Arithmetical hierarchy – Wikipedia

Hierarchy of complexity classes for formulas defining sets

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical.

The arithmetical hierarchy is important in computability theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.

The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines.

The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets.

The arithmetical hierarchy of formulas[edit]

The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted

${displaystyle Sigma _{n}^{0}}$

and

${displaystyle Pi _{n}^{0}}$

for natural numbers n (including 0). The Greek letters here are lightface symbols, which indicates that the formulas do not contain set parameters.

If a formula

${displaystyle phi }$

is logically equivalent to a formula without quantifiers, then

${displaystyle phi }$

is assigned the classifications

${displaystyle Sigma _{0}^{0}}$

and

${displaystyle Pi _{0}^{0}}$

. Since any formula with bounded quantifiers can be replaced by a formula without quantifiers^{[citation needed]} (for example,

${displaystyle exists x<2 phi (x)}$

is equivalent to

${displaystyle phi (0)vee phi (1)}$

), we can also allow

${displaystyle phi }$

to have bounded quantifiers.

The classifications

${displaystyle Sigma _{n}^{0}}$

and

${displaystyle Pi _{n}^{0}}$

are defined inductively for every natural number n using the following rules:

A

${displaystyle Sigma _{n}^{0}}$

formula is equivalent to a formula that begins with some existential quantifiers and alternates

${displaystyle n-1}$

times between series of existential and universal quantifiers; while a

${displaystyle Pi _{n}^{0}}$

formula is equivalent to a formula that begins with some universal quantifiers and alternates analogously.

Because every first-order formula has a prenex normal form, every formula is assigned at least one classification. Because redundant quantifiers can be added to any formula, once a formula is assigned the classification

${displaystyle Sigma _{n}^{0}}$

or

${displaystyle Pi _{n}^{0}}$

it will be assigned the classifications

${displaystyle Sigma _{m}^{0}}$

and

${displaystyle Pi _{m}^{0}}$

for every m > n. The only relevant classification assigned to a formula is thus the one with the least n; all the other classifications can be determined from it.

The arithmetical hierarchy of sets of natural numbers[edit]

A set X of natural numbers is defined by a formula φ in the language of Peano arithmetic (the first-order language with symbols “0” for zero, “S” for the successor function, “+” for addition, “×” for multiplication, and “=” for equality), if the elements of X are exactly the numbers that satisfy φ. That is, for all natural numbers n,

${displaystyle nin XLeftrightarrow mathbb {N} models varphi ({underline {n}}),}$

where

${displaystyle {underline {n}}}$

is the numeral in the language of arithmetic corresponding to

${displaystyle n}$

. A set is definable in first-order arithmetic if it is defined by some formula in the language of Peano arithmetic.

Each set X of natural numbers that is definable in first-order arithmetic is assigned classifications of the form

${displaystyle Sigma _{n}^{0}}$

,

${displaystyle Pi _{n}^{0}}$

, and

${displaystyle Delta _{n}^{0}}$

, where

${displaystyle n}$

is a natural number, as follows. If X is definable by a

${displaystyle Sigma _{n}^{0}}$

formula then X is assigned the classification

${displaystyle Sigma _{n}^{0}}$

. If X is definable by a

${displaystyle Pi _{n}^{0}}$

formula then X is assigned the classification

${displaystyle Pi _{n}^{0}}$

. If X is both

${displaystyle Sigma _{n}^{0}}$

and

${displaystyle Pi _{n}^{0}}$

then

${displaystyle X}$

is assigned the additional classification

${displaystyle Delta _{n}^{0}}$

.

