> On 20 Apr 2018, at 19:04, John Clark <johnkcl...@gmail.com> wrote: > > I never got past the first line of Bruno’s post because he said: > > "Consider any Turing universal machinery, for example the programming > language c++” > > C++ is Turing complete but is not a Turing machine because machines are > physical objects made of atoms but C++ is not nor is any language. >
That is not correct. There are many definition of Turing machine. Let me give you the most standard definition of Turing machine. You need two infinite set of symbols: Q = {q_1, q_2, q_3, q_4, …}. (Called set of internal configurations} S = {S_0, S_1, S_2, S_3, …}. (Called set of tape symbol} Along with the symbols R and L. (Right and Left symbols) A quadruplet is then define by a sequence of fours symbols having the shapes q_i S_j S_k q_r or q_i S_j R q_r or q_i S_j L q_r (Davis makes the general relativized theory at once, and so add q_i S_j q_k q_l, which is used to handle Turing’s Oracle, but I will ignore this here). Now a Turing machine is defined to be a finite set of quadruples. That’s all. It is set, so the order of the quadruples is irrelevant. If there is not two different quadruples with the same beginning q_i S_j, then we say that the machine is deterministic. If not it is non deterministic. The intuitive interpretation of q_i S_j S_k q_r is that if the machine is in the internal configuration q_i, in front of the symbol S_j, and if some quadruple in the machine contains that quadruplet, the next step of the computation will be obtained by overwriting S_k (on S_j) and getting the state q_r, or if q_i S_j R q_r was in the machine (which is a set of such quadruplets) moving to the right of the" tape” and getting the internal configuration q_r. (And similarly, for L = Left). Then we can define computation by finite sequence of “instantaneous tape description”, which are finite set of tape symbols, + a symbols of internal configuration, like S_4 S_5 q_22 S_2 S_6, which means that the machine is in front of S_2, in configuration q_22, and if q_22 S_2 begins one of its quadruplet, the machine “acts” accordingly, and if not, the machine stops. But the terms “tape”, “configuration” are just intuitive help, and not part of the definition. Note that most definition use only two tape symbols S_0 (called “blank”) and S_1 (written 1). Note that Turing’s original definition use quintuplet, and he allowed the machine to “move” and overwrite symbols simultaneously. But it is better to reason on the quadruplets (more general). As you can guess (and try to prove) a the quadruplet-TM can emulate the quintuplet -TM. Important remark: a universal Turing machine is a Turing machine, and thus a FINITE set of quadruplets. There is nothing infinite in the universal machine. It is an interpreter of Turing formalism coded in one (universal) program, that is here a FINITE set of quadruplets. That machine, assuming only the two symboles “.” (for blank, S_0), and 1 for S_1. The universal machine is thus a finite set of quadruplet, and it starts on q_1 1111111111111111111111111.11111111 And it interprets the first block “1111111111111111111111111” as some Gödel number of a Turing machine (set of quadruplet) acting on the input represented by the second bloc. Obviously it has q_1 1 as a beginning of some of its quadruplet to avoid stopping at the start! The key point is that the universal machine is a FINITE object. Turing’s talk on an infinite tape is only an aid to the intuition. It is better to consider the tape as being an environment, mental or physical, or even as a special oracle. > As for Löbian machines that is yet another term that Bruno made up and is > seen on this list but nowhere else. > It is a slightly more precise version of the “enough rich” machine/theory/set-of-beliefs notion, and it can be defined by for all p sigma_1, the machine can prove “p -> []p”, with “[]p” being the usual tedious arithmetical definition of provability. As I explained recently, with p sigma_1, The truth of all p -> []p is equivalent with “[]” (the machine’s arithmetical provability predicate) being a universal machine, in this sense or equivalent. Then a machine “[]" is Löbian if she can prove “p -> []p” As “[]” is itself sigma_1, (this should be obvious!!! Please tell me if you (anyone) don’t see this), we get []p -> [][]p, so Löbianity entails the axiom 4 (called self-awareness by Smullyan). A pure K4 reasoner has to go the Löb’s Island to become Löbian, but any enough rich machine can prove Gödel’s diagonal lemma, making them “born in the Löb’s Island” so to speak. The typical example are any sound machine believing in enough axioms of arithmetic, like PA, ZF, and many others (including us, as far as we are correct and agree with PA, say). > And Turing explained exactly precisely how to make one of his machines in the > real physical world > He is even the first to build one. But his machine is still a mathematical notion, as his notion of computations. Amazingly Robinson Arithmetic is Turing complete, and so we have very quickly that whatever you can do (mathematically or physically with Turing machine, physically incarnated/implemented or not), number relations provable in RA can mimic exactly any Turing machine. A Turing machine is unable to detect by its personal introspective feeling (we assume indexical computationalism) if she is incarnated relatively to an arithmetical base, or a physical base, but the point is that she can test this experimentally. I predicted Quantum Logic from this well before I knew about quantum physics (except as tool to study enzyme kinetic. > but Bruno has no idea how to even start to build one of his machines, which > means he doesn’t understand how it works or even exactly what it is he’s > talking about. > > I give often this as a simple exercice to my student. It takes not much line more than PA axiomatic if you use a language like Prolog. Note that to extract physics from arithmetic, you need only to prove the existence of the Löbian Machines’ computations in arithmetic, which can be done already in RA, as it is well known that RA is Turing complete, and so mimic all machines (even the Löbian one, which of coures assume more than RA). There is no need to implement a Löbian machine physically to get the logical point. Your remark above is thus only an annoying ad hominem insulting assertion, which does not help us to believe you ever tried to understand, and this means you have some prejudices or that you repeat some lies I have often been reported. It is pure direct or indirect bullying. You are either a liar, or a victim of liars. Bruno > > John K Clark > > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to everything-list+unsubscr...@googlegroups.com > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. 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