# Lexing and Parsing

You could code your own lexer and parser from scratch. But many languages include tools for automatically generating lexers and parsers from formal descriptions of the syntax of a language. The ancestors of many of those tools are lex and yacc, which generate lexers and parsers, respectively; lex and yacc were developed in the 1970s for C.

As part of the standard distribution, OCaml provides lexer and parser generators named ocamllex and ocamlyacc. There is a more modern parser generator named menhir available through opam; menhir is "90% compatible" with ocamlyacc and provides significantly improved support for debugging generated parsers.

## Lexers

Lexer generators such as lex and ocamllex are built on the theory of deterministic finite automata, which you learned in CS 2800. Recall that such automata accept regular languages, which can be described with regular expressions. So, the input to a lexer generator is a collection of regular expressions that describe the tokens of the language. The output is an automaton implemented in a high-level language, such as C (for lex) or OCaml (for ocamllex).

That automaton itself takes files (or strings) as input, and each character of the file becomes an input to the automaton. Eventually the automaton either recognizes the sequence of characters it has received as a valid token in the language, in which case the automaton produces an output of that token and resets itself to being recognizing the next token, or rejects the sequence of characters as an invalid token.

## Parsers

Parser generators such as yacc and menhir are similarly built on the theory of automata, but in this case, pushdown automata, which are not always covered in CS 2800. They are like finite automata except they also maintain a stack onto which they can push and pop symbols. The stack enables them to accept a bigger class of languages, which are known as context-free languages (CFLs). One of the big improvements of CFLs over regular languages is that CFLs can express the idea that delimiters must be balanced—for example, that every opening parenthesis must be balanced by a closing parenthesis.

Just as regular languages can be expressed with a special notation (regular expressions), so can CFLs. Context-free grammars are used to describe CFLs. A context-free grammar is a set of production rules of that describe how one symbol can be replaced by other symbols. For example, the language of balanced parentheses, which includes strings such as (()) and ()() and (()()), but not strings such as ) or ((), is generated by these rules:

• $S \rightarrow (S)$
• $S \rightarrow SS$
• $S \rightarrow \epsilon$

The symbols occurring in those rules are $S$, (, and ). The $\epsilon$ denotes the empty string. Every symbol is either a nonterminal or a terminal, depending on whether it is a token of the language being described. $S$ is a nonterminal in the example above, and ( and ) are terminals.

In the next section we'll study Backus-Naur Form (BNF), which is a standard notation for context-free grammars. The input to a parser generator is typically a BNF description of the language's syntax. The output of the parser generator is a program that recognizes the language of the grammar. As input, that program expects the output of the lexer. As output, the program produces a value of the AST type that represents the string that was accepted. The programs output by the parser generator and lexer generator are thus dependent upon on another and upon the AST type.