264:
254:
233:
200:
2084:, rather than just putting in a sentence about how the same concepts can be interpreted either on the reals or on the naturals. The thing is, I don't really think of this as computability, but as descriptive set theory, and I would have probably done the writeup for the concept on the naturals as descriptive set theory as well. But the computability-theory concepts are important and should be there, which complicates the presentation. --
191:
342:
3138:
The article previously contained the claim that any formula with bounded quantifiers is equivalent to a quantifier-free formula. There was a "citation needed" comment questioning this claim, and it is indeed false, so I removed it. I just wanted to record a counter-example here (there may be simpler
2343:
See Kozen - Theory of
Computation (pp. 239-246) for natural complete problems in the Arithmetic Hierarchy of sets of natural numbers. That is EMPTY for \Pi^0_1, the halting Problem for \Sigma^0_1, TOTAL for \Pi^0_2, FIN(ITE) for \Sigma^0_2 and COF(FINITE) for \Sigma^0_3. These are all index sets and
2313:
Examples: Sigma_1-complete to determine halting/looping of encoded Turing machines. Pi_2 complete to determine if encoded TMs accept an infinite set. Sigma_3-complete to determine if TM accepts a recursive set. Pi_3-complete to determine if TM encodes a walk that diverges properly to infinity.
3857:
It would be great if we had a NavBox of all the different hierarchies: polynomial hierarchy, weak exponential hierarchy, arithmethical hierarchy, hyperarithmetical hierarchy, and the ones shown here. The current NavBox (at the bottom) could be expanded. Assuming P≠NP≠PH, a NavBox could look like
2702:
The translation from the french is kind of rough, and the french version has either been taken offline (it was published as a printed book a couple years ago) or was never there, but anyway I can't compare the two as a result. Thanks for trying to decipher the english version. I thought maybe
2066:
Indeed, I made some typos which I corrected. But you are correct, I am a bit confused. I have not dealt with computability structure on uncountable sets before. What I was trying to get at was that there must be some sort of countable structure underneath the real number system (in some sense).
2189:
of the objects you're quantifying over: Type 0 objects are naturals, type 1 objects are sets of naturals, or functions from the naturals to the naturals, or reals; type 2 objects are sets of sets of naturals, or functions from the reals to the reals, or sets of reals, or.... In general a
2317:
I worked these out and they need checking. They are important to have around, because classifying the hierarchy level of a set almost invariably involves showing it complete for that level, and one needs a variety of target problems to work with just like with NP-completeness.
5197:
Wouldn't it be more useful to have a navbox of the complexity classes adjacent to the specific hierarchy in question (e.g., P ⊆ PH ⊆ PSPACE on the PH page)? Beyond the fact that they are all hierarchies, PH, EXPH, and AH belong to very different positions in complexity space.
5233:. Since the topological notion is probably more familiar to many readers, the sentence as it stands is likely to create confusion. Using the term "Baire space (set theory)" would alert readers to the fact that the topological notion is not the one being referred to here.
450:. I think MathMartin is thinking of sets of naturals, but the same concepts and notation are important for sets of reals (subsets of Baire space, Cantor space, general Polish spaces). Unfortunately finding good language for that is awkward. --
2966:
I have a general sense that the "-al" versions of "arithmetical hierarchy" and "hyperarithmetical hierarchy" are more common. I don't think "analytic hierarchy" is used at all. So I have restored the original endings here. — Carl
460:
I just noticed we were editing the page nearly in parallel :). Yes I am indeed thinking about sets of naturals and computable functions. What do you mean by arithmetical hierarchy for sets of reals ? Can you not just use a
2893:
153:
558:
Are you saying you want to put the sets in a certain topological space in some sort of hierarchy according to how difficult they are to construct using the countable basis of the space ? In your hierarchy what are the
2037:; this is all completely standard and is standardly called the arithmetical hierarchy for sets of reals (see Moschovakis). The problem is how to convey the concepts in the article without making a mess of it. --
3245:
2164:
Yep. There are all kinds of problems with this article. It's one of the ones I've got on my fix-up-someday list, but it looks like sort of an unpleasant job, so the expected time before I get to it is long.
401:
Hmmm. Actually, I think the article is pretty broken at the moment. It's on my "to fix" list, but it'll take time because I'll need to re-read material on the subject. Feel free to beat me to it :) --
1720:
2067:
Anyway thanks for the explanation. If you think the concepts you are talking about will mess up the article perhaps it would be best to start a new one. I would certainly like to read it. Cheers.
923:
800:
sets. That is, there's a computable way of putting indices for basic open sets into an infinite square matrix, intersecting the basic open sets in each row, and then unioning up all the resulting
3344:
1860:
1568:. That is, there's a computable way of assigning all at once a collection of "rows" of basic open sets whose union is the complement of each rational's singleton, and then unioning all those up.
320:
2344:
their names are rather self-explanatory. The halting problem is well-known and the names of the other ones refer to the property of the domains of the functions with the respective index.
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1485:}. (Note that not every singleton is effectively closed; for example, if 0' is a real that codes the halting problem, then its singleton is not effectively closed (I think)).
5229:
The
Knowledge entry on "Baire space" which is linked to here deals with the topological notion of a Baire space. Correct me if I am wrong, but I think the link should be to
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Furthermore, it seems that what Girard calls the "projective hierarchy" is actually the analytical hierarchy (i.e., lightface), but that does not make much difference here.—
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using the definition which is worked with in this article (it is not the case if one works with the other definition which might be why this result is quoted only for
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1481:, because there's a computable set of indices for basic open sets (these taken to be open intervals with rational endpoints) whose union is the complement of {
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what the superscript means, rather than merely complaining about something you know how to fix yourself, here's a quick-and-dirty explanation: When we write Σ
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I believe the "Extensions and variations" sections mentions two alternatives but it's not clear that these are in fact two different alternatives.
2310:
Would someone include some natural complete problems for the
Arithmetic Hierarchy (of sets of natural numbers)? Same goes for Analytic Hierarchy.
510:
set of reals is "effectively open"; that is, there's a computable collection of indices for basic open sets, the union of which is the given set. A
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3837:
Well done! Thanks for correcting the record. This claim looked very fishy to me as well but I didn't put in the time to find a counterexample.
