This article needs additional citations for verification. (August 2021) |
In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.^{[1]}
Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers.^{[1]}
Definition
editA pairing function is a bijection^{[verification needed]}
- ^{[1]}
More generally, a pairing function on a set A is a function that maps each pair of elements from A into an element of A, such that any two pairs of elements of A are associated with different elements of A,^{[2]} or a bijection from to A.^{[3]}
Hopcroft and Ullman pairing function
editHopcroft and Ullman (1979) define the following pairing function: , where .^{[1]} This is the same as the Cantor pairing function below, shifted to exclude 0 (i.e., , , and ).
Cantor pairing function
editThe Cantor pairing function is a primitive recursive pairing function
defined by
- ^{[1]}^{[verification needed]}
where .^{[1]}
It can also be expressed as .^{[2]}
It is also strictly monotonic w.r.t. each argument, that is, for all , if , then ; similarly, if , then .^{[citation needed]}
The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem.^{[1]}^{[verification needed]} Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to k_{1} and k_{2} we often denote the resulting number as ⟨k_{1}, k_{2}⟩.^{[citation needed]}
This definition can be inductively generalized to the Cantor tuple function^{[citation needed]}
for as
with the base case defined above for a pair: ^{[1]}
Inverting the Cantor pairing function
editLet be an arbitrary natural number. We will show that there exist unique values such that
and hence that the function π(x, y) is invertible. It is helpful to define some intermediate values in the calculation:
where t is the triangle number of w. If we solve the quadratic equation
for w as a function of t, we get
which is a strictly increasing and continuous function when t is non-negative real. Since
we get that
and thus
where ⌊ ⌋ is the floor function. So to calculate x and y from z, we do:
Since the Cantor pairing function is invertible, it must be one-to-one and onto.^{[2]}^{[additional citation(s) needed]}
Examples
editTo calculate π(47, 32):
- 47 + 32 = 79,
- 79 + 1 = 80,
- 79 × 80 = 6320,
- 6320 ÷ 2 = 3160,
- 3160 + 32 = 3192,
so π(47, 32) = 3192.
To find x and y such that π(x, y) = 1432:
- 8 × 1432 = 11456,
- 11456 + 1 = 11457,
- √11457 = 107.037,
- 107.037 − 1 = 106.037,
- 106.037 ÷ 2 = 53.019,
- ⌊53.019⌋ = 53,
so w = 53;
- 53 + 1 = 54,
- 53 × 54 = 2862,
- 2862 ÷ 2 = 1431,
so t = 1431;
- 1432 − 1431 = 1,
so y = 1;
- 53 − 1 = 52,
so x = 52; thus π(52, 1) = 1432.^{[citation needed]}
Derivation
editThe graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability.^{[a]} The algebraic rules of this diagonal-shaped function can verify its validity for a range of polynomials, of which a quadratic will turn out to be the simplest, using the method of induction. Indeed, this same technique can also be followed to try and derive any number of other functions for any variety of schemes for enumerating the plane.
A pairing function can usually be defined inductively – that is, given the nth pair, what is the (n+1)th pair? The way Cantor's function progresses diagonally across the plane can be expressed as
- .
The function must also define what to do when it hits the boundaries of the 1st quadrant – Cantor's pairing function resets back to the x-axis to resume its diagonal progression one step further out, or algebraically:
- .
Also we need to define the starting point, what will be the initial step in our induction method: π(0, 0) = 0.
Assume that there is a quadratic 2-dimensional polynomial that can fit these conditions (if there were not, one could just repeat by trying a higher-degree polynomial). The general form is then
- .
Plug in our initial and boundary conditions to get f = 0 and:
- ,
so we can match our k terms to get
- b = a
- d = 1-a
- e = 1+a.
So every parameter can be written in terms of a except for c, and we have a final equation, our diagonal step, that will relate them:
Expand and match terms again to get fixed values for a and c, and thus all parameters:
- a = 1/2 = b = d
- c = 1
- e = 3/2
- f = 0.
Therefore
is the Cantor pairing function, and we also demonstrated through the derivation that this satisfies all the conditions of induction.^{[citation needed]}
Other pairing functions
editThe function is a pairing function.
In 1990, Regan proposed the first known pairing function that is computable in linear time and with constant space (as the previously known examples can only be computed in linear time if multiplication can be too, which is doubtful).^{[4]} In fact, both this pairing function and its inverse can be computed with finite-state transducers that run in real time.^{[4]}^{[clarification needed]} In the same paper, the author proposed two more monotone pairing functions that can be computed online in linear time and with logarithmic space; the first can also be computed offline with zero space.^{[4]}^{[clarification needed]}
In 2001, Pigeon proposed a pairing function based on bit-interleaving, defined recursively as:
where and are the least significant bits of i and j respectively.^{[1]}^{[verification needed]}
In 2006, Szudzik proposed a "more elegant" pairing function defined by the expression:
Which can be unpaired using the expression:
(Qualitatively, it assigns consecutive numbers to pairs along the edges of squares.) This pairing function orders SK combinator calculus expressions by depth.^{[2]}^{[clarification needed]} This method is the mere application to of the idea, found in most textbooks on Set Theory,^{[5]} used to establish for any infinite cardinal in ZFC. Define on the binary relation
is then shown to be a well-ordering such that every element has predecessors, which implies that . It follows that is isomorphic to and the pairing function above is nothing more than the enumeration of integer couples in increasing order. (See also Talk:Tarski's theorem about choice#Proof of the converse.)
Notes
edit- ^ The term "diagonal argument" is sometimes used to refer to this type of enumeration, but it is not directly related to Cantor's diagonal argument.^{[citation needed]}
References
edit- ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} Steven Pigeon. "Pairing function". MathWorld. Retrieved 16 August 2021.
- ^ ^{a} ^{b} ^{c} ^{d} Szudzik, Matthew (2006). "An Elegant Pairing Function" (PDF). szudzik.com. Archived (PDF) from the original on 25 November 2011. Retrieved 16 August 2021.
- ^ Szudzik, Matthew P. (2017-06-01). "The Rosenberg-Strong Pairing Function". arXiv:1706.04129 [cs.DM].
- ^ ^{a} ^{b} ^{c} Regan, Kenneth W. (1992-12-01). "Minimum-complexity pairing functions". Journal of Computer and System Sciences. 45 (3): 285–295. doi:10.1016/0022-0000(92)90027-G. ISSN 0022-0000.
- ^ See for instance Thomas, Jech (2006). Set theory: the third millennium edition. Springer Monographs in Mathematics. Springer-Verlag. p. 30. doi:10.1007/3-540-44761-X. ISBN 3-540-44085-2.