In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with ). The sequence continues with the binary operations of addition (), multiplication (), and exponentiation (). After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the -th member of this sequence is named by Reuben Goodstein after the Greek prefix of suffixed with -ation (such as tetration (), pentation (), hexation (), etc.) and can be written using arrows in Knuth's up-arrow notation. Each hyperoperation may be understood recursively in terms of the previous one by:
It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function:
This can be used to easily show numbers much larger than those which scientific notation can, such as Skewes's number and googolplexplex (e.g. is much larger than Skewes's number and googolplexplex), but there are some numbers which even they cannot easily show, such as Graham's number and TREE(3).
This recursion rule is common to many variants of hyperoperations.
Definition The hyperoperation sequence is the sequence of binary operations defined recursively as follows:
For , this definition reproduces the basic arithmetic operations of successor (which is a unary operation), addition, multiplication, and exponentiation, respectively, as
for all nonnegative integers and . The hyperoperations can thus be seen as an answer to the question "what's next?" in the sequence of functions that begins successor, addition, multiplication, exponentiation. Just as integer multiplication is defined as iterated addition and integer exponentiation is defined by iterated multiplication, the next hyperoperation, tetration, is defined by iterated exponentiation; for example, is a power tower of three s, and . Likewise the fifth hyperoperation pentation, is defined by iterated tetration, so that .
The parameters of the hyperoperation hierarchy are sometimes referred to by their analogous exponentiation term; so is the base, is the exponent (or hyperexponent), and is the rank (or grade). In general, may be read as "the -th -ation of ", so that is read as "the 9th tetration of 7", and is read as "the 789th 123-ation of 456".
An alternative way of writing hyperoperations is the compact notation for . In this notation, exponentiation is denoted , tetration is denoted (so that , pentation is denoted , and so on. Hyperoperations can also be expressed using Knuth's up-arrow notation. In this notation, represents the exponentiation function , represents tetration, or represents the pentation , and more generally for Yet another alternative is Conway chained arrow notation. In this notation, one has , so that (for example) ..
Examples
Below is a list of the first seven (0th to 6th) hyperoperations (0⁰ is defined as 1).
Operation, Definition Names Domain 0 or Increment, successor, zeration, hyper0 Arbitrary 1 or Addition, hyper1 2 or Multiplication, hyper2 3 or Exponentiation, hyper3 real, with some multivalued extensions to complex numbers 4 or Tetration, hyper4 or an integer, an integer Let . By the recursive formula, . One solution is , because by definition when . This solution is unique because for all , (proof by recursion). (with some proposed extensions) 5 or Pentation, hyper5 , integers ≥ −1 6 Hexation, hyper6
Special cases
, when , when 0, when 1, when and For more details, see Powers of zero or Zero to the power of zero. 0, when and 1, when and is even (including 0) 0, when and is odd
, when 1, when
0, when 1, when , or , when
2, when , when , when
, when
0, when , or , when , when , when
3, when 4, when , easily demonstrable recursively.
History
One of the earliest discussions of hyperoperations was that of Albert Bennett in 1914, who developed some of the theory of commutative hyperoperations (see below). About 12 years later, Wilhelm Ackermann defined the function , which somewhat resembles the hyperoperation sequence.
In his 1947 paper, Reuben Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation (because they correspond to the indices 4, 5, etc.). As a three-argument function, e.g., , the hyperoperation sequence as a whole is seen to be a version of the original Ackermann function — recursive but not primitive recursive — as modified by Goodstein to incorporate the primitive successor function together with the other three basic operations of arithmetic (addition, multiplication, exponentiation), and to make a more seamless extension of these beyond exponentiation.
The original three-argument Ackermann function uses the same recursion rule as does Goodstein's version of it (i.e., the hyperoperation sequence), but differs from it in two ways. First, defines a sequence of operations starting from addition () rather than the successor function, then multiplication (), exponentiation (), etc. Secondly, the initial conditions for result in , thus differing from the hyperoperations beyond exponentiation. The significance of the in the previous expression is that = , where counts the number of operators (exponentiations), rather than counting the number of operands ("a"s) as does the in , and so on for the higher-level operations. (See the Ackermann function article for details.)
Notations
This is a list of notations that have been used for hyperoperations.
Name Notation equivalent to Comment Knuth's up-arrow notation Used by Knuth (for ), and found in several reference books.Hilbert's notationUsed by David Hilbert. Goodstein's notation Used by Reuben Goodstein. Original Ackermann function Used by Wilhelm Ackermann (for ) Ackermann–Péter function This corresponds to hyperoperations for base 2 () Nambiar's notation Used by Nambiar (for ) Superscript notation Used by Robert Munafo. Subscript notation (for lower hyperoperations) Used for lower hyperoperations by Robert Munafo. Operator notation (for "extended operations") Used for lower hyperoperations by John Doner and Alfred Tarski (for ). Square bracket notation Used in many online forums; convenient for ASCII. Conway chained arrow notation Used by John Horton Conway (for )
Variant starting from a
In 1928, Wilhelm Ackermann defined a 3-argument function which gradually evolved into a 2-argument function known as the Ackermann function. The original Ackermann function was less similar to modern hyperoperations, because his initial conditions start with for all . Also he assigned addition to , multiplication to and exponentiation to , so the initial conditions produce very different operations for tetration and beyond.
