Re: The seven step series
Bruno Marchal wrote: Ouh la la ... Mirek, You may be right, but I am not sure. You may verify if this was not in a intuitionist context. Without the excluded middle principle, you may have to use countable choice in some situation where classical logic does not, but I am not sure. Please see http://en.wikipedia.org/wiki/Countable_set the sketch of proof that the union of countably many countable sets is countable is in the second half of the article. I don't think it has anything to do with the law of excluded middle. Similar reasoning is described here http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2008;task=show_msg;msg=1545.0001 My opinion on choice axioms is that there are obviously true, and this despite I am not a set realist. OK, thanks. I am glad, nevertheless that ZF and ZFC have exactly the same arithmetical provability power, so all proof in ZFC of an arithmetical theorem can be done without C, in ZF. This can be seen through the use of Gödel's constructible models. I am sorry, but I have no idea what might an arithmetical provability power refer to. Just give me a hint ... I use set theory informally at the metalevel, and I will not address such questions. As I said, I use Cantor theorem for minimal purpose, and as a simple example of diagonalization. OK. Fair enough. I am far more puzzled by indeterminacy axioms, and even a bit frightened by infinite game theory I have no intuitive clues in such fields. Do you have some links please? Just to check it and write down few new key words. Cheers, Mirek On 01 Sep 2009, at 14:30, Mirek Dobsicek wrote: The reason why I am puzzled is that I was recently told that in order to prove that * the union of countably many countable sets is countable one needs to use at least the Axiom of Countable Choice (+ ZF, of course). The same is true in order to show that * a set A is infinite if and only if there is a bijection between A and a proper subset of A (or in another words, * if the set A is infinite, then there exists an injection from the natural numbers N to A) Reading the proofs, I find it rather subtle that some (weaker) axioms of choices are needed. The subtlety comes from the fact that many textbook do not mention it. In order to understand a little bit more to the axiom of choice, I am thinkig if it has already been used in the material you covered or whether it was not really needed at all. Not being able to answer it, I had to ask :-) Please note that I don't have any strong opinion about the Axiom of Choice. Just trying to understand it. May I ask about your opinion? Mirek Bruno Marchal wrote: Hi Mirek, On 01 Sep 2009, at 12:25, Mirek Dobsicek wrote: I am puzzled by one thing. Is the Axiom of dependent choice (DC) assumed implicitly somewhere here or is it obvious that there is no need for it (so far)? I don't see where I would have use it, and I don't think I will use it. Cantor's theorem can be done in ZF without any form of choice axioms. I think. Well, I may use the (full) axiom of choice by assuming that all cardinals are comparable, but I don't think I will use this above some illustrations. If you suspect I am using it, don't hesitate to tell me. But so far I don't think I have use it. Bruno --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step series
Hi Bruno, I am puzzled by one thing. Is the Axiom of dependent choice (DC) assumed implicitly somewhere here or is it obvious that there is no need for it (so far)? Thanks! mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step series
The reason why I am puzzled is that I was recently told that in order to prove that * the union of countably many countable sets is countable one needs to use at least the Axiom of Countable Choice (+ ZF, of course). The same is true in order to show that * a set A is infinite if and only if there is a bijection between A and a proper subset of A (or in another words, * if the set A is infinite, then there exists an injection from the natural numbers N to A) Reading the proofs, I find it rather subtle that some (weaker) axioms of choices are needed. The subtlety comes from the fact that many textbook do not mention it. In order to understand a little bit more to the axiom of choice, I am thinkig if it has already been used in the material you covered or whether it was not really needed at all. Not being able to answer it, I had to ask :-) Please note that I don't have any strong opinion about the Axiom of Choice. Just trying to understand it. May I ask about your opinion? Mirek Bruno Marchal wrote: Hi Mirek, On 01 Sep 2009, at 12:25, Mirek Dobsicek wrote: I am puzzled by one thing. Is the Axiom of dependent choice (DC) assumed implicitly somewhere here or is it obvious that there is no need for it (so far)? I don't see where I would have use it, and I don't think I will use it. Cantor's theorem can be done in ZF without any form of choice axioms. I think. Well, I may use the (full) axiom of choice by assuming that all cardinals are comparable, but I don't think I will use this above some illustrations. If you suspect I am using it, don't hesitate to tell me. But so far I don't think I have use it. Bruno --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Learning binary numbers
Hi Marty, thanks a lot for your reply. I was really interested in whether the lesson would work with you. I had the pleasure to teach the binary arithmetic to kids in summer school camps, in the grammar school and to university students as well. Some kids/students got it quite easily some did not. And then, recently, I have read that socratic article. Since I really like and enjoy teaching, I spent some time analyzing the article and one of the points I realized is that there was always some sort of satisfaction and accomplishment at each step taken. And you said you was missing these feelings during the introduction to the set theory. So if you have said that you got hooked-up to math by that socratic method... Bruno would definitely took the hint in his seveth step serii. Cheers, Mirek m.a. wrote: Mirek, My previous answer to your question was glib and evasive, I apologize for that, but I think your question was misleading as well. In an attempt to be kind, you asked my opinion, from a pedagogical POV, of a lesson designed to make binary arithmetic simple enough for third-grade students. I think that what you really wondered was whether the lesson would work with a math-challenged adult like me. So, because I believe you to be genuinely and unmaliciously curious and in the interest of science, I'll try to describe my experience of this lesson. Did it teach me about binary numbers? The answer must be Yes and No. As I studied the lesson step by step, I understood each point and felt at the end, that I had a solid grasp of the topic. Will it become part of my general knowledge? /No/. Will I remember it tomorrow? /No/. Why not? Because my sorry excuse for a brain won't try to absorb it; in fact it will try strenuously to forget it. I can think of several reasons for this. 1) I won't leave the safety and familiarity of base ten. After a lifetime of base-ten, base-two is disorienting and disturbing. If I were forced to live in a house with pyramidal rooms, I could do it; but as soon as I was released, I would return to a cubical house. Someone who is shaky in math to begin with, clings to the part that he finds to be solid and doesn't venture into the whirlwind of incomprehensible artifacts outside. 2) The space in my head set aside for mathematics is entirely occupied by base-ten. I use it constantly and value it as a trusty tool. I can see no way, since I don't design computers, that binary can be useful in my everyday life. 3) This is purely subjective, but perhaps worth mentioning. Binary arithmetic seems to me like a language of ants. I am not an entomologist or even a biologist. I don't want to know what the ants are saying. I /do/ want to know what the Russians and Italians and Spanish are saying and I study their literatures. My mind accepts and always finds more room for information about these languages even as it refuses/ /to accommodate binary. I know that computers and the modern world could not exist without the ants and I am grateful for all of it. But I am resigned to the sad fact that their language will always be inaccessible to me. Hope this helps, marty a . - Original Message - From: Mirek Dobsicek m.dobsi...@gmail.com mailto:m.dobsi...@gmail.com To: everything-list@googlegroups.com mailto:everything-list@googlegroups.com Sent: Saturday, August 22, 2009 11:05 AM Subject: Re: The seven step series m.a. wrote: a towel into the ring. I simply don't have the sort of mind that takes to juggling letters, numbers and symbols in increasingly fine-grained, complex arrangements. [...] Marty, If I can ask, I'd be really interested what do you think of this socratic experiment http://www.garlikov.com/Soc_Meth.html http://www.garlikov.com/Soc_Meth.html Cheers, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step series
m.a. wrote: a towel into the ring. I simply don't have the sort of mind that takes to juggling letters, numbers and symbols in increasingly fine-grained, complex arrangements. [...] Marty, If I can ask, I'd be really interested what do you think of this socratic experiment http://www.garlikov.com/Soc_Meth.html Cheers, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step series
3) compute { } ^ { } and card({ } ^ { })
If card(A) = n, and card(B) = m. What is
card(A^B)?
