# Context-Free Grammars

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1 Context-Free Grammars

2 Describing Languages We've seen two models for the regular languages: Automata accept precisely the strings in the language. Regular expressions describe precisely the strings in the language. Finite automata recognize strings in the language. Perform a computation to determine whether a specific string is in the language. Regular expressions match strings in the language. Describe the general shape of all strings in the language.

3 Context-Free Grammars A context-free grammar (or CFG) is an entirely different formalism for defining a class of languages. Goal: Give a procedure for listing off all strings in the language. CFGs are best explained by example...

4 Arithmetic Expressions Suppose we want to describe all legal arithmetic expressions using addition, subtraction, multiplication, and division. Here is one possible CFG: E int E E Op E E (E) Op + Op - Op * Op / E E Op E E Op (E) E Op (E Op E) E * (E Op E) int * (E Op E) int * (int Op E) int * (int Op int) int * (int + int)

5 Arithmetic Expressions Suppose we want to describe all legal arithmetic expressions using addition, subtraction, multiplication, and division. Here is one possible CFG: E int E E Op E E (E) Op + Op - Op * Op / E E Op E E Op int int Op int int / int

6 Context-Free Grammars Formally, a context-free grammar is a collection of four objects: A set of nonterminal symbols (also called variables), A set of terminal symbols (the alphabet of the CFG) A set of production rules saying how each nonterminal can be converted by a string of terminals and nonterminals, and A start symbol (which must be a nonterminal) that begins the derivation. E int E E Op E E (E) Op + Op - Op * Op /

7 Some CFG Notation Capital letters in Bold Red Uppercase will represent nonterminals. i.e. A, B, C, D Lowercase letters in blue monospace will represent terminals. i.e. t, u, v, w Lowercase Greek letters in gray italics will represent arbitrary strings of terminals and nonterminals. i.e. α, γ, ω

8 A Notational Shorthand E int E Op E (E) Op + - * /

9 Derivations E E Op E int (E) Op + * - / E E Op E E Op (E) E Op (E Op E) E * (E Op E) int * (E Op E) int * (int Op E) int * (int Op int) int * (int + int) A sequence of steps where nonterminals are replaced by the right-hand side of a production is called a derivation. If string α derives string ω, we write α * ω. In the example on the left, we see E * int * (int + int).

10 The Language of a Grammar If G is a CFG with alphabet Σ and start symbol S, then the language of G is the set L(G) = { ω Σ* S * ω } That is, L( G) is the set of strings derivable from the start symbol. Note: ω must be in Σ*, the set of strings made from terminals. Strings involving nonterminals aren't in the language.

11 More Context-Free Grammars Chemicals! C 19 H 14 O 5 S Cu 3 (CO 3 ) 2 (OH) 2 - MnO 4 S 2- Form Cmp Cmp Ion Cmp Term Term Num Cmp Cmp Term Elem (Cmp) Elem H He Li Be B C Ion + - IonNum + IonNum - IonNum Num 1 IonNum

12 CFGs for Chemistry Form Cmp Cmp Ion Cmp Term Term Num Cmp Cmp Term Elem (Cmp) Elem H He Li Be B C Ion + - IonNum + IonNum - IonNum Num 1 IonNum Form Cmp Ion Cmp Cmp Ion Cmp Term Num Ion Term Term Num Ion Elem Term Num Ion Mn Term Num Ion Mn Elem Num Ion MnO Num Ion MnO IonNum Ion MnO 4 Ion - MnO 4

13 CFGs for Programming Languages BLOCK STMT { STMTS } STMTS ε STMT STMTS STMT EXPR; if (EXPR) BLOCK while (EXPR) BLOCK do BLOCK while (EXPR); BLOCK EXPR var const EXPR + EXPR EXPR EXPR EXPR = EXPR... { } var = var * var; if (var) var = const; while (var) { var = var + const; }

14 Context-Free Languages A language L is called a context-free language (or CFL) iff there is a CFG G such that L = L( G). Questions: What languages are context-free? How are context-free and regular languages related?

15 From Regexes to CFGs CFGs don't have the Kleene star, parenthesized expressions, or internal operators. However, we can convert regular expressions to CFGs as follows: S a*b

16 From Regexes to CFGs CFGs don't have the Kleene star, parenthesized expressions, or internal operators. However, we can convert regular expressions to CFGs as follows: S Ab A Aa ε

17 From Regexes to CFGs CFGs don't have the Kleene star, parenthesized expressions, or internal operators. However, we can convert regular expressions to CFGs as follows: S a(b c*)

18 From Regexes to CFGs CFGs don't have the Kleene star, parenthesized expressions, or internal operators. However, we can convert regular expressions to CFGs as follows: S ax X b C C Cc ε

19 Regular Languages and CFLs Theorem: Every regular language is context-free. Proof Idea: Use the construction from the previous slides to convert a regular expression for L into a CFG for L.