Operation Comment 0 1 2 3 An offset form of tetration. The iteration of this operation is different than the iteration of tetration. 4 Not to be confused with pentation.
Another initial condition that has been used is (where the base is constant ), due to Rózsa Péter, which does not form a hyperoperation hierarchy.
Variant starting from 0 In 1984, C. W. Clenshaw and F. W. J. Olver began the discussion of using hyperoperations to prevent computer floating-point overflows. Since then, many other authors have renewed interest in the application of hyperoperations to floating-point representation. (Since are all defined for .) While discussing tetration, Clenshaw et al. assumed the initial condition , which makes yet another hyperoperation hierarchy. Just like in the previous variant, the fourth operation is very similar to tetration, but offset by one.
Operation Comment 0 1 2 3 4 An offset form of tetration. The iteration of this operation is much different than the iteration of tetration. 5 Not to be confused with pentation.
Lower hyperoperations An alternative for these hyperoperations is obtained by evaluation from left to right. Since
define (with ° or subscript)
with
This was extended to ordinal numbers by Doner and Tarski. They have index 0 rather than index 1 for addition. They extend the formulas to also handle each ordinal with no immediate predecessor by replacing in the above with the supremum over all ordinals less than , and they treat similarly. We use Greek letters to indicate that these are ordinal numbers not simply counting numbers.
With these definitions is addition, is multiplication, and is exponentiation. However, fails to form the "power tower" apparent with the corresponding (non-lower) hyperoperation.Ordinal addition is not commutative; see ordinal arithmetic for more information Instead,
Operation Comment 0 Increment, successor, zeration 1 2 3 4 Not to be confused with tetration. 5 Not to be confused with pentation.Similar to tetration.
Commutative hyperoperations
Commutative hyperoperations were considered by Albert Bennett as early as 1914, which is possibly the earliest remark about any hyperoperation sequence. Commutative hyperoperations are defined by the recursion rule
which is symmetric in and , meaning all hyperoperations are commutative. This sequence does not contain exponentiation, and so does not form a hyperoperation hierarchy.
Operation Comment 0 Smooth maximum (LogSumExp) 1 2 This is due to the properties of the logarithm. 3 In a finite field, this is the Diffie–Hellman key exchange operation. 4 Not to be confused with tetration.
Numeration systems based on the hyperoperation sequence
R. L. Goodstein used the sequence of hyperoperators to create systems of numeration for the nonnegative integers. The so-called complete hereditary representation of integer , at level and base , can be expressed as follows using only the first hyperoperators and using as digits only , together with the base itself:
For , is represented simply by the corresponding digit. For , the representation of is found recursively, first representing in the form
where are the largest integers satisfying (in turn)
...
Any exceeding is then re-expressed in the same manner, and so on, repeating this procedure until the resulting form contains only the digits , together with the base .
Unnecessary parentheses can be avoided by giving higher-level operators higher precedence in the order of evaluation; thus,
level-1 representations have the form , with also of this form;
level-2 representations have the form , with , also of this form;
level-3 representations have the form , with , , also of this form;
level-4 representations have the form , with , , , also of this form;
and so on.
In this type of base- hereditary representation, the base itself appears in the expressions, as well as "digits" from the set . This compares to ordinary base-2 representation when the latter is written out in terms of the base ; e.g., in ordinary base-2 notation, 6 = (110)2 = 2 2 1 2 1 1 2 0 0, whereas the level-3 base-2 hereditary representation is 6 = 2 (2 1 1 0) 1 (2 1 1 0). The hereditary representations can be abbreviated by omitting any instances of 0, 1, 1, 1, etc.; for example, the above level-3 base-2 representation of 6 abbreviates to 2 2 2.
Examples: The unique base-2 representations of the number 266, at levels 1, 2, 3, 4, and 5 are as follows: Level 1: 266 = 2 2 2 ... 2 (with 133 2s) Level 2: 266 = 2 (2 (2 (2 2 2 2 2 1)) 1) Level 3: 266 = 2 2 (2 1) 2 (2 1) 2 Level 4: 266 = 2 (2 1) 2 2 2 2 2 Level 5: 266 = 2 2 2 2 2 2 2
Computation The definitions of the hyperoperation sequence can naturally be transposed to term rewriting systems (TRS).
TRS based on definition sub 1.1 The basic definition of the hyperoperation sequence corresponds with the reduction rules
To compute one can use a stack, which initially contains the elements .