I find it neat to write | {} ^ {} | = | { {} } | = 1 :-)
It's almost like ASCII art. Just wanted to signal that I'm following.
mirek
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Re: The seven step series
Come on Mirek: Theaetetical is an adjective I have forged from Theatetus. Theatetus gives 195.000 results on Google. Theatetus wiki 4310. Of course, after all you reference the dialogue Theaetetus in your papers thus one can easily match the word Theaetetical agains it. Let me quickly summarize the experience I had with theatetical notion of knowledge while reading one of your papers for the first time. Maybe I am an ignorant, then shame on me, but I have not read the Theaetetus. So I took a look at the Wikipedia and read In this dialogue, Socrates and Theaetetus discuss three definitions of knowledge: knowledge as nothing but perception, knowledge as true judgment, and, finally, knowledge as a true judgment with an account. Each of these definitions are shown to be unsatisfactory. Hmm that really helps .., I told to myself and continued with reading. With an uneasy feeling of stepping into the water I eventually settled down to conclusion that you likely mean something as true justified belief. I really wished you wrote it more straightforwardly without turning your readers quite unnecessarily down to the Theaetetus and inventing new words such as Theaetetical. Anyway, I'd like to stop discussing this issue :-) since my only point was to give you a hint why I said that it is not easy to read your papers/letters. Feel free to ask for any clarification, position adjustments, question, at any level ...Do you understand what is the comp hypothesis? Let us see if I get it right. Your comp hypothesis is 1) I'm a machine, 2) Each possible computation is Turing-computable, 3) Natural numbers and their relations do exist. This should not be confused with other quite common comp hypothesis that the universe is a big computer. This hypothesis entails the existence of a physical computer. Ad 1) I take the position that I is only a convenient temporary pointer to a part of universe. The pointer Socrates' thoughts is of the same quality. Ad 2) Breath taking. While 1) and 3) are assumptions of the kind OK, let's think for a while that ..., 2) has the status of a thesis. I don't have any firm position on what could an objective reality be (and without a justification I tend to think it is inaccessible to us), but if there is any objective reality, 2) could be a statement about it. Ad 3) If natural numbers and their relations are the only entities which do exist then me, you, everything is a recipe of a Turing-computable number. OK, that is it. This is how I understand to your starting assumptions. Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step series
Hi Bruno, Bruno Marchal wrote: Hi Mirek, Long and perhaps key post. Thank you a lot for a prompt and long reply. I am digesting it :-) Just some quick comments. There is no shame in being ignorant. Only in staying ignorant :) I've ordered the dialogue from a second-hand book shop :-) The Stanford encyclopedia says Arguably, it is his (Plato) greatest work on anything. So I'll give it a try :-) judgment, and, finally, knowledge as a true judgment with an account. The remarkable thing is that if you accept to modelize account by sound machine provability, This is probably the key problem for me. I know next to nothing about provability, the logic of provability, PA/ZF provers. I know that quite often you reference Boolos 1993 - The Logic of Provability. I took a look at it at Google Books preview but ... there is something missing in my education. From the beginning I am puzzled with Why?, what?. What a headache :-) In french students are burned alive if they dare to create new adjective, and I thought that in English we have more freedom, but I may be wrong. Sorry. I'd grant this freedom to rational native speakers only :-) x divides y if and only if it exists a number z such that y = x*z. I don't dare to correct your english but there is/exists a number ... is what I would write. Ad 3) If natural numbers and their relations are the only entities which do exist then me, you, everything is a recipe of a Turing-computable number. No. Not at all. Sorry. Gosh, you will be very surprised if you follow the UDA-7. On the contrary. Arithmetical truth VASTLY extends the computable domain. Most relations between numbers are not Turing emulable. Aha! Then I really have a wrong mental picture of your work. I understood to arithmetical realism along the lines of this quotation from the Stanford article on realism: According to a platonist about arithmetic, the truth of the sentence '7 is prime' entails the existence of an abstract object, the number 7. This object is abstract because it has no spatial or temporal location, and is causally inert. A platonic realist about arithmetic will say that the number 7 exists and instantiates the property of being prime independently of anyone's beliefs, linguistic practices, conceptual schemes, and so on. So I thought that you essentially take a) Numbers and their properties and relations exists. b) Now, since you don't assume existence of anything else = your body, your bike and coffee must emerge as patterns in the world of numbers. c) Taking the Church-Turing thesis, these patterns are Turing-computable. d) Definitely, the vast majority of all patterns is not Turing-computable. This is how I have thought about your working framework. Notice, that I don't talk about what you try to show, argue for, want to end up with etc. Cheers, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step series
I am in a good mood and a bit picky :-) Do you know how many entries
google gave me upon entering
Theaetetical -marchal -bruno
Well 144?
Good way to find my papers on that. The pages refer quickly to this
list or the FOR list.
