Transcript Grammars - Columbus State University
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 → A → aA bA A → a }
Derivations
R = S → A → aA 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 q 0 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, q 0 the initial state and A the set of accepting states. Then a deterministic finite automaton (DFA) is a 5 tuple < Q , Σ , q 0 , 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 •
A Small Grammar
A Derivation
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
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
Two Parse Trees – Same Sentence
Derivation 1
Derivation 2
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
Associativity of Operators
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
Ambiguous If Grammar
Parse Trees for an If Statement
Unambiguous Grammar for If Statements
Extended BNF (EBNF)
• Optional part denoted by […]
BNF vs EBNF for Expressions
BNF: