2.3 Exercises for Section 2.3

2.3.1

Construct a syntax-directed translation scheme that translates arithmetic expressions from infix notation into prefix notation in which an operator appears before its operands; e.g. , -xy is the prefix notation for x - y. Give annotated parse trees for the inputs 9-5+2 and 9-5*2.

Answer

productions:

expr -> expr + term
      | expr - term
      | term
term -> term * factor
      | term / factor
      | factor
factor -> digit | (expr)

translation schemes:

expr -> {print("+")} expr + term
      | {print("-")} expr - term
      | term
term -> {print("*")} term * factor
      | {print("/")} term / factor
      | factor
factor -> digit {print(digit)}
        | (expr)

2.3.2

Construct a syntax-directed translation scheme that translates arithmetic expressions from postfix notation into infix notation. Give annotated parse trees for the inputs 95-2 and 952-.

Answer

productions:

expr -> expr expr +
      | expr expr -
      | expr expr *
      | expr expr /
      | digit

translation schemes:

expr -> expr {print("+")} expr +
      | expr {print("-")} expr -
      | {print("(")} expr {print(")*(")} expr {print(")")} *
      | {print("(")} expr {print(")/(")} expr {print(")")} /
      | digit {print(digit)}

Another reference answer

E -> {print("(")} E {print(op)} E {print(")"}} op | digit {print(digit)}

2.3.3

Construct a syntax-directed translation scheme that translates integers into roman numerals.

Answer

assistant function:

repeat(sign, times) // repeat('a',2) = 'aa'

translation schemes:

num -> thousand hundred ten digit
       { num.roman = thousand.roman || hundred.roman || ten.roman || digit.roman;
         print(num.roman)}
thousand -> low {thousand.roman = repeat('M', low.v)}
hundred -> low {hundred.roman = repeat('C', low.v)}
         | 4 {hundred.roman = 'CD'}
         | high {hundred.roman = 'D' || repeat('X', high.v - 5)}
         | 9 {hundred.roman = 'CM'}
ten -> low {ten.roman = repeat('X', low.v)}
     | 4 {ten.roman = 'XL'}
     | high {ten.roman = 'L' || repeat('X', high.v - 5)}
     | 9 {ten.roman = 'XC'}
digit -> low {digit.roman = repeat('I', low.v)}
       | 4 {digit.roman = 'IV'}
       | high {digit.roman = 'V' || repeat('I', high.v - 5)}
       | 9 {digit.roman = 'IX'}
low -> 0 {low.v = 0}
     | 1 {low.v = 1}
     | 2 {low.v = 2}
     | 3 {low.v = 3}
high -> 5 {high.v = 5}
      | 6 {high.v = 6}
      | 7 {high.v = 7}
      | 8 {high.v = 8}

2.3.4

Construct a syntax-directed translation scheme that trans­ lates roman numerals into integers.

Answer

productions:

romanNum -> thousand hundred ten digit
thousand -> M | MM | MMM | ε
hundred -> smallHundred | C D | D smallHundred | C M
smallHundred -> C | CC | CCC | ε
ten -> smallTen | X L | L smallTen | X C
smallTen -> X | XX | XXX  | ε
digit -> smallDigit | I V | V smallDigit | I X
smallDigit -> I | II | III | ε

translation schemes:

romanNum -> thousand hundred ten digit {romanNum.v = thousand.v || hundred.v || ten.v || digit.v; print(romanNun.v)}
thousand -> M {thousand.v = 1}
          | MM {thousand.v = 2}
          | MMM {thousand.v = 3}
          | ε {thousand.v = 0}
hundred -> smallHundred {hundred.v = smallHundred.v}
         | C D {hundred.v = smallHundred.v}
         | D smallHundred {hundred.v = 5 + smallHundred.v}
         | C M {hundred.v = 9}
smallHundred -> C {smallHundred.v = 1}
              | CC {smallHundred.v = 2}
              | CCC {smallHundred.v = 3}
              | ε {hundred.v = 0}
ten -> smallTen {ten.v = smallTen.v}
     | X L  {ten.v = 4}
     | L smallTen  {ten.v = 5 + smallTen.v}
     | X C  {ten.v = 9}
smallTen -> X {smallTen.v = 1}
          | XX {smallTen.v = 2}
          | XXX {smallTen.v = 3}
          | ε {smallTen.v = 0}
digit -> smallDigit {digit.v = smallDigit.v}
       | I V  {digit.v = 4}
       | V smallDigit  {digit.v = 5 + smallDigit.v}
       | I X  {digit.v = 9}
 smallDigit -> I {smallDigit.v = 1}
            | II {smallDigit.v = 2}
            | III {smallDigit.v = 3}
            | ε {smallDigit.v = 0}

2.3.5

Construct a syntax-directed translation scheme that translates postfix arithmetic expressions into equivalent prefix arithmetic expressions.

Answer

production:

expr -> expr expr op | digit

translation scheme:

expr -> {print(op)} expr expr op | digit {print(digit)}