I am sorry for the delay, I've just got back from my vacation.
Hmm. The above written search should not return any references to your
papers/letters as the minus sign in front of your name asks for an
exclusion.
Given that it works as supposed google then gives only 1 hit in my
location (Sweden). That hit is a translation of the word Theaetetical
into some eastern characters. Thus, I end up with zero meaningful hits
and a feeling that you might be the only one using this word.
That makes me insists a little bit more (in a very polite way) that,
occasionally, your work is
difficult to read unless one is willing to undertake long
discussions, clarifications and position adjustments.
I am writing this in a reference to your complains that sometimes you
have troubles to get enough relevant feedback to your work.
I let those interested to meditate on two questions (N is {0, 1, 2, 3,
4, ...}):
1) What is common between the set of all subsets of a set with n
elements, and the set of all finite sequences of 0 and 1 of length
n.
2) What is common between the set of all subsets of N, and the set of
all infinite sequences of 0 and 1.
Just some (finite and infinite) bread for surviving the day :)
I am going to catch up with the thread ...
Cheers,
mirek
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universal quantum TM
There has been progress in the direction of finding fully universal quantum Turing machine. Construction of a universal quantum computer Antonio A. Lagana, M. A. Lohe, and Lorenz von Smekal Physical Review A 79, 052322 (2009) (11 pages) http://link.aps.org/doi/10.1103/PhysRevA.79.052322 I'll read it and we can discuss then. Regards, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step series
Hi Bruno, I am in a good mood and a bit picky :-) Do you know how many entries google gave me upon entering Theaetetical -marchal -bruno Mirek for some people I think. It is just unusual. theorem and the Theaetetical definitions of knowledge. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step series
Hi Bruno, I'd like to let you know that I'm following the serie of your letters. While I have the background you are covering right now, I still enjoy your insights. I joined the list like two years ago and from that time I've read most of your key papers. Honestly, it is not the easiest stuff to read style-wise. You try to precise, define well, etc. yet it cannot really be compared to the quality of, let us say, Physical Review Letters and alike articles. In my opinion, that is why it is hard to either agree or disagree with your thesis. I can imagine that right now you are tempted to write something along the lines a\ I just propose to take Church thesis seriously b\ All I ask you is 'Do you say yes to the doctor? While valid proposal and question, there is really not much to agree with/disagree with/critize unless one is willing to undertake long discussions, clarifications and position adjustments. Anyway, your papers and letters are really a great source of ideas to think about and that is exactly what I do. From the day one on the list I keep myself busy with the question of Why should I believe in the Church thesis (you see, I don't write Why do I ...). I've got into the writing of Bernard Bolzano (I consider his work cruicial in order to keep an open mind about the Cantor diagonal argument) .. - and now back to the beginning of my letter - Bolzano (Cantor), your insights and thinking about alternatives at any moment make me pretty happy. Thanks! Mirek PS: I'd love to read a book by Bruno Marchal. Bruno Marchal wrote: Hi all, I suddenly feel sorry putting too much burden on just one correspondent in the list, and I would appreciate if someone else could propose answers or any remarks to the exercises. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
On Unruh effect
This blog post http://blog.sigfpe.com/2009/04/faster-than-speeding-photon.html outlines the basics about the Unruh effect, if there is any. From Wikipedia: The Unruh effect is the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none. You might like to think about how does this fit into your theories. For myself, it gives a compelling reason why one should avoid explaining things using the wave-particle duality concept. Cheers, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Wolfram Alpha
Günther Greindl wrote: Kim, great post, thanks! I second that! cheers, mirek Kim Jones wrote: Let's keep it simple. Schools and universities (globally identifiable as 'the education industry') have traditionally fulfilled the role of fountains of knowledge. .. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The Seventh Step 1 (Numbers and Notations)
I'm sorry but I can't resist to paste this short conversation between Lord Blackadder and his servant Baldrick. Maybe you know this british blackadder comedy. If you teach: III and III mean 3 and 7, then you said nothing, just named them. That was my point. To talk on notation. I just hope people understand enough the number so that if I ask them to give me 3 euros, they will not give me two or four. - Baldrick, if I have 2 beans and then I add 2 more beans, what do I have? - Some beans. - Yes... and no. Let's try again shall we? I have 2 beans, then I add 2 more beans. What does that make? - A very small casserole. - Baldrick, the ape creatures of the Indus have mastered this. Now try again. 1, 2, 3, 4. So how many are there? - 3 - What?! - And that one. - 3.. and that one. So if I add that 1 to the 3 what will I have? - Oh! Some beans. - Yes.. To you Baldrick the renaissance was just something that happened to other people wasn't it? mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
I would certainly like to read the book - I managed a bit the Lille thesis (with my French), but it was hard going and I think I only understood the stuff because we have had many discussions here on the list - so it was easy to translate. I am not so sure I can manage the huge Bruxelles work, but I will try someday when I have more time :-)) Maybe you can find a publisher who is prepared to translate the book into english? Excellent idea. For reason I don't want to bore you with, I am a bit stuck on those sort of issue. But I am sure a good publishing could make rich the publisher :) I've also tried to dig through both Bruno's thesis with the help of google translator. It works for a while but soon one hits a wall with a difficult sentence/paragraph which is hard to understand even if it stands as the author inteded - and extra hard to understand if its meaning is corrupted by the translation. Bruno, I'd love to read your thesis in english, but I fully understand how hard it must be to get a good translation that you would be happy with. At the end, it might be easier to start from scratch, take the essential from both thesis, update a little bit and write a book in english on your own directly. Is that an option for you? Cheers, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Goldblatt 1993, Mathematics of Modality this book is available online: http://standish.stanford.edu/bin/detail?fileID=458253745 mirek Goldblatt, Mathematics of Modality http://www.amazon.com/Mathematics-Modality-Center-Language-Information/dp/1881526240/ref=sr_1_1?ie=UTF8s=booksqid=1232402154sr=8-1 (the book contains the full paper) Not only that! It contains also his paper on the arithmetical intuitionist, alias the arithmetical knower, alias the universal first person, alias the arithmetical interpretation of Plotinus' third hypostase (the universal soul), alias the epistemical temporal arithmetical modal logic S4Grz (pronounce: S four Grzegorczyk). --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: QM Turing Universality
My question has perhaps no sense at all. Is there a notion of quantum computation done without any measurement? Quantum lambda calculus by Andre van Tonder does not containt measurement. http://arxiv.org/pdf/quant-ph/0307150v5 From the abstract, he proves equivalence between his quantum lambda calculus and quantum Turing machine (also without measurement). That's all I know in this respect for the moment. Is there a purely unitary transformation which augment the dimensionality of the initial quantum machine. Does the notion of universal quantum dovetailing makes sense. I am not too familiar with the process of dovetailing, but I'm fine with the general idea that there is program which systematically generates every possible C/Lisp code and in between steps of this generation it interprets parts of what is already generated. Can you sketch how should one think about such dovetailing in terms of classical logical gates, please? I don't find my Shi papers, but from what I remind, it gives some good argument about the difficulty of redefining the halting problem (halting in which universe? ...). Good, your note about the halting problem helped to refine my google search to the extend that I've found the Shi paper you are talking about. Hereby, I also apologize to the authors of QTM Revisited paper, their reference was correct. http://dx.doi.org/10.1016/S0375-9601(02)00015-4 I'll read it. Regards, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: QM Turing Universality
Hi Bruno,
I have finished the reading of the paper I mentioned (Deutsch's
Universal Quantum Turing Machine revisited) and I see they have very
similar problems, probably better described.
