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In [[logic]], a '''logical framework''' provides a means to define (or present) a logic as a signature in a higher-order [[type theory]] in such a way that [[provability]] of a formula in the original logic reduces to a [[type inhabitation]] problem in the framework type theory.<ref name="Jacobs2001">{{cite book|author=Bart Jacobs|title=Categorical Logic and Type Theory|year=2001|publisher=Elsevier|isbn=978-0-444-50853-9|page=598}}</ref><ref name="Gabbay1994">{{cite book|editor=Dov M. Gabbay|title=What is a logical system?|url=http://books.google.com/books?id=XqCu4XjHrIQC&pg=PA382|year=1994|publisher=Clarendon Press|isbn=978-0-19-853859-2|page=382}}</ref> This approach has been used successfully for (interactive) [[automated theorem proving]]. The first logical framework was [[Automath]], however the name of the idea comes from the more widely known Edinburgh Logical Framework, '''LF'''. Several more recent proof tools like [[Isabelle (theorem prover)|Isabelle]] are based on this idea.<ref name="Jacobs2001"/> Unlike a direct embedding, the logical framework approach allows many logics to be embedded in the same type system.<ref name="BoveBarbosa2009">{{cite book|author1=Ana Bove|author2=Luis Soares Barbosa|author3=Alberto Pardo|title=Language Engineering and Rigourous (sic) Software Development: International LerNet ALFA Summer School 2008, Piriapolis, Uruguay, February 24 - March 1, 2008, Revised, Selected Papers|url=http://books.google.com/books?id=YOqHiA5MYpEC&pg=PA48|year=2009|publisher=Springer|isbn=978-3-642-03152-6|pages=48}}</ref>

A logical framework is based on a general treatment of syntax, rules and proofs by means of a [[dependent type theory|dependently typed lambda calculus]]. Syntax is treated in a style similar to, but more general than [[Per Martin-Löf]]'s system of arities.

To describe a logical framework, one must provide the following:

# A characterization of the class of object-logics to be represented;
# An appropriate meta-language;
# A characterization of the mechanism by which object-logics are represented.

This is summarized by:

''‘Framework = Language + Representation’''.

== LF ==

In the case of the '''LF logical framework''', the meta-language is the [[λΠ-calculus|<math>\lambda\Pi</math>-calculus]]. This is a system of first-order dependent function types which are related by the [[propositions as types principle]] to first-order minimal logic. The key features of the <math>\lambda\Pi</math>-calculus are that it consists of entities of three levels: objects, types and families of types. It is [[Impredicativity | predicative]], all well-typed terms are [[strongly normalizing]] and [[Church-Rosser]] and the property of being well-typed is [[Decidability (logic)|decidable]]. However, [[type inference]] is [[Decidability (logic)|undecidable]].

A logic is represented in the '''LF logical framework''' by the judgements-as-types representation mechanism. This is inspired by [[Per Martin-Löf]]'s development of [[Immanuel Kant|Kant]]'s notion of [[judgement (mathematical logic)|judgement]], in the 1983 Siena Lectures. The two higher-order judgements, the hypothetical <math>J\vdash K</math> and the general, <math>\Lambda x\in J. K(x)</math>, correspond to the ordinary and dependent function space, respectively. The methodology of judgements-as-types is that judgements are represented as the types of their proofs. A [[logical system]] <math>{\mathcal L}</math> is represented by its signature which assigns kinds and types to a finite set of constants that represents its syntax, its judgements and its rule schemes. An object-logic's rules and proofs are seen as primitive proofs of hypothetico-general judgements <math>\Lambda x\in C. J(x)\vdash K</math>.

An implementation of the LF logical framework is provided by the [[Twelf]] system at [[Carnegie Mellon University]]. Twelf includes
:* a logic programming engine
:* meta-theoretic reasoning about logic programs (termination, coverage, etc.)
:* an inductive [[meta-logic]]al theorem prover

==References==
{{reflist}}

==See also==

* [[Grammatical Framework]]

