Another problem I see is "Clearly if R is provable but there is no proof of !R, then R is false, by definition of the oracle H ...".

Perhaps I glossed over this point: This is because in this case the oracle H differs only finitely from the oracle B which Baker, Gill, and Solovay demonstrated such that P^{B} != NP^{B}, and therefore it must also be true that P^{H} != NP^{H}.

Similarly if !R is provable, but R is not provable, then H differs only finitely from Baker, Gill, and Solovay's oracle A such that P^{A} == NP^{A}, and therefore P^{H} == NP^{H}.

The finite number of differences between the sets B and H, or A and H, respectively, can be managed as a finite number of exceptions by the Turing machine or primitive recursive algorithm that queries the oracle.

]]>I'm not going to waste my time on someone whose interest is in finding details which he can bend and abuse to attack someone personally.

I have made no comment on your person. (See the sentence quoted above for contrast.)

As for content you quite realiably don't care and fail. I asked you twice to tell us the html code for ==> and you offered but → and a unicode symbol which is or is not visible in a users browser depending on his code settings and installed fonts.

How much anyone cares is irrelevant.

As for "failing": I gave you **⇒** (in bold even) as a way of writing ⇒ in response to your question.

The question of availability of glyphs is independent of what we're talking about.

"HTML codes" (character references) are a notational device only. A browser will produce the same tokens for character references as for the characters they reference. (See e.g. tokenizing character references in HTML spec for the details.)

]]>I once more tried to reasonably talk with you. Now I'm stopping that non-discussion. I'm not going to waste my time on someone whose interest is in finding details which he can bend and abuse to attack someone personally.

As for content you quite realiably don't care and fail. I asked you twice to tell us the html code for ==> and you offered but → and a unicode symbol which is or is not visible in a users browser depending on his code settings and installed fonts.

"Moving the goal posts" - That's as much a self-accusation of you as it is an accusation of myself. You started it by arbitrarily focussing on an example rather than on the issue at hand.

No, it is not.

You made a claim about a recursively defined factorial function as an example of the supposed difference between the mathematical realm and the system realm.

I pointed out that your claim was bogus.

After implicitly acknowledging this, you immediately start talking about other functions that... [Goalposts are moved here.]

FYI, this time you used tu quoque.

Unfortunately things with Unicode are complicated and ugly. That even starts with entering Unicode symbols.

That's not Unicode. That's your keyboard and software. So use decent software and find out what's possible. Entering code points as hexadecimal numbers isn't the only option.

In my tools I do, of course, have everything needed (usually more or less Latex like) and I have special fonts, too. But as soon as I have to put things in HTML it gets ugly and painful and poor.

If your tools can produce characters like → directly, then why not use those characters directly in HTML?

We get ever new loads of BS like emoticons and whatnot but we still have no reasonable set of math symbols in HTML. Yuck!

(I think it's about adding glyphs for more emoticons/emojis, but whatever.)

Which math symbols are missing?

]]>In provability logic we have to consider that R might be ...

And indeed I did discern between R and PR (the latter being "R is provable"). However, I regard that as a niche case because: If R isn't true I need not propose it; "true" meaning "provably true". Pardon me if I don't care too much about exotics.

Suppose that either Q or !Q is provable ...

That's why I wrote the above. If Q isn't provable any statement with Q in it is a conjecture in the best of cases. Please keep in mind that while diverse niches of mathematics are certainly interesting and promising, it still holds that "provable" can not contradict well established math. Accordingly while I'm willing to discern and to write R and PR, I still consider "Q is provable" as a proposition, in this case one related to another one, namely R.

Back to your proposition:

First I think that your undertaking and attempt is interesting. Compliments. Way better than the usual delivery of bits and pieces found somewhere on the internet.

But still I do see at least some instability and loose ends in it. It is, for instance, not very useful to make a statement about an oracle without saying much more about it than that it's primitive recursive. Moreover, again, the very definition of an oracle isn't about its inner working but about its magic capability tp provide an answer to a problem of any complexity in a single step (nor is it about the Turing machine it provides with "answers"). Otherwise we know little; we merely consider it to be a mysterious part of (Turing) state machine because it's a useful assumption (or image) for input series that belong to a (however defined) set.