I finished a rather careful reading of that paper (QTM revisited) too,
http://arxiv.org/pdf/quant-ph/0701108v1
and if I got it right the main authors' point is:
Claims:
1) An Universal Probabilistic Turing Machine (PTM) can simulate the set
of PTMs with computable transition probabilities EXACTLY.
2) An Universal QTM can simulate the set of QTMs with computable
amplitudes only approximately.
Conclusion:
The notion of universality for Quantum TM is not of the same kind that
we have for Determinictic TM and Probabilistic TM.
Well, the first claim is correct and the corresponding algorithm for an
EXACT simulation is very simple. I think you know this well, but for the
sake of having a good reference, see for example Lemma 7.14 in
http://www.cs.princeton.edu/theory/complexity/bppchap.pdf
A tricky point, of course, is that in order to achive an EXACT
simulation your algorithm will potentionally never stop. For example,
trying to achieve ouput probability P=1/3 using UPTM with transitional
probabilities {0,1/2,1} is exact only in the limit.
In practice, such an EXACT simulation is not needed, and people prefer
to say that one machine CAN simulate other machine if properties in
question can be approximated with ARBITRARY accuracy. Yes, and it should
be reasonably fast. Typically the penalty for a better accuracy is
upper-bounded by a polylog factor.
Regarding the second claim, it is not true to my knowledge.
Approximation of amplitudes is a convergent process - set your accuracy,
suffer polylog slowdown factor, done. Wanna go to the limit, you get an
exact simulation.
The paper mentions (but
does not tackle) an old problem already described by Shi 2002, which
made me think at the time that the notion of Universality is a bit
dubious in the quantum realm.
I don't know which problem do you mean. In the QTM Revisited paper, the
authors do not suply a valid reference, the paper they refer to does not
exist and they don't get right even the first name of Dr. Shi.
Thus I may only assume that you/the authors refer to this paper
http://arxiv.org/pdf/quant-ph/0205115v2
I have read this paper few years ago and after a quick today's scan I'm
not aware of some explicitly described problem. On the contrary, the
message of the paper is that it is 'easy' to find universal set of
quantum gates (given that you start, for better or worse, from classical
universal set of gates).
To sum up: is there a (never stopping) quantum counting algorithm? I
think I can build a Quantum UD from it, well in case the Shi problem
is not too much devastating.
But here, and now, I got a feeling there is just no quantum counting
algorithm ...
Please be more specific about what do you mean by a quantum counting
algorithm. Sometimes I'm not too bright guy :-)
Is this what you mean?
step 1\ |0
step 2\ |0 + |1
step 3\ |0 + |1 + |2
or (a classical machine operated by quantum means)
step 1\ |0
step 2\ |1
step 3\ |2
or something different :-)
Best,
mirek
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Re: MGA 2
Hello Bruno, I think you are correct, but allowing the observer to be mechanically described as obeying the wave equation (which solutions obeys to comp), Hmm well if you have a basis, yes; - but naked infinite-dimensional Hilbert Space (the everything in QM)? You put the finger on a problem I have with QM. I ill make a confession: I don't believe QM is really turing universal. The universal quantum rotation does not generate any interesting computations! Could you please elaborate a bit on the two above sentences. I am missing a more context to understand where really points to. And with the second sentence, I simply don't understand it. I am open, say, to the idea that quantum universality needs measurement, and this could only exists internally. So the naked infinidimensional Hilbert space + the universal wave (rotation, unitary transformation) is a simpler ontology than arithmetical truth. Yet, even on the vaccum, from inside its gives all the non linearities you need to build arithmetic ... and consciousness. Cheers, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: QM Turing Universality (was: MGA 2)
Thank you for a quick answer! I'll take a look at it, my curiosity approves additional items on my TODO list :-) Best, mirek The classical universal dovetailer generates easily all the quantum computations, but I find hard to just define *one* unitary transformation, without measurement, capable of generating forever greater computational memory space. Other problems are more technical, and are related to the very notion of universality and are rather well discussed in the 2007 paper: Deutsch's Universal Quantum Turing Machine revisited. http://arxiv.org/pdf/quant-ph/0701108v1 I could relate this with technical problem with the BCI combinator algebra, that is those structure in which every process are reversible, and no cloning are possible (cf the No Kestrel, No Starling summary of physics(*)). Those algebra are easily shown being non turing universal, and pure unitarity seems to me to lead to such algebra. Could you implement with a quantum computer the really infinite counting algorithm by a purely unitary transformation? The one which generates without stopping 0, 1, 2, 3, ... That would already be a big help. Bruno (*) Marchal B., 2005, Theoretical computer science and the natural sciences http://www.sciencedirect.com/science?_ob=ArticleURL_udi=B75DC-4GX6J45-1_user=532047_coverDate=12%2F31%2F2005_rdoc=1_fmt=_orig=search_sort=dview=c_acct=C26678_version=1_urlVersion=0_userid=532047md5=e087a268f1a31acd7cd9ef629e6dc543, Physics of Life Reviews, Vol. 2 Issue 4 December 2005, pp. 251-289. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: UDA paper
Hi Bruno! I wish you the best to you and to your girlfriend. Thank you very much. I appreciate your wish. offered to you through the windows, like it happened to a friend of mine (and she threw the computer with!). I reassure you: I think that was exceptional! Presently I am not so lucky because I have been break in yesterday, and my home computer has been stolen with all the attached devices including the main backup disk. I will have to rewrite hundreds of unfinished papers... . Sorry to bother you with that, actually. Ohh, that is not good. I am sorry for you. Hopefully you can recover at least parts of the most important things. Sincerely, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: UDA paper
Hi Bruno,
yes, I am now a bit busy. Lecturing, seminars,.. wedding planning :-)
I am somewhere in the middle your paper. Regarding the very point of the
described 1-indeterminancy, I have no problem there at all. Anyone who
ever called a fork() unix function (read, cut, duplicate) followed by an
execve(read-destination-name-from-keyboard()) function, should not have
a problem here. A program, even with considerably good self-referential
skills, has no chance to know whether I will enter Warsaw or Moscow on
the keyboard.
