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Hierarchy of complexity classes for formulas defining sets
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 Kleene-Mostowski 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 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 numbers
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 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 , 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).
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 Xis arithmetical in a set Y, denoted , if X is definable as 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 space
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 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 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 notation
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.
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.
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 machines
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 results
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 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.