Menu. Problem 5: PRIMES. Problem 5: PRIMES. Please read the PS2 Comments thoroughly! cs3102: Theory of Computation

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enu cs302: heory of Computation Class 9: Contextually P2 Recap: Computability Classes, CFL Pumping Closure Properties of CFLs pring 200 University of Virginia David Evans Problem 5: PRIE Use the

1 enu cs302: heory of Computation Class 9: Contextually P2 Recap: Computability Classes, CFL Pumping Closure Properties of CFLs pring 200 University of Virginia David Evans Problem 5: PRIE Use the pumping lemma to prove the language, PRIE = { p p is a prime number is non-regular. Assume PRIE is regular. hen, there is a DFA ith pumping length p that recognizes PRIE. All RL pumping lemma proofs can start like this! Problem 5: PRIE Use the pumping lemma to prove the language, PRIE = { p p is a prime number is non-regular. Assume PRIE is regular. hen, there is a DFA ith pumping length p that recognizes PRIE. Choose s = r here r is some prime number p. s satisfies the requirements: s PRIE and s p Next: pick s. Next: sho for any choice of xyz here s = xyz, xy p and y, there is some i here xyiz PRIE. Why is this impossible? Broken definition of regular grammar: must also allo A ε. Please read the P2 Comments thoroughly!

b n c n Regular R b n Can be recognized by some DFA Pumping Lemma for Context Free : Player :picks p Player 2: picks s A, s p Player : picks u,v,x,y,z such that s =

Example: All b n c n R b n Regular Regular Can be recognized by some DFA All Ho many language classes are there?

All Even in theory, there are infinitely many different machine classes (but only a fe are interesting).

2 b n c n Regular R b n Can be recognized by some DFA Pumping Lemma for Context Free : Player :picks p Player 2: picks s A, s p Player : picks u,v,x,y,z such that s = uvxyzand vy > 0and vxy p. Player 2: picks i 0. Player 2 ins if uv i xy i z A. If Player 2 can alays in, Ais not context free! Example: All b n c n R b n Regular Regular Can be recognized by some DFA All Ho many language classes are there? Pirahã: one, to, many Computer ciencese: zero, one, infinity? All Even in theory, there are infinitely many different machine classes (but only a fe are interesting). Closure Properties of RLs If Aand Bare regular languages then: A R is a regular language: closed under reversal Construct the reverse NFA A * is a regular language Add a transition from accept states to start A is a regular language (complement) F' = Q F A Bis a regular language Construct an NFA that combines to DFAs A Bis a regular language Construct a DFA combining states from to DFAs that accepts if both accept

3 Closure Properties of CFLs If Aand Bare context freelanguages then: A R is a context-free language? A * is a context-free language? CFLs Closed Under Reverse? Given a CFL A, is A R a CFL? A is a context-free language (complement)? A B is a context-free language? A Bis a context-free language? ome of these are true. ome of them are false. CFLs Closed Under Reverse Given a CFL A, is A R a CFL? CFLs Closed Under *? Given a CFL A, is A * a CFL? Proof-by-construction: ince Ais a CFL, there is some CFG Gthat recognizes A. here is a CFG G R that recognizes A R. G = (V, Σ, R, ) G R = (V, Σ, R R, ) R R = { A α R A α R CFLs Closed Under * Given a CFL A, is A * a CFL? Closure Properties of CFLs If Aand Bare context freelanguages then: A R is a context-free language. rue A * is a context-free language. rue Proof-by-construction:ince Ais a CFL, there is some CFG G = (V, Σ, R, )that recognizes A. here is a CFG G*that recognizes A * : G * = (V { 0, Σ, R *, 0 ) R * = R { 0 { { 0 ε Is A context-free language (complement)? Is A Bis a context-free language? Is A Bis a context-free language? Is AB is a context-free language? Left for you on P3.