20 The Language of a Grammar Consider the following CFG G: S asb ε What strings can this generate? a a a a a a Sb b b b b b L(G) = { a n b n n N }

21 All Languages Regular Languages CFLs

22

23 Designing CFGs Like designing DFAs, NFAs, and regular expressions, designing CFGs is a craft. When thinking about CFGs: Think recursively: Build up bigger structures from smaller ones. Have a construction plan: Know in what order you will build up the string.

24 Designing CFGs Let Σ = {a, b} and let L = {w Σ* w is a palindrome } We can design a CFG for L by thinking inductively: Base case: ε, a, and b are palindromes. If ω is a palindrome, then aωa and bωb are palindromes. S ε a b asa bsb

25 Designing CFGs Let Σ = {(, )} and let L = {w Σ* w is a string of balanced parentheses } We can think about how we will build strings in this language as follows: The empty string is balanced. Any two strings of balanced parentheses can be concatenated. Any string of balanced parentheses can be parenthesized. S SS (S) ε

26 Designing CFGs: Watch Out! When designing CFGs, remember that each nonterminal can be expanded out independently of the others. Let Σ = {a, } and let L = {an a n n N }. Is the following a CFG for L? S X X S X X X ax ε ax X aax X aa X aa ax aa a

27 Finding a Build Order Let Σ = {a, } and let L = {an a n n N }. To build a CFG for L, we need to be more clever with how we construct the string. Idea: Build from the ends inward. Gives this grammar: S asa S asa aasaa aaasaaa aaa aaa

28 Designing CFGs: A Caveat Let Σ = {l, r} and let L = {w Σ* w has the same number of l's and r's } Is this a grammar for L? S lsr rsl ε Can you derive the string lrrl?

29 Designing CFGs: A Caveat When designing a CFG for a language, make sure that it generates all the strings in the language and never generates a string outside the language. The first of these can be tricky make sure to test your grammars! You'll design your own CFG for this language on the next problem set.

30 CFG Caveats II Is the following grammar a CFG for the language { a n b n n N }? S asb What strings can you derive? Answer: None! What is the language of the grammar? Answer: Ø When designing CFGs, make sure your recursion actually terminates!

31 Parse Trees

32 Parse Trees E E E Op E int Op E int * E int * (E) int * (E Op E) int * (int Op E) int * (int + E) int * (int + int) E Op int * E E E Op E ( int + int ) E E Op E int (E) Op + * - /

33 Parse Trees A parse tree is a tree encoding the steps in a derivation. Each internal node is labeled with a nonterminal. Each leaf node is labeled with a terminal. Reading the leaves from left to right gives the string that was produced.

34 Parsing Given a context-free grammar, the problem of parsing a string is to find a parse tree for that string. Applications to compilers: Given a CFG describing the structure of a programming language and an input program (string), recover the parse tree. The parse tree represents the structure of the program what's declared where, how expressions nest, etc.

35 Challenges in Parsing

36 A Serious Problem E E E Op E E Op E E Op E E Op E int * int + int int * (int + int) int * int + int (int * int) + int E E Op E int Op + * - /

37 Ambiguity A CFG is said to be ambiguous if there is at least one string with two or more parse trees. Note that ambiguity is a property of grammars, not languages: there can be multiple grammars for the same language, where some are ambiguous and some aren't. Some languages are inherently ambiguous: there are no unambiguous grammars for those languages.

38 Resolving Ambiguity Designing unambiguous grammars is tricky and requires planning from the start. It's hard to start with an ambiguous grammar and to manually massage it into an unambiguous one. Often, have to throw the whole thing out and start over.

39 Resolving Ambiguity We have just seen that this grammar is ambiguous: Goals: E E Op E int Op + - * / Eliminate the ambiguity from the grammar. Make the only parse trees for the grammar the ones corresponding to operator precedence.

40 Operator Precedence Can often eliminate ambiguity from grammars with operator precedence issues by building precedences into the grammar. Since * and / bind more tightly than + and -, think of an expression as a series of blocks of terms multiplied and divided together joined by +s and -s. int * int * int + int * int - int

41 Operator Precedence Can often eliminate ambiguity from grammars with operator precedence issues by building precedences into the grammar. Since * and / bind more tightly than + and -, think of an expression as a series of blocks of terms multiplied and divided together joined by +s and -s. int * int * int + int * int - int

42 Rebuilding the Grammar Idea: Force a construction order where First decide how many blocks there will be of terms joined by + and -. Then, expand those blocks by filling in the integers multiplied and divided together. One possible grammar: S T T + S T - S T int int * T int / T

43 An Unambiguous Grammar S S S S T T T T T T int * int + int int + int * int S T T + S T S T int int * T int / T

44 Summary Context-free grammars give a formalism for describing languages by generating all the strings in the language. Context-free languages are a strict superset of the regular languages. CFGs can be designed by finding a build order for a given string. Ambiguous grammars generate some strings with two different parse trees.

45 Next Time Turing Machines What does a computer with unbounded memory look like? How do you program them?

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