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 was invented independently by Kleene (1943) and Mostowski (1946). 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

The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted \Sigma^0_n and \Pi^0_n for natural numbers n (including 0). The Greek letters here are lightface symbols, which indicates that the formulas do not contain If a formula \phi is logically equivalent to a formula having no unbounded quantifiers, i.e. in which all quantifiers are bounded quantifiers then \phi is assigned the classifications \Sigma^0_0 and \Pi^0_0. The classifications \Sigma^0_n and \Pi^0_n are defined inductively for every natural number n using the following rules: A \Sigma^0_n formula is equivalent to a formula that begins with some existential quantifiers and alternates n-1 times between series of existential and universal quantifiers; while a \Pi^0_n 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 \Sigma^0_n or \Pi^0_n it will be assigned the classifications \Sigma^0_m and \Pi^0_m 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

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, where is the numeral in the language of arithmetic corresponding to 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 \Sigma^0_n, \Pi^0_n, and \Delta^0_n, where n is a natural number, as follows. If X is definable by a \Sigma^0_n formula then X is assigned the classification \Sigma^0_n. If X is definable by a \Pi^0_n formula then X is assigned the classification \Pi^0_n. If X is both \Sigma^0_n and \Pi^0_n then X is assigned the additional classification \Delta^0_n. Note that it rarely makes sense to speak of \Delta^0_n formulas; the first quantifier of a formula is either existential or universal. So a \Delta^0_n set is not necessarily defined by a \Delta^0_n formula in the sense of a formula that is both \Sigma^0_n and \Pi^0_n; rather, there are both \Sigma^0_n and \Pi^0_n formulas that define the set. For example, the set of odd natural numbers n is definable by either or. 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 first-order 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.

Meaning of the notation

The following meanings can be attached to the notation for the arithmetical hierarchy on formulas. The subscript n in the symbols \Sigma^0_n and \Pi^0_n indicates the number of alternations of blocks of universal and existential first-order quantifiers that are used in a formula. Moreover, the outermost block is existential in \Sigma^0_n formulas and universal in \Pi^0_n formulas. The superscript 0 in the symbols \Sigma^0_n, \Pi^0_n, and \Delta^0_n indicates the type of the objects being quantified over. Type 0 objects are natural numbers, and objects of type i+1 are functions that map the set of objects of type 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

Relativized arithmetical hierarchies

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 \Sigma^0_n, \Delta^0_n or \Pi^0_n in Y, denoted respectively, and \Pi^{0,Y}_n. 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 if it is defined by a \Sigma^0_n formula in this expanded language. In other words, X is if it is defined by a formula allowed to ask questions about membership of 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 natural numbers. 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 degrees

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 \Sigma^0_n or \Pi^0_n for some natural number n. A set X is arithmetical in a set Y, denoted X \leq_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 or \Pi^{0,Y}_n for some natural number n. A synonym for X \leq_A Y is: X is arithmetically reducible to Y. The relation X \leq_A Y is reflexive and transitive, and thus the relation \equiv_A defined by the rule is an equivalence relation. The equivalence classes of this relation are called the arithmetic degrees; they are partially ordered under \leq_A.

The arithmetical hierarchy of subsets of Cantor and Baire space

The Cantor space, denoted 2^{\omega}, is the set of all infinite sequences of 0s and 1s; the Baire space, denoted or \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 \Sigma^0_n if it is definable by a \Sigma^0_n formula. The set is assigned the classification \Pi^0_n if it is definable by a \Pi^0_n formula. If the set is both \Sigma^0_n and \Pi^0_n then it is given the additional classification \Delta^0_n. For example, let be the set of all infinite binary strings that aren't all 0 (or equivalently the set of all non-empty sets of natural numbers). As we see that O is defined by a \Sigma^0_1 formula and hence is a \Sigma^0_1 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 \Pi^0_n elements of the Cantor space are not (in general) the same as the elements X of the Cantor space so that {X} is a \Pi^0_n 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 is just the union of for all sets of natural numbers Y. Note that the boldface hierarchy is just the standard hierarchy of Borel sets.

Extensions and variations

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 \Sigma^0_0 by this definition are strictly in \Sigma^0_1 by the definition given in the beginning of this article. The class \Sigma^0_1 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 \Sigma^0_n and \Pi^0_n are defined inductively with the following rules. 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 \Delta^0_1 as the latter was formerly defined. It can be extended to cover finitary relations on the natural numbers, Baire space, and Cantor space.

Properties

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

Relation to Turing machines

Computable sets

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 \Sigma^0_1. This means that S is both in \Sigma^0_1 and in \Pi^0_1, and hence it is in \Delta^0_1. Similarly, for every set S in \Delta^0_1, both S and its complement are in \Sigma^0_1 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

The Turing computable sets of natural numbers are exactly the sets at level \Delta^0_1 of the arithmetical hierarchy. The recursively enumerable sets are exactly the sets at level \Sigma^0_1. 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 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 \Delta^0_1 of the arithmetical hierarchy.

Relation to other hierarchies