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is the collection of open sets which forms the countable basis of your topology. This numbering has to be choosen in such a way that your
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formulas, it thus corresponds to the projective hierarchy rather than the arithmetical hierarchy. According to Girard's examples, every
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3067:, which I think is something like "lightface projective", but the risk of confusion is so great that it's probably better avoided. --
474:
Not quite sure what you mean by that. The spaces I'm interested in are of course uncountable, though their topologies have countable
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The article doesn't explain the meaning of the zero superscript. Even if you are lucky enough to click through to the article on
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is either finite or has finite complement. Since this property is preserved by finite unions and intersections, it follows that
5256:
2231:
Thanks for the explanation -- no, I did not know the answer or how to fix it. I've tried to fix it based on your explanation.--
2154:, there's still not much explanation of the relationship between the two articles or the meaning of the superscripts 0 and 1.--
1768:
of your topology. This map is not a numbering because it is not surjective. The problem consists in finding a countable subset
99:
30:
2181:
tells you the number of alternating quantifiers (starting with existential if it's Σ or universal if it's Π). The superscript
465:
to transfer the computability concept (an the arithmetical hierarchy) defined on the natural numbers to any other structure ?
104:
20:
3021:, but there is an older usage (possibly still current in French?) that makes it synonymous with "projective" (i.e. boldface
1675:
74:
2363:
There's apparently an alternate expression called the "logic hierarchy", described in Girard's "The Blind Spot" p. 45-46
1243:
but a computability concept, which then should allow you to build a new hierarchy similar to the arithmetical hierarchy.
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is pronounced “sigma one zero” or “sigma zero one”. I think saying which it is would be a helpful and easy addition.
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Thanks to MathMartin for putting in some useful stuff. The big problem left with this article, and also with
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I took some time this morning to substantially rework this article. I tried to fix the following problems.
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1969:'s is itself uncountable. But I still have the nagging suspicion that you're thinking of trying to define
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24:
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Moreover for the recursive/computable definition I believe it only differs from the usual definition for
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When you do, please define \Delta _{n}^{0}, that would help make the article much more comprehensible.
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there are plenty. To make the discussion useful on the "real reals" it's better not to start with
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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2496:. I won't add this myself since probably I don't have it right, but maybe it's worth a mention.
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Please help fix the broken anchors. You can remove this template after fixing the problems. |
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868:'s, not just basic opens. So we actually need to start by computably filling up an infinite
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5121:(I've put the question mark above PSPACE since we could also consider ordinals larger than
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The article had factual errors, such as the claim that Sigma^0_0 is the same as recursive.
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The article was not precise about the fact that the AH classifies sets of natural numbers.
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sets are exactly the recursively enumerable set in the computability concept induced by
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No, at least not literally. Maybe you made a typo, but of course such a countable
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As someone with only passing familiarity of this area, I can't remember whether
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1219:. The numbering does not directly induce the arithmetical hierarchy on the set
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660:
259:
2638:(don't ask me why). In particular, all of arithmetical hierarchy fits inside
836:
Whoops--actually I skipped a step here. Along each row you need to intersect
3120:
2888:{\displaystyle \Sigma _{n}^{0}\cup \Pi _{n}^{0}\subsetneq \Delta _{n+1}^{0}}
663:; in the real numbers strictly speaking there are only two of these, but in
2519:—WTF???), but one thing is certain, namely that it is a stratification of
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403:
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2255:
The article was not precise about what language the formulas come from.
1672:
Hmm, now I think I understand the problem. You want to construct a map
362:
This article links to one or more target anchors that no longer exist.
1252:
I'm not sure if I'm understanding you. By a "computability concept on
396:
possibly conflating this article with any analytical hierarchy article
1040:. For example if you union up the basic open sets with indices in a
3247:
of all even numbers cannot be defined by a quantifier-free formula
2393:(with no subscript) in the logic hierarchy means the same thing as
4161:
2106:
I am not an expert on this topic so I might be missing something.
3240:{\displaystyle X=\{n\in \mathbb {N} \mid \exists k\leq n(2k=n)\}}
2080:. I'm coming to the conclusion that it probably needs a separate
5177:
884:
My understanding (what I meant to say) is you need a numbering
1133:
which you can use to construct the collection of "effectively F
336:
184:
15:
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with indices for basic opens. Details left as an exercise. --
393:
The hierarchy does not collapse; which I think is from Kleene
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sets are the ones that are the effective counterpart of the
1369:
Maybe an example would help focus things. Consider the set
446:, is that when it talks about sets, it's not clear sets of
366:] The anchor (#First-order theory of arithmetic) has been
3056:). So it wouldn't surprise me if someone somewhere uses
2280:
to mention both subsets of N and subsets of Baire space.
1304:? That's not what we want; we're categorizing subsets of
413:
I know it's a lot to read, but 13 years is a bit much ;)
2366:. I can't quite tell what it means, but it looks like
1920:
are exactly the recursively enumerable sets. Correct ?
1715:{\displaystyle f:2^{\mathbb {N} }\to 2^{\mathbb {R} }}
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is an open set in the usual topology on the space."
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918:{\displaystyle \nu :\mathbb {N} \to {\mathcal {B}}}
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1855:{\displaystyle \forall B\in {\mathcal {B}}:B\in X}
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5218:I was confused by the sentence "Every subset of
4040:
2517:but the first order quantifiers are not numerical
2135:I presume it means 'complement of A-recursive'.
33:for general discussion of the article's subject.
5277:Knowledge level-5 vital articles in Mathematics
2754:where the computable-definition version of the
1137:" sets (which corresponds to the collection of
2265:Please feel free to change the stuff I wrote.
1722:which maps the recursively enumerable sets in
3874:
174:
8:
3333:
3304:
3234:
3187:
2033:sets of reals. And there isn't actually any
1008:Pretty much, but that doesn't take you past
2125:, which doesn't actually define the term.--
3977:{\displaystyle {\overset {?}{\subseteq }}}
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3134:Bounded quantifiers versus quantifier-free
2703:"logic hierarchy" was some standard term.
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3049:{\displaystyle \Sigma _{<\omega }^{1}}
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2511:I don't understand Girard's description (
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2076:I hardly think it needs a whole separate
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2671:{\displaystyle \Pi ^{2}\cap \Sigma ^{2}}
2631:{\displaystyle \Pi ^{1}=\Sigma _{1}^{0}}
2593:1, while the next section suggests that
1391:of all rational numbers, as a subset of
5267:Knowledge vital articles in Mathematics
5015:{\displaystyle \Delta _{\omega }^{EXP}}
3789:does not have this property, and hence
3060:in the sense of "projective hierarchy".
1629:collection of basic open sets, because
229:
188:
4975:{\displaystyle \Delta _{\omega }^{P}=}
1889:{\displaystyle \nu :\mathbb {N} \to X}
1276:", do you mean a notion of computable
5282:B-Class vital articles in Mathematics
2276:I am planning to edit this page like
1109:induces a computability structure on
7:
3538:is equivalent either to the formula
2261:I added a reference to Rogers' book.
