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Grammars CPSC 5135 Formal Definitions • A symbol is a character. It represents an abstract entity that has no inherent meaning • Examples: a, A, 3, *, - ,= Formal Definitions • An alphabet is a finite set of symbols. • Examples: A = { a, b, c } B = { 0, 1 } Formal Definitions • A string (or word) is a finite sequence of symbols from a given alphabet. • Examples: S = { 0, 1 } is a alphabet 0, 1, 11010, 101, 111 are strings from S A = { a, b, c ,d } is an alphabet bad, cab, dab, d, aaaaa are strings from A Formal Definitions • A language is a set of strings from an alphabet. • The set can be finite or infinite. • Examples: A = { 0, 1} L1 = { 00, 01, 10, 11 } L2 = { 010, 0110, 01110,011110, …} Formal Definitions • A grammar is a quadruple G = (V, Σ, R, S) where 1) V is a finite set of variables (non-terminals), 2) Σ is a finite set of terminals, disjoint from V, 3) R is a finite set of rules. The left side of each rule is a string of one or more elements from V U Σ and whose right side is a string of 0 or more elements from V U Σ 4) S is an element of V and is called the start symbol Formal Definitions • Example grammar: • G = (V, Σ, R, S) V = { S, A } Σ = { a, b } R = { S → aA A → bA A→a } Derivations R = S → aA A → bA A→a • A derivation is a sequence of replacements , beginning with the start symbol, and replacing a substring matching the left side of a rule with the string on the right side of a rule S → aA → abA → abbA → abba Derivations • What strings can be generated from the following grammar? S → aBa B → aBa B→b Formal Definitions • The language generated by a grammar is the set of all strings of terminal symbols which are derivable from S in 0 or more steps. • What is the language generated by this grammar? • S→a S → aB B → aB B→a Kleene Closure • Let Σ be a set of strings. Σ* is called the Kleene closure of Σ and represents the set of all concatenations of 0 or more strings in Σ. • Examples Σ* = { 1 }* = { ø, 1, 11, 111, 1111, …} Σ* = { 01 }* = { ø, 01, 0101, 010101, …} Σ* = { 0 + 1 }* = set of all possible strings of 0’s and 1’s. (+ means union) Formal Definitions • A grammar G = (V,Σ, R, S) is right-linear if all rules are of the form: A → xB A→x where A, B ε V and x ε Σ* Right-linear Grammar • G = { V, Σ, R, S } V = { S, B } Σ = { a, b } R={ S → aS , S→B, B → bB , B→ε } What language is generated? Formal Definitions • A grammar G = (V,Σ, R, S) is left-linear if all rules are of the form: A → Bx A→x where A, B ε V and x ε Σ* Formal Definitions • A regular grammar is one that is either right or left linear. • Let Q be a finite set and let Σ be a finite set of symbols. Also let δ be a function from Q x Σ to Q, let q0 be a state in Q and let A be a subset of Q. We call each element of Q a state, δ the transition function, q0 the initial state and A the set of accepting states. Then a deterministic finite automaton (DFA) is a 5- tuple < Q , Σ , q0 , δ , A > • Every regular grammar is equivalent to a DFA Language Definition • Recognition – a machine is constructed that reads a string and pronounces whether the string is in the language or not. (Compiler) • Generation – a device is created to generate strings that belong to the language. (Grammar) Chomsky Hierarchy • Noam Chomsky (1950’s) described 4 classes of grammars 1) Type 0 – unrestricted grammars 2) Type 1 – Context sensitive grammars 3) Type 2 – Context free grammars 4) Type 3 – Regular grammars Grammars • Context-free and regular grammars have application in computing • Context-free grammar – each rule or production has a left side consisting of a single non-terminal Backus-Naur form (BNF) • BNF was used to describe programming language syntax and is similar to Chomsky’s context free grammars • A meta-language is a language used to describe another language • BNF is a meta-language for computer languages BNF • Consists of nonterminal symbols, terminal symbols (lexemes and tokens), and rules or productions • <if-stmt> → if <logical-expr> then <stmt> • <if-stmt> → if <logical-expr> then <stmt> else <stmt> • <if-stmt> → if <logical-expr> then <stmt> | if <logical-expr> then <stmt> else <stmt> A Small