Note that it rarely makes sense to speak of

${displaystyle Delta _{n}^{0}}$

formulas; the first quantifier of a formula is either existential or universal. So a

${displaystyle Delta _{n}^{0}}$

set is not necessarily defined by a

${displaystyle Delta _{n}^{0}}$

formula in the sense of a formula that is both

${displaystyle Sigma _{n}^{0}}$

and

${displaystyle Pi _{n}^{0}}$

; rather, there are both

${displaystyle Sigma _{n}^{0}}$

and

${displaystyle Pi _{n}^{0}}$

formulas that define the set. For example, the set of odd natural numbers

${displaystyle n}$

is definable by either

${displaystyle forall k(nneq 2times k)}$

or

${displaystyle exists k(n=2times k+1)}$

.

A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of the set of natural numbers. Instead of formulas with one free variable, formulas with k free number variables are used to define the arithmetical hierarchy on sets of k-tuples of natural numbers. These are in fact related by the use of a pairing function.

Relativized arithmetical hierarchies[edit]

Just as we can define what it means for a set X to be recursive relative to another set Y by allowing the computation defining X to consult Y as an oracle we can extend this notion to the whole arithmetic hierarchy and define what it means for X to be

${displaystyle Sigma _{n}^{0}}$

,

${displaystyle Delta _{n}^{0}}$

or

${displaystyle Pi _{n}^{0}}$

in Y, denoted respectively

${displaystyle Sigma _{n}^{0,Y}}$

,

${displaystyle Delta _{n}^{0,Y}}$

and

${displaystyle Pi _{n}^{0,Y}}$

. To do so, fix a set of natural numbers Y and add a predicate for membership of Y to the language of Peano arithmetic. We then say that X is in

${displaystyle Sigma _{n}^{0,Y}}$

if it is defined by a

${displaystyle Sigma _{n}^{0}}$

formula in this expanded language. In other words, X is

${displaystyle Sigma _{n}^{0,Y}}$

if it is defined by a

${displaystyle Sigma _{n}^{0}}$

formula allowed to ask questions about membership of Y. Alternatively one can view the

${displaystyle Sigma _{n}^{0,Y}}$

sets as those sets that can be built starting with sets recursive in Y and alternately taking unions and intersections of these sets up to n times.

For example, let Y be a set of natural numbers. Let X be the set of numbers divisible by an element of Y. Then X is defined by the formula

${displaystyle phi (n)=exists mexists t(Y(m)land mtimes t=n)}$

so X is in

${displaystyle Sigma _{1}^{0,Y}}$

(actually it is in

${displaystyle Delta _{0}^{0,Y}}$

as well, since we could bound both quantifiers by n).

Arithmetic reducibility and degrees[edit]

Arithmetical reducibility is an intermediate notion between Turing reducibility and hyperarithmetic reducibility.

A set is arithmetical (also arithmetic and arithmetically definable) if it is defined by some formula in the language of Peano arithmetic. Equivalently X is arithmetical if X is

${displaystyle Sigma _{n}^{0}}$

or

${displaystyle Pi _{n}^{0}}$

for some natural number n. A set X is arithmetical in a set Y, denoted

${displaystyle Xleq _{A}Y}$

, if X is definable as some formula in the language of Peano arithmetic extended by a predicate for membership of Y. Equivalently, X is arithmetical in Y if X is in

${displaystyle Sigma _{n}^{0,Y}}$

or

${displaystyle Pi _{n}^{0,Y}}$

for some natural number n. A synonym for

${displaystyle Xleq _{A}Y}$

is: X is arithmetically reducible to Y.

The relation

${displaystyle Xleq _{A}Y}$

is reflexive and transitive, and thus the relation

${displaystyle equiv _{A}}$

defined by the rule

${displaystyle Xequiv _{A}Yiff Xleq _{A}Yland Yleq _{A}X}$

is an equivalence relation. The equivalence classes of this relation are called the arithmetic degrees; they are partially ordered under

${displaystyle leq _{A}}$

.

