Arithmetical hierarchy

An illustration of how the levels of the hierarchy interact and where some basic set categories lie within it.

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or KleeneMostowski hierarchy 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 recursion 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 formulasEdit

The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted   and   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   is logically equivalent to a formula with only bounded quantifiers then   is assigned the classifications   and  .

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

  • If   is logically equivalent to a formula of the form  , where   is  , then   is assigned the classification  .
  • If   is logically equivalent to a formula of the form  , where   is  , then   is assigned the classification  .

Also, a   formula is equivalent to a formula that begins with some existential quantifiers and alternates   times between series of existential and universal quantifiers; while a   formula is equivalent to a formula that begins with some universal quantifiers and alternates similarly.

Because every formula is equivalent to a formula in prenex normal form, every formula with no set quantifiers is assigned at least one classification. Because redundant quantifiers can be added to any formula, once a formula is assigned the classification   or   it will be assigned the classifications   and   for every r greater than n. The most important classification assigned to a formula is thus the one with the least n, because this is enough to determine all the other classifications.

The arithmetical hierarchy of sets of natural numbersEdit

A set X of natural numbers is defined by 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,

 

where   is the numeral in the language of arithmetic corresponding to  . 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  ,  , and  , where   is a natural number, as follows. If X is definable by a   formula then X is assigned the classification  . If X is definable by a   formula then X is assigned the classification  . If X is both   and   then   is assigned the additional classification  .

Note that it rarely makes sense to speak of   formulas; the first quantifier of a formula is either existential or universal. So a   set is not defined by a   formula; rather, there are both   and   formulas that define the set.

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 hierarchiesEdit

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  ,   or   in Y, denoted respectively     and  . To do so, fix a set of integers Y and add a predicate for membership in Y to the language of Peano arithmetic. We then say that X is in   if it is defined by a   formula in this expanded language. In other words, X is   if it is defined by a   formula allowed to ask questions about membership in Y. Alternatively one can view the   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 integers. Let X be the set of numbers divisible by an element of Y. Then X is defined by the formula   so X is in   (actually it is in   as well since we could bound both quantifiers by n).

Arithmetic reducibility and degreesEdit

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   or   for some integer n. A set X is arithmetical in a set Y, denoted  , if X is definable a some formula in the language of Peano arithmetic extended by a predicate for membership in Y. Equivalently, X is arithmetical in Y if X is in   or   for some integer n. A synonym for   is: X is arithmetically reducible to Y.

The relation   is reflexive and transitive, and thus the relation   defined by the rule

 

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

The arithmetical hierarchy of subsets of Cantor and Baire spaceEdit

The Cantor space, denoted  , is the set of all infinite sequences of 0s and 1s; the Baire space, denoted   or  , is the set of all infinite sequences of natural numbers. Note that elements of the Cantor space can be identified with sets of integers and elements of the Baire space with functions from integers to integers.

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   if it is definable by a   formula. The set is assigned the classification   if it is definable by a   formula. If the set is both   and   then it is given the additional classification  . For example, let   be the set of all infinite binary strings which aren't all 0 (or equivalently the set of all non-empty sets of integers). As   we see that   is defined by a   formula and hence is a   set.

Note that while both the elements of the Cantor space (regarded as sets of integers) 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   elements of the Cantor space are not (in general) the same as the elements   of the Cantor space so that   is a   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 subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from   to   to the characteristic function of its graph. A subset of Baire space is given the classification  ,  , or   if and only if the corresponding subset of Cantor space has the same classification.
  • An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional version of second-order arithmetic; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space. This alternate definition gives exactly the same classifications as the first definition.

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 integers. In fact boldface   is just the union of   for all sets of integers Y. Note that the boldface hierarchy is just the standard hierarchy of Borel sets.

Extensions and variationsEdit

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  , since using primitive recursive functions in first-order Peano arithmetic requires, in general, an unbounded existential quantifier, and thus some sets that are in   by this definition are in   by the definition given in the beginning of this article.   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  . The classifications   and   are defined inductively with the following rules.

