Theoretical CS and Math - self-study recommendations

Question

I am a non-CS grad and my field of study is unrelated to CS. However, as part of a larger plan to become a computer scientist, I want to obtain a solid background in theoretical computer science and math as it relates to CS. I did a plenty of research and selected the following best/really good books on the topic of CS and math and would like to ask for your opinions on:

• Completeness of topics covered (please recommend anything I've missed)
• Overlap of material covered / overkill area (please recommend books that should be removed from the list)
• Order to study the books (I listed in the order that I think they should be studied)

The list feels excessively long, so I would appreciate recommendations to remove some books, without the loss of the core knowledge require for CS.

So, the books are:

1. Mathematician's Delight by W. W. Sawyer
2. How to Prove It: A Structured Approach by Daniel J. Velleman
3. How to Solve It: A New Aspect of Mathematical Method by G. Polya
4. An Introduction to Functional Programming Through Lambda Calculus by Greg Michaelson
5. Foundations of Computer Science by Al Aho and Jeff Ullman (http://i.stanford.edu/~ullman/focs.html)
6. Concrete Mathematics: A Foundation for Computer Science by Graham, Knuth, and Patashnik
7. Introduction to the Theory of Computation by Michael Sipser
8. Introduction to Automata Theory, Languages, and Computation by John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman
9. Computational Complexity: A Conceptual Perspective by Oded Goldreich
10. Computational Complexity: A Modern Approach by Sanjeev Arora, Boaz Barak
11. A Course in Combinatorics by J. H. van Lint, R. M. Wilson
12. Computability: An Introduction to Recursive Function Theory by Nigel Cutland
13. Computers and Intractability: A Guide to the Theory of NP-Completeness by M.R. Garey, D.S. Johnson
14. Theory of Recursive Functions and Effective Computability by Hartley Rogers
15. Inequalities by G.H. Hardy, J. E. Littlewood, G. Polya
16. Mathematical Logic: A Course With Exercises (Part I): Propositional Calculus, Bookean Algebras, Predicate Calculus by René Cori, Daniel Lascar
17. Mathematical Logic: A Course With Exercises (Part II): Recursion Theory, Godel's Theorems, Set Theory, Model Theory by René Cori, Daniel Lascar

Answer 1

Your list is extremely problematic.

To start with, I would flatly skip books 6,11,12,14,15,16,17: Books 6, 11 and 15 are general mathematics which is not really needed unless you become a theoretical researcher. Books 12 and 14 cover recursion theory which is not computer science (even though it deals with computability!). Books 16 and 17 cover advanced topics in logic, whereas you only need to know very basic logic.

Out of books 1,2,3, I would choose just one to serve as a general introduction to mathematics and proofs.

Books 5,7,8,9,10,13 cover several subjects: automata theory, algorithms, and complexity theory. Let me suggest that you follow Sipser (Book 7) for automata theory and complexity theory, and Introduction to Algorithms by Cormen, Leiserson, Rivest and Stein ("CLRS") for algorithms.

Book 4 deals with functional programming. While my computer science education has never included any courses on this subject, it is fair to say that many researchers in theoretical computer science count functional programming as part of the essential foundations.

Summarizing: what you remain with is

• One of books 1-3 (or any comparable "introduction to proof" text)
• CLRS
• Book 4 (functional programming)
• Book 7 (automata theory and complexity theory)

Answer 2

You might also consider taking advantage of some of the many online courses available. For example, both Stanford and MIT offer (free) online courses in computer science, and I think there are many others available as well.

As far as books go, I second most of Yuval's recommendations, except that CLRS is a great reference, but a little overwhelming as an introductory book to just sit down and read through. For a first pass at the algorithms portion, I might recommend Algorithms by Dasgupta et al.. The previous link is to the free online pre-print, but it's also pretty cheap to buy in paperback.

Answer 3

The references you are suggesting would make a "very" theoretical computer scientist. but I honestly dont find any benefit reading all these books if you are from a non-CS degree. This of course all depends on what you need.

I find some books such as Book 14, 15, 16, 17 are not intended to computer scientists. Book 3 is verbose. It is just general for any student. Therefore I assume book 1 and 2 are the same.

For me - being in your same situation as not originally a computer scientist (but instead an electrical/computer engineer) - I propose two initial directions:

• algorithms design and analysis, (many people suggest the CLRS Introduction to Algorithms. It is a great reference, but I am really not a fan of it and initially it was a bet harder to understand it and sometimes very verbose. I suggest though that you follow its table of contents, and from there you check online courses for clearer references).

--- make sure to master a programming language to IMPLEMENT the algorithms and data structures you learn - therefore, I suggest the series of Algorithms, of Sedgewick (amazing !)

--- ADDED: I also suggest this book: Combinatorial Algorithms: Generation, Enumeration, and Search by D. Kreher. This is a very nice book. Will give you a different prespective to many issues in algorithms.

• combinatorics (especially graph theory), A Course in Combinatorics by J. H. van Lint, R. M. Wilson, is good. There exist many other references. Usually any well-known combinatorics book is enough - everything else you'd get from additional references from the internet. I personally liked: peter j cameron combinatorics, and Bondy and Murty Graph Theory.

• a bet of probability (always necessary). It is striking that many fields in science do not use probability. But believe me, all you have to know are the basics.

Then: Many researchers calling themselves theoretical computer scientists focuses so much on theory of computation (automota and others). There are some good books for this (See Yuvul Filmus post),

Aho and Ullman is good (actually all books of Ullman are good). Make yourself comfortable with compiler design (see http://infolab.stanford.edu/~ullman/ullman-books.html).

After that: it all depends on what you want to do. Different directions you can take: 1) databases, 2) pattern recognition and data mining, 3) distributed algorithms, 4) foundation of programming languages, 4) randomize algorithms, and many others. [each of which requires another post] but try to have an idea about all !

*The general idea: if you are new to CS, then make yourself comfortable with as many CS sub-domains as possible. Restricting yourself to "theory" will make you lose much of CS creativity ! * (my opinion)