The arithmetical hierarchy of subsets of Cantor and Baire space[edit]

The Cantor space, denoted

${displaystyle 2^{omega }}$

, is the set of all infinite sequences of 0s and 1s; the Baire space, denoted

${displaystyle omega ^{omega }}$

or

${displaystyle {mathcal {N}}}$

, is the set of all infinite sequences of natural numbers. Note that elements of the Cantor space can be identified with sets of natural numbers and elements of the Baire space with functions from natural numbers to natural numbers.

The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantifiers can naturally be viewed as quantifying over Cantor space. A subset of Cantor space is assigned the classification

${displaystyle Sigma _{n}^{0}}$

if it is definable by a

${displaystyle Sigma _{n}^{0}}$

formula. The set is assigned the classification

${displaystyle Pi _{n}^{0}}$

if it is definable by a

${displaystyle Pi _{n}^{0}}$

formula. If the set is both

${displaystyle Sigma _{n}^{0}}$

and

${displaystyle Pi _{n}^{0}}$

then it is given the additional classification

${displaystyle Delta _{n}^{0}}$

. For example, let

${displaystyle Osubset 2^{omega }}$

be the set of all infinite binary strings which aren’t all 0 (or equivalently the set of all non-empty sets of natural numbers). As

${displaystyle O={Xin 2^{omega }|exists n(X(n)=1)}}$

we see that

${displaystyle O}$

is defined by a

${displaystyle Sigma _{1}^{0}}$

formula and hence is a

${displaystyle Sigma _{1}^{0}}$

set.

Note that while both the elements of the Cantor space (regarded as sets of natural numbers) and subsets of the Cantor space are classified in arithmetic hierarchies, these are not the same hierarchy. In fact the relationship between the two hierarchies is interesting and non-trivial. For instance the

${displaystyle Pi _{n}^{0}}$

elements of the Cantor space are not (in general) the same as the elements

${displaystyle X}$

of the Cantor space so that

${displaystyle {X}}$

is a

${displaystyle Pi _{n}^{0}}$

subset of the Cantor space. However, many interesting results relate the two hierarchies.

There are two ways that a subset of Baire space can be classified in the arithmetical hierarchy.

A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of Baire space or Cantor space, using formulas with several free variables. The arithmetical hierarchy can be defined on any effective Polish space; the definition is particularly simple for Cantor space and Baire space because they fit with the language of ordinary second-order arithmetic.

Note that we can also define the arithmetic hierarchy of subsets of the Cantor and Baire spaces relative to some set of natural numbers. In fact boldface

${displaystyle mathbf {Sigma } _{n}^{0}}$

is just the union of

${displaystyle Sigma _{n}^{0,Y}}$

for all sets of natural numbers Y. Note that the boldface hierarchy is just the standard hierarchy of Borel sets.

Extensions and variations[edit]

It is possible to define the arithmetical hierarchy of formulas using a language extended with a function symbol for each primitive recursive function. This variation slightly changes the classification of

${displaystyle Sigma _{0}^{0}=Pi _{0}^{0}=Delta _{0}^{0}}$

, since using primitive recursive functions in first-order Peano arithmetic requires, in general, an unbounded existential quantifier, and thus some sets that are in

${displaystyle Sigma _{0}^{0}}$

by this definition are in

${displaystyle Sigma _{1}^{0}}$

by the definition given in the beginning of this article.

${displaystyle Sigma _{1}^{0}}$

and thus all higher classes in the hierarchy remain unaffected.

A more semantic variation of the hierarchy can be defined on all finitary relations on the natural numbers; the following definition is used. Every computable relation is defined to be

${displaystyle Sigma _{0}^{0}=Pi _{0}^{0}=Delta _{0}^{0}}$

. The classifications

${displaystyle Sigma _{n}^{0}}$

and

${displaystyle Pi _{n}^{0}}$

are defined inductively with the following rules.