659:sets could be taken to be the basic
591:sets (is this your basis ?) and the
275:This article is within the scope of
3663:is either constant or tends toward
2431:from the arithmetic hierarchy, and
1212:{\displaystyle B\in {\mathcal {B}}}
218:It is of interest to the following
23:for discussing improvements to the
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4335:{\displaystyle \Sigma _{1}^{EXP}=}
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4020:{\displaystyle \Delta _{0}^{EXP}=}
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5292:Mid-priority mathematics articles
5044:{\displaystyle \Delta _{\omega }}
4816:{\displaystyle \Delta _{3}^{EXP}}
4642:{\displaystyle \Sigma _{2}^{EXP}}
4512:{\displaystyle \Delta _{2}^{EXP}}
4196:{\displaystyle \Delta _{1}^{EXP}}
2489:{\displaystyle \Sigma _{n-1}^{0}}
2001:sets of basic open sets. We want
1949:can't exist, because each of the
1434:{\displaystyle q\in \mathbb {Q} }
295:Knowledge:WikiProject Mathematics
5262:Knowledge level-5 vital articles
5169:{\displaystyle \omega _{1}^{CK}}
4115:{\displaystyle \Sigma _{1}^{P}=}
3946:{\displaystyle \Delta _{0}^{P}=}
3063:There is a separate meaning for
1810:{\displaystyle 2^{\mathbb {R} }}
463:numbering (computability theory)
340:
298:Template:WikiProject Mathematics
262:
252:
231:
198:
189:
45:Click here to start a new topic.
4744:{\displaystyle \Sigma _{3}^{P}}
4603:{\displaystyle \Delta _{2}^{P}}
4440:{\displaystyle \Sigma _{2}^{P}}
4293:{\displaystyle \Delta _{1}^{P}}
3815:{\displaystyle X_{\phi }\neq X}
3372:{\displaystyle X\neq X_{\phi }}
3167:{\displaystyle \Delta _{0}^{0}}
3112:{\displaystyle \Sigma _{1}^{0}}
3014:{\displaystyle \Sigma _{1}^{1}}
2811:{\displaystyle \Delta _{1}^{0}}
2779:{\displaystyle \Delta _{0}^{0}}
2026:{\displaystyle \Sigma _{n}^{0}}
1994:{\displaystyle \Sigma _{n}^{0}}
1622:{\displaystyle \Sigma _{2}^{0}}
1557:{\displaystyle \Sigma _{2}^{0}}
1162:{\displaystyle \Sigma _{2}^{0}}
1065:{\displaystyle \Sigma _{2}^{0}}
1033:{\displaystyle \Sigma _{1}^{0}}
977:{\displaystyle \Sigma _{1}^{0}}
861:{\displaystyle \Sigma _{1}^{0}}
763:{\displaystyle \Sigma _{3}^{0}}
728:{\displaystyle \Sigma _{1}^{0}}
696:{\displaystyle \Sigma _{0}^{0}}
652:{\displaystyle \Sigma _{0}^{0}}
616:{\displaystyle \Sigma _{3}^{0}}
584:{\displaystyle \Sigma _{0}^{0}}
535:{\displaystyle \Sigma _{2}^{0}}
503:{\displaystyle \Sigma _{1}^{0}}
315:This article has been rated as
5272:B-Class level-5 vital articles
5176:(for which it has been proven
4681:{\displaystyle \Pi _{2}^{EXP}}
4377:{\displaystyle \Pi _{1}^{EXP}}
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3129:10:16, 21 September 2020 (UTC)
2957:14:13, 30 September 2014 (UTC)
2554:{\displaystyle \Pi _{n-1}^{1}}
2424:{\displaystyle \Pi _{n-1}^{0}}
2089:22:08, 24 September 2005 (UTC)
2072:22:02, 24 September 2005 (UTC)
2042:21:42, 24 September 2005 (UTC)
1925:21:34, 24 September 2005 (UTC)
1913:{\displaystyle {\mathcal {B}}}
1880:
1761:{\displaystyle {\mathcal {B}}}
1744:to the open sets in the basis
1697:
1656:21:01, 24 September 2005 (UTC)
1355:20:47, 24 September 2005 (UTC)
1343:{\displaystyle {\mathcal {B}}}
1297:{\displaystyle {\mathcal {B}}}
1269:{\displaystyle {\mathcal {B}}}
1248:20:37, 24 September 2005 (UTC)
1236:{\displaystyle {\mathcal {B}}}
1126:{\displaystyle {\mathcal {B}}}
1081:18:45, 24 September 2005 (UTC)
945:{\displaystyle {\mathcal {B}}}
905:
877:18:53, 24 September 2005 (UTC)
832:18:45, 24 September 2005 (UTC)
551:17:19, 24 September 2005 (UTC)
470:17:14, 24 September 2005 (UTC)
455:17:10, 24 September 2005 (UTC)
1:
5243:06:39, 30 November 2023 (UTC)
4154:{\displaystyle \Pi _{1}^{P}=}
2989:has settled down to boldface
2713:13:32, 18 February 2010 (UTC)
2698:12:53, 18 February 2010 (UTC)
2686:12:49, 18 February 2010 (UTC)
2506:00:13, 18 February 2010 (UTC)
2337:04:03, 23 December 2006 (UTC)
2300:04:09, 8 September 2023 (UTC)
2236:00:05, 26 February 2006 (UTC)
2227:00:43, 25 February 2006 (UTC)
2216:), and mutatis mutandis for Π
2159:22:36, 24 February 2006 (UTC)
2140:22:49, 24 February 2006 (UTC)
2130:22:36, 24 February 2006 (UTC)
1862:and for a suitable numbering
289:and see a list of open tasks.
42:Put new text under old text.
5287:B-Class mathematics articles
4777:{\displaystyle \Pi _{3}^{P}}
4473:{\displaystyle \Pi _{2}^{P}}
4227:{\displaystyle \Sigma _{1}=}
3705:{\displaystyle n\to \infty }
1737:{\displaystyle \mathbb {N} }
1644:{\displaystyle \mathbb {Q} }
1586:{\displaystyle \mathbb {Q} }
1525:{\displaystyle \mathbb {R} }
1503:{\displaystyle \mathbb {Q} }
1474:{\displaystyle \Pi _{1}^{0}}
1406:{\displaystyle \mathbb {R} }
1384:{\displaystyle \mathbb {Q} }
1319:{\displaystyle \mathbb {R} }
1004:on your topological basis.
5208:21:42, 14 August 2023 (UTC)
5192:09:40, 14 August 2023 (UTC)
4844:{\displaystyle \Sigma _{3}}
4709:{\displaystyle \Delta _{2}}
4540:{\displaystyle \Sigma _{2}}
4405:{\displaystyle \Delta _{1}}
4076:{\displaystyle \Delta _{0}}
3853:hierarchies overview/NavBox
3766:has this property as well.
