Context-Free Grammars

Transcription

1 Context-Free Grmmrs

2 Describing Lnguges We've seen two models for the regulr lnguges: Finite utomt ccept precisely the strings in the lnguge. Regulr expressions describe precisely the strings in the lnguge. Finite utomt recognize strings in the lnguge. Perform computtion to determine whether specific string is in the lnguge. Regulr expressions mtch strings in the lnguge. Describe the generl shpe of ll strings in the lnguge.

3 Context-Free Grmmrs A context-free grmmr (or CFG) is n entirely different formlism for defining clss of lnguges. Gol: Give procedure for listing off ll strings in the lnguge. CFGs re best explined by exmple...

4 Arithmetic Expressions Suppose we wnt to describe ll legl rithmetic expressions using ddition, subtrction, multipliction, nd 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 wnt to describe ll legl rithmetic expressions using ddition, subtrction, multipliction, nd 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 Grmmrs Formlly, context-free grmmr is collection of four objects: A set of nonterminl symbols (lso clled vribles), A set of terminl symbols (the lphbet of the CFG) A set of production rules sying how ech nonterminl cn be replced by string of terminls nd nonterminls, nd A strt symbol (which must be nonterminl) tht begins the derivtion. By convention, it's the nonterminl tht's first in the list. E int E E Op E E (E) Op + Op - Op * Op /

7 Some CFG Nottion In tody's slides, cpitl letters in Bold Red Uppercse will represent nonterminls. i.e. A, B, C, D Lowercse letters in blue monospce will represent terminls. i.e. t, u, v, w Lowercse Greek letters in gry itlics will represent rbitrry strings of terminls nd nonterminls. i.e. α, γ, ω You don't need to use these conventions on your own; just mke sure whtever you do is redble.

8 A Nottionl Shorthnd E int E E Op E E (E) Op + Op - Op * Op /

9 A Nottionl Shorthnd E int E Op E (E) Op + - * /

10 Derivtions 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 derivtion is sequence of steps where in ech step single nonterminl is replced by one of its productions. If string α derives string ω, we write α * ω. In the exmple on the left, we see E * int * (int + int).

11 The Lnguge of Grmmr If G is CFG with lphbet Σ nd strt symbol S, then the lnguge of G is the set L(G) = { ω Σ* S * ω } Tht is, L( G) is the set of strings derivble from the strt symbol. Note tht ω must be in Σ*, the set of strings mde from terminls. Strings involving nonterminls ren't in the lnguge.

12 Context-Free Lnguges A lnguge L is clled context-free lnguge (or CFL) if there is CFG G such tht L = L( G). Questions: Wht lnguges re context-free? How re context-free nd regulr lnguges relted?

13 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S *b

14 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S Ab A A ε

15 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S (b c*)

16 From Regexes to CFGs CFGs consist purely of production rules of the form A ω. They do not hve the regulr expression opertors * or. However, we cn convert regulr expressions to CFGs s follows: S X X b C C Cc ε

17 Regulr Lnguges nd CFLs Theorem: Every regulr lnguge is context-free. Proof Ide: Use the construction from the previous slides to convert regulr expression for L into CFG for L. Problem Set 8 Exercise: Insted, show how to convert DFA/NFA into CFG.

18 The Lnguge of Grmmr Consider the following CFG G: S Sb ε Wht strings cn this generte? b b b b L(G) = { n b n n N }

19 All Lnguges Regulr Lnguges CFLs

20 Why the Extr Power? Why do CFGs hve more power thn regulr expressions? Intuition: Derivtions of strings hve unbounded memory. S Sb ε S b b b b

21 Time-Out for Announcements!

22 Problem Set Seven Problem Set Six ws due t the strt of tody's lecture. Wnt to use lte dys? Submit up to Mondy t 3:00PM. Problem Set Seven goes out now. It's due next Fridy. Build some nifty regulr expressions! Ply round with the Myhill-Nerode theorem nd the limits of regulr lnguges!

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25 Midterm II The second midterm exm is this upcoming Mondy. Plese feel free to sk us questions on Pizz over the weekend or to stop by our weekend office hours. We're hppy to help out! You cn do this! Best of luck on the exm!

26 Midterm Thoughts Some questions will be looking for intuition nd the feel of things. Think Midterm 1, Q1.i. To prep for these, sk questions like wht does n equivlence clss look like?, wht kinds of functions cn I mke?, or wht does biprtite grph look like? Some questions will be looking for proof setup nd proofwriting. Think Midterm 1, Q3 nd Q4. Q3 on the prctice midterm is n excellent testbed for these ides. These questions require you to be good t setting up nd executing proofs. To prep for questions like these, mke sure you hve good methodology for solving problems. Your gol is to not freeze up when you see something new. Set timer when working through the extr prctice problems tht re mrked s old midterm questions it's gret prctice. Consider putting sequence of steps on your notes sheet, like Step One: Set up the proof. Keep writing out wht to ssume nd wht to prove, nd repet this until there's nothing left to do.

