In the computer science course for mathematicians held at my university Turing Machines have been presented very briefly. So much so that I didn't quite get why they are relevant to mathematics. I did understand what a Turing Machine is, but I feel that the presentation given was (for schedule reasons) quite naive and simplified. So I would like to ask if you can point out a good reference for a mathematically mature introduction to turing machines and computability.
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What computability theory book claims the existence of a universal machine is clear?! Anyway, I recommend Shen-Vereshchagin, Computable Functions as a rigorous introduction. Most of the book is, as advertised, about computable functions in the abstract, with one chapter specifically about Turing machines, but I think for a mathematician that's the right approach: there are many models of computation but computation itself doesn't depend on them. – Kevin Carlson Mar 27 '14 at 22:48
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@Kevin: If downplaying Turing machines is the right thing for a mathematician (as it might well be, for all I know), then I wonder who they are for at all. Computer scientists will be much more comfortable thinking about RAMs, lambda calculus, or actual toy programming languages. – hmakholm left over Monica Mar 27 '14 at 22:52
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One example is the following "classic" written by Martin Davis http://www.amazon.com/Computability-Unsolvability-Prof-Martin-Davis/dp/0486614719 – boumol Mar 27 '14 at 23:02
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@HenningMakholm I guess I would say that Turing machines really aren't as central a concept to computability as they might have appeared in the '30s, before people had experience of programming. Certainly one wants some model, but I'm not sure what exactly Turing machines are best at. It's pleasing to make some explicit complexity calculations with them, I suppose. – Kevin Carlson Mar 27 '14 at 23:02
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I am a fan of John C. Martin's Introduction to Languages and the Theory of Computation. It has about $30$ pages on Turing machines and $40$ exercises. – A.E Dec 20 '14 at 22:17
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Computational Complexity: A Modern Approach, by Sanjeev Arora? (less on computatibility, much more on computational complexity) – Clement C. Dec 20 '14 at 22:17
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This question was a duplicate of this earlier one: https://math.stackexchange.com/questions/729583/mathematical-intro-to-turing-machines, but the older one was "merged", technically closed as duplicate of this one, more than 4 years later, by a single moderator @davidlowryduda♦ . – YCor Mar 29 '19 at 18:09
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Classics:
Introduction to Metamathematics by S.C. Kleene. One of the first books about computation theory. General introduction to the mathematical logic. Includes very basic set theory, first order logic, formal number theory (including Gödel), recursive functions and Turing machines. Centered around the logic. See Teach Yourself Logic, #8. The Big Books — starting with Kleene 1952.
Computability and Unsolvability by Martin Davis. Turing machines, computable/recursive functions, several applications of the general theory: the words problem for semigroups, Hilbert's 10th probrem, incompleteness, classification of unsolvable decision problems... More centered in computer science.
Modern:
The Pillars of Computation Theory by Arnold L.Rosenberg. Finite automata, Turing machines, formal languages, the halting problem, nondeterministic Turing machynes, introduction to the compexity theory.
Added by popular demand:
Computability and logic by Boolos, Burgess and Jeffrey.
Martín-Blas Pérez Pinilla
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Thanks for your answer. I would appreciate if you could elaborate on it a little more and add some remarks on these references. – Dal Dec 20 '14 at 22:31
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I wanted to upvote your answer , but then you forgot "computability and logic" Boolos, Burgess and Jeffrey , add it quic, before anybody notice :) – Willemien Dec 20 '14 at 22:55
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@Willemien, I've included the three best books about computability that I have. I can add Computability and logic under your own responsibility. ;) – Martín-Blas Pérez Pinilla Dec 20 '14 at 23:17
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get the book soon (but the 3th edition), i heard that is that this edition was the best one. see http://www.logicmatters.net/tyl/ the pdf itself – Willemien Dec 20 '14 at 23:55
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I'm partial to Ullman, Hopcroft's text. Jeff Ullman runs a Coursera course on Automata theory, and I have heard very good things from someone I know who is a linguist. The text gives a very comprehensive overview of formal languages and automata, as well as issues of computability and complexity. I think it is very thorough and well done, plus having a Coursera course to work through can be helpful. More here: http://infolab.stanford.edu/~ullman/pub/ialctoc.txt
Michael Sipser is another standard text on the subject, and he is one of the foremost experts in the field. I haven't used this book, but I've heard good things. It also looks pretty comprehensive. http://www.amazon.com/Introduction-Theory-Computation-Michael-Sipser/dp/113318779X
ml0105
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Upvote for Sipser. While Martin is a good book too, if you're looking for a good introduction to Turing machines, Sipser has to be in the conversation. – starflyer Dec 21 '14 at 21:49
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I have found that this pdf http://www.staff.science.uu.nl/~ooste110/syllabi/compthmoeder.pdf helps give a mature & brief theoretical introduction. Oosten also includes detailed proofs of the main theorems in recursive function theory that are necessary for carrying out rigorous details in every domain of mathematical logic related to the use of recursive functions, including the $s^m_n$ theorem acting on the programs of Turing machines, the recursion theorem, and the fixed point theorem (all located in section 2.4 of the pdf).
A more encyclopedic direction which covers a great deal is Odifreddi's Classical Recursion Theory (2 volumes). A quick search on amazon or even google may have an online pdf, but I have not checked into this.
Hope this helps!
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