3679:{\displaystyle \pm \infty }
3631:with integer coefficients.
2386:{\displaystyle \Sigma ^{n}}
820:{\displaystyle G_{\delta }}
429:16:12, 1 October 2016 (UTC)
50:New to Knowledge? Welcome!
5308:
4048:{\displaystyle {\Bigg \}}}
3847:21:57, 28 April 2023 (UTC)
3832:06:32, 28 April 2023 (UTC)
2354:20:11, 15 April 2011 (UTC)
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3759:{\displaystyle X_{\phi }}
3732:{\displaystyle X_{\psi }}
3473:{\displaystyle X_{\psi }}
3446:{\displaystyle X_{\phi }}
3290:{\displaystyle X_{\phi }}
2786:sets are the same as the
2721:Extensions and variations
2285:11:32, 26 June 2006 (UTC)
2212:(which is a stage of the
2202:formula in the structure
2196:set can be defined by a Σ
1189:-computable sets for any
438:article on the move again
314:
247:
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80:Be welcoming to newcomers
5231:Baire space (set theory)
4872:{\displaystyle \Pi _{3}}
4568:{\displaystyle \Pi _{2}}
4258:{\displaystyle \Pi _{1}}
3602:{\displaystyle p(n): -->
3531:{\displaystyle \psi (n)}
3502:{\displaystyle \psi (n)}
3379:for all quantifier-free
3077:20:35, 24 May 2018 (UTC)
2980:20:14, 24 May 2018 (UTC)
2581:{\displaystyle \Pi ^{n}}
2451:{\displaystyle \Pi ^{n}}
1564:, that is, effectively F
321:project's priority scale
5134:{\displaystyle \omega }
5109:{\displaystyle \vdots }
5088:{\displaystyle \vdots }
5067:{\displaystyle \vdots }
4937:{\displaystyle \vdots }
4916:{\displaystyle \vdots }
4895:{\displaystyle \vdots }
3903:(Strong) Exponential H.
3424:disjunctive normal form
3346:, we are claiming that
2985:I think the meaning of
2940:{\displaystyle n\geq 1}
2895:property also hold for
406:14:15 12 Jun 2003 (UTC)
278:WikiProject Mathematics
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2244:Major edit 2006-6-13
2167:Just in case you're
2152:Analytical hierarchy
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1169:of naturals) as the
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105:No original research
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270:Mathematics portal
214:content assessment
86:dispute resolution
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3782:{\displaystyle X}
3624:{\displaystyle p}
3270:That is, writing
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2121:is a redirect to
1962:{\displaystyle B}
1942:{\displaystyle X}
1781:{\displaystyle X}
1510:, as a subset of
1182:{\displaystyle B}
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357:in most browsers.
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4071:
4054:
4052:
4051:
4046:
4044:
4043:
4026:
4024:
4023:
4018:
4012:
4001:
3983:
3981:
3980:
3975:
3973:
3965:
3952:
3950:
3949:
3944:
3938:
3933:
3914:
3905:
3896:
3889:
3888:
3883:
3876:
3869:
3863:
3821:
3819:
3818:
3813:
3805:
3804:
3788:
3786:
3785:
3780:
3765:
3763:
3762:
3757:
3755:
3754:
3738:
3736:
3735:
3730:
3728:
3727:
3711:
3709:
3708:
3703:
3685:
3683:
3682:
3677:
3662:
3660:
3659:
3654:
3630:
3628:
3627:
3622:
3610:
3607:
3606:
3600:
3572:
3570:
3569:
3564:
3537:
3535:
3534:
3529:
3508:
3506:
3505:
3500:
3479:
3477:
3476:
3471:
3469:
3468:
3452:
3450:
3449:
3444:
3442:
3441:
3421:
3419:
3418:
3413:
3398:
3396:
3395:
3390:
3378:
3376:
3375:
3370:
3368:
3367:
3345:
3343:
3342:
3337:
3317:
3296:
3294:
3293:
3288:
3286:
3285:
3266:
3264:
3263:
3258:
3246:
3244:
3243:
3238:
3200:
3173:
3171:
3170:
3165:
3162:
3157:
3118:
3116:
3115:
3110:
3107:
3102:
3055:
3053:
3052:
3047:
3044:
3039:
3020:
3018:
3017:
3012:
3009:
3004:
2968:
2946:
2944:
2943:
2938:
2920:
2918:
2917:
2912:
2894:
2892:
2891:
2886:
2883:
2878:
2859:
2854:
2841:
2836:
2817:
2815:
2814:
2809:
2806:
2801:
2785:
2783:
2782:
2777:
2774:
2769:
2753:
2751:
2750:
2745:
2677:
2675:
2674:
2669:
2667:
2666:
2654:
2653:
2637:
2635:
2634:
2629:
2626:
2621:
2609:
2608:
2587:
2585:
2584:
2579:
2577:
2576:
2560:
2558:
2557:
2552:
2549:
2544:
2495:
2493:
2492:
2487:
2484:
2479:
2457:
2455:
2454:
2449:
2447:
2446:
2430:
2428:
2427:
2422:
2419:
2414:
2392:
2390:
2389:
2384:
2382:
2381:
2322:
2321:Thanks, Andy D.