• If the relation
  ${displaystyle R(n_{1},ldots ,n_{l},m_{1},ldots ,m_{k}),}$

  is

  ${displaystyle Sigma _{n}^{0}}$

  then the relation

  ${displaystyle S(n_{1},ldots ,n_{l})=forall m_{1}cdots forall m_{k}R(n_{1},ldots ,n_{l},m_{1},ldots ,m_{k})}$

  is defined to be

  ${displaystyle Pi _{n+1}^{0}}$

  

• If the relation
  ${displaystyle R(n_{1},ldots ,n_{l},m_{1},ldots ,m_{k}),}$

  is

  ${displaystyle Pi _{n}^{0}}$

  then the relation

  ${displaystyle S(n_{1},ldots ,n_{l})=exists m_{1}cdots exists m_{k}R(n_{1},ldots ,n_{l},m_{1},ldots ,m_{k})}$

  is defined to be

  ${displaystyle Sigma _{n+1}^{0}}$

  

This variation slightly changes the classification of some sets. In particular,

${displaystyle Sigma _{0}^{0}=Pi _{0}^{0}=Delta _{0}^{0}}$

, as a class of sets (definable by the relations in the class), is identical to

${displaystyle Delta _{1}^{0}}$

as the latter was formerly defined. It can be extended to cover finitary relations on the natural numbers, Baire space, and Cantor space.

Meaning of the notation[edit]

The following meanings can be attached to the notation for the arithmetical hierarchy on formulas.

The subscript

${displaystyle n}$

in the symbols

${displaystyle Sigma _{n}^{0}}$

and

${displaystyle Pi _{n}^{0}}$

indicates the number of alternations of blocks of universal and existential number quantifiers that are used in a formula. Moreover, the outermost block is existential in

${displaystyle Sigma _{n}^{0}}$

formulas and universal in

${displaystyle Pi _{n}^{0}}$

formulas.

The superscript

${displaystyle 0}$

in the symbols

${displaystyle Sigma _{n}^{0}}$

,

${displaystyle Pi _{n}^{0}}$

, and

${displaystyle Delta _{n}^{0}}$

indicates the type of the objects being quantified over. Type 0 objects are natural numbers, and objects of type

${displaystyle i+1}$

are functions that map the set of objects of type

${displaystyle i}$

to the natural numbers. Quantification over higher type objects, such as functions from natural numbers to natural numbers, is described by a superscript greater than 0, as in the analytical hierarchy. The superscript 0 indicates quantifiers over numbers, the superscript 1 would indicate quantification over functions from numbers to numbers (type 1 objects), the superscript 2 would correspond to quantification over functions that take a type 1 object and return a number, and so on.

Examples[edit]

• The
  ${displaystyle Sigma _{1}^{0}}$

  sets of numbers are those definable by a formula of the form

  ${displaystyle exists n_{1}cdots exists n_{k}psi (n_{1},ldots ,n_{k},m)}$

  where

  ${displaystyle psi }$

  has only bounded quantifiers. These are exactly the recursively enumerable sets.

• The set of natural numbers that are indices for Turing machines that compute total functions is
  ${displaystyle Pi _{2}^{0}}$

  . Intuitively, an index

  ${displaystyle e}$

  falls into this set if and only if for every

  ${displaystyle m}$

  “there is an

  ${displaystyle s}$

  such that the Turing machine with index

  ${displaystyle e}$

  halts on input

  ${displaystyle m}$

  after

  ${displaystyle s}$

  steps”. A complete proof would show that the property displayed in quotes in the previous sentence is definable in the language of Peano arithmetic by a

  ${displaystyle Sigma _{1}^{0}}$

  formula.

• Every
  ${displaystyle Sigma _{1}^{0}}$

  subset of Baire space or Cantor space is an open set in the usual topology on the space. Moreover, for any such set there is a computable enumeration of Gödel numbers of basic open sets whose union is the original set. For this reason,

  ${displaystyle Sigma _{1}^{0}}$

  sets are sometimes called effectively open. Similarly, every

  ${displaystyle Pi _{1}^{0}}$

  set is closed and the

  ${displaystyle Pi _{1}^{0}}$

  sets are sometimes called effectively closed.