  • If the relation   is   then the relation   is defined to be  
  • If the relation   is   then the relation   is defined to be  

This variation slightly changes the classification of some sets. In particular,  , as a class of sets (definable by the relations in the class), is identical to   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 notationEdit

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

The subscript   in the symbols   and   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   formulas and universal in   formulas.

The superscript   in the symbols  ,  , and   indicates the type of the objects being quantified over. Type 0 objects are natural numbers, and objects of type   are functions that map the set of objects of type   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.

ExamplesEdit

  • The   sets of numbers are those definable by a formula of the form   where   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  . Intuitively, an index   falls into this set if and only if for every   "there is an   such that the Turing machine with index   halts on input   after   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   formula.
  • Every   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,   sets are sometimes called effectively open. Similarly, every   set is closed and the   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   subset of Cantor or Baire space is a   set (that is, a set which equals the intersection of countably many open sets). Moreover, each of these open sets is   and the list of Gödel numbers of these open sets has a computable enumeration. If   is a   formula with a free set variable X and free number variables   then the   set   is the intersection of the   sets of the form   as n ranges over the set of natural numbers.
  • The   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  ); 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  , which allow the use of primitive recursive functions, as now the quantifiers may be bound by any primitive recursive function of the arguments.
  • The   formulas under an alternative definition, that allows the use of primitive recursive functions with bounded quantifiers, correspond to sets of integers of the form   for a primitive recursive function f. This is because allowing bounded quantifier adds nothing to the definition: for a primitive recursive f,   is the same as  , and   is the same as  ; with course-of-values recursion each of these can be defined by a single primitive recursion function.

PropertiesEdit

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

  • The collections   and   are closed under finite unions and finite intersections of their respective elements.
  • A set is   if and only if its complement is  . A set is   if and only if the set is both   and  , in which case its complement will also be  .
  • The inclusions   and   hold for all  . Thus the hierarchy does not collapse. This is a direct consequence of Post's theorem.
  • The inclusions  ,   and   hold for  .
  • 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   (being computable with an oracle to the halting problem) but not in  , .
  •  . The inclusion is strict by the definition given in this article, but an identity with   holds under one of the variations of the definition given above.

Relation to Turing machinesEdit

Computable setsEdit

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  . This means that S is both in   and in  , and hence it is in  .

Similarly, for every set S in  , both S and its complement are in   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 resultsEdit

The Turing computable sets of natural numbers are exactly the sets at level   of the arithmetical hierarchy. The recursively enumerable sets are exactly the sets at level  .

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

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 set   (the nth Turing jump of the empty set) is many-one complete in  .
  • The set   is many-one complete in  .
  • The set   is Turing complete in  .

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   of the arithmetical hierarchy.

Relation to other hierarchiesEdit

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
= Π0
= Δ0
= Σ1
0
= Π1
0
= Δ1
0
= arithmetical
Σ0
= Π0
= Δ0
= Σ1
0
= Π1
0
= Δ1
0
= boldface arithmetical
Δ0
α
recursive)
Δ0
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
= Π1
= Δ1
= Σ2
0
= Π2
0
= Δ2
0
= analytical
Σ1
= Π1
= Δ1
= Σ2
0
= Π2
0
= Δ2
0
= P = projective


See alsoEdit

ReferencesEdit

  • Japaridze, Giorgie (1994), "The logic of arithmetical hierarchy", Annals of Pure and Applied Logic, 66 (2): 89–112, doi:10.1016/0168-0072(94)90063-9, Zbl 0804.03045.
  • Moschovakis, Yiannis N. (1980), Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics, 100, North Holland, ISBN 0-444-70199-0, Zbl 0433.03025.
  • Nies, André (2009), Computability and randomness, Oxford Logic Guides, 51, Oxford: Oxford University Press, ISBN 978-0-19-923076-1, Zbl 1169.03034.
  • Rogers, H., jr (1967), Theory of recursive functions and effective computability, Maidenhead: McGraw-Hill, Zbl 0183.01401.