Another problem I see is "Clearly if R is provable but there is no proof of !R, then R is false, by definition of the oracle H ...". You should make clear how a complexity statement is defined by an oracle; an oracle is assumed to properly work over a certain set of values in a certain problem constellation. An oracle for "Is Hm a valid SHA512 hash value?" is capable to decide that question for any member of the set of all strings that meet the condition for an output of SHA512. The fact that the problem of determining that is of complexity (say) NP is completely independent of the oracle. You, however, seem to posit a (even well definable) relation.

Also note that "R is provable but ¬R is not" does not imply that R is false. Let's for a moment use standard logic and formulate:

Let P(R) be the fact that R is provable. Then we can simplify to R, because if R (whatever R may happen to be) is not provable it is not true. If it is provable, however, then P(R) is equivalent to R. Also note that "is provable" is a statement with a mathematical meaning.

If ¬P(¬R), i.e. if ¬R is not provable then it may or may not be true and can anyway not be used for the purpose of proving no matter what. R is (provably) true isn't any less true because we can not prove that ¬R is true.

No! In provability logic we have to consider that R might be proven but not true, true but not proven, both true and proven, or neither proven nor true. And then we can talk about whether a proof that a proof of R exists implies that such a proof really does exist...

Anyways, back to the proof...of my last unproven claim above.

Suppose that either Q or !Q is provable and furthermore we can prove that if Q and R are equivalent, then our system of logic is inconsistent. This is equivalent to proving that either R or !R, respectively, is provable. By Lob's theorem, this will not help us prove R or !R if we cannot already do so without the supposition that Q and R are equivalent.

I will have to think through the implications, but my intention here is to have proven what I set out to do in my first post in this thread.

]]>Moving the goalposts. You claimed, with a smidgen of hyperbole...

No, no hyperbole, just "loosely speaking" to make my point clear, a point that we encounter painfully often, namely the difference between the mathematical realm and the physical system realm.

"Moving the goal posts" - That's as much a self-accusation of you as it is an accusation of myself. You started it by arbitrarily focussing on an *example* rather than on the issue at hand.

*HTML and math*

There is a reason why I want HTML codes: smallest common demoninator. I can reasonably assume that anyone with a properly configured OS and browser will be able to see what I write.

Unfortunately things with Unicode are complicated and ugly. That even starts with entering Unicode symbols. On Linux (seemingly at least gtk based) one presses and holds Ctrl-Shift, then types 'u' and the unicode in hex while still having Ctrl-Shift pressed; finally a single 'space' ends the sequence. I have read that KDE works differently (or not at all; don't know, don't use it). Staying within unix but switching to, say, FreeBSD changes things. Let alone apple which, as I understand, does not even have a Ctrl key (but presumably other ways to achieve the same), and so on.

In my tools I do, of course, have everything needed (usually more or less Latex like) and I have special fonts, too. But as soon as I have to put things in HTML it gets ugly and painful and poor.

As for the HTML "rarr" code, I know that and in fact I used it ("PR and ¬¬PR → ¬ PH = NPH", which btw. contains an error; I should have put parantheses around "PH = NHP" but was too distressed by the HMTL sh*t) but after encountering problems like these often enough with HTML I'm unnerved. We get ever new loads of BS like emoticons and whatnot but we still have no reasonable set of math symbols in HTML. Yuck!

In case you really know more about Unicode and all the problems around it than I do (which is quite easily possible), congratulations.

]]>Moving the goalposts. You claimed, with a smidgen of hyperbole, that if factorial is implemented in a recursive manner the world production of RAM isn't sufficient for the needed stack.

As for your immensely smart remark re. HTML and math symbols just give me the proper HTML code for what in ASCII is typically written as "==>" (implication). Good luck.

Who needs an "HTML code" when you can input any character you please using a keyboard? (You don't use an "HTML code" to input the letter "a", do you? Or maybe you do. But then, how do you enter the ampersand without using the ampersand?)

Have a ⇒. If you feel your really need a special "HTML code", you can use **⇒** instead. Right single arrow is **→** as you can see: →. Or you just use your keyboard and type →.

Congrats to your yellow belt.

I don't do belts.

