Defining syntax using CFGs


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1 Defining syntax using CFGs
2 Roadmap Last time Defined contextfree grammar This time CFGs for specifying a language s syntax Language membership List grammars Resolving ambiguity
3 CFG Review G = (N,Σ,P,S) " means derives in 1 or more steps CFG generates a string by applying productions until no nonterminals remain Example: Nested parens N = { Q } Σ = { (, ) } P = Q ( Q ) ε S = Q
4 Formal Definition of a CFG s Language Let G = (N,Σ,P,S) be a CFG. Then L(G) = w S " w where S is the start nonterminal of G, and w is a sequence that consists of (only) terminal symbols or ε
5 A CFG Defines a Language CFG productions define the syntax of a language 1. Prog begin end 2. semicolon Stmt 3. Stmt 4. Stmt id assign 5. id 6. plus id We call this notation BNF (for BackusNaur Form ) or extended BNF HTTP grammar using BNF:
6 List Grammars Useful to repeat a structure arbitrarily often semicolon Stmt Stmt ; Stmt List skews left ; Stmt ; Stmt ; Stmt
7 List Grammars Useful to repeat a structure arbitrarily often Stmt semicolon Stmt Stmt ; Stmt ; List skews right Stmt ; Stmt ;
8 List Grammars What if we allowed both skews? semicolon Stmt ; ; ; Stmt ;
9 Derivation Order Leftmost Derivation: always expand the leftmost nonterminal Rightmost Derivation: always expand the rightmost nonterminal Prog 1. Prog begin end 2. semicolon Stmt begin end 3. Stmt 4. Stmt id assign semicolon Stmt 5. id 6. plus id Leftmost expands this nonterminal Rightmost expands this nonterminal
10 Ambiguity Even with a fixed derivation order, it is possible to derive the same string in multiple ways For Grammar G and string w G is ambiguous if >1 leftmost derivation of w >1 rightmost derivation of w > 1 parse tree for w
11 Exercise Give a grammar G and a word w that has more than 1 leftmost derivation in G
12 Example: Ambiguous Grammars minus times lparen rparen Derive the string 47 * 3 (assume tokenization) Parse Tree 1 Parse Tree 2 minus times times minus
13 Why is Ambiguity Bad?
14 Why is Ambiguity Bad? Eventually, we ll be using CFGs as the basis for our parser Parsing is much easier when there is no ambiguity in the grammar The parse tree may mismatch user understanding! 47 * 3 Operator precedence minus times times minus
15 Resolving Grammar Ambiguity: Precedence minus times lparen rparen Intuitive problem Nonterminals are the same for both operators To fix precedence 1 nonterminal per precedence level Parse lowest level first
16 Resolving Grammar Ambiguity: Precedence minus times lparen rparen lowest precedence level first 1 nonterminal per precedence level Derive the string 47 * 3 minus minus times Factor times Factor Factor Factor Factor lparen rparen 4 7 3
17 Resolving Grammar Ambiguity: Precedence Fixed Grammar expr minus expr times Factor Factor lparen rparen Derive the string 47 * 3 Let s try to rebuild the wrong parse tree times minus Factor times 4 Factor Factor 7 3 We ll never be able to derive minus without parens Factor 3
18 Did we fix all ambiguity? Fixed Grammar minus times Factor Factor lparen rparen Derive the string minus Factor minus Factor Factor NO! These subtrees could have been swapped!
19 Where we are so far Precedence We want correct behavior on 4 7 * 9 A new nonterminal for each precedence level Associativity We want correct behavior on Minus should be left associative: a b c = (a b) c Problem: the recursion in a rule like minus
20 Definition: Recursion in Grammars A grammar is recursive in (nonterminal) X if X " αxγ for nonempty strings of symbols α and γ A grammar is leftrecursive in X if X " Xγ for nonempty string of symbols γ A grammar is rightrecursive in X if X " αx for nonempty string of symbols α
21 4 Resolving Grammar Ambiguity: Associativity Recognize leftassoc operators with leftrecursive productions Recognize rightassoc operators with rightrecursive productions minus Factor times Factor Factor lparen rparen Example: E E  E  T T F F 7 T F 9
22 Resolving Grammar Ambiguity: Associativity minus times Factor Factor Factor lparen rparen E Example: Let s try to rebuild the wrong parse tree again E  T T F 4 We ll never be able to derive minus without parens
23 Example Language of Boolean expressions bexp TRUE bexp FALSE bexp bexp OR bexp bexp bexp AND bexp bexp NOT bexp bexp LPAREN bexp RPAREN Add nonterminals so that OR has lowest precedence, then AND, then NOT. Then change the grammar to reflect the fact that both AND and OR are left associative. Draw a parse tree for the expression: true AND NOT true.
24 Another ambiguous example Stmt if Cond then Stmt if Cond then Stmt else Stmt Consider this word in this grammar: if a then if b then s else s2 How would you derive it?
25 Summary To understand how a parser works, we start by understanding contextfree grammars, which are used to define the language recognized by the parser. terminal symbol (non)terminal symbol grammar rule (or production) derivation (leftmost derivation, rightmost derivation) parse (or derivation) tree the language defined by a grammar ambiguous grammar
Defining syntax using CFGs
Defining syntax using CFGs Roadmap Last 8me Defined contextfree grammar This 8me CFGs for syntax design Language membership List grammars Resolving ambiguity CFG Review G = (N,Σ,P,S) means derives derives
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