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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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