Talk:Arithmetical hierarchy

message: We need the following interesting facts:
* The hierarchy does not collapse; which I think is from Kleene
* possibly conflating this article with any analytical hierarchy article
: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 :) -- Pde 14:15 12 Jun 2003 (UTC)
:: I know it's a lot to read, but 13 years is a bit much ;) 91.66.2.94 (talk)  —Preceding undated comment added 16:12, 1 October 2016 (UTC)
thread_title:  Untitled 

message: The link to co-A-recursive set is a redirect to Relative computability, which doesn't actually define the term.--Bcrowell 22:36, 24 February 2006 (UTC)
replies: text: I presume it means 'complement of A-recursive'. Charles Matthews 22:49, 24 February 2006 (UTC)
thread_title: co-A-recursive set

message: I am planning to edit this page like analytical hierarchy to mention both subsets of N and subsets of Baire space. CMummert 11:32, 26 June 2006 (UTC)
replies: text: When you do, please define \Delta _{n}^{0}, that would help make the article much more comprehensible. 76.254.27.203 (talk) 04:09, 8 September 2023 (UTC)
thread_title:  See [[analytical hierarchy]] 

message: Would someone include some natural complete problems for the Arithmetic Hierarchy (of sets of natural numbers)?  Same goes for Analytic Hierarchy. 
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.
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.
 Thanks, Andy D. —The preceding unsigned comment was added by 66.117.149.88 (talk) 04:03, 23 December 2006 (UTC)
thread_title:  Content Request 

message: There's apparently an alternate expression called the "logic hierarchy", described in Girard's "The Blind Spot" p. 45-46 .  I can't quite tell what it means, but it looks like \Sigma^n (with no subscript) in the logic hierarchy means the same thing as \Pi^0_{n-1} from the arithmetic hierarchy, and \Pi^n means \Sigma^0_{n-1}.  I won't add this myself since probably I don't have it right, but maybe it's worth a mention. 66.127.55.192 (talk) 00:13, 18 February 2010 (UTC)
replies: text: I don't understand Girard's description (Same as before [presumably referring to the projective hierarchy described in the previous paragraph], but the first order quantifiers are not numerical—WTF???), but one thing is certain, namely that it is a stratification of second-order formulas, it thus corresponds to the projective hierarchy rather than the arithmetical hierarchy. According to Girard's examples, every \Pi^1_{n-1}-formula is \Pi^n for n > 1, while the next section suggests that \Pi^1=\Sigma^0_1 (don't ask me why). In particular, all of arithmetical hierarchy fits inside \Pi^2\cap\Sigma^2.—Emil J. 12:49, 18 February 2010 (UTC)

text: 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.—Emil J. 12:53, 18 February 2010 (UTC)
replies: text: 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 "logic hierarchy" was some standard term. 66.127.55.192 (talk) 13:32, 18 February 2010 (UTC)
thread_title:  logic hierarchy 

message: I believe the "Extensions and variations" sections mentions two alternatives but it's not clear that these are in fact two different alternatives. 
Moreover for the recursive/computable definition I believe it only differs from the usual definition for n=0 where the computable-definition version of the \Delta^0_0 sets are the same as the \Delta^0_1 ones.
Lastly doesn't the \Sigma^0_n \cup \Pi^0_n \subsetneq \Delta^0_{n+1} property also hold for n =0 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 n\geq 1. Catrincm (talk) 14:13, 30 September 2014 (UTC)
thread_title:  Extensions and variations 

message: As someone with only passing familiarity of this area, I can't remember whether \Sigma^0_1 is pronounced “sigma one zero” or “sigma zero one”. I think saying which it is would be a helpful and easy addition. N4m3 (talk) 10:16, 21 September 2020 (UTC)
thread_title:  Pronunciation 

