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Specifying Languages

CS 480/680 – Comparative Languages. Specifying Languages. Specifying a Language. Informal methods Textbooks, tutorials, etc. Formal definitions Needed for exactness Compiler writers, etc. Like technical specifications for design Syntax – what expressions are legal?

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Specifying Languages

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  1. CS 480/680 – Comparative Languages Specifying Languages

  2. Specifying a Language • Informal methods • Textbooks, tutorials, etc. • Formal definitions • Needed for exactness • Compiler writers, etc. • Like technical specifications for design • Syntax – what expressions are legal? • Semantics – what should they do? Language Specification

  3. Context Free Grammars • Definition: A context-free grammar (CFG) is a 4-tuple, G = (V, , R, S) • V = variables, non-terminal symbols •  = terminal symbols (alphabet) • R = production rules • S = start symbol, S  V • V, , R, S are all finite Language Specification

  4. A Context Free Grammar • V = A, B •  = (a, b) • R = A  aAa A  B B  bBb B  A B   • S = A A  aAa A  aAa  aaAaa A  aAa  aaBaa A  B  aabBbaa B  bBb  aabbBbbaa B  bBb  aabbbba B   What language does this grammar specify? Language Specification

  5. Another Example CFG • V = A •  = (a, b) • R = A  aAa A  bAb A  a A  b A   • S = A What language does this grammar specify? Language Specification

  6. More examples • Write a CFG for the following languages:“All strings consisting of one or more a’s, followed by twice as many b’s.”“Strings with more a’s than b’s.” • There is an entire class devoted to formal specifications of languages: CS 466/666 – Introduction to Formal Languages Language Specification

  7. A CFG for Integer Arithmetic Expressions • V = <num>, <digit>, <op>, <expr> •  = [(, ), 0…9, , , , ] • R = <expr>  <num>  <expr> <op> <expr> (<expr>) <num>  <digit><num> | <digit> <digit>  0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 <op>   |  |  |  • S = <expr> Language Specification

  8. Derivation of an Expression • <expr>  <expr> <op><expr>  (<expr>) <op><expr>  (<expr>) + <expr>  (<expr> <op> <expr>) + <expr>  (<expr>  <expr>) + <expr>  (<num>  <expr>) + <expr>  (<digit><num>  <expr>) + <expr>  (<digit><digit>  <num>) + <expr>  (7<digit>  <num>) + <expr>  (73 <num>) + <expr>  (73 <digit>) + <expr>  (73 4) + <expr> (73 4) + <num> (73 4) + <digit>  (73 4) + 9 Language Specification

  9. Parse Trees • The derivation of an expression can also be expressed as a tree • This parse tree can help to resolve the interpretation of an expression • A compiler reads in the source code, and produces a parse tree before generating code. Language Specification

  10. Example Parse Tree • A simple CFG: E  E  E | 0 | 1 • E  E  E  E  E  E  1 E  E  1 0 E  1 0 1 E E   E E E E   E E E E 1 1 0 1 0 1 (1 – 0) – 1 1 – (0 – 1) Since there aretwo parse trees for this expression, the grammar is ambiguous. (Note: the order of substitution is not the issue.) Language Specification

  11. Ambiguity • If there are two parse trees for any expression, the grammar is syntactically ambiguous • Programming languages should be specified by unambiguous grammars • Otherwise it is difficult to determine the semantics of a syntactically correct statement • a = b + c * d; • Conventions (like operator precedence) can be used to clarify syntactically ambiguous grammars Language Specification

  12. Disambiguating a grammar • We can disambiguate our simple grammar by adding explicit parentheses: E  (E  E) | 0 | 1 • E  E  E  (E  E) E  (1 E) E  (1 0) E  (1 0) 1 • In general, you can remove ambiguity in a grammar by imposing state in the derivation. Language Specification

  13. An ambiguous grammar • S  aSb | aSbb |  • Language: L = {anbm | 0  n  m  2n} • The number of b’s is between the number of a’s and twice the number of a’s • aabbb can be generated two ways • Disambiguating: • Step 1: Produce all a’s with matching b’s • Step 2: Produce all extra b’s. • S  aSb | A | A  aAbb | abb Language Specification

  14. BNF • Backus-Naur Form • A standard notation for CFG’s, often used in specifying languages • Non-terminals (variables) are enclosed in <> • <expression>, <number> • <empty> =  •  is the production symbol () • | is used for “or” Language Specification

  15. BNF Example • <real-number> ::= <integer-part> . <fraction> • <integer-part> ::= <digit> | <integer-part> <digit> • <fraction> ::= <digit> | <digit><fraction> • <digit> ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 Can we generate the number “.7” from this grammar? Language Specification

  16. Extended BNF • Makes some constructs easier to specify • No more powerful than BNF • Rules: • { } = “zero or more” • [ ] = “optional” or, equivalently “zero or one” • | = “or” • ( ) are used for grouping Language Specification

  17. Arithmetic Expressions • <expression> ::= <expression> + <term> | <expression> – <term> | <term> • <term> ::= <term> * <factor> | <term> / <factor> | <factor> • <factor> ::= number | name | | (<expression>) • <expression> ::= <term> { (+| – ) <term> } • <term> ::= <factor> { (*| / ) <factor> } • <factor> ::= ‘(’ <expression> ‘)’ | number | name Language Specification

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