]]>Oh, in fact, they invented even loops replacing recursive functions on my planet. Unfortunately we also found out that not every recursive function is implementable in a tail-recursive fashion. Moreover we found out that those functions may need local variables which are used both before and after the subroutine call.

As for your immensely smart remark re. HTML and math symbols just give me the proper HTML code for what in ASCII is typically written as "==>" (implication). Good luck.

My karate teacher once explained to a worried local politician "I don't worry about my black belts. While they certainly could kill someone they also have the quality not to do so. What I'm worried about is idiotic wannabees with a fresh yellow belt who lack both the capability to survice what they might start and the wisdom to not start it in the first place" - Congrats to your yellow belt.

]]>I see, implementing proper tail calls has yet to happen on your planet.

Sidenote: F*ck [HT|x]ML! We're in the 21. century and still have to either fight with unicode intricacies or use distastingly poor HTML symbols like a single arrow for implication.

Yeah. let's blame HTML and Unicode for the difficulties you have with your own keyboard and software. (How's it go? A poor something blames his something?)

Have a double turnstile: ⊨

]]>language classes, ladies shoes, or software, no matter, mathematical logic always holds true.

One classical problem one should avoid are the traps when transposing verbal facts (like "X is proven", "on tuesday I have a compsci test" or whatever) to mathematical propositions. In some cases, for example, it is perfectly fine to state "R is proven" as simply "R" while in other cases one should rather differentiate between "R" (R itself) and "PR" ("R is proven").

Let me put it in a (well known) funny way:

A politician, a doctor, and a mathematician are having a walk in the beautiful scottish hills when the politician spots a black sheep in a small valley and says "Funny. The scottish sheep are all black". The doctor remarks "well, there are indeed some black sheep in Scottland". Finally, the mathematician dryly states "It can be considered proven that in at least one small valley in this region of Scottland at least one sheep has at least one black side".

More background on Godel-Lob system of provability logic: http://plato.stanford.edu/entries/logic-provability/

Sorry whatever letters or symbols I use I'll clash with some animal or another in that zoo.

]]>Lol nothing to do with the *language classes* R, PR, or PH.

"Clearly if R is provable but there is no proof of !R, then R is false..."

I assume you mistyped? Let me suggest to properly introduce, discern, and use "being provable", "being proven", in particular "¬R" and the binding of R proven.

Otherwise your proposition is hard to discuss and invites misunderstandings like, for instance:

Let PR be the fact of R being proven. PR and ¬¬PR → ¬ PH = NPH. Also kindly clearly define what kind of implication (material, ...) "then" is meant to be.

"So if R is provable, then it is also provable that a proof of !R exists" is a reductio ad absurdum of your very proposition.

Moreover it seems desirable to tell us more about where you want to go with that and why.

Sidenote: F*ck [HT|x]ML! We're in the 21. century and still have to either fight with unicode intricacies or use distastingly poor HTML symbols like a single arrow for implication.

]]>First, references:

Baker, Gill, and Solovay http://epubs.siam.org/doi/abs/10.1137/0204037

Hartmanis and Hopcroft http://dl.acm.org/citation.cfm?id=1008336

Second, a proof for one of my claims in the last paragraph of my post above. For brevity let Q be the proposition that P == NP and R be the proposition that P^{H} == NP^{H}.

Clearly if R is provable but there is no proof of !R, then R is false, by definition of the oracle H. So if R is provable, then it is also provable that a proof of !R exists, which would be an inconsistency in our system of logic and set theory, (say ZFC.) But in such a case we have also proven H to be a finite set, yielding a proof of the unrelativized proposition Q. Likewise with any proof of !R.

Now suppose it is the case that Q and !R. Clearly H is infinite in this case, and therefore R is provable but false. But then Q is both provable and true. Likewise if !Q and R, then !Q is both provable and true. Therefore if the propositions Q and R are not of equivalent truth value, then if either Q or !Q is provable, it must be true. However by Lob's theorem, if we are able to prove a proposition assuming that some proof for it already exists, then our proof still suffices without this assumption. The only remaining case is when Q and R are equivalent. A proof of Q or !Q in this case only (i.e. a proof of Q || R or of !Q || !R) therefore suffices as an unqualified proof of P == NP or P != NP.

