I originally planned this summer to finish the work on my Introduction to Theoretical Computer Science book, and in particular write the two missing chapters on space complexity and interactive proof systems. Needless to say, this summer did not go as planned and I won’t be able to write these chapters. However, I still intend to go over the existing chapters, fixing typos, adding examples, exercises, and generally making it friendlier to beginning undergraduate students.
Toward this end, I would be grateful for people posting bugs, typos, and suggestions as GitHub issues (I currently have 267 closed and 14 open issues which I hope to get to soon). Of course, if you are technically inclined and there’s a simple local fix, you can also make a pull request.
Aside from these fixes, I am making two more “global” changes to the book. First, I am adding a “non mathy overview” for each chapter. While some students got a lot from reading the book prior to lectures, others were intimidated by the mathematical notation, and so I hope this more gentle introduction will be helpful. I am also adding more examples & solved exercises toward this end.
Another change is that I now follow the more traditional way of presenting deterministic finite automata before Turing machines – DFAs are still optional and can be skipped without missing anything, but some instructors find them as a good introduction to Turing Machines. Thus the order of presentation of materials in the book is roughly as follows:
- Introduction, representing objects as strings – Representing numbers, lists, etc. Specifying computational tasks as functions mapping binary strings to binary strings, Cantor’s theorem.
- Finite functions and Boolean circuits – Every function can be computed by some circuit, circuits as straightline programs, representing circuits as strings, universal circuit evaluator, counting lower bound.
- Computing on unbounded inputs – DFAs (optional), Turing Machines, equivalence between Turing machines, RAM machines and programming languages, λ calculus (optional), cellular automata (optional)
- Uncomputability – Universal Turing machine, Halting problem, reductions, Rice’s Theorem. Optional: Gödel’s incompleteness theorem, uncomputability of quantified arithmetic statements, context free grammars.
- Efficient computation – Modeling running time, time hierarchy theorem, P and EXP
- NP and NP completeness – Polynomial-time reductions, Cook-Levin Theorem (using circuits), definition of NP using “proof system”/”verifying algorithms” (no non-deterministic TMs), NP completeness, consequences of P=NP: search to decision, optimal machine learning, etc..
- Randomized computation: Worst-case randomized computation, defining BPP, Sipser-Gács, does BPP=P? (a little on derandomization)
- Cryptography: One time pad, necessity of long keys for information theoretic crypto, pseudorandom generators and stream ciphers, taste of public key and “magic” (ZKP, FHE, MPC)
- Quantum computing: Some quantum mechanics background – double slit experiment, Bell’s inequality. Modeling quantum computation. Bird’s eye view of Shor’s algorithm and quantum Fourier transform.