1. Automata and Formal Languages (Final Review)
Hongwei Xi
University of Cincinnati
2 December 2018
Final Review

2. Regular Languages
Deterministic Finite Automata D = (Q, S, d, q0, F) d: Q x S -> Q
Nondeterministic Finite Automata N = (Q, S, D, q0, F) D: Q x S -> P(Q)
Regular Expressions R = a | e | f | R1 o R2 | R1 u R2 | R*
2 December 2018
Final Review

3. Regular Languages
Every NFA N can be converted to a DFA D, where L(N) = L(D)
Every regular expression R corresponds to an NFA N, where L(R) = L(N)
Every DFA D can be converted to a regular expression R, where L(D) = L(R)
2 December 2018
Final Review

4. Nonregular Languages
Pumping Lemma: For any regular language L, there exists a number p (pumping length) such that for each s in L with length at least p, we have s=xyz, where
xy^iz in L for i = 0,1,2,…
|y| > 0
|xy| <= p
2 December 2018
Final Review

5. Context-free Languages
Context-free grammars G = (V, S, R, S)
Ambiguous CFGs
Chomsky Normal Form
Pushdown Automata
L is a context-free language if and only if it is recognized by a PDA
2 December 2018
Final Review

6. Non-context-free Languages
Pumping Lemma: For any CFL A, there exists a number p (pumping length) such that for each s in A, we have s = uvxyz, where
uv^ixy^iz in L for i = 0,1,2,…
|vy| > 0
|vxy| <= p
2 December 2018
Final Review

7. Computability Theory
Turing machines: TM = (Q,S,G,d,q0,qaccept,qreject) d: Q x G -> Q x G x {L,R}
Multitape Turing machines
Nondeterministic Turing machines NTM = (Q,S,G,D,q0,qaccept,qreject) D: Q x G -> P (Q x G x {L,R})
2 December 2018
Final Review

8. Computability Theory Enumerator Decider
Church-Turing Thesis: Intuitive notion of algorithms = Turing machine algorithms
2 December 2018
Final Review

9. Decidability
Decidable problems concerning regular languages: ADFA, EDFA, EQDFA, ALLDFA, INFINITEDFA, etc.
Decidable problems concerning context-free languages: ACFG, ECFG, etc.
2 December 2018
Final Review

10. Undecidability Halting Problem: ATM = {<M, w> | M accepts w}
Countability, Uncountablity
Diagnalization
The complement of ATM is not Turing-recognizable
2 December 2018
Final Review

11. More Undecidable Problems
EQTM
EQCFG
ALLCFG (but ECFG is decidable!)
Post correspondence problem
2 December 2018
Final Review

12. Time Complexity Big-O and Small-o notation
Time(t(n)) = { L | L is a language decided by a O(t(n)) time Turing Machine }
2 December 2018
Final Review

13. Time Complexity
Let t(n) be a function, where t(n) >= n for all natural numbers n.
Then every t(n) time multitape TM has an equivalent O(t^2(n)) time single-tape TM
Then every t(n) time Nondeterministic TM single-tape TM has an equivalent O(2^(t(n))) time single-tape TM
2 December 2018
Final Review

14. The Class P P = the union of time(n^k) for all natural numbers k
Examples in P
PATH
RELPRIME
Every context-free language is in P
Dynamic programming
2 December 2018
Final Review

15. The Class NP NP stands for nondeterministic polynomial time Verifier
Every language in NP has a polynomial time verifier
Every language in NP is decided by a polynomial time nondeterministic TM
2 December 2018
Final Review

16. The Class NP Examples in NP HAMPATH CLIQUE SUBSET-SUM 2 December 2018
Final Review

17. P vs. NP
One of the greatest questions in theoretical computer science:
P = NP?
In general, people believe P is a proper subclass of NP
In general, it is difficult to find a lower time bound for a given problem
2 December 2018
Final Review

18. NP-completeness Polynomial time reducibility: A <=P B
B is NP-complete if
B is in NP
A <=P B for all A in NP
Cook-Levin Theorem: SAT is NP-complete
2 December 2018
Final Review

19. More NP-Complete Problems
3SAT
HAMPATH
CLIQUE
SUBSET-SUM
2 December 2018
Final Review

20. Final Exam Date: December 5, 2000 Place: 501 Swift Hall
Time: 1:30 – 3:30 PM
Choose 10 out of 15 problems
2 December 2018
Final Review