Grammar <program> begin <stmt_list> end <stmt_list> <stmt> | <stmt> ; <stmt_list> <stmt> <var> = <expression> <var> A | B | C <expression> <var> + <var> | <var> - <var> | <var> A Derivation <program> begin <stmt_list> end begin <stmt> end begin <var> = <expression> end begin A = <expression> end begin A = <var> + <var> end begin A = B + <var> end begin A = B + C end Terms • Each of the strings in a derivation is called a sentential form. • If the leftmost non-terminal is always the one selected for replacement, the derivation is a leftmost derivation. • Derivations can be leftmost, rightmost, or neither • Derivation order has no effect on the language generated by the grammar Derivations Yield Parse Trees <program> begin <stmt_list> end <Program> begin <stmt> end begin <var> = <expression> end begin <stmt_list> end begin A = <expression> end begin A = <var> + <var> end begin A = B + <var> end <stmt> begin A = B + C end <var> = <expression> A <var> + <var> B C Parse Trees • Parse trees describe the hierarchical structure of the sentences of the language they define. • A grammar that generates a sentence for which there are two or more distinct parse trees is ambiguous. An Ambiguous Grammar <assign> <id> = <expr> <id> A | B | C <expr> <expr> + <expr> | <expr> * <expr> | ( <expr> ) | <id> Two Parse Trees – Same Sentence <assign> <assign> <id> = <expr> <id> = <expr> A <expr> + <expr> A <expr> * <expr> <id> <expr> * <expr> <expr> + <expr> <id> B <id> <id> <id> <id> A C A B C Derivation 1 <assign> <id> = <expr> A = <expr> A = <expr> + <expr> A = <id> + <expr> A = B + <expr> A = B + <expr> * <expr> A = B + <id> * <expr> A = B + C * <expr> A = B + C * <id> A=B+C*A Derivation 2 <assign> <id> = <expr> A = <expr> A = <expr> * <expr> A = <expr> + <expr> * <expr> A = <id> + <expr> * <expr> A = B + <expr> * <expr> A = B + <id> * <expr> A = B + C * <expr> A = B + C * <id> A=B+C*A Ambiguity • Parse trees are used to determine the semantics of a sentence • Ambiguous grammars lead to semantic ambiguity - this is intolerable in a computer language • Often, ambiguity in a grammar can be removed Unambiguous Grammar <assign> <id> = <expr> <id> A | B | C <expr> <expr> + <term> | <term> <term> <term> * <factor> | <factor> <factor> ( <expr> ) | <id> • This grammar makes multiplication take precedence over addition Associativity of Operators <assign> <id> = <expr> <assign> <id> A | B | C <expr> <expr> + <term> | <id> = <expr> <term> <term> <term> * <factor> | A <expr> + <term> <factor> <factor> ( <expr> ) | <id> <expr> + <term> <factor> Addition operators associate from left to right <term> <factor> <id> <factor> <id> A <id> C B BNF • A BNF rule that has its left hand side appearing at the beginning of its right hand side is left recursive . • Left recursion specifies left associativity • Right recursion is usually used for associating exponetiation operators <factor> <exp> ** <factor> | <exp> <exp> ( <expr> ) | <id> Ambiguous If Grammar <stmt> <if_stmt> <if_stmt> if <logic_expr> then <stmt> | if <logic_expr> then <stmt> else <stmt> • Consider the sentential form: if <logic_expr> then if <logic_expr> then <stmt> else <stmt> Parse Trees for an If Statement <if_stmt> <if_stmt> If <logic_expr> then <stmt> else <stmt> If <logic_expr> then <stmt> <if_stmt> <if_stmt> if <logic_expr> then <stmt> if <logic_expr> then <stmt> else <stmt> Unambiguous Grammar for If Statements <stmt> <matched> | <unmatched> <matched> if <logic_expr> then <matched> else <matched> | any non-if statement <unmatched> if <logic_expr> then <stmt> | if <logic_expr> then <matched> else <unmatched> Extended BNF (EBNF) • Optional part denoted by […] <selection> if ( <expr> ) <stmt> [ else <stmt> ] • Braces used to indicate the enclosed part can be repeated indefinitely or left out <ident_list> <identifier> { , <identifier> } • Multiple choice options are put in parentheses and separated by the or operator | <for_stmt> for <var> := <expr> (to | downto) <expr> do <stmt> BNF vs EBNF for Expressions BNF: EBNF: <expr> <expr> + <term> <expr> <term> { (+ | - ) <term> } | <expr> - <term> | <term> <term> <factor> { ( * | / ) <factor> <term> <term> * <factor> | <term> / <factor> | <factor>