The document discusses converting a pushdown automaton (PDA) to a context-free grammar (CFG) by constructing the CFG from the PDA's transition function. It also discusses deterministic PDAs and properties of languages accepted by deterministic PDAs, including that regular languages are accepted by deterministic PDAs and unambiguous CFGs correspond to languages of deterministic PDAs. The document also summarizes steps for eliminating useless symbols, epsilon productions, and unit productions from a CFG to put it in a normal form.

1. Convert PDA to CFG
• Given a PDA M = (Q, €, F, 6, qO, ZO, {}), we construct a CFG G = (V,
€, P, S)
– V = {[q, A, p] | q, p are in Q, A is in F+{S}}
– P is a set of productions constructed as follows:
• S c [qO, ZO, q] for all q in Q
• for each (p, s) in 6(q, a, A), where a is in € + {s}, we define
production:
[q, A, p] c a € + {s} and B are
in
• for each (p, B1B2…Bm) in 6(q, a, A), where a is
in
F [q, A, qm+1] -> a [q1, B1, q2] [q2, B2, q3] …[qm, Bm, qm+1]
for all q, q1, q2, …, qm+1 in Q

2. Deterministic PDA’s
• A PDA is deterministic if there is at most one move for any
input symbol, any top stack symbol, and at any state.
• A no move or move without advancing input string are
possible in a deterministic PDA.
• A move without advancing input string implies that no
other move exists.
• The pushdown automaton for wcwR is deterministic.

3. Deterministic PDA’s
• The set of languages accepted by deterministic PDD’s is
only a subset of the languages accepted by
nondeterministic PDA’s.
• A regular language is language accepted by a DPDA.
• Not all languages accepted by DPAD are regular.
• All languages accepted by DPAD have an unambiguous
CFG.
• Not all unambiguous languages are accepted by DPAD.

4. Grammatical Format
• Eliminate useless symbols
– derivable from S
– can derive a string of terminals
• Eliminate s-productions
– A -> s is an s-production.
• Eliminate unit productions
– A -> B is a unit production if both A and B are variables.
• For any CFG G and s is not in L(G), there exists
an equivalent context -free grammar with no
useless symbol, s-production, and unit production.

5. Eliminating Useless Symbols
• Given a CFG G = (V, T, P, S), withL(G ) is not empty, we
can effectively find an equivalent CFG G’ = (V’, T, P’, S)
such that for each A in V’ there is some w in T* for which
A=>*w.
• Given a CFG G = (V, T, P, S), we can effectively find an
equivalent CFG G’ = (V’, T’, P’, S) such that for each X in
(V’+T’) there exist a and þ in (V’+T’)* for which
S =>* aXþ.

6. Eliminatingssss-productions
• If s is not in L(G), we can eliminate all s-productions.
• A variable is nullable if A =>* s.
• For A -> X1X2…Xn, add the following productions A ->
Y1Y2…Yn
– If Xi is not nullable, then Yi = Xi.
– If Xi is nullable, then Yi = Xi or Yi = s.
– If Xi is nullable for (i = 1, …, n), don’t add A => s.

7. Eliminating Unit Productions
• If A=>*B and for all nonunit production B->X, add
– A->X
• Eliminate all unit productions.

8. Eliminating Useless Symbols and sss
and Unit Productions
1. Eliminate s-productions
2. Eliminate unit productions
3. Eliminate useless symbols.

9. Chomsky Normal Form (CNF)
• If aCFG has only productions of the form, it is called
Chomsky Normal Form (CNF)
– A -> BC, or
– A -> a
where A, B, and C are variables and a is a terminal.
• Any context-free language with e can be generated by a
context-free grammar in which all productions are in CNF.

10. Chomsky Normal Form (CNF)
• If aCFG has only productions of the form, it is called
Chomsky Normal Form (CNF)
– A -> BC, or
– A -> a
where A, B, and C are variables and a is a terminal.
• Any context-free language with e can be generated by a
context-free grammar in which all productions are in CNF.