I have more to say concerning possible proofs of R or !R but that shall have to wait for another post.

]]>I object. For different reasons, among them:

Scheduling algorithms are exponential or super-exponential (and typically handled by CLP). While trivial (low |P|) networks can be solved in very little time, they rapidly go towards NP. Additionally, there is almost always a need for optimization, i.e., for instance, that train A shall, staying within a rather tight speed range, meet train B no more than x minutes before or after B is scheduled to arrive. Additionally this optimization is not singular but for a significant number of network nodes.

You make an assertion without any proof, even without so much as a lemma.

Baker, Gill, Solovay made a very interesting step but there is much more behind it. Moreover the whole field isn't exact. Note, for instance Kolmogorov's "almost sure" being simply taken as probability 1.

Moreover the role of an oracle is to be properly understood. One might say, and basically does in crypto, from a pragmatic perspective that an oracle stands for the maximal opponent the current state of science can reasonably imagine.

More pragmatically speaking "NP" is mathematically irrelevant (for the given problem domain of crypto) as it is anyway not a fixed point but a region whose borders may move to a degree. In other words: What was NP for all practical purposes in, say, 1985 may be not NP today. And there I speak sloppily because that doesn't mean that that algorithm somehow magically converted into polynomial time; it merely means that its NP properties are within computational reach today.

While P vs NP is understandably regarded as attractive for many mindgames we should be careful for two reasons:

a) In the field of crypto it is often used in a sloppy way to simply mean "well within or way beyond current (and assumed near future) computationaly reachabilty".

b) computability theory != complexity theory. Moreover complexity theory still is in rather early stage; this can be seen by many cases of approximation being used.

Practically the whole issue is a non issue as we will simply continue to increase our defenses by increasing insanely big numbers to even more insanely big numbers while the attackers will increase computational power.

As you mentioned "recursive": Kindly note that while mathematicians usually find recursive functions lovely and elegant they are also one of the more hurting examples of the difference between the mathematical realm and the system realm. In math I can find it elegant and simple to make statements about 3000000 digits factorials; On a system, however, the world production of RAM isn't sufficient for the needed stack.

]]>There exists a primitive recursive oracle H such that x € H if and only if

[≤x] P^{H} != NP^{H} but not [≤x] P^{H} == NP^{H} and x € A

or

[≤x] P^{H} == NP^{H} but not [≤x] P^{H} != NP^{H} and x € B,

where I use the notation "[≤x]" to denote the existence of a valid proof with Godel number not exceeding x for the proposition immediately following, and A and B are the recursive (in fact primitive recursive) oracles that Baker, Gill, and Solovay demonstrated such that P^{A} == NP^{A} but P^{B} != NP^{B}.

To prove this, note that every primitive recursive language is effectively constructible by a primitive recursive algorithm, every one of which has a Godel number. There is a primitive recursive function which given the Godel number of a (p.r.) algorithm that decides a language H, returns the Godel number of a p.r. algorithm that computes a function of x yielding the truth value of the proposition after the "if and only if" in the definition above. A fixed-point theorem guarantees the existence of a fixed point for this function satisfying the "if and only if" in the definition above.

(This oracle is very similar to one introduced by Hartmanis and Hopcroft.)

My claims are that neither P^{H} == NP^{H} nor P^{H} != NP^{H} is provable without proving one's entire system of logic inconsistent, and that proving the corresponding unrelativized proposition (P == NP or P != NP) is both necessary and sufficient to prove one of these.

The "security theater" that is often mentioned and the groping and raping which Clive Robinson mentioned above are generally part of a pro-prostitution shakedown and intrusion into personal privacy that goes under the guise of anti-terrorism.

They aren't looking for "terrorists" which by all accounts are few and far between for all this theater. They are looking for men with money and loose morals, and no, they aren't looking to arrest *them* per se, but to provide them with other "services" in exchange for their money.

Those who interfere with and expose this business of prostitution and human trafficking on the other hand are subject to arrest and prosecution under the post-9/11/2001 new world order.

Open up MS Word and type the letters "NYC" in all caps, select that text, and change the font to "Wingdings." You will see a skull and crossbones, a star of David, and a thumbs-up symbol. Put together, this symbolizes rejoicing over dead Jews. This is the true terrorist agenda which was planned long ago ... and all too many of those who claim to fight against it are actually fighting for it.