27 Need Help? Some of the TAs hve grciously volunteered to hold extr office hours over the weekend. Check the OH clendr for detils. Ask questions on Pizz even conceptul questions! We're hppy to help out.

28 Your Questions

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32 Bck to CS103!

33 Designing CFGs Like designing DFAs, NFAs, nd regulr expressions, designing CFGs is crft. When thinking bout CFGs: Think recursively: Build up bigger structures from smller ones. Hve construction pln: Know in wht order you will build up the string. Store informtion in nonterminls: Hve ech nonterminl correspond to some useful piece of informtion.

34 Designing CFGs Let Σ = {, b} nd let L = {w Σ* w is plindrome } We cn design CFG for L by thinking inductively: Bse cse: ε,, nd b re plindromes. If ω is plindrome, then ω nd bωb re plindromes. S ε b S bsb

35 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of blnced prentheses } Some smple strings in L: ((())) (())() (()())(()()) ((((()))(()))) ε ()()

36 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of blnced prentheses } Let's think bout this recursively. Bse cse: the empty string is string of blnced prentheses. Recursive step: Look t the closing prenthesis tht mtches the first open prenthesis. (( ( ) ( ( ) ) )( ( ) ))(())(( ( ) ))

37 Designing CFGs Let Σ = {(, )} nd let L = {w Σ* w is string of blnced prentheses } Let's think bout this recursively. Bse cse: the empty string is string of blnced prentheses. Recursive step: Look t the closing prenthesis tht mtches the first open prenthesis. Removing the first prenthesis nd its mtching prenthesis forms two new strings of blnced prentheses. S (S)S ε

38 Designing CFGs: A Cvet Let Σ = {, b} nd let L = {w Σ* w hs the sme number of 's nd b's } Is this CFG for L? S Sb bs ε Cn you derive the string bb?

39 Designing CFGs: A Cvet When designing CFG for lnguge, mke sure tht it genertes ll the strings in the lnguge nd never genertes string outside the lnguge. The first of these cn be tricky mke sure to test your grmmrs! You'll design your own CFG for this lnguge on PS8.

40 CFG Cvets II Is the following grmmr CFG for the lnguge { n b n n N }? S Sb Wht strings cn you derive? Answer: None! Wht is the lnguge of the grmmr? Answer: Ø When designing CFGs, mke sure your recursion ctully termintes!

41 CFG Cvets III When designing CFGs, remember tht ech nonterminl cn be expnded out independently of the others. Let Σ = {, } nd let L = {n n n N }. Is the following CFG for L? S X X X X ε S X X X X X X X X

42 Finding Build Order Let Σ = {, } nd let L = { n n n N }. To build CFG for L, we need to be more clever with how we construct the string. If we build the strings of 's independently of one nother, then we cn't enforce tht they hve the sme length. Ide: Build both strings of 's t the sme time. Here's one possible grmmr bsed on tht ide: S S S S S S

43 Function Prototypes Let Σ = {void, int, double, nme, (, ),,, ;}. Let's write CFG for C-style function prototypes! Exmples: void nme(int nme, double nme); int nme(); int nme(double nme); int nme(int, int nme, int); void nme(void);

44 Function Prototypes Here's one possible grmmr: S Ret nme (Args); Ret Type void Type int double Args ε void ArgList ArgList OneArg ArgList, OneArg OneArg Type Type nme Fun question to think bout: wht chnges would you need to mke to support pointer types?

45 Summry of CFG Design Tips Look for recursive structures where they exist: they cn help guide you towrd solution. Keep the build order in mind often, you'll build two totlly different prts of the string concurrently. Usully, those prts re built in opposite directions: one's built left-to-right, the other right-to-left. Use different nonterminls to represent different structures.

46 Applictions of Context-Free Grmmrs

47 CFGs for Progrmming Lnguges BLOCK STMT { STMTS } STMTS ε STMT STMTS STMT EXPR; if (EXPR) BLOCK while (EXPR) BLOCK do BLOCK while (EXPR); BLOCK EXPR identifier constnt EXPR + EXPR EXPR EXPR EXPR * EXPR...

48 Grmmrs in Compilers One of the key steps in compiler is figuring out wht progrm mens. This is usully done by defining grmmr showing the high-level structure of progrmming lnguge. There re certin clsses of grmmrs (LL(1) grmmrs, LR(1) grmmrs, LALR(1) grmmrs, etc.) for which it's esy to figure out how prticulr string ws derived. Tools like ycc or bison utomticlly generte prsers from these grmmrs. Curious to lern more? Tke CS143!

49 Nturl Lnguge Processing By building context-free grmmrs for ctul lnguges nd pplying sttisticl inference, it's possible for computer to recover the likely mening of sentence. In fct, CFGs were first clled phrse-structure grmmrs nd were introduced by Nom Chomsky in his seminl work Syntctic Structures. They were then dpted for use in the context of progrmming lnguges, where they were clled Bckus- Nur forms. Stnford's CoreNLP project is one plce to look for n exmple of this. Wnt to lern more? Tke CS124 or CS224N!

50 Next Time Turing Mchines Wht does computer with unbounded memory look like? How do you progrm them? Good luck on the exm!