As I have said, I have not finished reading the paper yet. But sometime
I have a problem with a bit of feeling of circularity of arguments, or
described in better words, given assumptions A={..}, conjectures B={...}
are true, where Bs feels like rephrased As, and therefore Bs are
trivially true. No disrespect here!
It just how do I feel now. Bs are overwhelming, but As are pretty strong
assumptions, so Bs are not surprising anymore, yet an hour later Bs are
overwhelming again.
Best,
Mirek
Bruno Marchal wrote:
Hi Mirek,
I guess you are busy.
I would just like to insist that when I say (14-febr.-08):
Please note that the 1-indeterminacy I am talking about in the third
step is really a pure classical indeterminacy. It arises from the fact
that my classical state is duplicable, and then I cannot predict which
*experience* I will *feel* after a self-duplication: mainly Washington
OR Moscow (or Sidney *or* Beijing), ...
This is really a key point, if not *the* key point. I think it is
almost trivial, but sometimes some people have a problem with this. In
that case it helps to imagine the same experiment done with some
inference inductive machine in place of a human or you, and this in
an iterated self-duplication. In that case the result amount to saying
that no robot, when duplicated iteratively (in Washington and Moscow,
say) can predict its future sequence of results of first person
self-localization. This becomes equivalent with the fact that most
finite bit-strings, like WMMMWWMWWWM ... are not compressible.
Someone told me (out-of-line) that he *can* predict with certainty his
future in that situation: for example he can predict W..., but
this means he is not taking into account the saying of the other
reconstituted people, which, *assuming comp* are genuine descendant
of the original. Those people will acknowledge that their prediction
with certainty was false, and they have the same right and reason to
be taken seriously, again when we *assume* the comp hypothesis.
Have you a problem with this? I think most on this list grasp this
point, but don't hesitate to tell me if you don't. Without a clear
understanding of what happens here we can't really proceed ... (nor can
we grasp Everett formulation of QM I could argue ...).
Bruno
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UDA paper
Hi Bruno, The UDA, in english, can be found here: */The Origin of Physical Laws and Sensations/*, (Invited Talk SANE 2004). Click on that title, or copy the following in your browser: http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html (if you study it I would suggest you print the slider too, so that you could perhaps tell me which step you would find hard to go through ). I have started reading this paper. Just a quick question. At the first step of UDA it seems you restrict yourself to classical bits. That is fine. I can imagine that somebody deliberately read and cut my running computer so that the computer goes on with its job after being 'reincarnated' in Helsinky. Even the substitution level is more or less clear. Noise on transistors is definitely not important. However, at the third step you mention quantum mechanics. It is not clear to me how would you classically teleport my quantum computer. What are the read cut operations? Yes, there exists a classical Turing machine which can simulate my quantum computer, but I am not giving the running simulator to you. I don't have it. Please, make a short clarification about your framework. I might be just misinterpreting you. What is the page reference to Gruska's book? Sincerely, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Key Post 1, toward Church Thesis and Lobian machine
But thanks to that crashing, *Church thesis remains consistent*. I would just say An existence of a universal language is not ruled out. I am ok with you. Consistent (in math) means basically not rule out. Formally consistent means not formally ruled out, or not refutable. That is: Consistent(p) is the same as ~ Provable(~ p) ~ = negation like Provable(p) is equivalent with ~ Consistent( ~ p) All right... Now, let me just rephrase few points of the key post one more time. I will try to be picky with wording. Points which are not mentioned - I fill comfortable with. 1\ There is no language/algorithm/procedure/machine which can describe/execute all total computable functions. 2\ There exists non-empty set of all machine computable functions (inevitably includes both total and strict partial functions). Let us call this set MC (machine computable). 3\ Church himself *defined* the set of so-far-known-computable-functions as those computable by lambda calculus. 4\ What we use to call a *Church thesis* today says, that MC is in fact equal to the set of functions computable by lambda calculus. 5\ Church thesis is refutable. * * * Something else: to us verify MM = SII(SII) does crash the system: SII(SII) = I(SII)(I(SII)) = SII(SII) = I(SII)(I(SII)) = SII(SII) = I(SII)(I(SII)) = SII(SII) = I(SII)(I(SII)) = SII(SII) = ... (crashing). Working with SK combinators, I had a bit problems with omitted parenthesis. Also it was not clear to me what is the meaning of eg. (SKK) since S is expected to take three arguments. What helped me was the unlambda language (http://www.madore.org/~david/programs/unlambda/) So here is my crashing sequence (yep, yours is shorter) (I don't expand the I term to keep it short) SII(SI(S(KI)I)) a reference implementation in unlambda: ```sii``si``s`kii the ` stands for 'apply' operation, aka left parenthesis. with a small modification ```sii``si``s`k.ui we can achieve the computer to print u in an endless loop. .u is a special function with a side effect of printing the character u. Best, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: SUMMARY
Time for the Kleene diagonal argument. Opps, a language L that I dreamt of does not exist. I have to relax from the condition that M on E_i always return a number in a finite time. Well, what to return if not a number ... nothing - M experiences an infinite loop. What a world, ok, my language has to describe total functions from N to N and as well as strict partial functions from N to N. And it is clear that I cannot know whether E_i corresponds to a total function or a strict partial function. f' stands for any function descriable by L. 0 --- E_0 ~ f'_0 1 --- E_1 ~ f'_1 2 --- E_2 ~ f'_2 3 --- E_3 ~ f'_3 N, E and C are enumerable, moreover obviously effectively enumerable. Any subset of C is at least enumerable. A subset of C corresponding to total functions is no effectively enumerable. It cannot be. Correction: N and E are enumerable, moreover obviously effectively enumerable. C is enumerable and thus any subset of C is at least enumerable. A subset of C corresponding to total functions is not effectively enumerable. It cannot be. Neither C as such is effectively enumerable. It cannot be. Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: coffee-bar machine exercises
5) describe informally the coffee-bar language, and, choosing an order
on its alphabet, write the first 7 jobs in the lexicographical order.