==Further reading==
* {{cite book|editor=Helmut Schwichtenberg, Ralf Steinbrüggen|title=Proof and system-reliability|chapter=Logical frameworks—a brief introduction|year=2002|publisher=Springer|isbn=978-1-4020-0608-1|author=[[Frank Pfenning]]|url=http://www.cs.cmu.edu/~fp/papers/mdorf01.pdf}}
*[[Robert Harper (computer scientist)|Robert Harper]], Furio Honsell and [[Gordon Plotkin]]. ''A Framework For Defining Logics''. Journal of the Association for Computing Machinery, 40(1):143-184, 1993
*Arnon Avron, Furio Honsell, Ian Mason and Randy Pollack. ''Using typed lambda calculus to implement formal systems on a machine''. Journal of Automated Reasoning, 9:309-354, 1992.
*Robert Harper. ''An Equational Formulation of LF''. Technical Report, University of Edinburgh, 1988. LFCS report ECS-LFCS-88-67.
*Robert Harper, Donald Sannella and Andrzej Tarlecki. ''Structured Theory Presentations and Logic Representations''. Annals of Pure and Applied Logic, 67(1-3):113-160, 1994.
*Samin Ishtiaq and David Pym. ''A Relevant Analysis of Natural Deduction''. Journal of Logic and Computation 8, 809-838, 1998.
* Samin Ishtiaq and David Pym. ''Kripke Resource Models of a Dependently-typed, Bunched <math>\lambda</math>-calculus''. Journal of Logic and Computation 12(6), 1061-1104, 2002.
* Per Martin-Löf. "[http://www.hf.uio.no/filosofi/njpl/vol1no1/meaning/meaning.html On the Meanings of the Logical Constants and the Justifications of the Logical Laws]." "[[Nordic Journal of Philosophical Logic]]", 1(1): 11-60, 1996.
* Bengt Nordström, Kent Petersson, and Jan M. Smith. ''Programming in Martin-Löf's Type Theory''. Oxford University Press, 1990. (The book is out of print, but [http://www.cs.chalmers.se/Cs/Research/Logic/book/ a free version] has been made available.)
*David Pym. ''A Note on the Proof Theory of the <math>\lambda\Pi</math>-calculus''. Studia Logica 54: 199-230, 1995.
*David Pym and Lincoln Wallen. ''Proof-search in the <math>\lambda\Pi</math>-calculus''. In: G. Huet and G. Plotkin (eds), Logical Frameworks, Cambridge University Press, 1991.
*Didier Galmiche and David Pym. ''Proof-search in type-theoretic languages:an introduction''. Theoretical Computer Science 232 (2000) 5-53.
*Philippa Gardner. ''Representing Logics in Type Theory''. Technical Report, University of Edinburgh, 1992. LFCS report ECS-LFCS-92-227.
*Gilles Dowek. ''The undecidability of typability in the lambda-pi-calculus''. In M. Bezem, J.F. Groote (Eds.), Typed Lambda Calculi and Applications. Volume 664 of ''Lecture Notes in Computer Science'', 139-145, 1993.
*David Pym. ''Proofs, Search and Computation in General Logic''. Ph.D. thesis, University of Edinburgh, 1990.
*David Pym. ''A Unification Algorithm for the <math>\lambda\Pi</math>-calculus.'' Int. J. of Foundations of Computer Science 3(3), 333-378, 1992.

== External links ==
* [http://www.cs.cmu.edu/~fp/lfs-impl.html Specific Logical Frameworks and Implementations] (a list maintained by [[Frank Pfenning]])

[[Category:Logic in computer science]]
[[Category:Type theory]]
[[Category:Proof assistants]]
[[Category:Dependently typed programming]]

Ordered exponential

2013-10-24T01:56:03Z

134.84.140.54: /* Definition */ alginment of punctuation

In [[mathematics]], '''Knuth's up-arrow notation''' is a method of notation for [[large number|very large]] [[integer]]s, introduced by [[Donald Knuth]] in 1976.<ref>{{cite journal |author=Knuth, Donald E. |title=Mathematics and Computer Science: Coping with Finiteness |journal=Science |volume=194 |issue=4271 |year=1976 |pages=1235–1242 |doi=10.1126/science.194.4271.1235 |pmid=17797067}}</ref> It is closely related to the [[Ackermann function]] and especially to the [[hyperoperation]] sequence. The idea is based on the fact that [[multiplication]] can be viewed as [[iteration|iterated]] [[addition]] and [[exponentiation]] as iterated [[multiplication]]. Continuing in this manner leads to [[Tetration|iterated exponentiation]] (tetration) and to the remainder of the hyperoperation sequence, which is commonly denoted using Knuth arrow notation.

==Introduction==

The ordinary arithmetical operations of [[addition]], [[multiplication]] and [[exponentiation]] are naturally extended into a sequence of [[hyperoperations]] as follows.

