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 <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  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 <Program> begin <stmt_list> end <stmt> <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> <id> = <expr> A <expr> + <expr> <id> <expr> * <expr> B <id> <id> C A <assign> <id> = <expr> A <expr> * <expr> <expr> + <expr> <id> <id> <id> 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> A <expr> + <term> <expr> + <term> <factor> <term> <factor> <id> <factor> <id> A <id> C B <assign>  <id> = <expr> <id>  A | B | C <expr>  <expr> + <term> | <term> <term>  <term> * <factor> | <factor> <factor>  ( <expr> ) | <id> Addition operators associate from left to right

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 <logic_expr> then <stmt> else <stmt> if <logic_expr> then <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 <expr>  <expr> + <term> | <expr> - <term> | <term> <term>  <term> * <factor> | <term> / <factor> | <factor> EBNF: <expr>  <term> { (+ | - ) <term> } <term>  <factor> { ( * | / ) <factor>