• Every arithmetical subset of Cantor space or Baire space is a Borel set. The lightface Borel hierarchy extends the arithmetical hierarchy to include additional Borel sets. For example, every
  ${displaystyle Pi _{2}^{0}}$

  subset of Cantor or Baire space is a

  ${displaystyle G_{delta }}$

  set (that is, a set which equals the intersection of countably many open sets). Moreover, each of these open sets is

  ${displaystyle Sigma _{1}^{0}}$

  and the list of Gödel numbers of these open sets has a computable enumeration. If

  ${displaystyle phi (X,n,m)}$

  is a

  ${displaystyle Sigma _{0}^{0}}$

  formula with a free set variable X and free number variables

  ${displaystyle n,m}$

  then the

  ${displaystyle Pi _{2}^{0}}$

  set

  ${displaystyle {Xmid forall nexists mphi (X,n,m)}}$

  is the intersection of the

  ${displaystyle Sigma _{1}^{0}}$

  sets of the form

  ${displaystyle {Xmid exists mphi (X,n,m)}}$

  as n ranges over the set of natural numbers.

• The
  ${displaystyle Sigma _{0}^{0}=Pi _{0}^{0}=Delta _{0}^{0}}$

  formulas can be checked by going over all cases one by one, which is possible because all their quantifiers are bounded. The time for this is polynomial in their arguments (e.g. polynomial in n for

  ${displaystyle varphi (n)}$

  ); thus their corresponding decision problems are included in E (as n is exponential in its number of bits). This no longer holds under alternative definitions of

  ${displaystyle Sigma _{0}^{0}=Pi _{0}^{0}=Delta _{0}^{0}}$

  , which allow the use of primitive recursive functions, as now the quantifiers may be bound by any primitive recursive function of the arguments.

• The
  ${displaystyle Sigma _{0}^{0}=Pi _{0}^{0}=Delta _{0}^{0}}$

  formulas under an alternative definition, that allows the use of primitive recursive functions with bounded quantifiers, correspond to sets of natural numbers of the form

  ${displaystyle {n:f(n)=0}}$

  for a primitive recursive function f. This is because allowing bounded quantifier adds nothing to the definition: for a primitive recursive f,

  ${displaystyle forall k$

  is the same as

  ${displaystyle f(0)+f(1)+…f(n)=0}$

  , and

  ${displaystyle exists k$

  is the same as

  ${displaystyle f(0)*f(1)*…f(n)=0}$

  ; with course-of-values recursion each of these can be defined by a single primitive recursion function.

Properties[edit]

The following properties hold for the arithmetical hierarchy of sets of natural numbers and the arithmetical hierarchy of subsets of Cantor or Baire space.

• For example, for a universal Turing machine T, the set of pairs (n,m) such that T halts on n but not on m, is in
  ${displaystyle Delta _{2}^{0}}$

  (being computable with an oracle to the halting problem) but not in

  ${displaystyle Sigma _{1}^{0}cup Pi _{1}^{0}}$

  , .

• 
  ${displaystyle Sigma _{0}^{0}=Pi _{0}^{0}=Delta _{0}^{0}=Sigma _{0}^{0}cup Pi _{0}^{0}subset Delta _{1}^{0}}$

  . The inclusion is strict by the definition given in this article, but an identity with

  ${displaystyle Delta _{1}^{0}}$

  holds under one of the variations of the definition given above.

Relation to Turing machines[edit]

Computable sets[edit]

If S is a Turing computable set, then both S and its complement are recursively enumerable (if T is a Turing machine giving 1 for inputs in S and 0 otherwise, we may build a Turing machine halting only on the former, and another halting only on the latter).

By Post’s theorem, both S and its complement are in

${displaystyle Sigma _{1}^{0}}$

. This means that S is both in

${displaystyle Sigma _{1}^{0}}$

and in

${displaystyle Pi _{1}^{0}}$

, and hence it is in

${displaystyle Delta _{1}^{0}}$

.

