Context-Free Languages Wen-Guey Tzeng Department of Computer - - PowerPoint PPT Presentation

Context-Free Languages

Wen-Guey Tzeng Department of Computer Science National Chiao Tung University

Context-Free Grammars

• A grammar G=(V, T, S, P) is context-free if all

productions in P are of form Ax, where AV, x(VT)*

– The left side has only one variable.

• Example,

G = ({S,A,B}, {a,b}, S, {SaAb|bBa, AaAb|, BbBb|})

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• Derivation:

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• L(G) = {w* | S * w}
• A language L is context-free if and only if

there is a context-free grammar G such that L=L(G).

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Examples

• G=({S}, {a, b}, S, P), with P={SaSa|bSb|}

– S  aSa  aaSaa  aabSbaa  aabbaa=aabbaa – L(G) = {wwR : w{a, b}*}

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• S abB, AaaBb|, BbbAa

– L(G) = {ab(bbaa)nbba(ba)n : n0} ?

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Design cfg’s

• Give a cfg for L={anbn : n0}

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• Give a cfg for L={anbm : n>m}

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• Give a cfg for L={anbm : nm0}

– Idea1:

• parse L into two cases (not necessarily disjoint)

L1={anbm : n>m}  L2={anbm : n<m}.

• Then, construct productions for L1 and L2, respectively.

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• Give a cfg for L={anbm : nm0}

– Idea2:

• produce the same amount of a’s and b’s, then extra a’s
• r b’s

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• Give a cfg for L={anbmck : m=n+k}

– Match ‘a’ and ‘b’, ‘b’ and ‘c’

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• Give a cfg for L={anbmck : m>n+k}

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• Give a cfg for L={w{a,b}* : na(w)=nb(w)}

– Find the “recursion”

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• Give a cfg for L={w{a,b}* : na(w)>nb(w)}

– Find relation with other language – Consider starting with ‘a’ and ‘b’, respectively

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Leftmost and rightmost derivation

• G=({A, B, S}, {a, b}, S, P), where P contains

SAB, AaaA, A, BBb, B 

– L(G)={a2nbm : n, m0}

• For string aab

– Rightmost derivation – Leftmost derivation

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Derivation (parse) tree

• AabABc

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• SaAB, AbBb, BA|

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Some comments

• Derivation trees represent no orders of

derivation

• Leftmost/rightmost derivations correspond to

depth-first visiting of the tree

• Derivation tree and derivation order are very

important to “programming language” and “compiler design”

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Grammar for C

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main() { int i=1; printf("i starts out life as %d.", i); i = add(1, 1); /* Function call */ printf(" And becomes %d after function is executed.\n", i); }

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Parsing and ambiguity

• Parsing of wL(G): find a sequence of

productions by which wL(G) is derived.

• Questions: given G and w

– Is wL(G) ? (membership problem) – Efficient way to determine whether wL(G) ? – How is wL(G) parsed ? (build the parsing tree) – Is the parsing unique ?

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Exhaustive search/top down parsing

• SSS|aSb|bSa|
• Determine aabbL(G) ?

– 1st round: (1) SSS; (2) SaSb; (3) SbSa; (4) S – 2nd round:

• From (1), SSSSSS, SSSaSbS, SSSbSaS,

SSS S

• From (2), SaSbaSSb, SaSbaaSbb, SaSbabSab,

SaSbab

– 3rd round: …

• Drawback: inefficiency
• Other ways ?

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• If no productions of form A or AB, the

exhaustive search for wL(G) can be done in |P|+|P|2+…+|P|2|w| = O(|P|2|w|+1)

– Consider the leftmost parsing method. – w can be obtained within 2|w| derivations.

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Bottom up parsing

• To reduce a string w to the start variable S
• SaSb|

– w=aabb  aaSbb  aSb  S

• Efficiency: O(|w|3)

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Linear-time parsing

• Simple grammar (s-grammar)

– All productions are of form Aax, where x(V T)* – Any pair (A, a) occurs at most once in P.

• Example: SaS|bSS|c

– Parsing for ababccc

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Ambiguous grammars

• G is ambiguous if some wL(G) has two

derivation trees.

• Example: SaSb|SS|

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Example from programming languages

• C-like grammar for arithmetic expressions.

G=({E, I}, {a, b, c, +, x, (, )}, E, P), where P contains EI EE+E EExE E(E) Ia|b|c

• w=a+bxc has two derivation trees

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Ambiguous languages

• A cfl L is inherently ambiguous if any cfg G

with L(G)=L is ambiguous. Otherwise, it is unambiguous.

• Note: an unambiguous language may have

ambiguous grammar.

• Example: L={anbncm} {anbmcm} is inherently

ambigous.

– Hard to prove.

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CFG and Programming Languages

• Programming language: syntax + semantics
• Syntax is defined by a grammar G

– <expression> ::= <term> | <expression> + <term> <term> ::= <factor> | <term> * <factor> – <while_statement> ::= while <expression><statement>

• Syntax checking in compilers is done by a parser

– Is a program p grammatically correct ? – Is pL(G) ? – We need efficient parsers.

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Restricted CFG Programming Languages

• Goal:

– Its expression power is enough. – It has no ambiguity. if then if then else

If then “if then else” If then “if then” else

– There exists an efficient parser.

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• C -- LR(1)
• PASCAL -- LL(1)
• Hierarchy of classes of context-free languages

– LL(1)  LR(0)  LR(1)=DCFL  LR(2)  …  CFL

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Syntactic Correctness

• Lexical analyzer produces a stream of tokens

x = y +2.1  <id> <op> <id> <op> <real>

• Parser (syntactic analyzer) verifies that this

token stream is syntactically correct by constructing a valid parse tree for the entire program

– Unique parse tree for each language construct – Program = collection of parse trees rooted at the top by a special start symbol

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CFG For Floating Point Numbers

::= stands for production rule; <…> are non-terminals; | represents alternatives for the right-hand side of a production rule

Sample parse tree:

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CFG For Balanced Parentheses

Sample derivation: <balanced>  ( <balanced> )  (( <balanced> ))  (( <empty> ))  (( ))

Could we write this grammar using regular expressions or DFA? Why?

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CFG For Decimal Numbers (Redux)

Sample top-down leftmost derivation: <num>  <digit> <num>  7 <num>  7 <digit> <num>  7 8 <num>  7 8 <digit>  7 8 9

This grammar is right-recursive

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Compiler-compiler

• A compiler-compiler is a program that

generates a compiler from a defined grammar

• Parser can be built automatically from the BNF

description of the language’s CFG

• Tools: yacc, Bison

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Compiler- compiler

Compiler: parser + code generator Execution code Programming language grammar G=(V, T, S, P) program Input data result

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