptotic density to quantify how close a set is to being computable.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Definition Let r be a real number.

1

r is left Σ0

n if

{q ∈ Q : q < r} is Σ0

n.

2

r is right Σ0

n if

{q ∈ Q : q > r} is Σ0

n.

3

etc.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Theorem (Downey, Jockusch, Schupp, 2012) Suppose A ⊆ N.

1

The density of a Σ0

n set, if defined, is a left Π0 n+1 real.

2

Every left Π0

n+1 real is the density of a Σ0 n set.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Problem Classify the densities of the 2-c.e. sets. Question Is there a 2-c.e. set whose density is not the density of any c.e. set? Question Is there a 3-c.e. set whose density is not the density of any 2-c.e. set?

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Theorem (Downey,Jockusch, Schupp, McNicholl 2013) Suppose n ≥ 2 and that A is an n-c.e. whose density is

• defined. Then, its density is a difference of left Π0

2-reals.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Lemma (Downey,Jockusch, Schupp, McNicholl 2013) If M ≥ an ≥ bn ≥ L for all n, and if limn→∞(an − bn) exists, then limn→∞(an − bn) = lim supn an − lim supn bn. Proof sketch: Let aj,s = ρs(Aj). ρs(A) =  

j odd

aj,s   −  

j even

aj,s   .

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Theorem (Downey, Jockusch, Schupp, McNicholl 2013) If r ∈ [0, 1] is a difference of left Π0

2 reals, then r is the density of

a 2-c.e. set.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Theorem (Downey, Jockusch, Schupp, McNicholl 2013) Let f be a computable, nondecreasing, unbounded function. If A is a ∆0

2 set that has a density, then the density of A is that of

an f-c.e. set. Corollary (Downey, Jockusch, Schupp, McNicholl 2013) If A is a ∆0

2 set whose density is defined, then its density is that

• f an ω-c.e. set.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Theorem (Ambos-Spies, Weihrauch, Zheng 2000 ) There is a real r ∈ [0, 1] that is a difference of left Π0

2 reals but is

not left-Π0

2 nor left-Σ0 2.

Corollary There is a 2-c.e. set whose density is not the density of any c.e. set.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Theorem (Ambos-Spies, Weihruach, Zheng 2000) There is a ∆0

3 real that is not a difference of left Π0 2-reals.

Corollary There is a ∆0

2 set whose density is not the density of any n-c.e.

set.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Definition

1

If A ⊆ ω and if r ∈ [0, 1], an r-description of A is any set B such that the lower density of {n : A(n) = B(n)} is at least r.

2

A set A is coarsely computable at density r if there is a computable r-description B of A.

3

A set A is coarsely computable if it is coarsely computable at density 1. Clearly every computable set is coarsely computable; converse fails.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Definition If A ⊆ ω, the coarse computability bound of A is γ(A) = sup{r : A is coarsely computable at density r}.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Definition When A, B ⊆ N, let d(A, B) = ρ(A△B) It follows that d is a metric on the power set of N (modulo an equivalence relation). When A ⊆ N, γ(A) = 1 − d(A, Comp).

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Definition A set is extremal for coarse computability if it is coarsely computable at density γ(A). i.e. there is a computable set C so that d(A, C) = d(A, Comp).

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Theorem (Hirschfeldt, Jockusch, McNicholl, Schupp 2014) Every real in [0, 1] is the coarse computability bound of a set that is extremal for coarse computability. Every real in (0, 1] is the coarse computability bound of a set that is not extremal for coarse computability.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Definition If A ⊆ N, then Γ(A) is defined to be the infimum of the coarse computability bounds of the sets computed by A. That is, Γ(A) = inf{γ(B) : A computes B}. Theorem (Andrews, Cai, Diamondstone, Jockusch, Lempp 2013) There is a ∆0

2 set A so that Γ(A) = 1/2.

Theorem (Hirschfeldt, Jockusch, McNicholl, Schupp 2014)

1

There is a set A so that Γ(A) = 0.

2

If Γ(A) > 1/2, then Γ(A) = 1.

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Question Is there a set A so that 0 < Γ(A) < 1/2? Theorem (Monin 2017) If Γ(A) < 1/2, then Γ(A) = 0

Timothy H. McNicholl

Computability Theory Approximating the incomputable- the Ershov hierarchy Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation

Summary

Two themes of computability theory: complexity of incomputable sets, approximation of incomputable sets. Asymptotic density has given us new perspectives on these themes and a wealth of new results and problems. Thank you Paul Schupp!

Timothy H. McNicholl