CSE 105 THEORY OF COMPUTATION

Transcription

1 CSE 105 THEORY OF COMPUTATION Spring

2 Today's learning goals Sipser Ch 2 Define push down automata Trace the computation of a push down automaton Design a push down automaton recognizing a given language

3 Recap: Context-free languages Context-free grammar One-step derivation Derivation Language generated by grammar

4 Designing CFGs Sipser p. 104 Given a language L over Σ, to prove L is a context-free language need to build a CFG that generates it. How? Express L as union of simpler languages If L is regular, design DFA and convert into CFG Exploit any recursive structure in L.

5 Example Design a CFG recognizing the language {a n b m n m} On Thursday, we showed that the language {a n b n n 0} is CFL. How can we use its CFG for this example? A. Use the same CFG but reverse the role of variables and terminals. B. Use the same set of variables, terminals, and start state, but reverse the RHS and LHS of each rule. C. We can't use it directly: the two sets are not related. D. We can't use it directly: the class of CFLs is not closed under complementation. E. I don't know.

6 Example Design a CFG recognizing the language {a n b m n m} = {a n b m n > m} U {a n b m n < m} Simpler problem: Design a CFG generating the language {a n b m n > m}

7 Example Design a CFG recognizing the language {a n b m n m} = {a n b m n > m} U {a n b m n < m} Simpler problem: Design a CFG generating the language {a n b m n > m} G 1 = ( {S 1 }, {a,b}, R, S 1 ) with rules S 1 à as 1 b as 1 a

8 Example Design a CFG recognizing the language {a n b m n m} = {a n b m n > m} U {a n b m n < m} CFG generating the language {a n b m n > m} is G 1 = ( {S 1 }, {a,b}, R, S 1 ) with rules S 1 à as 1 b as 1 a CFG generating the language {a n b m n < m} is G 2 = ( {S 2 }, {a,b}, R, S 2 ) with rules S 2 à as 2 b S 2 b b

9 Example For {a n b m n > m}: G 1 = ( {S 1 }, {a,b}, R, S 1 ) with S 1 à as 1 b as 1 a For {a n b m n < m}: G 2 = ( {S 2 }, {a,b}, R, S 2 ) with S 2 à as 2 b S 2 b b What is CFG generating {a n b m n m}? A. G 1 U G 2 B. ( {S 1, S 2 }, {a,b}, R, S 1 ) with S 1 à as 1 b as 1 a, S 2 à as 2 b S 2 b b C. ( {S 1, S 2 }, {a,b}, R, S 2 ) with S 1 à as 1 b as 1 as 2, S 2 à as 2 b S 2 b b D. ( {S, S 1, S 2 }, {a,b}, R, S) with S à S 1 S 2, S 1 à as 1 b as 1 a, S 2 à as 2 b S 2 b b E. I don't know.

10 An alternative Sipser p. 109 NFA + stack

11 Pushdown automata Sipser p. 109 NFA + stack At each step 1. Transition to new state based on current state, letter read, and top letter of stack. 2. (Possibly) push or pop a letter to (or from) top of stack

12 Pushdown automata Sipser p. 109 NFA + stack Accept a string if there is some sequence of states and some sequence of stack contents which processes the entire input string and ends in an accepting state.

13 State diagram for PDA Sipser p. 109 If hand-drawn or in Sipser State transition labelled a, b à c means "when machine reads an a from the input and the top symbol of the stack is a b, it may replace the b with a c." In JFLAP: use ; instead of à

14 State diagram for PDA Sipser p. 109 If hand-drawn or in Sipser State transition labelled a, b à c means "when machine reads an a from the input and the top symbol of the stack is a b, it may replace the b with a c." What edge label would indicate "Read a 0, don't pop anything from stack, don t push anything to the stack"? A. 0, ε à ε B. ε, 0 à ε C. ε, ε à 0 D. ε à ε, 0 E. I don't know.

15 Useful trick What would ε, ε à $ mean? A. Without reading any input or popping any symbol from stack, push $ B. If the input string and stack are both empty strings, push $ C. At the end of reading the input string, push $ to top of stack D. I don't know.

16 Useful trick What would ε, ε à $ mean? A. Without reading any input or popping any symbol from stack, push $ B. If the input string and stack are both empty strings, push $ C. At the end of reading the input string, push $ to top of stack D. I don't know. Why is this useful? Commonly used from initial state (at start of computation) to record top of stack with a special symbol) we'll see applications soon!

17 Formal definition of PDA Sipser Def 2.13 p. 111

18 Designing a PDA L = { 0 i 1 i+1 i 0 } Informal description of PDA: Read symbols from the input. As each 0 is read, push it onto the stack. As soon as 1s are seen, pop a 0 off the stack for each 1 read. If the stack becomes empty and there is exactly one 1 left to read, read that 1 and accept the input. If the stack becomes empty and there are either zero or more than one 1s left to read, or if the 1s are finished while the stack still contains 0s, or if any 0s appear in the input following 1s, reject the input.

19 Designing/Tracing a PDA L = { 0 i 1 i+1 i 0 } What are the contents of the stack after processing 001? A. (TOP) $00 B. (TOP) 00$ C. (TOP) 0$ D. (TOP) 100$ E. I don't know.

20 Designing/Tracing a PDA L = { 0 i 1 i+1 i 0 } Which of the following strings are not accepted by this PDA? A. 0 B. 1 C. 01 D. 011 E. I don't know.

21 Designing/Tracing a PDA L = { 0 i 1 i+1 i 0 } Which of these CFGs generate L? Assume V = set of variables mentioned in rules; Σ={0,1}, S start A. S à ε 0S1 B. S à ε 0S11 C. S à T1 0S1, T à ε D. S à ε 0T1, T à ε 0T1 E. I don't know.

22 PDAs and CFGs are equivalently expressive Theorem 2.20: A language is context-free if and only some nondeterministic PDA recognizes it. Consequences - Quick proof that every regular language is context free - To prove closure of class of CLFs under a given operation, can choose two modes of proof (via CFGs or PDAs) depending on which is easier

23 Overview??? Context-free languages Regular languages

24 Reminders HW4 due Wednesday April 27 Gradescope link now available