[[Multiplication]] by a [[natural number]] is defined as iterated [[addition]]:

:<math>
\begin{matrix}
a\times b & = & \underbrace{a+a+\dots+a} \\
& & b\mbox{ copies of }a
\end{matrix}
</math>

For example,

:<math>
\begin{matrix} 4\times 3 & = & \underbrace{4+4+4} & = & 12\\
& & 3\mbox{ copies of }4
\end{matrix}
</math>

[[Exponentiation]] for a natural power <math>b</math> is defined as iterated multiplication, which Knuth denoted by a single up-arrow:

:<math>
\begin{matrix}
a\uparrow b= a^b = & \underbrace{a\times a\times\dots\times a}\\
& b\mbox{ multiplied copies of }a
\end{matrix}
</math>

For example,

:<math>
\begin{matrix}
4\uparrow 3= 4^3 = & \underbrace{4\times 4\times 4} & = & 64\\
& 3\mbox{ multiplied copies of }4
\end{matrix}
</math>

To extend the sequence of operations beyond exponentiation, Knuth defined a “double arrow” [[Operator (mathematics)|operator]] to denote iterated exponentiation ([[tetration]]):

:<math>
\begin{matrix}
a\uparrow\uparrow b & = {\ ^{b}a} = & \underbrace{a^{a^{{}^{.\,^{.\,^{.\,^a}}}}}} &
= & \underbrace{a\uparrow (a\uparrow(\dots\uparrow a))}
\\
& & b\mbox{ multiplied copies of }a\uparrow
& & b\mbox{ multiplied copies of }a\uparrow
\end{matrix}
</math>

For example,

:<math>
\begin{matrix}
4\uparrow\uparrow 3 & = {\ ^{3}4} = & \underbrace{4^{4^4}} &
= & \underbrace{4\uparrow (4\uparrow 4)} & = & 4^{256} & \approx & 1.34078079\times 10^{154}&
\\
& & 3\mbox{ multiplied copies of }4\uparrow
& & 3\mbox{ multiplied copies of }4\uparrow
\end{matrix}
</math>

Here and below evaluation is to take place from right to left, as Knuth's arrow operators (just like exponentiation) are defined to be [[Right associative operator|right-associative]].

According to this definition,
:<math>3\uparrow\uparrow 2=3^3=27 </math>
:<math>3\uparrow\uparrow 3=3^{3^3}=3^{27}=7625597484987 </math>
:<math>3\uparrow\uparrow 4=3^{3^{3^3}}=3^{3^{27}}=3^{7625597484987}\approx 1.2580143\times 10^{3638334640024}
</math>
:<math>3\uparrow\uparrow 5=3^{3^{3^{3^3}}}=3^{3^{3^{27}}}=3^{3^{7625597484987}} </math>
:etc.

This already leads to some fairly large numbers, but Knuth extended the notation. He went on to define a “triple arrow” operator for iterated application of the “double arrow” operator (also known as [[pentation]]):

:<math>
\begin{matrix}
a\uparrow\uparrow\uparrow b= &
\underbrace{a_{}\uparrow\uparrow (a\uparrow\uparrow(\dots\uparrow\uparrow a))}\\
& b\mbox{ multiplied copies of }a
\end{matrix}
</math>

followed by a 'quadruple arrow' operator (also known as hexation):

:<math>
\begin{matrix}
a\uparrow\uparrow\uparrow\uparrow b= &
\underbrace{a_{}\uparrow\uparrow\uparrow (a\uparrow\uparrow\uparrow(\dots\uparrow\uparrow\uparrow a))}\\
& b\mbox{ multiplied copies of }a
\end{matrix}
</math>

and so on. The general rule is that an <math>n</math>-arrow operator expands into a right-associative series of (<math>n - 1</math>)-arrow operators. Symbolically,

:<math>
\begin{matrix}
a\ \underbrace{\uparrow_{}\uparrow\!\!\dots\!\!\uparrow}_{n}\ b=
\underbrace{a\ \underbrace{\uparrow\!\!\dots\!\!\uparrow}_{n-1}
\ (a\ \underbrace{\uparrow_{}\!\!\dots\!\!\uparrow}_{n-1}
\ (\dots
\ \underbrace{\uparrow_{}\!\!\dots\!\!\uparrow}_{n-1}
\ a))}_{b\text{ multiplied copies of }a}
\end{matrix}
</math>

