Theory of Computation
Spring 2007
Qualifying Exam Information

Suggested Reading List

1. [HMU] John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, second edition, Addison-Wesley, 2001.
2. [SIPSER] Michael Sipser, Introduction to the Theory of Computation, second edition, Thomson, 2006.

Topics

1. Basic Concepts
   A. Proof techniques: direct, mathematical induction, proof-by-contradiction
   B. Sets, functions, relations, equivalence relations and equivalence classes
   C. Strings and languages
   D. Countability and enumerability
   
   Reading: HMU - Chapter 1; SIPSER - Chapter 0; Section 4.2
    
2. Finite Automata (FA) and Regular Languages (RL)
   A. DFA, NFA, FA with epsilon transitions
   B. NFA to DFA conversion
   C. Regular expressions (RE)
   D. Relationship between DFAs and REs
   E. Algebraic laws of REs
   F. Pumping lemma for and closure properties of RLs
   G. Decision properties of REs
   H. Equivalence and minimization of automata
   
   Reading: HMU - Chapter 2, Chapter 3 and Chapter 4; SIPSER - Chapter 2
    
3. Context-Free Languages/Grammars (CFL/CFG) and Pushdown Automata (PDA)
   A. Parse trees and parsing
   B. The language and properties of a PDA
   C. Equivalence of PDAs and CFGs
   D. Deterministic PDAs
   E. Normal forms for CFGs
   F. The pumping lemma for and closure properties of CFLs
   G. Decision properties of CFLs
   
   Reading: HMU - Chapter 5, Chapter 6 and Chapter 7; SIPSER - Chapter 3
   
    
4. Turing Machines (TMs)
   A. Definitions of and notions for various Turing machines
   B. Extensions of TM s (e.g., multi-tape, equiv. of one-tape and multi-tape TMs)
   C. Non-deterministic TMs (NTMs) and running time of various simulations
   D. Restricted TMs (e.g., multistack TMs, counter machines, etc.)
   E. Relations (and power) among DFAs, PDAs and TMs
   
   Reading: HMU - Chapter 8; SIPSER - Chapter 3
   
    
5. Computability
   A. Church-Turing Thesis and its meaning
   B. Encoding of TMs
   C. Diagonalization
   D. Undecidable languages (or problems)
   E. Recursive and recursively enumerable languages
   F. Universal machines
   G. Many-one reducibility
   H. Important theorems: the s-m-n theorem, recursion and fixed point theorems, various forms and extensions of Rice s theorem, and the isomorphism theorem
   I. Other equivalent systems/models: the RAM model, Post systems, Thue systems and Lambda calculus.
   J. Oracles and Turing reducibility
   
   Reading: HMU - Chapter 9; SIPSER - Chapter 4, Chapter 5 and Chapter 6
   
    
6. Computational Complexity
   A. Classes P, NP and NP-complete
   B. Polynomial reduction of NP-complete problems
   C. NP-complete problems
   D. Classes co-NP and co-NP-complete
   E. Classes NSPACE, NTIME, PSPACE, NPSPACE, P-complete and PSPACE-complete
   F. Classes NL and NC
   G. Relations among the above mentioned classes
   
   Reading: HMU - Chapter 10, and Section 11.1, 11.2 and 11.3; SIPSER - Chapter 7 and Chapter 8