CMPS 260 Spring 2022
HW #7: Chomsky Normal Form, Pushdown Automtata
Due: 2pm, Monday, May 2

1. Provide a precise description of the language accepted by the PDA M1 shown below. That is, describe those properties by which it can be determined whether a given string is, or is not, a member of L(M1). Also indicate whether or not M1 is deterministic, and justify your answer.

M1

2. Let G2 be the q-grammar shown below. Present a DPDA (deterministic PDA) that accepts L(G2).

G2
S ⟶ aBb  |  ba  |  cB (1) (2) (3)
B ⟶ aS  |  λ (4) (5)

3. Present a PDA that accepts the language L3 = { ambm+ncn  |  m≥0, n≥1 }

4. Present a Chomsky Normal Form grammar that is equivalent to (i.e., generates the same language as) this context-free grammar:

S⟶ AbA  |  c
A⟶ aAaSc | ba

5. Consider the following Chomsky Normal Form grammar G5:

S⟶ AB  |  a  |  c    (1) (2) (3)
A⟶ AS  |  a    (4) (5)
B⟶ SA  |  b    (6) (7)

Use the CYK algorithm to determine whether or not w = acaba is a member of L(G5). Specifically, fill in the relevant cells of the matrix pictured below so that the cell in row i and column j contains Vi,j = { X ∈ {S,A,B} : X ⟹+ wi,j }, where wi,j is the substring of w beginning at its i-th symbol and ending with its j-th symbol. (Your answer is expected to include not only the answer to the question "Is acaba ∈ L(G)?" but also the correctly filled in table.)

Recall that, for i satisfying 1≤i≤|w|, X ∈ Vi,i iff X → wi,i is a production in the grammar. Meanwhile, for i and j satisfying 1≤i<j≤|w|, X ∈ Vi,j iff there exists k, where i≤k<j, and nonterminals Y and Z such that Y ∈ Vi,k, Z ∈ Vk+1,j, and X → YZ is a production in the grammar. 

   1      2      3      4      5
+------+------+------+------+------+
|      |      |      |      |      |
|      |      |      |      |      | 1
|      |      |      |      |      |
+------+------+------+------+------+
       |      |      |      |      |
       |      |      |      |      | 2
       |      |      |      |      |
       +------+------+------+------+
              |      |      |      |
              |      |      |      | 3
              |      |      |      |
              +------+------+------+
                     |      |      |
                     |      |      | 4
                     |      |      |
                     +------+------+
                            |      |
                            |      | 5
                            |      |
                            +------+

6. A linear grammar is a CFG in which every production's right-hand side contains at most one nonterminal symbol. A right-linear grammar is a linear grammar in which any such occurrence of a nonterminal symbol must be the last symbol of that right-hand side.

(a) Describe a construction that, given a right-linear grammar GR and a linear grammar GL, produces a linear grammar G such that L(G) = L(GR) · L(GL).

(b) Apply your construction to these particular grammars and show the resulting linear grammar.

SR ⟶aM | b
M ⟶bSR | aaN
N ⟶bM | λ
SL ⟶cAdc | Ac
A ⟶dcSLd | c
GR GL