Examples:

:<math>3\uparrow\uparrow\uparrow2 = 3\uparrow\uparrow3 = 3^{3^3} = 3^{27}=7,625,597,484,987</math>

:<math>
\begin{matrix}
3\uparrow\uparrow\uparrow3 = 3\uparrow\uparrow3\uparrow\uparrow3 = 3\uparrow\uparrow(3\uparrow3\uparrow3) = &
\underbrace{3_{}\uparrow 3\uparrow\dots\uparrow 3} \\
& 3\uparrow3\uparrow3\mbox{ multiplied copies of }3
\end{matrix}
\begin{matrix}
= & \underbrace{3_{}\uparrow 3\uparrow\dots\uparrow 3} \\
& \mbox{7,625,597,484,987 multiplied copies of 3}
\end{matrix}=\underbrace{3^{3^{3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}}}}_{7,625,597,484,987}
</math>

The notation <math>a \uparrow^n b</math> is commonly used to denote <math>a \uparrow\uparrow \dots \uparrow b</math> with ''n'' arrows.

==Notation==
In expressions such as <math>a^b</math>, the notation for exponentiation is usually to write the exponent <math>b</math> as a superscript to the base number <math>a</math>. But many environments — such as [[programming language]]s and plain-text [[e-mail]] — do not support superscript typesetting. People have adopted the linear notation <math>a \uparrow b</math> for such environments; the up-arrow suggests 'raising to the power of'. If the character set doesn't contain an up arrow, the [[caret]] ^ is used instead.

The superscript notation <math>a^b</math> doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation <math>a \uparrow b</math> instead.

<math>a \uparrow^n b</math> is a shorter alternative notation for n uparrows. Thus <math>a \uparrow^4 b = a \uparrow \uparrow \uparrow \uparrow b</math>.

===Writing out up-arrow notation in terms of powers===
Attempting to write <math>a \uparrow \uparrow b</math> using the familiar superscript notation gives a power tower.

:For example: <math>a \uparrow \uparrow 4 = a \uparrow (a \uparrow (a \uparrow a)) = a^{a^{a^a}}</math>

If ''b'' is a variable (or is too large), the power tower might be written using dots and a note indicating the height of the tower.

:<math>a \uparrow \uparrow b = \underbrace{a^{a^{.^{.^{.{a}}}}}}_{b}</math>

Continuing with this notation, <math>a \uparrow \uparrow \uparrow b</math> could be written with a stack of such power towers, each describing the size of the one above it.

:<math>a \uparrow \uparrow \uparrow 4 = a \uparrow \uparrow (a \uparrow \uparrow (a \uparrow \uparrow a)) =
\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{a} }}</math>

Again, if ''b'' is a variable or is too large, the stack might be written using dots and a note indicating its height.

:<math>a \uparrow \uparrow \uparrow b =
\left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\} b</math>

Furthermore, <math>a \uparrow \uparrow \uparrow \uparrow b</math> might be written using several columns of such stacks of power towers, each column describing the number of power towers in the stack to its left:

:<math>a \uparrow \uparrow \uparrow \uparrow 4 = a \uparrow \uparrow \uparrow (a \uparrow \uparrow \uparrow (a \uparrow \uparrow \uparrow a)) =
\left.\left.\left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
a</math>

And more generally:

:<math>a \uparrow \uparrow \uparrow \uparrow b =
\underbrace{
\left.\left.\left. \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
\underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{a^{a^{.^{.^{.{a}}}}}}_{ \underbrace{\vdots}_{a} }} \right\}
\cdots \right\}
a
}_{b}</math>

This might be carried out indefinitely to represent <math>a \uparrow^n b</math> as iterated exponentiation of iterated exponentiation for any ''a'', ''n'' and ''b'' (although it clearly becomes rather cumbersome).

====Using tetration====

The [[tetration]] notation <math>^{b}a</math> allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call these ''tetration towers'').

: <math> a \uparrow \uparrow b = { }^{b}a</math>

: <math> a \uparrow \uparrow \uparrow b = \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{b}</math>

: <math> a \uparrow \uparrow \uparrow \uparrow b =
\left. \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \underbrace{^{^{^{^{^{a}.}.}.}a}a}_{ \underbrace{\vdots}_{a} }} \right\} b</math>

Finally, as an example, the fourth Ackermann number <math>4 \uparrow^4 4</math> could be represented as:
: <math>\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{4} }} =
\underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ \underbrace{^{^{^{^{^{4}.}.}.}4}4}_{ ^{^{^{4}4}4}4 }}</math>

==Generalizations==
Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an ''n''-arrow operator <math>\uparrow^n</math> is useful (and also for descriptions with a variable number of arrows), or equivalently, [[hyper operator]]s.

