Unary numeral system

Addition and subtraction are particularly simple in the unary system, as they involve little more than string concatenation.{{citation

| last = Sazonov

| first = Vladimir Yu.

| contribution = On feasible numbers

| doi = 10.1007/3-540-60178-3_78

| mr = 1449655

| pages = [https://archive.org/details/logiccomputation0000unse/page/30 30–51]

| publisher = Springer, Berlin

| series = Lecture Notes in Comput. Sci.

| title = Logic and computational complexity (Indianapolis, IN, 1994)

| volume = 960

| year = 1995

| isbn = 978-3-540-60178-4

| url = https://archive.org/details/logiccomputation0000unse/page/30

}}. See in particular p. 48. The Hamming weight or population count operation that counts the number of nonzero bits in a sequence of binary values may also be interpreted as a conversion from unary to binary numbers.{{citation

| last = Blaxell | first = David

| editor1-last = Hogben | editor1-first = David

| editor2-last = Fife | editor2-first = Dennis W.

| contribution = Record linkage by bit pattern matching

| pages = 146–156

| publisher = U.S. Department of Commerce / National Bureau of Standards

| series = NBS Special Publication

| title = Computer Science and Statistics--Tenth Annual Symposium on the Interface

| url = https://books.google.com/books?id=-MrJPUqTPh8C&pg=PA146

| volume = 503

| year = 1978}}. However, multiplication is more cumbersome and has often been used as a test case for the design of Turing machines.{{citation|at=[https://archive.org/details/introductiontoau00hopc/page/ Example 7.7, pp. 158–159]|title=Introduction to Automata Theory, Languages, and Computation|first1=John E.|last1=Hopcroft|author1-link=John Hopcroft|first2=Jeffrey D.|last2=Ullman|author2-link=Jeffrey Ullman|publisher=Addison Wesley|year=1979|isbn=978-0-201-02988-8|url-access=registration|url=https://archive.org/details/introductiontoau00hopc/page/}}.{{citation|title=The New Turing Omnibus: Sixty-Six Excursions in Computer Science|first=A. K.|last=Dewdney|author-link=A. K. Dewdney|publisher=Computer Science Press|year=1989|page=209|url=https://books.google.com/books?id=NDiU62j7jeMC&pg=PA209|isbn=9780805071665}}.{{citation|title=Turing Machine Universality of the Game of Life|volume=18|series=Emergence, Complexity and Computation|first=Paul|last=Rendell|publisher=Springer|year=2015|isbn=9783319198422|url=https://books.google.com/books?id=px8_CgAAQBAJ&pg=PA83|pages=83–86|contribution=5.3 Larger Example TM: Unary Multiplication}}.

Compared to standard positional numeral systems, the unary system is inconvenient and hence is not used in practice for large calculations. It occurs in some decision problem descriptions in theoretical computer science (e.g. some P-complete problems), where it is used to "artificially" decrease the run-time or space requirements of a problem. For instance, the problem of integer factorization is suspected to require more than a polynomial function of the length of the input as run-time if the input is given in binary, but it only needs linear runtime if the input is presented in unary.{{citation

| last1 = Arora | first1 = Sanjeev | author-link1 = Sanjeev Arora

| last2 = Barak | first2 = Boaz

| year = 2007

| chapter = The computational model —and why it doesn't matter

| title = Computational Complexity: A Modern Approach

| publisher = Cambridge University Press

| at = §17, pp. 32–33

| edition = January 2007 draft

| chapter-url = http://www.cs.princeton.edu/theory/complexity/modelchap.pdf

| access-date = May 10, 2017

}}. However, this is potentially misleading. Using a unary input is slower for any given number, not faster; the distinction is that a binary (or larger base) input is proportional to the base 2 (or larger base) logarithm of the number while unary input is proportional to the number itself. Therefore, while the run-time and space requirement in unary looks better as function of the input size, it does not represent a more efficient solution.{{citation|title=The Nature of Computation|first1=Cristopher|last1=Moore|author1-link=Cristopher Moore|first2=Stephan|last2=Mertens|publisher=Oxford University Press|year=2011|isbn=9780199233212|page=29|url=https://books.google.com/books?id=z4zMiZyAE1kC&pg=PA29}}.

In computational complexity theory, unary numbering is used to distinguish strongly NP-complete problems from problems that are NP-complete but not strongly NP-complete. A problem in which the input includes some numerical parameters is strongly NP-complete if it remains NP-complete even when the size of the input is made artificially larger by representing the parameters in unary. For such a problem, there exist hard instances for which all parameter values are at most polynomially large.{{citation

| last1 = Garey | first1 = M. R. | author1-link = Michael R. Garey

| last2 = Johnson | first2 = D. S. | author2-link = David S. Johnson

| doi = 10.1145/322077.322090

| issue = 3

| journal = Journal of the ACM

| mr = 478747

| pages = 499–508

| title = 'Strong' NP-completeness results: Motivation, examples, and implications

| volume = 25

| year = 1978| s2cid = 18371269 | doi-access = free

}}.

In addition to the application in tally marks, unary numbering is used as part of some data compression algorithms such as Golomb coding.{{citation

| last = Golomb | first = S.W. | author-link = Solomon W. Golomb

| issue = 3

| journal = IEEE Transactions on Information Theory

| pages = 399–401

| title = Run-length encodings

| url = http://urchin.earth.li/~twic/Golombs_Original_Paper/

| volume = IT-12

|doi =10.1109/TIT.1966.1053907

| year = 1966}}. It also forms the basis for the Peano axioms for formalizing arithmetic within mathematical logic.{{citation

| last1 = Magaud | first1 = Nicolas

| last2 = Bertot | first2 = Yves

| contribution = Changing data structures in type theory: a study of natural numbers

| doi = 10.1007/3-540-45842-5_12

| mr = 2044538

| pages = 181–196

| publisher = Springer, Berlin

| series = Lecture Notes in Comput. Sci.

| title = Types for proofs and programs (Durham, 2000)

| volume = 2277

| year = 2002| isbn = 978-3-540-43287-6

}}.

A form of unary notation called Church encoding is used to represent numbers within lambda calculus.{{citation

| last = Jansen | first = Jan Martin

| title = The Beauty of Functional Code

| contribution = Programming in the λ-calculus: from Church to Scott and back

| doi = 10.1007/978-3-642-40355-2_12

| pages = 168–180

| publisher = Springer-Verlag

| series = Lecture Notes in Computer Science

| volume = 8106

| year = 2013| isbn = 978-3-642-40354-5

}}.

Some email spam filters tag messages with a number of asterisks in an e-mail header such as X-Spam-Bar or X-SPAM-LEVEL. The larger the number, the more likely the email is considered spam. Using a unary representation instead of a decimal number lets the user search for messages with a given rating or higher. For example, searching for **** yield messages with a rating of at least 4.{{citation | url=http://answers.uillinois.edu/illinois/page.php?id=49002 | title=Email, Spam Control, How to get service for departmental email servers }}

• Unary coding
• One-hot encoding

{{reflist|30em}}

{{commons|Unary numeral|Unary numeral system}}

• {{OEIS el|sequencenumber=A000042|name=Unary representation of natural numbers}}

Category:Numeral systems

Category:1 (number)

Category:Elementary mathematics

Category:Coding theory

Category:Formal languages