Similarly, for every set S in

${displaystyle Delta _{1}^{0}}$

, both S and its complement are in

${displaystyle Sigma _{1}^{0}}$

and are therefore (by Post’s theorem) recursively enumerable by some Turing machines T₁ and T₂, respectively. For every number n, exactly one of these halts. We may therefore construct a Turing machine T that alternates between T₁ and T₂, halting and returning 1 when the former halts or halting and returning 0 when the latter halts. Thus T halts on every n and returns whether it is in S, So S is computable.

Summary of main results[edit]

The Turing computable sets of natural numbers are exactly the sets at level

${displaystyle Delta _{1}^{0}}$

of the arithmetical hierarchy. The recursively enumerable sets are exactly the sets at level

${displaystyle Sigma _{1}^{0}}$

.

No oracle machine is capable of solving its own halting problem (a variation of Turing’s proof applies). The halting problem for a

${displaystyle Delta _{n}^{0,Y}}$

oracle in fact sits in

${displaystyle Sigma _{n+1}^{0,Y}}$

.

Post’s theorem establishes a close connection between the arithmetical hierarchy of sets of natural numbers and the Turing degrees. In particular, it establishes the following facts for all n ≥ 1:

The polynomial hierarchy is a “feasible resource-bounded” version of the arithmetical hierarchy in which polynomial length bounds are placed on the numbers involved (or, equivalently, polynomial time bounds are placed on the Turing machines involved). It gives a finer classification of some sets of natural numbers that are at level

${displaystyle Delta _{1}^{0}}$

of the arithmetical hierarchy.

Relation to other hierarchies[edit]

Lightface		Boldface
Σ0 0 = Π0 0 = Δ0 0 (sometimes the same as Δ0 1)		Σ0 0 = Π0 0 = Δ0 0 (if defined)
Δ0 1 = recursive		Δ0 1 = clopen
Σ0 1 = recursively enumerable	Π0 1 = co-recursively enumerable	Σ0 1 = G = open	Π0 1 = F = closed
Δ0 2		Δ0 2
Σ0 2	Π0 2	Σ0 2 = Fσ	Π0 2 = Gδ
Δ0 3		Δ0 3
Σ0 3	Π0 3	Σ0 3 = Gδσ	Π0 3 = Fσδ
⋮		⋮
Σ0 <ω = Π0 <ω = Δ0 <ω = Σ1 0 = Π1 0 = Δ1 0 = arithmetical		Σ0 <ω = Π0 <ω = Δ0 <ω = Σ1 0 = Π1 0 = Δ1 0 = boldface arithmetical
⋮		⋮
Δ0 α (α recursive)		Δ0 α (α countable)
Σ0 α	Π0 α	Σ0 α	Π0 α
⋮		⋮
Σ0 ωCK 1 = Π0 ωCK 1 = Δ0 ωCK 1 = Δ1 1 = hyperarithmetical		Σ0 ω1 = Π0 ω1 = Δ0 ω1 = Δ1 1 = B = Borel
Σ1 1 = lightface analytic	Π1 1 = lightface coanalytic	Σ1 1 = A = analytic	Π1 1 = CA = coanalytic
Δ1 2		Δ1 2
Σ1 2	Π1 2	Σ1 2 = PCA	Π1 2 = CPCA
Δ1 3		Δ1 3
Σ1 3	Π1 3	Σ1 3 = PCPCA	Π1 3 = CPCPCA
⋮		⋮
Σ1 <ω = Π1 <ω = Δ1 <ω = Σ2 0 = Π2 0 = Δ2 0 = analytical		Σ1 <ω = Π1 <ω = Δ1 <ω = Σ2 0 = Π2 0 = Δ2 0 = P = projective
⋮		⋮

See also[edit]

References[edit]