Some numbers are so large that even that notation is not sufficient. [[Graham's number]] is an example. The [[Conway chained arrow notation]] can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful.

:<math>
\begin{matrix}
a\uparrow^n b & = & \mbox{hyper}(a,n+2,b) & = & a\to b\to n \\
\mbox{(Knuth)} & & & & \mbox{(Conway)}
\end{matrix}
</math>

It is generally suggested that Knuth's arrow should be used for smaller magnitude numbers, and the chained arrow or hyper operators for larger ones.

==Definition==

The up-arrow notation is formally defined by

:<math>
a\uparrow^n b=
\left\{
\begin{matrix}
a\times b, & \mbox{if }n=0; \\
a^b, & \mbox{if }n=1; \\
1, & \mbox{if }b=0; \\
a\uparrow^{n-1}(a\uparrow^n(b-1)), & \mbox{otherwise}
\end{matrix}
\right.
</math>
for all integers <math>a, b, n</math> with <math>b \ge 0, n \ge 1</math>.

All up-arrow operators (including normal exponentiation, <math>a \uparrow b</math>) are [[Associativity#Non-associativity|right associative]], i.e. evaluation is to take place from right to left in an expression that contains two or more such operators. For example, <math>a \uparrow b \uparrow c = a \uparrow (b \uparrow c)</math>, not <math>(a \uparrow b) \uparrow c</math>; for example
 <math>3\uparrow\uparrow 3=3^{3^3}</math> is <math>3^{(3^3)}=3^{27}=7625597484987</math> not <math>\left(3^3\right)^3=27^3=19683.</math>

There is good reason for the choice of this right-to-left order of evaluation. If we used left-to-right evaluation, then <math>a \uparrow\uparrow b</math> would equal
<math>a \uparrow (a \uparrow (b - 1))</math>, so that <math>\uparrow\uparrow</math> would not be an essentially new operation.
Right associativity is also natural because we can rewrite the iterated arrow expression <math>a\uparrow^n\cdots\uparrow^na</math> that appears
in the expansion of <math>a \uparrow^{n + 1}b</math> as
<math>a\uparrow^n\cdots\uparrow^na\uparrow^n1</math>, so that all the <math>a</math>s appear
as left operands of arrow operators. This is significant since the arrow operators are not [[commutativity|commutative]].

Writing <math>(a\uparrow ^m)^b</math> for the ''b''th [[functional power]] of the function <math>f(x)=a\uparrow ^m x</math> we have <math>a\uparrow ^n b = (a\uparrow ^{n-1})^b 1</math>.

The definition could be extrapolated one step, starting with <math>a\uparrow^n b= ab</math> if ''n'' = 0, because exponentiation is repeated multiplication starting with 1. Extrapolating one step more, writing multiplication as repeated addition, is not as straightforward because multiplication is repeated addition starting with 0 instead of 1. "Extrapolating" again one step more, writing addition of ''n'' as repeated addition of 1, requires starting with the number ''a''. Compare the [[Hyperoperation#Definition|definition of the hyper operator]], where the starting values for addition and multiplication are also separately specified.

==Tables of values==

===Computing 2↑''m''''n''===

Computing <math>2\uparrow^m n</math> can be restated in terms of an infinite table. We place the numbers <math>2^n</math> in the top row, and fill the left column with values 2. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>2\uparrow^m n</math> = [[Hyper operator|hyper]](2, ''m'' + 2, ''n'') = [[Conway chained arrow notation|2 → n → m]]
|-
! ''m''\''n''
! 1
! 2
! 3
! 4
! 5
! 6
! formula
|-
! 1
| 2 || 4 || 8 || 16 || 32 || 64 || <math>2^n</math>
|-
! 2
| 2 || 4 || 16 || 65536 || <math>2^{65\,536}\approx 2.0 \times 10^{19\,728}</math> || <math>2^{2^{65\,536}}\approx 10^{6.0 \times 10^{19\,727}}</math> || <math>2\uparrow\uparrow n</math>
|-
! 3
| 2 || 4 || 65536 || <math>
\begin{matrix}
\underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}} \\
65536\mbox{ multiplied copies of }2 \end{matrix}
</math> ||   ||   || <math>2\uparrow\uparrow\uparrow n</math>
|-
! 4
| 2 || 4 || <math>
\begin{matrix}
\underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}}\\
65536\mbox{ multiplied copies of }2
\end{matrix}</math> ||   ||   ||  
| <math>2\uparrow\uparrow\uparrow\uparrow n</math>
|}

