# Context-Free Grammars

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1 Context-Free Grammars

2 Describing Languages We've seen two models for the regular languages: Finite 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 replaced 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 E Op E E (E) Op + Op - Op * Op /

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

10 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).

11 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.

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

13 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not have the regular expression operators * or. However, we can convert regular expressions to CFGs as follows: S a*b

14 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not have the regular expression operators * or. However, we can convert regular expressions to CFGs as follows: S Ab A Aa ε

15 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not have the regular expression operators * or. However, we can convert regular expressions to CFGs as follows: S a(b c*)

16 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not have the regular expression operators * or. However, we can convert regular expressions to CFGs as follows: S ax X b C C Cc ε

17 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. Problem Set Exercise: Instead, show how to convert a DFA/NFA into a CFG.

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

19 All Languages Regular Languages CFLs

20 Why the Extra Power? Why do CFGs have more power than regular expressions? Intuition: Derivations of strings have unbounded memory. S asb ε a a a a b b b b

21 Time-Out for Announcements!

22 Problem Set Seven Problem Set Six was due at the start of today's lecture. Want to use late days? Submit up to Monday at 3:00PM. Problem Set Seven goes out now. It's due next Friday. Play around with the Myhill-Nerode theorem and the limits of regular languages! Play around with your very own CFGs!

23 Midterms Graded Midterms have been graded. They're available for pickup in the Gates building. SCPD students: we've sent the exams back to the SCPD office. You should hear back from them soon. Solutions and stats are available in the Gates building in the normal handout filing cabinet.

25 Back to CS103!

26 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. Store information in nonterminals: Have each nonterminal correspond to some useful piece of information.

27 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

28 Designing CFGs Let Σ = {(, )} and let L = {w Σ* w is a string of balanced parentheses } Some sample strings in L: ((())) (())() (()())(()()) ((((()))(()))) ε ()()

29 Designing CFGs Let Σ = {(, )} and let L = {w Σ* w is a string of balanced parentheses } Let's think about this recursively. Base case: the empty string is a string of balanced parentheses. Recursive step: Look at the closing parenthesis that matches the first open parenthesis. ((()(()))(()))(())((()))

30 Designing CFGs Let Σ = {(, )} and let L = {w Σ* w is a string of balanced parentheses } Let's think about this recursively. Base case: the empty string is a string of balanced parentheses. Recursive step: Look at the closing parenthesis that matches the first open parenthesis. (( ( ) ( ( ) ) )( ( ) ))(())(( ( ) ))

31 Designing CFGs Let Σ = {(, )} and let L = {w Σ* w is a string of balanced parentheses } Let's think about this recursively. Base case: the empty string is a string of balanced parentheses. Recursive step: Look at the closing parenthesis that matches the first open parenthesis. ( ( ) ( ( ) ) )( ( ) ) (())(( ( ) ))

32 Designing CFGs Let Σ = {(, )} and let L = {w Σ* w is a string of balanced parentheses } Let's think about this recursively. Base case: the empty string is a string of balanced parentheses. Recursive step: Look at the closing parenthesis that matches the first open parenthesis. Removing the first parenthesis and the matching parenthesis forms two new strings of balanced parentheses. S (S)S ε

33 Designing CFGs: A Caveat Let Σ = {a, b} and let L = {w Σ* w has the same number of a's and b's } Is this a CFG for L? S asb bsa ε Can you derive the string abba?

34 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.

35 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!

36 CFG Caveats III 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 X ax ε S X X ax X aax X aa X aa ax aa a

37 Finding a Build Order Let Σ = {a, } and let L = {a n a n n N }. To build a CFG for L, we need to be more clever with how we construct the string. If we build the strings of a's independently of one another, then we can't enforce that they have the same length. Idea: Build both strings of a's at the same time. Here's one possible grammar based on that idea: S asa S asa aasaa aaasaaa aaa aaa

38 Function Prototypes Let Σ = {void, int, double, name, (, ),,, ;}. Let's write a CFG for C-style function prototypes! Examples: void name(int name, double name); int name(); int name(double name); int name(int, int name, int); void name(void);

39 Function Prototypes Here's one possible grammar: S Ret name (Args); Ret Type void Type int double Args ε void ArgList ArgList OneArg ArgList, OneArg OneArg Type Type name Fun question to think about: what changes would you need to make to support pointer types?

40 Summary of CFG Design Tips Look for recursive structures where they exist: they can help guide you toward a solution. Keep the build order in mind often, you'll build two totally different parts of the string concurrently. Usually, those parts are built in opposite directions: one's built left-to-right, the other right-to-left. Use different nonterminals to represent different structures.

41 Applications of Context-Free Grammars

42 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 identifier constant EXPR + EXPR EXPR EXPR EXPR * EXPR...

43 Grammars in Compilers One of the key steps in a compiler is figuring out what a program means. This is usually done by defining a grammar showing the high-level structure of a programming language. There are certain classes of grammars (LL(1) grammars, LR(1) grammars, LALR(1) grammars, etc.) for which it's easy to figure out how a particular string was derived. Tools like yacc or bison automatically generate parsers from these grammars. Curious to learn more? Take CS143!

44 Natural Language Processing By building context-free grammars for actual languages and applying statistical inference, it's possible for a computer to recover the likely meaning of a sentence. In fact, CFGs were first called phrase-structure grammars and were introduced by Noam Chomsky in his seminal work Syntactic Structures. They were then adapted for use in the context of programming languages, where they were called Backus- Naur forms. Stanford's CoreNLP project is one place to look for an example of this. Want to learn more? Take CS124 or CS224N!

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

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