![Arithmetic hierarchy](https://upload.wikimedia.org/wikipedia/commons/5/53/Arithmetic_hierarchy.svg)
In
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians
Stephen Cole Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...
and
Andrzej Mostowski) classifies certain
sets based on the
complexity
Complexity characterises the behaviour of a system or model whose components interaction, interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generall ...
of formulas that define them. Any set that receives a classification is called arithmetical.
The arithmetical hierarchy is important in
recursion theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since e ...
,
effective descriptive set theory
Effective descriptive set theory is the branch of descriptive set theory dealing with sets of reals having lightface definitions; that is, definitions that do not require an arbitrary real parameter (Moschovakis 1980). Thus effective descriptive ...
, and the study of formal theories such as
Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...
.
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 In recursion theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an importa ...
and the
analytical hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers ...
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
In first-order logic, a first-order theory is given by a set of axioms in some
language. This entry lists some of the more common examples used in model theory and some of their properties.
Preliminaries
For every natural mathematical structure ...
. The classifications are denoted
and
for natural numbers ''n'' (including 0). The Greek letters here are
lightface
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a ''point'' is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by ...
symbols, which indicates that the formulas do not contain set parameters.
If a formula
is logically equivalent to a formula without quantifiers, then
is assigned the classifications
and
. Since any formula with
bounded quantifier
In the study of formal theories in mathematical logic, bounded quantifiers (a.k.a. restricted quantifiers) are often included in a formal language in addition to the standard quantifiers "∀" and "∃". Bounded quantifiers differ from "∀" and " ...
s can be replaced by a formula without quantifiers (for example,
is equivalent to
), we can also allow
to have bounded quantifiers.
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
.
A
formula is equivalent to a formula that begins with some
existential quantifier
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
s and alternates
times between series of existential and
universal quantifier
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other w ...
s; while a
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
A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in prop ...
, every formula 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'' > ''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
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...
(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. For example, the set of odd natural numbers
is definable by either
or
.
A parallel definition is used to define the arithmetical hierarchy on finite
Cartesian power
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\tim ...
s 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''-
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s of natural numbers. These are in fact related by the use of a
pairing function
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.
Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural ...
.
Relativized arithmetical hierarchies
Just as we can define what it means for a set ''X'' to be
recursive
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
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 degrees
Arithmetical reducibility is an intermediate notion between
Turing reducibility
In computability theory, a Turing reduction from a decision problem A to a decision problem B is an oracle machine which decides problem A given an oracle for B (Rogers 1967, Soare 1987). It can be understood as an algorithm that could be used to s ...
and
hyperarithmetic reducibility In recursion theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an importa ...
.
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 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 In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
, denoted
, is the set of all infinite sequences of 0s and 1s; the
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
, 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
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.
A precurs ...
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 In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
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 In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in effective descriptive set theory and in constructive analysis. In particular, standard examples of ...
; 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
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named a ...
.
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
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined ...
. 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
In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers ...
. 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
* The
sets of numbers are those definable by a formula of the form
where
has only bounded quantifiers. These are exactly the
recursively enumerable set
In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if:
*There is an algorithm such that the ...
s.
* 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
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
. 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'',