Assignment on Pushdown Automata and Context-free Grammars

Context-free Grammars

1. Let CFG W be

   S → x S y | x Q | Q y
   Q → x Q | y Q | ε
   1. Describe L(w) in simple English, and justify your answer.
   2. Use that description to give a CFG for L(W), the complement of L(W).

2. if Statements in Pascal: Suppose the symbols Boolean-expression and statement have been defined for the Pascal language through a context-free grammar. Then, the Pascal Standard specifies the following context-free grammar for "if" statements:

   if-statement → "if"  Boolean-expression  "then"  statement
                      |   "if"  Boolean-expression  "then"  statement else-part
     
    else-part →   "else"  statement
           

   Assume both (x < 1.5) and (y > 4) are valid Boolean-expressions, and assume assignments a := 6.7 and b := -x are valid statements. Consider the Pascal code:

   if (x < 1.5)
        then if (y > 4)
          
           then   a := 6.7
   
     else 
          b := -x
           

   1. Show the formal grammar is ambiguous by giving two different derivations of this code segment.
   2. For each derivation, show the corresponding parse tree.
   3. Convert the above grammar to Chomsky Normal Form.

   Note that the Pascal standard added a note to resolve this issue for compilers. Thus, while the formal grammar was ambiguous, the interpretation in actual programs was not.

3. Conditional Statements in Some Programming Languages: In some programming languages, a cond statement is roughly defined as a type of generalized if statement in C/C++. This exercise explores a variation of this construction.

   • This statement begins with the word cond.
   • A sequence of Boolean expressions follows.
     • One or more executable statements follow each Boolean expression
     • For clarity, parentheses are placed around each Boolean expression and its associated executable statements.
   • After the Boolean expressions and associated statements, an optional default option may be given.
     • When a default is explicitly given, the word else is written, followed by one or more executable statements.
     • For clarify, parentheses are placed around the else and its associated statements.
   • An overall set of parentheses is placed around the entire cond statement.
     • A left parenthesis is written before cond
     • A right parenthesis is written after the final executable statement list.

   Notes:

   • Many of the programming languages with this type of statement utilize a prefix notation.
     • Comparison of variables x and y and z might be written
       • (< x y)
       • (<= x y z)
     • If f is a function of 1 variable and g is a function of 2 variables, then these functions might be called with the syntax
       • (f x)
       • (g y z)
     • a statement may be a literal number or string, such as
       "no parentheses needed for a literal string---no operation or function involved"
   • Four examples of cond statements follows:

        (cond ((< x 0) "negative")
              ((< x 10) "single-digit")
              ((< x 100) "double-digit")
              (else "big")
        )
     
        (cond ((< x 0) "negative" (f x))
              ((>= x 0) "nonnegative")
        )
                                 
        (cond ((< x 0) "negative" (f x))
              (else "nonnegative" (g (f y) z) "nice example")
        )
     
        (cond (else "any value")
               

   Suppose the symbols condition and statement have been defined through a context-free grammar for Boolean expressions and statements, respectively, within the programming language.

   1. Give a context-free grammar that generates exactly the cond statements of this language.
   2. Write a state diagram of a PDA that accepts exactly the cond statements of this programming language (using condition and statement as simple input symbols).

Pushdown Automata

5. Using the alphabet Σ = {a, b}, consider the language L of strings with the same number of a's and b's.

   1. Is language L regular? Either give a DFA that accepts this language, or prove that no such DFA exists.
   2. Describe a pushdown automaton that accepts exactly those strongs in the language L, or explain why such a PDA cannot exist.

6. Consider the problem of checking whether an expression has balanced parentheses. For example, (())() is balanced, but ())()), ((), and ())(() are not. (In the last case, there are the same number of left and right parentheses, but the second right parenthesis appears before the a corresponding left parenthesis appears—that is, the left and right parentheses appear in the wrong order to balance.)

   The following diagram is a proposed to accepted expressions with matching parentheses:

   
   1. Does the proposed PDF accept strings with balanced parentheses or not? Explain.
   2. Write a state diagram for a PDA over the alphabet Σ = { {, }, <, >, (, ) } , which accepts exactly those strings where parentheses match and are properly nested,

      For example, the string ({<>}) is properly matched, but neither ({><}) nor ({<)}>) are.

created Fall, 2023 revised Fall, 2023
For more information, please contact Henry M. Walker at walker@cs.grinnell.edu.