The alphabet contains all symbols needed in the jobs, including
commas,
parentheses, etc. + some grammatical rules making clear that Z(23) is
a
good instruction, but 23(Z) is not, ...
Program ::= BEGIN commands END
commands::= line-no instruction comment next
line-no ::= num:
instruction ::= Z(num) | S(num) | T(num,num) | J(num,num,num)
comment ::= # text `new-line` | `new-line`
next::= commands | END
num ::= [0,1,2,3,...]
text::= [ascii text without the `new-line` character]
First 7 jobs
1)
BEGIN
0: J(0,0,0)
END
Hmmm First I strongly suggest no putting the commentaries in the
program (# ...). You just make your life harder.
Second, if you accept, like above a empty instruction (like the 8: in
your addition program above, you have to agree that
BEGIN
0:
END
is shorter, no? But I disallowed it.
But then why is not the following program
BEGIN
0: S(0)
END
shorter?
Well :-) my lexicographical ordering abilities have shown a weak point.
I give two new exercises:
1) the alphabet of the coffee-bar language (accepting your :
improvement) is
{:, (, ), J, S, T, Z, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9} (we write the
number in decimal).
Now a program, like
BEGIN
1: S(1)
END
is the same as BEGIN1:S(1)END (less readable, but it is the same
program: we are not in PYTHON where the carriage return is in the
alphabet!).
But the expression BEGIN1:S(1)END denotes a number in base 17. All
right?
Yes, it is clear.
The question is: does a program U exists which is such that if you put
n, written in base 17, output nothing on table 1 in case the number is
not a program, and output the result of the running of that program in
the other case. Would you say that such a U program is universal?
Yes. We call such a program an interpreter. And the existence of a
program called 'interpreter' is a consequence of the fact that a machine
capable of executing a universal language L, must be descriable itself by L.
What happens when we feed U with its own code? Doing something, doing
something and .. hanging ... waiting for an input.
Best,
Mirek
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Re: SUMMARY
Hi Bruno and everybody, I hope to send my comments and/or 'OK' sign :-) on Monday. Take it easy. There is no deadline on the list. Making a declaration helps me to get things done. Yet I'm late. Whenever you see such sentences in my posts, you can skip it, they are mostly for me :-) --- I'll try to write a summary in my own words. Let's see how much I did understand. Prepositions: A finite alphabet A* finite words over A (it is enumerable, moreover effectively enumerable) L . a language over A. E a subset of A*, a set of valid expressions in L (obviously, it is at least enumerable) M . a machine which understands to L f . a total function from N to N. Goal: I want to develop a universal language L which describes all and only all functions f. Given an expression from E, M computes the result in finite time. Given the restrictions on L, the result is a number and nothing else. The set of all functions f from N to N is not enumerable (by Cantor's diagonal). Thus there is no bijection to E. Thus, I have to find a smaller set of functions. I will call this set a set of computable functions, C. Inevitably, this is computability by definition, by definition of L. So L should be 'really good' in order to encompass as much functions f from N to N as possible. Now, I think of a bijection between E and C. 0 --- E_0 ~ f_0 1 --- E_1 ~ f_1 2 --- E_2 ~ f_2 3 --- E_3 ~ f_3 Since E are efficiently enumerable, C are efficiently enumerable as well. Yes, it might happen that f_i = f_j for i != j, but is does not matter as long as all unique f_i are in the enumeration. Time for the Kleene diagonal argument. Opps, a language L that I dreamt of does not exist. I have to relax from the condition that M on E_i always return a number in a finite time. Well, what to return if not a number ... nothing - M experiences an infinite loop. What a world, ok, my language has to describe total functions from N to N and as well as strict partial functions from N to N. And it is clear that I cannot know whether E_i corresponds to a total function or a strict partial function. f' stands for any function descriable by L. 0 --- E_0 ~ f'_0 1 --- E_1 ~ f'_1 2 --- E_2 ~ f'_2 3 --- E_3 ~ f'_3 N, E and C are enumerable, moreover obviously effectively enumerable. Any subset of C is at least enumerable. A subset of C corresponding to total functions is no effectively enumerable. It cannot be. Well, a language L which describes both total and strict partial functions and hereby defines a set of computable functions is not refuted by Kleene's diagonal. It is not obvious to me how strong sort of evidence that universal L exists, it is to survive Kleene's diagnonal. Droping an apple on the floor is in favor of Newton's law but not very convincing :-) Oh, now I realize that my question is actually weird. Since the set of computable functions is defined by L, and L is said to be universal if it describes exactly these functions - it is simple to develop a trivial L - it defines a set of computable functions ... and of course universal L exists. In this sence a universal language L always exists. So I write it rather like this: If we develop many sufficiently different programming languages and they turn out to be all equivalent, it might convince me that the set of 'computable functions' is fixed. Although, written like this I can think of educated (math) people who will tell me: This is all you have So, what are like 2-3 most direct consequences of CT which make CT to seem rock-solid? Here I assume that CT basically says that the set of functions descriable by the lambda calculus is all what I can ever compute. - Regarding points 3)-5) of your summary, I am lost on terms such as Absolute ontic TOE, Observer Moments, Aristotelian principles, Machine theology, ... - I wrote down a list of short-term goals on what I would like to have some background/knowledge with a help from this list: 1\ I saw somewhere a sentence saying approximately this: so universe is performing a computation. Is then universe a big computer? No. I would like to know in a broad sense what it tries to say a why one shoud rather accept it or reject. 2\ Bruno, you recently wrote that you do not agree with Wolfram's Principle of Computational Equivalence. As I understand to that principle, Wolfram says that universe is a big cellular automata. What is the evidence that it is unlikely this way? Sincerely, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: SUMMARY
Bruno Marchal wrote: Title: SUMMARY (was: OM = SIGMA_1) I send to David Nyman (the 06 Nov 2007) a little planning: 1) Cantor's diagonal 2) Does the universal digital machine exist? 3) Lobian machines, who and what are they? 4) The 1-person and the 3- machine. 5) Lobian machines' theology 6) Lobian machines' physics 7) Lobian machines' ethics Let me summarize what has been done and what remains to be done. Hi Bruno, just want to let you know that I am still following your CT posts. I hope to send my comments and/or 'OK' sign :-) on Monday. Nice weekend to everyone, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Quantum Interference and the Plentitude