message: 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 ones):
We claim that the \Delta_0^0 set X=\{n\in\mathbb{N}\mid \exists k\le n (2k=n)\} of all even numbers cannot be defined by a quantifier-free formula \phi.
That is, writing X_\phi for \{n\in\mathbb{N}\mid\phi(n)\}, we are claiming that X\ne X_\phi for all quantifier-free \phi.
By putting \phi in disjunctive normal form, we have that X_\phi is a finite union of finite intersections of sets X_\psi with \psi(n) atomic.
Such a \psi(n) is equivalent either to the formula p(n)=0 or to p(n)>0 for some polynomial p with integer coefficients.
Such a p(n) is either constant or tends toward \pm\infty as n\to\infty.
It follows that X_\psi is either finite or has finite complement.
Since this property is preserved by finite unions and intersections, it follows that X_\phi has this property as well.
But X does not have this property, and hence X_\phi\ne X, as claimed. Jslam (talk) 06:32, 28 April 2023 (UTC)
replies: text: 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. Caleb Stanford (talk) 21:57, 28 April 2023 (UTC)
thread_title:  Bounded quantifiers versus quantifier-free 

message: 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 this: 
                        	                        	                            	                            	                            	        	               	
\Delta_0^P=  P          	\Delta_0^P=  P          	\overset{?}{\subseteq}PSPACE	\Delta_0^{EXP}=  EXPTIME    	\Delta_0^{EXP}=  EXPTIME    	\Bigg \}	\Delta_0  =    	\Delta_0  =
\Sigma_1^P=  NP         	\Pi_1^P=  CoNP          	\overset{?}{\subseteq}PSPACE	\Delta_1^{EXP}              	\Delta_1^{EXP}              	\Bigg \}	\Sigma_1=  c.e.	\Pi_1
\Delta_1^P              	\Delta_1^P              	\overset{?}{\subseteq}PSPACE	\Sigma_1^{EXP}=NEXP         	\Pi_1^{EXP}                 	\Bigg \}	\Delta_1       	\Delta_1
\Sigma_2^{P}            	\Pi_2^{P}               	\overset{?}{\subseteq}PSPACE	\Delta_2^{EXP}              	\Delta_2^{EXP}              	\Bigg \}	\Sigma_2       	\Pi_2
\Delta_2^{P}            	\Delta_2^{P}            	\overset{?}{\subseteq}PSPACE	\Sigma_2^{EXP}              	\Pi_2^{EXP}                 	\Bigg \}	\Delta_2       	\Delta_2
\Sigma_3^{P}            	\Pi_3^{P}               	\overset{?}{\subseteq}PSPACE	\Delta_3^{EXP}              	\Delta_3^{EXP}              	\Bigg \}	\Sigma_3       	\Pi_3
\vdots                  	\vdots                  	\overset{?}{\subseteq}PSPACE	\vdots                      	\vdots                      	\Bigg \}	\vdots         	\vdots
\Delta^{P}_{\omega}=  PH	\Delta^{P}_{\omega}=  PH	\overset{?}{\subseteq}PSPACE	\Delta^{EXP}_{\omega}  =EXPH	\Delta^{EXP}_{\omega}  =EXPH	\Bigg \}	\Delta_\omega  	\Delta_\omega
\vdots                  	\vdots                  	\overset{?}{\subseteq}PSPACE	\vdots                      	\vdots                      	\Bigg \}	\vdots         	\vdots
(I've put the question mark above PSPACE since we could also consider ordinals larger than \omega or \omega_1^{CK} (for which it has been proven here)
CaffeineWitcher (talk) 09:40, 14 August 2023 (UTC)
replies: text: 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. Caleb Stanford (talk) 21:42, 14 August 2023 (UTC)
thread_title:  hierarchies overview/NavBox 

message: I was confused by the sentence "Every  subset of Baire space or Cantor space is an open set in the usual topology on the space." 
The Wikipedia 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 Baire space (set theory). 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. Jrvarma (talk) 06:39, 30 November 2023 (UTC)
thread_title:  Baire Space