midterm 2 review jundong liu school of eecs [email protected]
TRANSCRIPT
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
101010 cba
ppp cba 2
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
ppp 101 4
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.