Russell Standish wrote: On Mon, Jan 07, 2008 at 02:48:35PM +0100, Mirek Dobsicek wrote: If quantum mechanics was done using a real-valued Hilbert space, you simply don't get wavelike interference patterns. To my knowledge, you don't get interference patterns for *positive* real-valued Hilbert space, but for real-valued Hilbert space you do. Check http://mina4-49.mc2.chalmers.se/~dobsicek/PhDThesis.pdf on page 39 for a quick review and references. Sincerely, Mirek Thanks for the info - I will take a look, when I'm on top of a few things. However, I'm not sure what you mean by positive real Hilbert space, as the positive real numbers do not form a field. I can only guess you mean some kind of non-Hilbert space generalisation, a bit like my non-Hilbert space non-commutative division ring gadgets I've alluded to in the past. Hi Russel, you are right, a *positive* real Hilbert space is a wrong term. However, the point of my comment was to express a belief that your sentence If quantum mechanics was done using a real-valued Hilbert space, you simply don't get wavelike interference patterns. is not correct. But of course, QM as physical framework and as derived from experiments goes with complex numbers. Best, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Quantum Interference and the Plentitude
Very interesting thesis Mirek. I have download it, and will certainly try to dig a bit more on it some week-ends. Thanks, hopefully you will find something interesting in there. I see you don't cite Everett, which indeed is not necessary for the practice of quantum computing. But your presence in this list could mean that you are open to many-worlds ideas, or everything-like theories. I would like to know, if you don't mind, your opinion on Everett, Deutsch ..., and perhaps about interpretation of QM in general Well yes, my opinion about QM is not well established yet. I have jumped to the field of quantum computing three years ago. I had no previous knowledge on the topic nor I did know any quantum physics. I had been playing with the C compiler mostly in the past. Due to relatively short time in the quantum world, I have so far used QM only as a tool - 'shut up and calculate' interpretation. It happened relatively recently, that I was searching the web for buddhist philosophy reading and found James Higgo's Four reasons why you don’t exist. Approximately at the same time I saw a lot of Everett related articles thanks to the 50th anniversary. So I bought Deutsch's Fabric of Reality. It led me to the Fabric-of-Reality mailing list and that took me to the Everything-list. Here I saw your UDA - (QM) - MWI and all these things together attracted me a lot. I think, I can say that I am pretty much open minded. It often seems to me that I keep a 'superposition' of all what I see and read. I don't abandom some forks, I just change the amplitudes. I hardly ever insist on something, I am rarely sure, I don't try to push 'good' and 'bad' far apart from each other. I like when I climb on a steep vertical wall and life is hard and simple at the same time. Regarding MWI ... I have not read the original Everett paper yet. I expected to get most of his ideas from the Fabric of Reality, but alas in this book I got stuck at the strange frog seeing individual photons. It seemed to me too much scientific-popular reading and I did not get back to the book so far. For christmass, I asked my girlfrind to give me The Wisdom of Insecurity, Computability: An Introduction to Recursive Function Theory, and Godel's Theorem: An Incomplete Guide to Its Use and Abuse, so these books might precede the Fabric :-) About the S K programming language, I guess some people have recognize the Shoenfinkel-Curry combinators. More on this in my old combinators Uuu, new excercises and I am still reading the DIAGONAL post from 31.12.2007. It means I am still living in the previous year, it is good, who wants to be old :-) Have a nice weekend, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Quantum Interference and the Plentitude
If quantum mechanics was done using a real-valued Hilbert space, you simply don't get wavelike interference patterns. To my knowledge, you don't get interference patterns for *positive* real-valued Hilbert space, but for real-valued Hilbert space you do. Check http://mina4-49.mc2.chalmers.se/~dobsicek/PhDThesis.pdf on page 39 for a quick review and references. Sincerely, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: coffee-bar machine excerices
The Shepherdson Sturgis coffee-bar formal definition of computability. (A variant by Cutland). Here is a job offer in an (infinite) coffee bar in Platonia. (Infinite, just for making things a bit simpler.) The basic instructions are the following 3 types + 1. a. - Please add a coffee cup on table 17 (say) b.- Please put on table 24 as many coffee cups than there are coffee cups on table 42 (say) c. - Please make sure there is no more coffee cups on table 56 The last instruction is a bit more difficult. To do the job you need minimal ability to read a language in which the preceding instructions can be described. Also, to economize paper (yes in Platonia Forest are protected too!), the instruction a. - Please add a coffee cup on table 17 (say) is written S(17) b.- Please put on table 24 as many coffee cups than there are coffee cups on table 42 (say) is written T(42, 24) c. - Please make sure there is no more coffee cups on table 56 is written Z(56) (Z is for zero cup of coffee) For the last instruction you have to know that a job is the given of an ordered, even numbered instructions. For example, a typical Job is 1) Z(1) 2) S(1) 3) S(2) Here the job consists in making sure there are no more coffee cups on table one. Then to add a cup of coffe on table one, and then add a cup of coffee on table 2. Now here is the last instruction: d. - if the number of coffee cups on table 14 (say) is equal to the number of coffee cups on table 45 (say) then proceed from the instruction 5 (say) described in your job. In case there are no instruction numbered 5, stop (the job will be said to be completed); in case the number of coffee cups on table 14 is not equal to the number of coffee cups on table 45, then proceed from the next instruction. It is written: J(14, 45, 5). DEFINITION: A function f from N to N is said to be Shepherdson Sturgis coffee bar computable, if there is a job (a list of numbered instructions) such that when putting n cups of coffee on table one, then, after the job is completed there is f(n) cups of coffee on table one. Similarly, a function h from NXN to N is said to be Shepherdson Sturgis coffee bar computable if there is a job such that, after having put n cups of coffee on table one and m cups of coffee on table two, then, after the job is completed there is h(n,m) cups of coffee on table one. I have to