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 Computability Theory 1 Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy Approximating the incomputable- the Ershov hierarchy 2 3 Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation Timothy H. McNicholl
Computability on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density Complexity via the arithmetic hierarchy Outline Computability Theory 1 Computability on N Computability on other structures Complexity via Turing degrees Complexity via the arithmetic hierarchy Approximating the incomputable- the Ershov hierarchy 2 3 Asymptotic density Another perspective on complexity Coarse computability: another perspective on approximation Timothy H. McNicholl
Computability on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density 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 on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density 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 on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density Complexity via the arithmetic hierarchy Fundamental definition Let N = { 0 , 1 , 2 , . . . } . Definition A function f : ⊆ N m → N is computable if there is an algorithm that given any n 1 , . . . , n m ∈ N as input, halts with output f ( n 1 , . . . , n m ) if ( n 1 , . . . , n m ) ∈ dom ( f ) and does not halt if ( n 1 , . . . , n m ) �∈ dom ( f ) . Examples: Addition, multiplication, division gcd’s Probably any function you can think of. Timothy H. McNicholl
Computability on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density 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 on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density Complexity via the arithmetic hierarchy Notation When A ⊆ N , χ A is the characteristic function of A. Definition A ⊆ N is computable if χ A is computable. Timothy H. McNicholl
Computability on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density Complexity via the arithmetic hierarchy Definition A ⊆ N is computably enumerable if either A = ∅ or if there is a sequence { a n } n ∈ N such that A = { a 0 , a 1 , a 2 , . . . } and such that n �→ a n 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 on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density 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 on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density Complexity via the arithmetic hierarchy Definition A coding of a set X is a function c : ⊆ N → X that is onto. Definition Suppose c j is a coding of X j for j = 1 , 2 and that f : ⊆ X 1 → X 2 . With respect to these codings, we say that f is computable if there is a computable function F : ⊆ N → N such that dom ( F ) = c 1 [ dom ( f )] and F ( c 1 ( x )) = c 2 ( f ( x )) for all x ∈ dom ( f ) . Timothy H. McNicholl
Computability on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density Complexity via the arithmetic hierarchy F ! ! c 2 c 1 X 2 X 1 f Timothy H. McNicholl
Computability on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density 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 on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density 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 Timothy H. McNicholl and only if A .
Computability on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density 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 on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density 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 on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density Complexity via the arithmetic hierarchy Definition Suppose A ⊆ N . A is Σ 0 1 if there is a computable R ⊆ N × N such that 1 A = { x ∈ N | ∃ y ∈ N ( x , y ) ∈ R } . A is Π 0 1 if there is a computable R ⊆ N × N such that 2 A = { x ∈ N | ∀ y ∈ N ( x , y ) ∈ R } . A is ∆ 0 1 if it is Σ 0 1 and Π 0 1 . 3 Timothy H. McNicholl
Computability on N Computability Theory Computability on other structures Approximating the incomputable- the Ershov hierarchy Complexity via Turing degrees Asymptotic density Complexity via the arithmetic hierarchy Definition Suppose A ⊆ N . 2 if there is a computable R ⊆ N 3 such that A is Σ 0 1 A = { x ∈ N | ∃ y 1 ∈ N ∀ y 2 ∈ N ( x , y 1 , y 2 ) ∈ R } . 2 if there is a computable R ⊆ N 3 such that A is Π 0 2 A = { x ∈ N | ∀ y 1 ∈ N ∃ y 2 ∈ N ( x , y 1 , y 2 ) ∈ R } . A is ∆ 0 2 if it is Σ 0 2 and Π 0 2 . 3 Timothy H. McNicholl
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