4 CFLs Closed Under Union Given to CFLs Aand Bis A Ba CFL? CFLs Closed Under Union Proof-by-construction: here is a CFG G AUB that recognizes A B. ince Aand Bare CFLs, there are CFGs G A = (V A, Σ A, R A, A )and G B = (V B, Σ B, R B, B ) that generate Aand B. G AUB = (V A V B, Σ A Σ B, R AUB, 0 ) R AUB = R A R B { 0 A { 0 B (Assumes V A and V B are disjoint hich is easy to arrange by changing variable names.) CFLs Closed Under Complement? {0 i i i 0 is a CFL. Is its complement? CFLs Closed Under Complement? {0 i i i 0 is a CFL. Is its complement? Yes. We can make a DPDA that recognizes it: sap accepting states of DPDA that recognizes 0 i i. Not a counterexample but not a proof either. Complementing Non-CFLs { Σ* is nota CFL. Is its complement? CFG for L (L ) All odd length strings are in L Odd Even Odd P Odd 0 P Even XY YX X ZXZ 0 Y ZYZ Z 0

b n c n R b n Regular All Engineering Where is Java?

public class est { public static void main(tring [] a) { println("hello World!

$tring) // C:\users\luser\est.$ java public class est { public static void main(tring [] a) { ystem.out.

$java:: illegal unicode escape // C:\users\luser\est.$

5 b n c n R b n Regular All Engineering Where is Java? What is the Java Programming Language? public class est { public static void main(tring [] a) { println("hello World!"); s JAVA > javac est.java est.java:3: cannot resolve symbol symbol : method println (java.lang.tring) // C:\users\luser\est.java public class est { public static void main(tring [] a) { ystem.out.println("hello Universe!"); > javac est.java est.java:: illegal unicode escape // C:\users\luser\est.java s JAVA Defining the Java Language JAVA = { can be generated by the CFG for Java in the Java Language pecification JAVA = { a correct Java compiler can build a parse tree for + () number Programming languages are (should be) designed to make parsing easy, efficient, and unambiguous. Derivation + * Unambiguous + * () number 2 * *

6 Ambiguity Ho can one determine if a CFG is ambiguous? uper-duper-challenge problem (automatic A++): create a program that solve the is this CFG ambiguous problem: Input: any CFG Output: Yes (ambiguous)/ No (unambiguous) + () number Programming languages are (should be) designed to make parsing easy, efficient, and unambiguous. Derivation + * Warning: Undecidable Problem Alert! Don t slack off on the rest of the course thinking you can solve this. It is knon to be impossible! 3 2 Easy and Efficient Easy: e can automate the process of building a parser from a description of a grammar Efficient:the resulting parser can build a parse tree quickly (linear time in the length of the input) Recursive Descent Parse() { (); () { try { (); expect( + ); (); catch { backup(); try { (); catch {backup(); error(); () { try { (); expect( * ); (); catch { backup(); try { (); catch { backup(); error (); () { try { expect( ( ); (); expect( ) ); catch { backup(); try { number(); catch { backup(); error (); + () number Easy to produce and understand Works for any CFG Inefficient(might not even finish) LL(k) (Lookahead-Left) A CFG is an LL(k) grammar if it can be parser deterministically ith k tokens lookahead + () number LL() grammar Parse() { (); Look-ahead Parser () { if(lookahead(, + )){ (); eat( + ); (); else{ (); () { if(lookahead(, * )){ (); eat( * ); (); else{ (); () { if(lookahead(0, ( )){ eat( ( ); (); eat( ) ); else{ number(); + () number Fairly easy to produce automatically Efficient (for lo lookahead) Doesn t ork for all CFGs

7 JavaCC Input: Grammar specification Output:A Java program that is a recursive descent parser for the specified grammar b n c n LL(k) Python JAVA Python Regular Doesn t ork for all CFGs: only for LL(k) grammars imilar tools exist for all major programming languages: Lex/Flex + YACC/Bison (C): Yet another compiler compiler PLY (Python): Python lex/yacc ANLR Language Classes All jth2ey (James Harrison) - pmc8p Return P2 front of room ras3kd (Robyn hort) yyz5 Read P2 Comments P3 due uesday Charge afg2s (Arthur Gordon) dk8p dr7jx (David Renardy) jmd9xk

Complementing Non-CFLs If A and B are context free languages then: AR is a context-free language TRUE

Complementing Non-CFLs If A and B are context free languages then: AR is a context-free language TRUE Lecture : Parsimonious Menu Fix proof from last class Interpretive Dance! Parsimonious (Parsimoniously) P Comments Available oday P will be returned uesday cs0: heory of Computation University of Virginia