Languages and Grammars MSU CSE 260

Outline Introduction: E xample Phrase-Structure Grammars: Terminology, Definition, Derivation, Language of a Grammar, Examples –Exercise 10.1 (1) Types of Phrase-Structure Grammars Derivation Trees: Example, Parsing –Exercise 10.1 (2, 3) Backus-Naur Form

Introduction In the English language, the grammar determines whether a combination of words is a valid sentence. Are the following valid sentences? –The large rabbit hops quickly.Yes –The frog writes neatly.Yes –Swims quickly mathematician.No Grammars are concerned with the syntax (form) of a sentence, and NOT its semantics (or meaning.)

English Grammar Sentence: noun phrase followed by verb phrase; Noun phrase: article adjective noun, or article noun; Verb phrase: verb adverb, or verb; Article: a, or the; Adjective: large, or hungry; Noun: rabbit, or mathematician, or frog; Verb: eats, or hops, or writes, or swims; Adverb: quickly, or wildly, or neatly;

Example Sentence Noun phrase verb phrase Article adjective noun verb phrase Article adjective noun verb adverb the adjective noun verb adverb the large noun verb adverb the large rabbit verb adverb the large rabbit hops adverb the large rabbit hops quickly

Grammars and Computation Grammars are used as a model of computation. Grammars are used to: –generate the words of a language, and –determine whether a word is in a language.

Phrase-Structure Grammars Terminology Definitions. A vocabulary (or alphabet) V is a finite, nonempty set of elements called symbols. A word (or sentence) over V is a string of finite length of elements of V. The empty string (or null string,) denoted by, is the string containing no symbols. The set of all words over V is denoted by V*. A language over V is a subset of V*.

Phrase-Structure Grammars A language can be specified by: –listing all the words in the language, or –giving a set of criteria satisfied by its words, or –using a grammar. A grammar provides: –a set of symbols, and –a set of rules, called productions, for producing words by replacing strings by other strings: w 0  w 1.

Phrase-Structure Grammar Definition A phrase-structure grammar G = (V, T, S, P) consists of: –a vocabulary V, –a subset T of V consisting of terminal elements, –a start symbol S from V, and –a set P of productions. The set N = V-T consists of nonterminal symbols. Every production in P must contain at least one nonterminal on its left side.

Phrase-structure Grammar Example G = {V, T, S, P}, where –V = {a, b, A, B, S}, –T = {a, b}, –S is the start symbol, and –P = {S  Aba, A  BB, B  ab, AB  b}.

Phrase-Structure Grammars Derivation Definition. Let G = (V, T, S,P) be a phrase-structure grammar. Let w 0 = lz 0 r and w 1 = lz 1 r be strings over V. –If z 0  z 1 is a production of G, we say that: w 1 is directly derivable from w 0 (denoted: w 0  w 1.) –If w 0, w 1, …, w n are strings over V such that: w 0  w 1, w 1  w 2, …, w n-1  w n, we say that: w n is derivable from w 0 (denoted: w 0  * w n.) Note. * should be on top of . –The sequence of all steps used to obtain w n from w 0 is called a derivation.

Example In the previous example grammar, the production: B  ab makes the string Aaba directly derivable from string ABa. –ABa  Aaba Also Aaba  BBaba  Bababa  abababa –using: A  BB, B  ab, and B  ab. So: ABa  * abababa  abababa is derivable from ABa.

Language of a Grammar Definition. Let G = (V, T, S, P) be a phrase-structure grammar. The language generated by G (or the language of G), denoted by L(G), is the set of all strings of terminals that are derivable from the start symbol S. L(G) = {w  T* | S  * w}.

Example Let G = {V, T, S, P} be the grammar where: –V = {S, 0, 1}, –T = {0, 1}, –P = {S  11S, S  0}. What is L(G)? –At any stage of the derivation we can either: add two 1s at the end of the string, or terminate the derivation by adding a 0 at the end of the string. –L(G)={0, 110, 11110, , …} = Set of all strings that begin with an even number of 1s and end with 0.

Exercise 10.1 (1)

Types of Grammars A type 0 (phrase-structure) grammar has no restrictions on its productions. A type 1 (or context-sensitive) grammar has productions only of forms: –w 1  w 2 with length of w 2  length of w 1, or –w 1 . A type 2 (or context-free) grammar has productions only of the form A  w 2, where A is a single nonterminal symbol.

Types of Grammars – cont. A type 3 (or regular) grammar has productions only of the form: –A  aB, or A  a, where A and B are nonterminal symbols, and a is a terminal symbol, or –S . Note. –Every type 3 grammar is a type 2 grammar –Every type 2 grammar is a type 1 grammar –Every type 1 grammar is a type 0 grammar

Types of Grammars - Summary Type Restrictions on productions w 1  w 2 0No restrictions 1l(w 1 )  l(w 2 ), or w 2 = 2w 1 =A where A  N 3w 1 =A, and w 2 =aB or w 2 =a, where A  N, B  N, a  T, or w 1 =S and w 2 =

Derivation Trees For type 2 (context-free) grammars: A derivation (or parse) tree, is an ordered rooted tree that represents a derivation in the language generated by a context-free grammar, where: –the root represents the starting symbol; –the internal vertices represent nonterminal symbols; –the leaves represent the terminal symbols; –for a production A  w, the vertex representing A will have children vertices that represent each symbol in w.

Example Derivation tree for: the hungry rabbit eats quickly sentence noun phrase verb phrase article adjective noun verb adverb the hungry rabbit eats quickly

Exercise 10.1 (2, 3)

Parsing To determine whether a string is in the language generated by a grammar, use: –Top-down parsing: Begin with S and attempt to derive the word by successively applying productions, or –Bottom-up parsing: Work backward: Begin by inspecting the word and apply productions backward.

Example Let G = {V, T, S, P} be the grammar where: –V = {a, b, c, A, B, C, S}, T = {a, b, c}, –Productions: Determine whether cbab is in L(G)? S  AB Top-down parsing: A  Ca S  AB B  Ba S  AB  CaB B  Cb S  AB  CaB  cbaB B  b S  AB  CaB  cbaB  cbab C  cb Bottom-up parsing: C  b Cab  cbab Ab  Cab  cbab AB  Ab  Cab  cbab S  AB  Ab  Cab  cbab

Backus-Naur Form Used with type 2 (context-free) grammars; like for specification of programming languages: –Use ::= instead of  –Enclose nonterminal symbols within –Group productions with same left side with symbol | Example. – ::= – ::= + | - – ::= | – ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9