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Complexity class
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==Summary of relationships between complexity classes== The following table shows some of the classes of problems that are considered in complexity theory. If class '''X''' is a strict [[subset]] of '''Y''', then '''X''' is shown below '''Y''' with a dark line connecting them. If '''X''' is a subset, but it is unknown whether they are equal sets, then the line is lighter and dotted. Technically, the breakdown into decidable and undecidable pertains more to the study of [[computability theory]], but is useful for putting the complexity classes in perspective. {| cellpadding="0" cellspacing="0" border="0" style="margin:auto;" |- style="text-align:center;" | colspan=2 | | colspan=4 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightBlue; width:100%; height:100%;" |- | style="text-align:center;" | [[decision problem|Decision Problem]] |} |- style="text-align:center;" | colspan=2 | | [[File:solidLine.png]] | colspan=2 | | [[File:solidLine.png]] |- style="text-align:center;" | colspan=3 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightBlue; width:100%; height:100%;" |- | style="text-align:center;" | [[recursively enumerable language|Type 0 (Recursively enumerable)]] |} | | colspan=4 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightBlue; width:100%; height:100%;" |- | style="text-align:center;" | [[List of undecidable problems|Undecidable]] |} |- style="text-align:center;" | colspan=3 | [[File:solidLine.png]] |- style="text-align:center;" | colspan=3 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightBlue; width:100%; height:100%;" |- | style="text-align:center;" | [[recursive language|Decidable]] |} |- style="text-align:center;" | colspan=3 | [[File:solidLine.png]] |- style="text-align:center;" | colspan=3 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[EXPSPACE]] |} |- style="text-align:center;" | colspan=3 | [[File:dottedLine.png]] |- style="text-align:center;" | colspan=3 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[NEXPTIME]] |} |- style="text-align:center;" | colspan=3 | [[File:dottedLine.png]] |- style="text-align:center;" | colspan=3 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[EXPTIME]] |} |- style="text-align:center;" | colspan=3 | [[File:dottedLine.png]] |- style="text-align:center;" | colspan=8 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[PSPACE]] |} |- style="text-align:center;" | [[File:solidLine.png]] | width=40 | [[File:solidLine.png]] | [[File:dottedLine.png]] | [[File:dottedLine.png]] | | [[File:dottedLine.png]] |- style="text-align:center;" | {| cellpadding="0" cellspacing="0" border="1" style="background:lightBlue; width:100%; height:100%;" |- | style="text-align:center;" | [[context-sensitive grammar|Type 1 (Context Sensitive)]] |} | [[File:solidLine.png]] | [[File:dottedLine.png]] | border="1" | [[File:dottedLine.png]] | | [[File:dottedLine.png]] |- style="text-align:center;" | [[File:solidLine.png]] | [[File:solidLine.png]] | [[File:dottedLine.png]] | [[File:dottedLine.png]] | | [[File:dottedLine.png]] |- style="text-align:center;" | [[File:solidLine.png]] | [[File:solidLine.png]] | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[co-NP]] |} | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[BQP]] |} | | colspan=2 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[NP (complexity)|NP]] |} |- style="text-align:center;" | [[File:solidLine.png]] | [[File:solidLine.png]] | [[File:dottedLine.png]] | [[File:dottedLine.png]] | | [[File:dottedLine.png]] |- style="text-align:center;" | [[File:solidLine.png]] | [[File:solidLine.png]] | [[File:dottedLine.png]] | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[Bounded-error probabilistic polynomial|BPP]] |} | width=10 | | [[File:dottedLine.png]] |- style="text-align:center;" | [[File:solidLine.png]] | [[File:solidLine.png]] | [[File:dottedLine.png]] | [[File:dottedLine.png]] | | [[File:dottedLine.png]] |- style="text-align:center;" | [[File:solidLine.png]] | [[File:solidLine.png]] | colspan=5 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[P (complexity)|P]] |} |- style="text-align:center;" | [[File:solidLine.png]] | [[File:solidLine.png]] | [[File:dottedLine.png]] |- style="text-align:center;" | [[File:solidLine.png]] | colspan=2 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightGreen; width:100%; height:100%;" |- | style="text-align:center;" | [[NC (complexity)|NC]] |} |- style="text-align:center;" | [[File:solidLine.png]] | colspan=2 | [[File:solidLine.png]] |- style="text-align:center;" | colspan=3 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightBlue; width:100%; height:100%;" |- | style="text-align:center;" | [[context-free grammar|Type 2 (Context Free)]] |} |- style="text-align:center;" | colspan=3 | [[File:solidLine.png]] |- style="text-align:center;" | colspan=3 | {| cellpadding="0" cellspacing="0" border="1" style="background:lightBlue; width:100%; height:100%;" |- | style="text-align:center;" | [[regular grammar|Type 3 (Regular)]] |} |}
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