By Gregory. J. Chaitin
Chaitin, the inventor of algorithmic details thought, offers during this ebook the most powerful attainable model of Gödel's incompleteness theorem, utilizing a knowledge theoretic procedure in response to the scale of desktop courses. One 1/2 the publication is worried with learning the halting chance of a common computing device if its application is selected through tossing a coin. the opposite part is worried with encoding the halting likelihood as an algebraic equation in integers, a so-called exponential diophantine equation.
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That’s how big q is. How about the register variables? Well, they are vectors giving a chronological history of the contents of a register (in reverse order). , each register variable is a vector of time elements, each of which is (8time)-bits long, for a total of 8time 2 bits altogether. Thus 2 register variable ≈ 28time . And how about the variables that arise when ⇒’s are expanded into equations? Well, very roughly speaking, they can be of the order of 2 raised to a power which is itself a register variable!
On the large 370 mainframe that I use, the elapsed time per million simulated register machine cycles is usually from 1 to 5 seconds, depending on the load on the machine. Fortunately, this same LISP can be directly implemented in 370 machine language using standard LISP implementation techniques. Then it runs extremely fast, typically one, two, or three orders of magnitude faster than on the register machine simulator. How much faster depends on the size of the character strings that the register machine LISP interpreter is constantly sweeping through counting parentheses in order to break lists into their component elements.
7. EXPANSION OF ⇒’S 63 Variables added to expand =>’s: r1 s1 t1 u1 v1 w1 x1 y1 z1 ... 485622 seconds.
Algorithmic Information Theory by Gregory. J. Chaitin
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