Template:Infobox Complexity Class

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Infobox Complexity Class
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{{Infobox Complexity Class
PSPACE (example)
Complexity subsets pspace.svg
Complete class PSPACE-complete
Complement class self
Equalities AP[1], BPPSPACE[2], IP[3], NPSPACE[4], PPSPACE[5], SAPTIME[5]
Proper supersets EXPSPACE[6]
Improper supersets AlmostPSPACE[7], EXPTIME, RG, QPSPACE[8]
Inequalities P-close, P/log
Improper subsets CH[9], P^PP[10], P^#P[10], QSZK, RG[1]
Proper subsets NL[6]
Canonical problems QSAT
Properties Syntactic
Low with self
Low for self
Closed reductions Poly-time
Models Alternating Turing machine, Turing machine

Parameter Explanation[edit]

Regarding the meaning of "minimal" and "maximal", see the paragraph of Inclusion Guidelines.

class: The (short) name of the class. Example: "NP" but not "Polynomial Time". Default: "[[{{PAGENAME}}]]".

image: Code for image, if any. Example: "[[File:Complexity_subsets_pspace.svg|200px]]" but not "File:Complexity_subsets_pspace.svg".

long-name: Paraphrased/full name or short description of the class. Example: "Non-deterministic polynomial time" but not "the set of all decision problems for which ...".

description: Longer description of the class. Not used.

wheredefined: First definitions of the class. If not available, any suitable place. Secondly, internet sources preferred. Leave this field out rather than linking to the wikipedia page of the class. Example: "<ref>Christos H. Papadimitriou, Games against nature, FOCS 1983</ref>" (definition of SAPTIME) or "<ref>Scott Aaronsson, [https://www.blogger.com/comment.g?blogID=17392898&postID=114560148725169634&pli=1 Shetl-Optimized, What is the name of this post?], 2006</ref>" (definition of YP - Yaroslav-Percival).

dtime: Functions f such that the class is in DTIME[f(n)], if applicable. Example: "n^{\Theta(1)}" (PTIME) or "n^{\Omega(1)}, 2^{n^{O(1)}}" (NP).

complete-class: The complete class under a suitable reduction or none.

complement-class: The complement class.

proper-supersets: (Minimal) classes containing our class and are known not to equal our class. Example: (for NP) "[[NEXP]], [[NP_(complexity)|NP]]/one" but not "[[PSPACE]]".

improper-supersets: (Minimal) classes containing our class but not known not to equal our class. Example: (for NP) "[[NE]], [[PSPACE]], [[MA]]".

equals: Classes equaling our class. Example: (for NP) "[[Probabilistically_checkable_proof_(complexity)|PCP]](<math>O(log n)</math>, 3), Existential [[Second-order logic]]".

related: Interesting related classes that does not fall in one of the other categories. Example: (for NP) "[[coNP]], [[FNP_(complexity)|FNP]]".

notequals: Classes that are known to not to equal our class but not known to be either supersets or subsets. Example: (for NP) "[[PTIME|P]]/log".

improper-subsets: (Maximal) classes contained by our class but not known to be different. Example: (for NP) "[[PTIME|P]]".

proper-subsets: (Maximal) classes contained by our class and known to be different. Example: (for PSPACE) "[[NL_(complexity)|NL]]".

properties: Interesting properties of the class. Example: (for NP) "syntactic".

low-with: Classes that are low to the class, i.e. A such that C^A = C.

low-to: Classes the class is low to, i.e. A such that A^C = A.

closed-reductions: (Maximal) reductions under which the class is closed. Example: (for NP) "[[Polynomial-time reduction|Poly-time]]" but not "Log-space" since NP is closed under polynomial-time reductions and log-space reductions are also polynomial-time reductions.

closed-operations: Notable operations under which the class is closed. Not used.

canonical-problems: A few select canonical problems. These should typically be complete problems whenever applicable.

models: Most noteworthy models of computation for which the class can be naturally defined. Example: (for NP) "[[Non-Deterministic Turing machine]], [[Descriptive complexity]]". As a guide, list only the three most important models for the class in the info box but any number in the article. For example, the descriptive complexity characterization of PH (PH = languages expressible with second-order logic) is much more natural than that of NP (second-order logic with only first-order universal quantification) and PSPACE (second-order logic with a transitive closure operator). It should therefore take less to omit "descriptive complexity" as a natural model of computation for NP than for PH.

Inclusion Guidelines[edit]

These guidelines are still in the making. WP:BB

If all inclusions were listed in the relevant field, then most classes would have boxes with hundreds of names. The following suggestions are made to deal with this issue.

  • Cite sources, even if taken from e.g. the Complexity Zoo.
  • Inference based on basic set theory is considered "routine calculation" and is acceptable (see WP:NOR). In other words, if prof. X has published AB and prof. Y has published BC, then we may correctly infer that AC by referring to the two sources. Any inference that is not considered "routine" should be justified with a credible source explicating the inference.
  • Do not include relations that can be reasonably inferred by following relations on the relevant pages, or relations that are of little interest and involves a class not defined on Wikipedia. For instance, if the page for class B does not exist but the page of class A contains the relation AB and the page of class C contains BC, then it would still be reasonable to add AC to the pages of both A and C. This would not be the case if B also listed the two relations. If B was defined but did not list these relations, then the page of B should be extended, not A or C. Even if the page of B does not exists, the relation involving B may be of interest. If there is another class D which has a relation to C that implies BC, then one may reasonably replace the relation involving B with the relation involving D.
  • Classes especially important for the class may certainly violate the above guideline. For instance, one may very well include the relations of the complement, non-deterministic, or quantum equivalences even if the status of these classes are implied by other classes. This does not mean that P or NP should be included on every page.

Example References[edit]

  1. ^ Chandra, A.K. and Kozen, D.C. and Stockmeyer, L.J., 'Alternation', Journal of the ACM, Volume 28, Issue 1, pp. 114-133, 1981.
  2. ^ Complexity Zoo, [1], accessed Mars 25, 2009
  3. ^ Adi Shamir. IP = PSPACE. Journal of the ACM, volume 39, issue 4, p.869–877. October 1992.
  4. ^ Savitch's theorem
  5. ^ a b Christos Papadimitriou (1985). ""Games against Nature"". "Journal of Computer and System Sciences" 31. 
  6. ^ a b Space hierarchy theorem
  7. ^ Definition of Almost-PSPACE. PSPACE ⊆ PSPACE^A for every A.
  8. ^ Greg Kuperberg, Complexity Zoology: Active Inclusion Diagram, 2006, http://www.math.ucdavis.edu/~greg/zoology/diagram.xml
  9. ^ K. W. Wagner (1986). "The complexity of combinatorial problems with succinct representation". Informatica 23: 325–356. 
  10. ^ a b S. Toda (1989). "On the computational power of PP and ⊕P". FOCS 1989: 514–519. 

See also[edit]