CSCE 355 Foundations of Computation Exam Review Topics: Mathematical preliminaries Finite Automata Context free languages Pushdown Automata Turing Machines Undecidability December 10, 2008

Relations: properties and examples Proof Techniques: Induction, mutual induction, contradiction. From English description to DFA or NFA. For DFA, NFA, ε NFA give L(M) ε NFA DFA using subset construction Given language L find a regular expression r such that L(r) = L Given regular expression r find L(r) Regular Expressions to ε NFA Minimization of DFA Non-regular languages: Distinguished Prefixes, Pumping Lemma

Context Free Grammars Given language L find pushdown automata with L(P)=L, or N(P)=L. Show language is not context free, using pumping lemma for CFLs. Flex/YACC Design TM to recognize a language L. Convert a CFG to Chomsky-normal form. What does it mean for a language to be: regular, context-free, context sensitive, recursively enumerable, recursive Give example languages of each set of languages in the hierarchy that is not in the next set, except for context sensitive. Design Turing machine to compute f(x) = 2x

Relations: properties and examples R1 = {(a,a), (a,b), (a,c), (a,d)} Reflexive closure? Transitive closure? Reflexive-transitive closure, R1*

x

Proof Techniques: Induction, mutual induction, contradiction.

From English description to DFA or NFA

For DFA, NFA, ε NFA give L(M)

ε NFA DFA using subset construction

Given language L find a regular expression

Given regular expression r find L(r)

Regular Expressions to ε NFA (Thompson)

Minimization of DFA

Non-regular languages: Distinguished Prefixes Pumping Lemma

Context Free Grammars

Given language L find pushdown automata with L(P)=L, or N(P)=L

Show language is not context free, using pumping lemma for CFLs.

Flex/YACC/Ruby

Design TM to recognize a language L.

Convert a CFG to Chomsky-normal form.

What does it mean for a language to be: regular, context-free, context sensitive, recursively enumerable, recursive

Give example languages Chomsky Hierarchy

Design Turing machine to compute Algorithm for computing function f(x,y) = x – y (max {0, x-y}) f(x,y) = x + y f(x) = 2x

Recursive vs R.E.

Halting Problem