Saya tahu bahwa masalah penghentian tidak dapat diputuskan secara umum tetapi ada beberapa mesin Turing yang jelas berhenti dan beberapa yang jelas tidak. Dari semua mesin turing yang mungkin, apa yang terkecil di mana tidak ada yang punya bukti apakah itu berhenti atau tidak?
halting-problem
Harun
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Jawaban:
Mesin Turing terbesar yang masalah penghentiannya dapat ditentukan adalah:
(di mana T M ( k , l ) adalah himpunan mesin Turing denganstatus k dansimbol l ).TM(2,3),TM(2,2),TM(3,2) TM(k,l) k l
The decidability dari dan T M ( 3 , 3 ) adalah pada batas dan sulit untuk menetap karena tergantung pada dugaan Collatz yang merupakan masalah terbuka.TM(2,4) TM(3,3)
Lihat juga jawaban saya pada cstheory tentang mesin Turing seperti Collatz dan " mesin Turing Kecil dan kompetisi berang-berang yang digeneralisasi secara umum " oleh P. Michel (2004) (di mana diduga bahwa juga dapat dipilih).TM(4,2)
Komentar Kaveh dan jawaban Mohammad benar, jadi untuk definisi formal dari mesin Turing standar / non-standar yang digunakan dalam hasil seperti ini lihat Turlough Neary dan Damien Woods bekerja pada mesin Turing universal kecil, misalnya Kompleksitas mesin Turing universal kecil: sebuah survei (Aturan 110 TM sangat universal).
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I would like to add that there are some Turing Machines for which the Halting problem is independent of ZFC.
For instance take a Turing machine which looks for a proof of contradiction in ZFC. Then if ZFC is consistent, it won't halt, but you cannot prove it in ZFC (because of Gödel's second incompleteness theorem).
So it is not only a matter of not having found a proof yet, sometimes proofs don't even exist.
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No one has a proof whether Universal Turing machine halts or not. In fact, such proof is impossible as a result of the undecidability of the the Halting problem . The smallest is a 2-state 3-symbol universal Turing machine which was found by Alex Smith for which he won a prize of $25,000.
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an inexactly phrased but reasonable general question that can be studied in several particular technical ways. there are many "small" machines measured by states/symbols where halting is unknown but no "smallest" machine is possible unless one comes up with some justifiable/quantifiable metric of the complexity of a TM that takes into account both states and symbols (apparently nobody has proposed one so far).
actually research into this problem related to Busy Beavers suggests that there are are many such "small" machines lying on a hyperbolic curve wherex×y , x states and y symbols, is small. in fact it appears to be a general phase transition/boundary between decidable and undecidable problems.
this new paper Problems in number theory from busy beaver competition 2013 by Michel a leading authority exhibits many such cases for lowx,y and shows the connection to general number theoretic sequences similar to the Collatz conjecture.
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