go, so I give some Exercise for the week-end (I provide solution monday) 1) find a short job crashing the coffee bar computer. Such a job will never be completed. BEGIN 1: Z(1) 2: Z(2) 3: J(1,2,3) # true condition, loop END 2) find a job which computes addition (which is of course a function from NXN to N) BEGIN 1: Z(3) # clean temp tables 2: Z(4) 3: J(2,3,8) # are we done? get out of the loop 4: S(1) 5: S(3) 6: S(4) 7: J(3,4,3) # always true condition, continue with adding 8: END 3) using the preceding job, find a job which computes multiplication. BEGIN 1: Z(5) # clean temp tables 2: Z(6) 3: T(1,5)# copy 1 - 5 4: J(5,6,14) # are we done? get out of the multipl. loop 5: S(6) 6: Z(3) # adding loop start 7: Z(4) 8: J(2,3,13) 9: S(7) 10: S(3) 11: S(4) 12: J(3,4,8) # adding loop end 13: J(3,4,4) # always true condition, continue with multipl. loop 14: Z(1) 15: T(7,1)# copy 7 - 1, table one contains the final result END 4) is the following proposition plausible: a function from N to N is intuitively computable if and only if it can be computed by some coffee bar job. yes 1) instructions of the coffee-bar language coincide with important instructions of nowadays central processing units in computers 2) coffee-bar instructions are sufficient for constructing logical AND, OR, NOT operations 3) I should be able to find a better reason, damn... 5) describe informally the coffee-bar language, and, choosing an order on its alphabet, write the first 7 jobs in the lexicographical order. The alphabet contains all symbols needed in the jobs, including commas, parentheses, etc. + some grammatical rules making clear that Z(23) is a good instruction, but 23(Z) is not, ... Program ::= BEGIN commands END commands::= line-no instruction comment next line-no ::= num: instruction ::= Z(num) | S(num) | T(num,num) | J(num,num,num) comment ::= # text `new-line` | `new-line` next::= commands | END num ::= [0,1,2,3,...] text::= [ascii text without the `new-line` character] First 7 jobs 1) BEGIN 0: J(0,0,0) END 2) BEGIN 0: J(0,0,1) END 3) BEGIN 0: J(0,1,0) END 4) BEGIN 0: J(0,1,1) END 5) BEGIN 0: J(1,0,0) END 6) BEGIN 0: J(1,0,1) END 7) BEGIN 0: J(1,1,0) END Assuming empty tables, programs 1,3,5,7 never reach END. Sincerely, Mirek --~--~-~--~~~---~--~~ You received this
Re: Key Post 1, toward Church Thesis and Lobian machine
Hi Bruno, From what you told me, I think you have no problem with Cantor 's diagonal. Yep, no problem. Are you ok with the key post, that is with the two supplementary uses of the diagonal in the enumerable context? 95% grasped, and for the rest I'm lacking time to do a sufficient amount of scribes in order to get it completely. But nothing fundamental ... Now, I'm very busy with finishing my phd thesis (study and simulations of a certain two-qubit procedure, a sort of benchmark). I guess, I'll be fine at the beginning of the New Year. Sincerely, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Key Post 1, toward Church Thesis and Lobian machine
Hi Bruno, thank you for your post. I read it a couple of times in order to more or less grasp it, but it worth it. I have some questions... Suppose there is a secure universal machine M. The set of expressions it can compute provide a secure universal language L. That set is not only enumerable (given that it is a subset of an enumerable set) but above all, it can be enumerated effectively (by the ashole). What do you mean here by effectively? I understand it as you just want to emphasize that the set is really enumerable. So, as you know, Church did *define* a computable function by a function computable by a lambda expression, in its conversion calculus. Why do you introduce here the term 'lambda expression'? I'm asking this because in the sequel you work just with 'a well specified language which is promised to be universal' and you prove that such a promise is not ruled out. I do not see how you reached the conlusion: But thanks to that crashing, *Church thesis remains consistent*. I would just say An existence of a universal language is not ruled out. Each expression like that denotes now either a computable function from N to N, or as we have to expect something else. And we have to expect they are no computable means to distinguish which U_i represents functions from N to N, and which represents the other beast. Can I say that the other beasts are only and only infinite loops? I assume that the machine cannot destroy itself, so it either stops after computing a computable function or enters some silly loop. Indeed, the diagonalisation above, where now the f_i are the *partial computable functions*, meaning they are from N to N, OR from a subset of N to N, does no more lead to a contradiction. I have a problem with this paragraph, could you please write more on this? I understand to partial computable functions as to functions which are not *defined* for every possible input (total functions are defined for all inputs, in my limited understanding). Do you say in that paragraph that beasts are only and only these partial functions? Huh, now what do I mean by *defined* ... maybe I should say 'which are not computable for every possible input'. I am really lost here... And my last question, consider the profound function f such that f(n) = 1 if there is a sequence of n consecutive fives in the decimal expansion of PI, and f(n) = 0 otherwise Is this an example of a partial computable function? Or is this function as such already considered as un-computable function? With best regards, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Last post before the key post (was OM = SIGMA_1) 1
Hi Bruno, I'm ready. Luckily, it is not long time ago, I've received my university degree in CS, so it was rather easy to follow :-) Sincerely, Mirek Bruno Marchal wrote: Le 27-nov.-07, à 17:27, Günther Greindl a écrit : Dear Bruno, thanks for your posts! I like them very much! Looking forward to further stuff, Thanks for telling. In this post I recall Cantor's proof of the non enumerability of the set of infinite binary sequences. First with a drawing, and then with mathematical notation. The goal is to train you with the notations. Then I do the same with the almost exactly identical proof that the set of functions from N to N is also non enumerable. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Rép : Observer Moment = Sigma1-Sentences
Bruno Marchal wrote: Question to David, and others who could be interested: is the notion of enumerable and non enumerable set clear? Can you explain why the set of functions from N to N is not enumerable? Let us go slow and deep so that everybody can understand, once and for all. OK? Hello Bruno ! I am a freshman to this list and it seems to me that some kind of a 'course' is going to happen. I checked a couple of last messages and it looks interesting. Please, would you mind to repeat what is approximately the starting point of your explanations and where do you aim? Hopefully, I'll be able to follow. Best regards, Mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
