Context-Free Languages:Review

Chapter 16

Languages and Machines

Regular and CF Languages

Regular Languages Context-Free Languages

● regular exprs. ● context-free grammars ● or● regular grammars● = DFSMs ● = NDPDAs● recognize ● parse● minimize FSMs ● find unambiguous grammars

● reduce nondeterminism in PDAs ● find efficient parsers

● closed under: ● closed under: ♦ concatenation ♦ concatenation ♦ union ♦ union ♦ Kleene star ♦ Kleene star ♦ complement ♦ intersection ♦ intersection w/ reg. langs● pumping theorem ● pumping theorem● D = ND ● D ND

Example

{bam1bam2bam3… bamn : n 2, m1, m2, …, mn 0, and

mi mj for some i, j}

A PDA:

A CFG:

Is L regular?

L = {aibj: j = 4i + 2}

Is L Regular, Context Free, or Neither?

L = {x xneg : x {0, 1}*}.

xneg = x with all 0’s replaced by 1’s and 1’s replaced by 0’s.

Is L Regular, Context Free, or Neither?

L = {w {0, 1}* : k (w is a binary encoding, leading zeros allowed, of 2k+1)}

Is L Regular, Context Free, or Neither?

L = {w {a, b, c}* : every a has a matching b and amatching c somewhere in w (and no b or c is consideredto match more than one a)}

Is L Regular, Context Free, or Neither?

L = {anbm : n + m 3 2, n 1, m 1}

An Example

L = {aibjck : k i or k j}

Construct a context-free grammar for L.

Functions on Languages

Again, let L = {aibjck : k i or k j}.

Describe precisely the language L = maxstring(L), where:

maxstring(L) =

{x: x L and (y * (y ) (xy L))}

Is L = maxstring(L) context free?

Are the context-free languages closed under maxstring?

Regular, Context Free or Neither?

L1 = {anbmck: m min(n, k)}.

L2 = {anbmck: n = m + k or m = n + k}.