2177:, the subscript
2137:Charles Matthews
2032:
2030:
2029:
2024:
2021:
2016:
2000:
1998:
1997:
1992:
1989:
1984:
1968:
1966:
1965:
1960:
1948:
1946:
1945:
1940:
1919:
1917:
1916:
1911:
1909:
1908:
1895:
1893:
1892:
1887:
1879:
1861:
1859:
1858:
1853:
1839:
1838:
1816:
1814:
1813:
1808:
1806:
1805:
1804:
1787:
1785:
1784:
1779:
1767:
1765:
1764:
1759:
1757:
1756:
1743:
1741:
1740:
1735:
1733:
1721:
1719:
1718:
1713:
1711:
1710:
1709:
1696:
1695:
1694:
1650:
1648:
1647:
1642:
1640:
1628:
1626:
1625:
1620:
1617:
1612:
1592:
1590:
1589:
1584:
1582:
1563:
1561:
1560:
1555:
1552:
1547:
1531:
1529:
1528:
1523:
1521:
1509:
1507:
1506:
1501:
1499:
1480:
1478:
1477:
1472:
1469:
1464:
1440:
1438:
1437:
1432:
1430:
1412:
1410:
1409:
1404:
1402:
1390:
1388:
1387:
1382:
1380:
1349:
1347:
1346:
1341:
1339:
1338:
1325:
1323:
1322:
1317:
1315:
1303:
1301:
1300:
1295:
1293:
1292:
1275:
1273:
1272:
1267:
1265:
1264:
1242:
1240:
1239:
1234:
1232:
1231:
1218:
1216:
1215:
1210:
1208:
1207:
1188:
1186:
1185:
1180:
1168:
1166:
1165:
1160:
1157:
1152:
1132:
1130:
1129:
1124:
1122:
1121:
1108:
1106:
1105:
1100:
1076:set of reals. --
1071:
1069:
1068:
1063:
1060:
1055:
1039:
1037:
1036:
1031:
1028:
1023:
1003:
1001:
1000:
995:
983:
981:
980:
975:
972:
967:
951:
949:
948:
943:
941:
940:
924:
922:
921:
916:
914:
913:
904:
867:
865:
864:
859:
856:
851:
826:
824:
823:
818:
816:
815:
799:
797:
796:
791:
789:
788:
769:
767:
766:
761:
758:
753:
734:
732:
731:
726:
723:
718:
702:
700:
699:
694:
691:
686:
658:
656:
655:
650:
647:
642:
622:
620:
619:
614:
611:
606:
590:
588:
587:
582:
579:
574:
546:", and so on. --
541:
539:
538:
533:
530:
525:
509:
507:
506:
501:
498:
493:
376:Reporting errors
344:
343:
337:
303:
302:
299:
296:
293:
272:
267:
266:
256:
249:
248:
243:
235:
228:
211:
202:
201:
194:
193:
185:
179:
178:
164:
95:Article policies
16:
5307:
5306:
5302:
5301:
5300:
5298:
5297:
5296:
5247:
5246:
5216:
5184:CaffeineWitcher
5143:
5142:
5123:
5122:
5098:
5097:
5077:
5076:
5056:
5055:
5031:
5026:
5025:
4986:
4985:
4949:
4948:
4926:
4925:
4905:
4904:
4884:
4883:
4859:
4854:
4853:
4831:
4826:
4825:
4787:
4786:
4754:
4753:
4721:
4720:
4696:
4691:
4690:
4652:
4651:
4613:
4612:
4580:
4579:
4555:
4550:
4549:
4527:
4522:
4521:
4483:
4482:
4450:
4449:
4417:
4416:
4392:
4387:
4386:
4348:
4347:
4303:
4302:
4270:
4269:
4245:
4240:
4239:
4211:
4206:
4205:
4167:
4166:
4128:
4127:
4089:
4088:
4063:
4058:
4057:
4033:
4032:
3988:
3987:
3959:
3958:
3920:
3919:
3912:Arithmetical H.
3910:
3901:
3892:
3887:
3861:
3855:
3796:
3791:
3790:
3771:
3770:
3746:
3741:
3740:
3719:
3714:
3713:
3688:
3687:
3665:
3664:
3636:
3635:
3613:
3612:
3575:
3574:
3540:
3539:
3511:
3510:
3509:atomic. Such a
3482:
3481:
3460:
3455:
3454:
3433:
3428:
3427:
3426:, we have that
3404:
3403:
3381:
3380:
3359:
3348:
3347:
3299:
3298:
3277:
3272:
3271:
3249:
3248:
3176:
3175:
3144:
3143:
3136:
3089:
3088:
3085:
3023:
3022:
2991:
2990:
2964:
2923:
2922:
2897:
2896:
2823:
2822:
2788:
2787:
2756:
2755:
2730:
2729:
2723:
2658:
2645:
2640:
2639:
2600:
2595:
2594:
2568:
2563:
2562:
2525:
2524:
2460:
2459:
2438:
2433:
2432:
2395:
2394:
2373:
2368:
2367:
2361:
2359:logic hierarchy
2346:129.206.103.183
2341:
2323:—The preceding
2308:
2306:Content Request
2274:
2246:
2221:
2211:
2201:
2195:
2176:
2148:
2115:
2003:
2002:
1971:
1970:
1951:
1950:
1931:
1930:
1898:
1897:
1864:
1863:
1819:
1818:
1795:
1790:
1789:
1770:
1769:
1746:
1745:
1724:
1723:
1700:
1685:
1674:
1673:
1651:is not open. --
1631:
1630:
1599:
1598:
1597:the union of a
1573:
1572:
1567:
1534:
1533:
1512:
1511:
1490:
1489:
1451:
1450:
1415:
1414:
1393:
1392:
1371:
1370:
1328:
1327:
1306:
1305:
1282:
1281:
1254:
1253:
1221:
1220:
1191:
1190:
1171:
1170:
1139:
1138:
1136:
1111:
1110:
1091:
1090:
1042:
1041:
1010:
1009:
986:
985:
954:
953:
930:
929:
889:
888:
838:
837:
807:
802:
801:
777:
772:
771:
740:
739:
705:
704:
673:
672:
629:
628:
593:
592:
561:
560:
545:
512:
511:
480:
479:
440:
387:
378:
360:
359:
358:
341:
300:
297:
294:
291:
290:
268:
261:
241:
212:on Knowledge's
209:
199:
121:
116:
115:
114:
91:
61:
12:
11:
5:
5305:
5303:
5295:
5294:
5289:
5284:
5279:
5274:
5269:
5264:
5259:
5249:
5248:
5215:
5212:
5211:
5210:
5200:Caleb Stanford
5163:
5160:
5155:
5151:
5130:
5117:
5116:
5105:
5095:
5084:
5074:
5063:
5052:
5051:
5038:
5034:
5023:
5009:
5006:
5003:
4998:
4994:
4983:
4971:
4966:
4961:
4957:
4945:
4944:
4933:
4923:
4912:
4902:
4891:
4880:
4879:
4866:
4862:
4851:
4838:
4834:
4823:
4810:
4807:
4804:
4799:
4795:
4784:
4771:
4766:
4762:
4751:
4738:
4733:
4729:
4717:
4716:
4703:
4699:
4688:
4675:
4672:
4669:
4664:
4660:
4649:
4636:
4633:
4630:
4625:
4621:
4610:
4597:
4592:
4588:
4576:
4575:
4562:
4558:
4547:
4534:
4530:
4519:
4506:
4503:
4500:
4495:
4491:
4480:
4467:
4462:
4458:
4447:
4434:
4429:
4425:
4413:
4412:
4399:
4395:
4384:
4371:
4368:
4365:
4360:
4356:
4345:
4331:
4326:
4323:
4320:
4315:
4311:
4300:
4287:
4282:
4278:
4266:
4265:
4252:
4248:
4237:
4223:
4218:
4214:
4203:
4190:
4187:
4184:
4179:
4175:
4164:
4150:
4145:
4140:
4136:
4125:
4111:
4106:
4101:
4097:
4085:
4084:
4070:
4066:
4055:
4042:
4030:
4016:
4011:
4008:
4005:
4000:
3996:
3985:
3971:
3968:
3956:
3942:
3937:
3932:
3928:
3916:
3915:
3908:
3906:
3899:
3897:
3886:
3885:
3878:
3871:
3860:
3854:
3851:
3850:
3849:
3839:Caleb Stanford
3822:, as claimed.