]]>For the forthcoming kids' vacation decided against a US trip - many other places are far more welcoming.

Will only come for business trips.

]]>As objectionable as Internet snooping might be, it's nothing as compared to being accosted by someone with a badge and a gun simply for looking brown. . . er. . . "suspicious."

Internet snooping is annoying. An encounter with someone with a gun can get you dead.

Of the two, I'll take annoying any day.

]]>Personally, I started liking rail transport when I was commuting to work. Even with the occasional breakdown and mishap, the train was on the whole a far better proposition than getting stuck in commute traffic for hours a day, virtually every day.

I've taken one longer trip by rail from San Diego to San Luis Obispo and my only complaint was that it was about five six hours longer than if I had driven and there was no easily available car rental when I got to my destination.

I wish that there was more service. . . and faster trains.

I'm still cautiously rooting for our California "bullet train" project.

]]>it's a money pit, and has been that way for decades. not a big fan of rails, and most privatization efforts seemed to have been focused on the fact that govt grants won't stop aftwards. I don't see the advantage privatization when the govt still got to bail them out as the last line of defense, and this isn't limited to the service sector.

Since a recent noted economics pundit said that

the US rail system would embarass Romania, maybe

the edition of 1950s eastern europe security is

a step upwards into the glorious future.

However I expect to be regaled with horrors by

the elected officials about the orgy of public

infrastructure expenditures as we see the same

stuff crumbling from neglect around us.

You'd think that the Net would be better at

exposing the real numbers to a wider audience.

How about spending a few tax dollars on the

infrastructure and a few less on bombs and

crooked bankers ? That might make us all a bit

safer in the long run.

You want improved railway safety, build a triple

mainline track system and maintain it properly.

-- And it didn't matter what race or color --

You sure about that? I'm pretty sure that Google and Amazon at least have a pretty good idea of my ethnic makeup. OK, maybe not as good as if I ever visit ancestry.com or 23andme, but enough for their profiling, and I am _damn_ sure they share their guesses with at least a couple agencies, who may very well politely request an "enhanced user experience" for those who tick the right (wrong?) boxes

]]>If you surf the web, so happens that you are/were, you've already been stop-and-frisked a godzillion times. And it didn't matter what race or color. :)

]]>> Hopefully the US citizenry will wake up to this thieving maw of an apparently insatiable guard labour beast that --endlessly bites the hand that feeds it and just to add insult to the injury-- defecates all over --all but a few of the citizenry--, befor the citizenry is smothered.

I couldn't help but read this in the voice of Zack de la Rocha, initially whispered, escalating to screaming "the citizenry is smothered" over and over again as the the crazy Tom Morello solo kicks off.

]]>And thus the US lose tourist and visitor dollars, that often gave the poorest of US workers a job, self respect and an income even if meager. But importantly their earnings unlike those of the top 1% paid into the communal pot of taxation giving emoloyment to others including those who educate the next generations.

Thus the US beggers it's self in oh so many ways, because that 1% know what the lie behind the "trickle down" effect is, after all they did pay the idiots that gave rise to the myth they so easily benifit by at every one elses expense.

But those greedy US agencies, with their secret tax take and unacountability, that pass your hard erned tax dollars back to the 1%, what do they actually do... After all "terrorism still happens" so I'm guessing that's not where the money is going other than in name.

Hopefully the US citizenry will wake up to this thieving maw of an apparently insatiable guard labour beast that --endlessly bites the hand that feeds it and just to add insult to the injury-- defecates all over --all but a few of the citizenry--, befor the citizenry is smothered.

But I somehow think it will not happen in this generation or the next, without the blood shed on notable American once noted was rrquired to refresh the tree of liberty.

]]>Here Here! I agree! The TSA has successfully dissuaded me from traveling as much by air as I used to as well. Mission accomplished. The skies are much safer now from the burden of my money being spent on them.

]]>Even if these ads are somebody's art project, I can fully imagine Amtrak using sane security measures as a marketing tool. For people with looser travel schedules, the lack of airport "security" is an easy sell.

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