The table is the same as [[Ackermann function#Table of values|that of the Ackermann function]], except for a shift in <math>m</math> and <math>n</math>, and an addition of 3 to all values.

===Computing 3↑''m''''n''===

We place the numbers <math>3^n</math> in the top row, and fill the left column with values 3. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>3\uparrow^m n</math> = [[Hyper operator|hyper]](3, ''m'' + 2, ''n'') = [[Conway chained arrow notation|3 → n → m]]
|-
! ''m''\''n''
! 1
! 2
! 3
! 4
! 5
! formula
|-
! 1
| 3 || 9 || 27 || 81 || 243 || <math>3^n</math>
|-
! 2
| 3 || 27 || 7,625,597,484,987 || <math>3^{7{,}625{,}597{,}484{,}987}</math> ||   || <math>3\uparrow\uparrow n</math>
|-
! 3
| 3 || 7,625,597,484,987 || <math>
\begin{matrix}
\underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^3}}}}}}\\
7{,}625{,}597{,}484{,}987\mbox{ multiplied copies of }3
\end{matrix}</math> ||   ||   || <math>3\uparrow\uparrow\uparrow n</math>
|-
! 4
| 3 || <math>\begin{matrix}
\underbrace{3_{}^{3^{{}^{.\,^{.\,^{.\,^3}}}}}}\\
7{,}625{,}597{,}484{,}987\mbox{ multiplied copies of }3
\end{matrix}</math> ||   ||   ||  
| <math>3\uparrow\uparrow\uparrow\uparrow n</math>
|}

===Computing 10↑''m''''n''===

We place the numbers <math>10^n</math> in the top row, and fill the left column with values 10. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken.

{| class="wikitable"
|+ Values of <math>10\uparrow^m n</math> = [[Hyper operator|hyper]](10, ''m'' + 2, ''n'') = [[Conway chained arrow notation|10 → n → m]]
|-
! ''m''\''n''
! 1
! 2
! 3
! 4
! 5
! formula
|-
! 1
| 10 || 100 || 1,000 || 10,000 || 100,000 || <math>10^n</math>
|-
! 2
| 10 || 10,000,000,000 || <math>10^{10,000,000,000}</math> || <math>10^{10^{10,000,000,000}}</math> || <math>10^{10^{10^{10,000,000,000}}}</math> || <math>10\uparrow\uparrow n</math>
|-
! 3
| 10 || <math>
\begin{matrix}
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> || <math>
\begin{matrix}
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> || <math>
\begin{matrix}
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
\underbrace{10_{}^{10^{{}^{.\,^{.\,^{.\,^{10}}}}}}}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> ||   || <math>10\uparrow\uparrow\uparrow n</math>
|-
! 4
| 10 || <math>
\begin{matrix}
\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> || <math>
\begin{matrix}
\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
\underbrace{^{^{^{^{^{10}.}.}.}10}10}\\
10\mbox{ multiplied copies of }10
\end{matrix}</math> ||   ||  
| <math>10\uparrow\uparrow\uparrow\uparrow n</math>
|}

Note that for 2 ≤ ''n'' ≤ 9 the numerical order of the numbers <math>10\uparrow^m n</math> is the [[lexicographical order]] with ''m'' as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line. The same applies for the numbers in the 97 columns with 3 ≤ ''n'' ≤ 99, and if we start from ''m'' = 1 even for 3 ≤ ''n'' ≤ 9,999,999,999.