3811:
3808:
3803:
3799:
3778:
3753:
3749:
3726:
3722:
3701:
3698:
3695:
3675:
3672:
3652:
3649:
3646:
3643:
3620:
3598:
3595:
3592:
3589:
3586:
3583:
3562:
3559:
3556:
3553:
3550:
3547:
3527:
3524:
3521:
3518:
3498:
3495:
3492:
3489:
3467:
3463:
3440:
3436:
3411:
3388:
3366:
3362:
3358:
3355:
3335:
3332:
3329:
3326:
3323:
3320:
3316:
3312:
3309:
3306:
3284:
3280:
3256:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3199:
3195:
3192:
3189:
3186:
3183:
3161:
3156:
3152:
3135:
3132:
3106:
3101:
3097:
3084:
3081:
3080:
3079:
3061:
3043:
3038:
3035:
3031:
3008:
3003:
2999:
2963:
2960:
2936:
2933:
2930:
2910:
2907:
2904:
2882:
2877:
2874:
2871:
2867:
2863:
2858:
2853:
2849:
2845:
2840:
2835:
2831:
2805:
2800:
2796:
2773:
2768:
2764:
2743:
2740:
2737:
2722:
2719:
2718:
2717:
2716:
2715:
2688:
2665:
2661:
2657:
2652:
2648:
2625:
2620:
2616:
2612:
2607:
2603:
2575:
2571:
2548:
2543:
2540:
2537:
2533:
2513:Same as before
2483:
2478:
2475:
2472:
2468:
2445:
2441:
2418:
2413:
2410:
2407:
2403:
2380:
2376:
2360:
2357:
2320:
2307:
2304:
2303:
2302:
2273:
2267:
2263:
2262:
2259:
2256:
2253:
2245:
2242:
2241:
2240:
2239:
2238:
2217:
2206:
2197:
2191:
2185:tells you the
2172:
2165:
2147:
2144:
2143:
2142:
2114:
2111:
2104:
2103:
2102:
2101:
2100:
2099:
2098:
2097:
2096:
2095:
2094:
2093:
2092:
2091:
2053:
2052:
2051:
2050:
2049:
2048:
2047:
2046:
2045:
2044:
2020:
2015:
2011:
1988:
1983:
1979:
1958:
1938:
1907:
1885:
1882:
1878:
1874:
1871:
1851:
1848:
1845:
1842:
1837:
1832:
1829:
1826:
1803:
1798:
1777:
1755:
1732:
1708:
1703:
1699:
1693:
1688:
1684:
1681:
1663:
1662:
1661:
1660:
1659:
1658:
1639:
1616:
1611:
1607:
1581:
1569:
1565:
1551:
1546:
1542:
1520:
1498:
1486:
1468:
1463:
1459:
1449:set, that is,
1429:
1425:
1422:
1401:
1379:
1362:
1361:
1360:
1359:
1358:
1357:
1337:
1314:
1291:
1263:
1230:
1206:
1201:
1198:
1178:
1156:
1151:
1147:
1134:
1120:
1098:
1084:
1083:
1059:
1054:
1050:
1027:
1022:
1018:
993:
971:
966:
962:
939:
926:
925:
912:
907:
903:
899:
896:
882:
881:
880:
879:
855:
850:
846:
814:
810:
787:
784:
780:
757:
752:
748:
736:
722:
717:
713:
690:
685:
681:
646:
641:
637:
610:
605:
601:
578:
573:
569:
556:
555:
554:
553:
543:
529:
524:
520:
497:
492:
488:
439:
436:
435:
434:
433:
432:
427:comment added
408:
407:
398:
397:
394:
386:
383:
380:
379:
373:
372:
371:
355:case-sensitive
349:
348:
347:
345:
333:
332:
329:
328:
325:
324:
313:
307:
306:
304:
287:the discussion
274:
273:
257:
245:
244:
236:
224:
223:
217:
195:
181:
180:
118:
117:
113:
112:
107:
102:
93:
92:
90:
89:
82:
77:
68:
62:
60:
59:
48:
39:
38:
35:
34:
28:
13:
10:
9:
6:
4:
3:
2:
5304:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5254:
5252:
5245:
5244:
5240:
5236:
5232:
5227:
5225:
5221:
5213:
5209:
5205:
5201:
5196:
5195:
5194:
5193:
5189:
5185:
5181:
5179:
5161:
5158:
5153:
5149:
5128:
5103:
5082:
5061:
5053:
5036:
5007:
5004:
5001:
4996:
4969:
4964:
4959:
4946:
4931:
4910:
4889:
4881:
4864:
4852:
4836:
4824:
4808:
4805:
4802:
4797:
4769:
4764:
4752:
4736:
4731:
4719:
4718:
4701:
4673:
4670:
4667:
4662:
4650:
4634:
4631:
4628:
4623:
4611:
4595:
4590:
4577:
4560:
4548:
4532:
4520:
4504:
4501:
4498:
4493:
4465:
4460:
4448:
4432:
4427:
4415:
4414:
4397:
4369:
4366:
4363:
4358:
4346:
4344:
4329:
4324:
4321:
4318:
4313:
4301:
4285:
4280:
4267:
4250:
4238:
4236:
4221:
4216:
4204:
4188:
4185:
4182:
4177:
4163:
4148:
4143:
4138:
4126:
4124:
4109:
4104:
4099:
4087:
4086:
4068:
4029:
4014:
4009:
4006:
4003:
3998:
3969:
3966:
3955:
3940:
3935:
3930:
3917:
3913:
3907:
3904:
3898:
3895:
3894:Polynomial H.
