Midterm 2 review Jundong Liu School of EECS

True of false: The pumping lemma for CFLs can be used to prove that a language is Context-free. Define the language EQUAL3 over {a, b, c} to be the language with an equal number of a’s, b’s and c’s. Can the string be used in a pumping lemma proof? Can the string be used for EQUAL3? Pumping Lemma for CFLs

Pump lemma for CFLs You can’t control The Pumping length p You can control The selection of s S should be a string | S |>= p, for example, Then S can be Next step: based on analysis to derive contradiction

Closure Properties of CFLs, How to prove them?

Turing Machines

Turing Machine: Formal definition

Turing Machines: Concepts, definitions transitions of a TM accepting, rejecting states A language of a TM = set of strings it accepts Turing recognizable language = language of certain TM decidable language

Skills Understand the difference between a decidable and a recognizable language Given a configuration and the transition table, what will be the next configuration? How would one design TM that decides {02^n}, {w#w}, {a^n b^n c^n}?

Turing Machines Variants Stay put Turing Machine Multi-tape Turing Machine Doubly infinite Turing Machine Enumerator Nondeterministic Turing Machine The notion of simulation

Church-Turing Thesis

Algorithm in real world = Turing Machine algorithm 3 levels of descriptions

The languages A-DFA, A-NFA, A-REX, A-CFG are decidable. The languages E-DFA, E-NFA, E-REX, E-CFG are decidable. The languages EQ-DFA, EQ-NFA, EQ-REX are decidable. The symmetric difference of S1 and S2. The languages EQ-CFG, A-TM, E-TM, EQ-TM are not decidable. (Un)solvable problems in the real world  (un)decidable languages under Turing Machine framework

How to? Prove that the languages A-DFA, A-NFA, A- REX, A-CFG are decidable. Prove that the languages E-DFA, E-NFA, E- REX, E-CFG are decidable. Prove that the languages EQ-DFA, EQ-NFA, EQ-REX are decidable.

Understand … The diagonalization technique Why E (the set of even numbers), Z (set of integer numbers) are countable. Why R (the set of real numbers) is uncountable. Why there are more languages than Turing machines.