==Numeration systems based on the hyperoperation sequence==

[[Reuben Goodstein|R. L. Goodstein]],<ref>{{cite journal |author=Goodstein, R. L.|title=Transfinite ordinals in recursive number theory |jstor=2266486 |journal=Journal of Symbolic Logic |volume=12 |year=1947 |issue=4 |doi=10.2307/2266486 |pages=123–129}}</ref> with a system of notation different from Knuth arrows, used the sequence of [[hyperoperation|hyperoperators]] here denoted by <math>( +, \ \times, \ \uparrow, \ \uparrow\uparrow, \ \dots)\,\!</math> to create systems of numeration for the nonnegative integers. Letting superscripts <math> \quad ( ^{(1)}, \ ^{(2)}, \ ^{(3)}, \ ^{(4)}, \ \dots )\,\!</math> denote the respective hyperoperators <math>( +, \ \times, \ \uparrow, \ \uparrow\uparrow, \ \dots)\,\!</math>, the so-called ''complete hereditary representation'' of integer ''n'', at level ''k'' and base ''b'', can be expressed as follows using only the first ''k'' hyperoperators and using as digits only 0, 1, ..., ''b''-1:

* For 0 ≤ ''n'' ≤ ''b''-1, ''n'' is represented simply by the corresponding digit.
* For ''n'' > ''b''-1, the representation of ''n'' is found recursively, first representing ''n'' in the form
:<math>b^{(k)}{x_k}^{(k-1)}{x_{k-1}}^{(k-2)} \dots {x_2}^{(1)}x_1</math>
:where ''x''''k'', ..., ''x''1 are the largest integers satisfying (in turn)

:<math>b^{(k)}x_k \le n</math>

:<math>b^{(k)}{x_k}^{(k-1)}x_{k-1} \le n</math>

:...

:<math>b^{(k)}{x_k}^{(k-1)}{x_{k-1}}^{(k-2)} \dots {x_2}^{(1)}x_1 \le n</math>.

:Any ''x''''i'' exceeding ''b''-1 is then re-expressed in the same manner, and so on, repeating this procedure until the resulting form contains only the digits 0, 1, ..., ''b''-1.

The remainder of this section will use <math>( +, \ \times, \ \uparrow, \ \uparrow\uparrow, \ \uparrow\uparrow\uparrow, \ \dots)</math>, rather than superscripts, to denote the hyperoperators.

Unnecessary parentheses can be avoided by giving higher-level operators higher precedence in the order of evaluation; thus,

level-1 representations have the form <math>b + X</math>, with ''X'' also of this form;

level-2 representations have the form <math>b \times X + Y</math>, with ''X'',''Y'' also of this form;

level-3 representations have the form <math>b \uparrow X \times Y + Z</math>, with ''X'',''Y'',''Z'' also of this form;

level-4 representations have the form <math>b \uparrow\uparrow X \uparrow Y \times Z + T</math>, with ''X'',''Y'',''Z'',''T'' also of this form;

and so on.

The representations can be abbreviated by omitting any instances of <math>+0, \ \times1, \uparrow1, \ \uparrow\uparrow1, </math> etc.; for example, the level-3 base-2 representation of the number 6 is <math>2\uparrow(2\uparrow1\times1+0)\times1+(2\uparrow1\times1+0)</math>, which abbreviates to <math>2 \uparrow 2 + 2</math>.

Examples:
The unique base-2 representations of the number 266, at levels 1, 2, 3, 4, and 5 are as follows:

:<math>\text{Level 1:} \ \ 266 = 2 + 2 + \dots + 2 \ \ \text{(with 133 2s)}</math>
:<math>\text{Level 2:} \ \ 266 = 2 \times (2 \times (2 \times (2 \times 2 \times 2 \times 2 \times 2 + 1)) + 1)</math>
:<math>\text{Level 3:} \ \ 266 = 2 \uparrow 2 \uparrow (2 + 1) + 2 \uparrow (2 + 1) + 2</math>
:<math>\text{Level 4:} \ \ 266 = 2 \uparrow\uparrow (2 + 1) \uparrow 2 + 2 \uparrow\uparrow 2 \times 2 + 2</math>
:<math>\text{Level 5:} \ \ 266 = 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 2 + 2 \uparrow\uparrow\uparrow 2 \times 2 + 2</math>.

==See also==

*[[Primitive recursion]]
*[[Hyper operator]]
*[[Busy beaver]]
*[[Cutler's bar notation]]
*[[Tetration]]

==References==
{{reflist}}

==External links==
* {{mathworld|urlname=ArrowNotation|title=Arrow Notation}}
* Robert Munafo, ''[http://www.mrob.com/pub/math/largenum.html Large Numbers]''

[[Category:Mathematical notation]]
[[Category:Large numbers]]
[[Category:Donald Knuth]]

Surgery obstruction

2011-10-18T20:27:32Z

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