lexers vs parsers

What parsers and lexers have in common:

1. They read symbols of some alphabet from their input.

   • Hint: The alphabet doesn’t necessarily have to be of letters. But it
     has to be of symbols which are atomic for the language
     understood by parser/lexer.
   • Symbols for the lexer: ASCII characters.
   • Symbols for the parser: the particular tokens, which are terminal symbols of their grammar.

2. They analyse these symbols and try to match them with the grammar of the language they understood.
   

   • Here’s where the real difference usually lies. See below for more.
   • Grammar understood by lexers: regular grammar (Chomsky’s level 3).
   • Grammar understood by parsers: context-free grammar (Chomsky’s level 2).

3. They attach semantics (meaning) to the language pieces they find.

   • Lexers attach meaning by classifying lexemes (strings of symbols from the input) as the particular tokens. E.g. All these lexemes: *, ==, <=, ^ will be classified as “operator” token by the C/C++ lexer.
   • Parsers attach meaning by classifying strings of tokens from the input (sentences) as the particular nonterminals and building the parse tree. E.g. all these token strings: [number][operator][number], [id][operator][id], [id][operator][number][operator][number] will be classified as “expression” nonterminal by the C/C++ parser.

4. They can attach some additional meaning (data) to the recognized elements.

   • When a lexer recognizes a character sequence constituting a proper number, it can convert it to its binary value and store with the “number” token.
   • Similarly, when a parser recognize an expression, it can compute its value and store with the “expression” node of the syntax tree.

5. They all produce on their output a proper sentences of the language they recognize.

   • Lexers produce tokens, which are sentences of the regular language they recognize. Each token can have an inner syntax (though level 3, not level 2), but that doesn’t matter for the output data and for the one which reads them.
   • Parsers produce syntax trees, which are representations of sentences of the context-free language they recognize. Usually it’s only one big tree for the whole document/source file, because the whole document/source file is a proper sentence for them. But there aren’t any reasons why parser couldn’t produce a series of syntax trees on its output. E.g. it could be a parser which recognizes SGML tags sticked into plain-text. So it’ll tokenize the SGML document into a series of tokens: [TXT][TAG][TAG][TXT][TAG][TXT]....

As you can see, parsers and tokenizers have much in common. One parser can be a tokenizer for other parser, which reads its input tokens as symbols from its own alphabet (tokens are simply symbols of some alphabet) in the same way as sentences from one language can be alphabetic symbols of some other, higher-level language. For example, if * and - are the symbols of the alphabet M (as “Morse code symbols”), then you can build a parser which recognizes strings of these dots and lines as letters encoded in the Morse code. The sentences in the language “Morse Code” could be tokens for some other parser, for which these tokens are atomic symbols of its language (e.g. “English Words” language). And these “English Words” could be tokens (symbols of the alphabet) for some higher-level parser which understands “English Sentences” language. And all these languages differ only in the complexity of the grammar. Nothing more.

So what’s all about these “Chomsky’s grammar levels”? Well, Noam Chomsky classified grammars into four levels depending on their complexity:

• Level 3: Regular grammars

  They use regular expressions, that is, they can consist only of the symbols of alphabet (a,b), their concatenations (ab,aba,bbb etd.), or alternatives (e.g. a|b).
  They can be implemented as finite state automata (FSA), like NFA (Nondeterministic Finite Automaton) or better DFA (Deterministic Finite Automaton).
  Regular grammars can’t handle with nested syntax, e.g. properly nested/matched parentheses (()()(()())), nested HTML/BBcode tags, nested blocks etc. It’s because state automata to deal with it should have to have infinitely many states to handle infinitely many nesting levels.

• Level 2: Context-free grammars

  They can have nested, recursive, self-similar branches in their syntax trees, so they can handle with nested structures well.
  They can be implemented as state automaton with stack. This stack is used to represent the nesting level of the syntax. In practice, they’re usually implemented as a top-down, recursive-descent parser which uses machine’s procedure call stack to track the nesting level, and use recursively called procedures/functions for every non-terminal symbol in their syntax.
  But they can’t handle with a context-sensitive syntax. E.g. when you have an expression x+3 and in one context this x could be a name of a variable, and in other context it could be a name of a function etc.

• Level 1: Context-sensitive grammars

• Level 0: Unrestricted grammars
  Also called recursively enumerable grammars.