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Understanding the Limits of Regular Expressions and the Role of Context-Free Languages

This note discusses the limitations of regular expressions in pattern recognition, particularly their inability to handle strings of the form anbn. It introduces context-free languages, as proposed by Noam Chomsky, as a more powerful method for pattern description. The Backus-Naur Form (BNF) is presented as a notation to express context-free languages. The document includes examples and conventions for BNF, along with a sample PL/0 program illustrating its use. Lastly, it guides the reader in writing a BNF grammar to validate specific string types.

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Understanding the Limits of Regular Expressions and the Role of Context-Free Languages

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  1. COP 3402 System Software Recitation Notes 21st September 2007

  2. Limits Of Regular Expressions • Regular expressions are very useful for recognizing patterns • However, there are some patterns they cannot handle. • An Example is strings of the form anbn

  3. Context Free Languages • Since regular expressions cannot handle some important patterns, we need a more powerful method of describing patterns. • Context Free Languages are one such mechanism • Proposed by Noam Chomsky in the 1950s.

  4. BNF • BNF is a way to express Context free languages. • First used in ALGOL 58. • BNF stands for Backus-Naur Form • Named after two members of the ALGOL design team.

  5. BNF Example for PL/0 • Next slide contains a BNF description that can verify the correctness of any PL/0 program. • Conventions: • Variables/Tokens are written between “<“ and ”>”. • The “→” symbol can be read as “is defined as”. • “ε “ is the empty string.

  6. BNF Grammer for PL/0

  7. Sample PL/0 Programme • const m = 7, n = 85; • var i,x,y,z,q,r; • procedure mult; • var a, b; • begin • a := x; b := y; z := 0; • while b > 0 do • begin • if odd x then z := z+a; • a := 2*a; • b := b/2; • end • end; • begin • x := m; • y := n; • call mult; • end.

  8. Syntax Graphs Visual Way to represent BNF Grammars

  9. Syntax Graphs Cont’d

  10. Extended BNF • Conventions • Optional Constructions are enclosed in square braces ‘[‘ and ‘]’. • Constructs enclosed in curly braces ‘{‘ and ‘}’ are repeated zero or more times. • Example on next slide

  11. EBNF Example for PL/0

  12. Another Example • Try to write a BNF Grammar that validates strings of the type anbn. • Hint: anbn can be rewritten as aan-1bn-1b.

  13. Solution • <string> → a<string>b | ε • Will ensure that • string always has the same number of a’s and b’s • The a’s always come before the b’s in the string. • As to keeping track of the ‘n’, we don’t care in this example because the number of a’s and b’s will always be matched.s

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