From Context-Free Grammars to Pushdown Automata: Understanding the Process, Slides of Computer Science

Download From Context-Free Grammars to Pushdown Automata: Understanding the Process and more Slides Computer Science in PDF only on Docsity! 1 Pushdown Stack Automata Docsity.com 2 Agenda Pushdown Automata  Stacks and recursiveness  Formal Definition Docsity.com 19 From CFG’s to Stack Machines A: Palindromes in {a,b,#}* containing exactly one #-symbol. Q: Using a stack, how can we recognize such strings? Docsity.com 20 From CFG’s to Stack Machines A: Use a three phase process: 1. Push mode: Before reading “#”, push everything on the stack. 2. Reading “#” switches modes. 3. Pop mode: Read remaining symbols making sure that each new read symbol is identical to symbol popped from stack. Accept if able to empty stack completely. Otherwise reject, and reject if could not pop somewhere. Docsity.com 21 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? Docsity.com 24 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? a a Input: aaab#baa Docsity.com 25 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? a a a Input: aaab#baa Docsity.com 26 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? b a a a Input: aaab#baa Docsity.com 29 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? a a Input: aaab#baa Docsity.com 30 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? a Input: aaab#baa REJECT (nonempty stack) Docsity.com 31 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? a a Input: aaab#baaa Docsity.com 34 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? Input: aaab#baaa ACCEPT Docsity.com 35 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? a a Input: aaab#baaaa Docsity.com 36 From CFG’s to Stack Machines (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? a Input: aaab#baaaa Docsity.com 39 PDA’s à la Sipser To aid analysis, theoretical stack machines restrict the allowable operations. Each text- book author has his own version. Sipser’s machines are especially simple: Push/Pop rolled into a single operation: replace top stack symbol No intrinsic way to test for empty stack Epsilon’s used to increase functionality, result in default nondeterministic machines. Docsity.com 40 Sipser’s Version Becomes: (1) PUSH (3) POP read a or b ? Push it ACCEPT (2) read # ? (ignore stack) read == peek ? Pop Else: CRASH! empty stack? e , e$ a , ea b , eb #, ee e , $e a , ae b , be Docsity.com 41 Sipser’s Version Meaning of labeling convention: If at p and next input x and top stack y, then go to q and replace y by z on stack. x = e: ignore input, don’t read y = e: ignore top of stack and push z z = e: pop y p q x, y z Docsity.com 44 Sipser’s Version e , e$ a , ea b , eb #, ee e , $e a , ae b , be $ Input: aaab#baaa Docsity.com 45 Sipser’s Version e , e$ a , ea b , eb #, ee e , $e a , ae b , be a $ Input: aaab#baaa Docsity.com 46 Sipser’s Version e , e$ a , ea b , eb #, ee e , $e a , ae b , be a a $ Input: aaab#baaa Docsity.com 49 Sipser’s Version e , e$ a , ea b , eb #, ee e , $e a , ae b , be b a a a $ Input: aaab#baaa Docsity.com 50 Sipser’s Version e , e$ a , ea b , eb #, ee e , $e a , ae b , be a a a $ Input: aaab#baaa Docsity.com 51 Sipser’s Version e , e$ a , ea b , eb #, ee e , $e a , ae b , be a a $ Input: aaab#baaa Docsity.com 54 Sipser’s Version e , e$ a , ea b , eb #, ee e , $e a , ae b , be Input: aaab#baaa ACCEPT! Docsity.com 55 PDA Formal Definition DEF: A pushdown automaton (PDA) is a 6-tuple M = (Q, S, G, d, q0, F ). Q, S, and q0, are the same as for an FA. G is the tape alphabet. d is as follows: So given a state p, an input letter x and a tape letter y, d(p,x,y) gives all (q,z) where q is a target state and z a stack replacement for y. )(Σ:δ εεε GG QPQ Docsity.com 56 PDA Formal Definition Q: What is d(p,x,y) in each case? 1. d(0,a,b) 2. d(0,e,e) 3. d(1,a,e) 4. d(3,e,e) 0 1 2 e , e$ 3 a , ea b , eb b, e$ e , $e a , ae b , be a , e e Docsity.com