3890:
3884:
3879:
3877:
3872:
3870:
3865:
3864:
3859:
3852:
3848:
3844:
3840:
3836:
3835:
3834:
3833:
3829:
3825:
3809:
3806:
3801:
3797:
3776:
3767:
3751:
3747:
3724:
3720:
3693:
3670:
3647:
3641:
3632:
3618:
3596:
3593:
3587:
3581:
3560:
3557:
3551:
3545:
3522:
3516:
3493:
3487:
3465:
3461:
3438:
3434:
3425:
3409:
3400:
3386:
3364:
3360:
3356:
3353:
3327:
3321:
3318:
3310:
3307:
3282:
3278:
3268:
3254:
3228:
3225:
3222:
3219:
3213:
3210:
3207:
3201:
3193:
3190:
3184:
3181:
3159:
3154:
3140:
3133:
3131:
3130:
3126:
3122:
3104:
3099:
3083:Pronunciation
3082:
3078:
3074:
3070:
3066:
3062:
3059:
3041:
3036:
3033:
3006:
3001:
2988:
2984:
2983:
2982:
2981:
2976:
2972:
2961:
2959:
2958:
2954:
2950:
2934:
2931:
2928:
2908:
2905:
2902:
2880:
2875:
2872:
2869:
2861:
2856:
2851:
2843:
2838:
2833:
2819:
2803:
2798:
2771:
2766:
2741:
2738:
2735:
2726:
2720:
2714:
2710:
2706:
2705:66.127.55.192
2701:
2700:
2699:
2696:
2693:
2689:
2687:
2684:
2681:
2663:
2655:
2650:
2623:
2618:
2610:
2605:
2591:
2573:
2546:
2541:
2538:
2535:
2522:
2518:
2514:
2510:
2509:
2508:
2507:
2503:
2499:
2498:66.127.55.192
2481:
2476:
2473:
2470:
2443:
2416:
2411:
2408:
2405:
2378:
2365:
2358:
2356:
2355:
2351:
2347:
2338:
2334:
2330:
2329:66.117.149.88
2326:
2319:
2315:
2311:
2305:
2301:
2297:
2293:
2292:76.254.27.203
2289:
2288:
2287:
2286:
2283:
2279:
2272:
2268:
2266:
2260:
2257:
2254:
2251:
2250:
2249:
2243:
2237:
2234:
2230:
2229:
2228:
2225:
2220:
2215:
2210:
2205:
2200:
2194:
2188:
2184:
2180:
2175:
2170:
2166:
2163:
2162:
2161:
2160:
2157:
2153:
2146:superscript 0
2145:
2141:
2138:
2134:
2133:
2132:
2131:
2128:
2124:
2120:
2112:
2110:
2109:
2090:
2087:
2083:
2079:
2075:
2074:
2073:
2070:
2065:
2064:
2063:
2062:
2061:
2060:
2059:
2058:
2057:
2056:
2055:
2054:
2043:
2040:
2036:
2018:
2013:
1986:
1981:
1956:
1936:
1928:
1927:
1926:
1923:
1883:
1872:
1869:
1849:
1846:
1843:
1840:
1830:
1827:
1796:
1775:
1701:
1686:
1682:
1679:
1671:
1670:
1669:
1668:
1667:
1666:
1665:
1664:
1657:
1654:
1614:
1609:
1596:
1570:
1549:
1544:
1487:
1484:
1466:
1461:
1448:
1444:
1423:
1420:
1368:
1367:
1366:
1365:
1364:
1363:
1356:
1353:
1279:
1251:
1250:
1249:
1246:
1199:
1196:
1176:
1154:
1149:
1096:
1088:
1087:
1086:
1085:
1082:
1079:
1075:
1057:
1052:
1025:
1020:
1007:
1006:
1005:
991:
969:
964:
897:
894:
887:
886:
885:
878:
875:
871:
853:
848:
835:
834:
833:
830:
812:
808:
785:
782:
778:
755:
750:
737:
720:
715:
688:
683:
670:
666:
662:
644:
639:
626:
625:
624:
608:
603:
576:
571:
552:
549:
527:
522:
495:
490:
477:
473:
472:
471:
468:
464:
459:
458:
457:
456:
453:
449:
445:
437:
430:
426:
420:
416:
412:
411:
410:
409:
405:
400:
399:
395:
392:
391:
390:
384:
377:
369:
365:
364:
363:
356:
352:
346:
339:
338:
322:
318:
312:
309:
308:
305:
288:
284:
280:
279:
271:
265:
260:
258:
255:
251:
250:
246:
240:
237:
234:
230:
225:
221:
215:
207:
206:
196:
192:
187:
186:
177:
173:
170:
167:
163:
159:
155:
152:
149:
146:
143:
140:
137:
134:
131:
127:
124:
123:Find sources:
120:
119:
111:
110:Verifiability
108:
106:
103:
101:
98:
97:
96:
87:
83:
81:
78:
76:
72:
69:
67:
64:
63:
57:
53:
52:Learn to edit
49:
46:
41:
40:
37:
36:
32:
26:
22:
18:
17:
5228:
5224:Cantor space
5217:
5182:
5120:
3856:
3768:
3633:
3401:
3269:
3141:
3137:
3086:
3064:
3057:
2986:
2965:
2820:
2727:
2724:
2589:
2561:-formula is
2521:second-order
2520:
2516:
2512:
2362:
2342:
2316:
2312:
2309:
2275:
2264:
2247:
2218:
2208:
2203:
2198:
2192:
2186:
2182:
2178:
2173:
2168:
2149:
2117:The link to
2116:
2105:
2081:
2077:
2034:
1594:
1482:
1446:
1442:
1277:
1073:
927:
883:
869:
669:Cantor space
557:
447:
441:
388:
361:
353:Anchors are
350:
317:Mid-priority
316:
276:
242:Mid‑priority
220:WikiProjects
203:
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
5220:Baire space
5214:Baire Space
3862:This box:
3402:By putting
703:, but with
665:Baire space
661:clopen sets
423:—Preceding
292:Mathematics
283:mathematics
239:Mathematics
148:free images
31:not a forum
5251:Categories
3609:0}" /: -->
3065:analytical
2108:MathMartin
2069:MathMartin
1922:MathMartin
1245:MathMartin
467:MathMartin
415:91.66.2.94
3069:Trovatore
2224:Trovatore
2086:Trovatore
2039:Trovatore
1653:Trovatore
1352:Trovatore
1326:, not of
1078:Trovatore
874:Trovatore
829:Trovatore
548:Trovatore
452:Trovatore
208:is rated
88:if needed
71:Be polite
21:talk page
3577:0}": -->
2987:analytic
2949:Catrincm
2325:unsigned
2282:CMummert
2233:Bcrowell
2156:Bcrowell
2127:Bcrowell
1817:so that
1488:The set
1445:} is an
1089:Ok, but
623:sets ?
385:Untitled
56:get help
29:This is
27:article.
5235:Jrvarma
4028:EXPTIME
3984:PSPACE
3858:this:
3634:Such a
3139:ones):
2082:section
2078:article
2035:problem
1278:subsets
425:undated
370:before.
319:on the
210:B-class
154:WP refs
142:scholar
5022:=EXPH
3573:or to
2818:ones.
