10.1.1 A simple expression grammar

In this section, we extend the grammar example with code to represent a complete grammar, a parser and an evaluator. The reader may skip this section and the subsequent ones on the expression parser and the abstract syntax tree during the first reading of this book.

We use a grammar for describing aritemhetic expressions of digits using '+', '*' and parentheses:

Start: <Exp>
<Exp> ::= <Exp> "+" <Term> | <Term>
<Term> ::= <Term> "*" <Primary> | <Primary>
<Primary> ::= <Number> | "(" <Exp> ")"
<Number> ::= <Number> <Digit> | <Digit>
<Digit> ::= "0" | "1" | "2" | "3" | "4" | "5" | "6" | "7" | "8" | "9" 

The start symbol of the grammar is the nonterminal <Exp>.

The nonterminal symbols are: <Exp>, <Term>, <Primary>, <Number>, and <Digit>.

The terminal symbols are: "+", "*", "(", ")" and the digits "0" – "9".

We declare an ExpressionGrammar as a sub of Grammar:

ExpressionGrammar: obj Grammar
   -- declaration of symbols
   -- declaration of rules
   -- specification of start symbol

The symbols are declared as follows:

   expSy: obj Nonterminal("Exp")
   termSy: obj Nonterminal("Term")
   primarySy: obj Nonterminal("Primary")
   numberSy: obj Nonterminal("Number")
   digitSy: obj Nonterminal("Digit")
   add: obj Terminal("+")
   mult: obj Terminal("*")
   leftB: obj Terminal("(")
   rightB: obj Terminal(")")

To generate the rules, we introduce a method for each nonterminal. The one for <Exp> looks as follows:

   mkExpRule: addRule(expSy)
      S: ref SymbolList
      S := SymbolList.insertList((ExpSy,add,termSy))
      R.alternatives.insert(Alternative(S))
      S := SymbolList.insertList(termSy)
      alt := Alternative(S)
      R.alternatives.insert(alt)

As can bee seen, mkExpRule is a sub of addRule, which we have added to class Grammar. In addition, we have added class SymbolList with a print method:

class Grammar:
   ...
   class SymbolList: OrderedList(#Symbol)
       print:
          scan
             current.print
   addRule(L: ref Nonterminal): 
      R: ref Rule
      R := Rule
      R.leftSide := L
      inner(addRule)
      rules.insert(R) 

The method addRule has the leftside of the rule to be added a s a parameter. It generates a Rule-object and assign its reference to R, and assigns the leftSide of R. Then it executes inner(addRule) implying that the mainpart of mkExpRule is executed. When returning from inner, the Rule R is inserted into the list of rules.

The expression SymbolList.insertList((ExpSy,add,termSy)) may need an explanation:

1. First SymboList is evaluated creating a SymbolList-object returning a reference to this newly created object.
2. Then insertList((ExpSy,add,termSy)) is invoked on the reference to the new SymbolList-object.
3. The parameter of insertList is an array and the argument of the invocation is the array-literal (???) (ExpSy,add,termSy). InsertList inserts each element of the array in the newly generated SymbolList-object.

To complete the generation of ExpressionGrammer, we may add methods similar to mkExpRule for the other nonterminals, but leaves this an exercise for the reader. The ExpressionGrammer then looks as follows:

ExpressionGrammar: obj Grammar
   -- declaration of symbols
   ExpSy: obj Nontermial("Exp")
   ...
   -- declaration of rules
   mkExpRule: addRule(ExpSy)
      ...
   mkTermRule: addRule(TermSy)
      ...
   mkPrimaryRule: addRule(PrimarySy)
      ...
   mkNumberRule: addRule(NumberSy)
      ...
   mkDigitRule: addRule(DigitSy)
      ...
   -- specification of start symbol
   start := ExpSy
   mkExpRule
   ...

Parser and abstract syntax tree

Next we show how to write a parser for the expressions of our grammar and how to represent an expression by means of an abstract syntax tree. The reader may skip theses sections during a first reading.