2458:means
2169:asking
928:where
827:'s. --
216:scale.
126:Google
3824:Jslam
3594:: -->
3480:with
2592:: -->
1532:, is
476:bases
197:This
169:JSTOR
130:books
84:Seek
5239:talk
5204:talk
5188:talk
5178:here
4343:NEXP
4235:c.e.
4162:CoNP
3882:edit
3875:talk
3868:view
3843:talk
3828:talk
3769:But
3297:for
3174:set
3125:talk
3121:N4m3
3073:talk
3034:<
2975:talk
2953:talk
2709:talk
2692:Emil
2680:Emil
2588:for
2502:talk
2350:talk
2333:talk
2296:talk
2269:See
2187:type
1571:But
1350:. --
1074:open
870:cube
738:The
627:The
448:what
419:talk
351:Tip:
162:FENS
136:news
73:and
5222:or
5141:or
4982:PH
3686:as
3422:in
2971:CBM
2962:-al
1788:of
1595:not
1593:is
1441:, {
1280:of
667:or
421:)
404:Pde
311:Mid
176:TWL
5253::
5241:)
5206:)
5190:)
5180:)
5150:ω
5129:ω
5104:⋮
5083:⋮
5062:⋮
5037:ω
5033:Δ
4997:ω
4993:Δ
4960:ω
4956:Δ
4932:⋮
4911:⋮
4890:⋮
4861:Π
4833:Σ
4794:Δ
4761:Π
4728:Σ
4698:Δ
4659:Π
4620:Σ
4587:Δ
4557:Π
4529:Σ
4490:Δ
4457:Π
4424:Σ
4394:Δ
4355:Π
4310:Σ
4277:Δ
4247:Π
4213:Σ
4174:Δ
4135:Π
4123:NP
4096:Σ
4083:=
4065:Δ
3995:Δ
3967:⊆
3927:Δ
3845:)
3830:)
3807:≠
3802:ϕ
3752:ϕ
3725:ψ
3700:∞
3697:→
3674:∞
3671:±
3603:0}
3517:ψ
3488:ψ
3466:ψ
3439:ϕ
3410:ϕ
3399:.
3387:ϕ
3365:ϕ
3357:≠
3322:ϕ
3319:∣
3311:∈
3283:ϕ
3267:.
3255:ϕ
3211:≤
3205:∃
3202:∣
3194:∈
3151:Δ
3127:)
3096:Σ
3075:)
3037:ω
3030:Σ
2998:Σ
2973:·
2955:)
2947:.
2932:≥
2866:Δ
2862:⊊
2848:Π
2844:∪
2830:Σ
2795:Δ
2763:Δ
2711:)
2695:J.
2683:J.
2678:.—
2660:Σ
2656:∩
2647:Π
2615:Σ
2602:Π
2570:Π
2539:−
2532:Π
2515:,
2504:)
2474:−
2467:Σ
2440:Π
2409:−
2402:Π
2375:Σ
2352:)
2335:)
2298:)
2207:ω+
2010:Σ
1978:Σ
1881:→
1870:ν
1847:∈
1831:∈
1825:∀
1698:→
1606:Σ
1541:Σ
1458:Π
1424:∈
1200:∈
1146:Σ
1097:ν
1049:Σ
1017:Σ
992:ν
961:Σ
906:→
895:ν
845:Σ
813:δ
786:σ
783:δ
747:Σ
712:Σ
680:Σ
636:Σ
600:Σ
568:Σ
519:Σ
487:Σ
156:)
54:;
5237:(
5202:(
5186:(
5162:K
5159:C
5154:1
5008:P
5005:X
5002:E
4970:=
4965:P
4865:3
4837:3
4809:P
4806:X
4803:E
4798:3
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4737:P
4732:3
4702:2
4674:P
4671:X
4668:E
4663:2
4635:P
4632:X
4629:E
4624:2
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4591:2
4561:2
4533:2
4505:P
4502:X
4499:E
4494:2
4466:P
4461:2
4433:P
4428:2
4398:1
4370:P
4367:X
4364:E
4359:1
4330:=
4325:P
4322:X
4319:E
4314:1
4286:P
4281:1
4251:1
4222:=
4217:1
4189:P
4186:X
4183:E
4178:1
4149:=
4144:P
4139:1
4110:=
4105:P
4100:1
4069:0
4041:}
4015:=
4010:P
4007:X
4004:E
3999:0
3970:?
3954:P
3941:=
3936:P
3931:0
3841:(
3826:(
3810:X
3798:X
3777:X
3748:X
3721:X
3694:n
3651:)
3648:n
3645:(
3642:p
3619:p
3597:0
3591:)
3588:n
3585:(
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3558:=
3555:)
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3546:p
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3523:n
3520:(
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3462:X
3435:X
3361:X
3354:X
3334:}
3331:)
3328:n
3325:(
3315:N
3308:n
3305:{
3279:X
3235:}
3232:)
3229:n
3226:=
3223:k
3220:2
3217:(
3214:n
3208:k
3198:N
3191:n
3188:{
3185:=
3182:X
3160:0
3155:0
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3105:0
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3071:(
3042:1
3007:1
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2969:(
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2870:n
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2839:0
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2804:0
2799:1
2772:0
2767:0
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2739:=
2736:n
2707:(
2664:2
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2611:=
2606:1
2590:n
2574:n
2547:1
2542:1
2536:n
2500:(
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2339:.
2331:(
2294:(
2219:m
2209:n
2204:V
2199:m
2193:m
2190:Σ
2183:n
2179:m
2174:m
2019:0
2014:n
1987:0
1982:n
1957:B
1937:X
1906:B
1884:X
1877:N
1873::
1850:X
1844:B
1841::
1836:B
1828:B
1802:R
1797:2
1776:X
1754:B
1731:N
1707:R
1702:2
1692:N
1687:2
1683::
1680:f
1638:Q
1615:0
1610:2
1580:Q
1566:σ
1550:0
1545:2
1519:R
1497:Q
1483:q
1467:0
1462:1
1443:q
1428:Q
1421:q
1400:R
1378:Q
1336:B
1313:R
1290:B
1262:B
1229:B
1205:B
1197:B
1177:B
1155:0
1150:2
1135:σ
1119:B
1058:0
1053:2
1026:0
1021:1
970:0
965:1
938:B
911:B
902:N
898::
854:0
849:1
809:G
779:G
756:0
751:3
735:.
721:0
716:1
689:0
684:0
645:0
640:0
609:0
604:3
577:0
572:0
544:σ
528:0
523:2
496:0
491:1
417:(
323:.
222::
172:·
166:·
158:·
151:·
145:·
139:·
133:·
128:(
58:.
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