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Fall 2006Costas Busch - RPI1 Pushdown Automata PDAs
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Fall 2006Costas Busch - RPI2 Pushdown Automaton -- PDA Input String Stack States
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Fall 2006Costas Busch - RPI3 Initial Stack Symbol Stack bottomspecial symbol stack head top Appears at time 0
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Fall 2006Costas Busch - RPI4 The States Input symbol Pop symbol Push symbol
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Fall 2006Costas Busch - RPI5 top input stack Replace
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Fall 2006Costas Busch - RPI6 Push top input stack
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Fall 2006Costas Busch - RPI7 Pop top input stack
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Fall 2006Costas Busch - RPI8 No Change top input stack
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Fall 2006Costas Busch - RPI9 Pop top input stack Pop from Empty Stack Automaton halts! If the automaton attempts to pop from empty stack then it halts and rejects input
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Fall 2006Costas Busch - RPI10 Non-Determinism PDAs are non-deterministic Allowed non-deterministic transitions
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Fall 2006Costas Busch - RPI11 Example PDA PDA :
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Fall 2006Costas Busch - RPI12 Basic Idea: 1.Push the a’s on the stack 2. Match the b’s on input with a’s on stack 3. Match found
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Fall 2006Costas Busch - RPI13 Execution Example: Input current state Time 0 Stack
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Fall 2006Costas Busch - RPI14 Input Time 1 Stack
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Fall 2006Costas Busch - RPI15 Input Stack Time 2
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Fall 2006Costas Busch - RPI16 Input Stack Time 3
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Fall 2006Costas Busch - RPI17 Input Stack Time 4
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Fall 2006Costas Busch - RPI18 Input Stack Time 5
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Fall 2006Costas Busch - RPI19 Input Stack Time 6
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Fall 2006Costas Busch - RPI20 Input Stack Time 7
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Fall 2006Costas Busch - RPI21 Input Time 8 accept Stack
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Fall 2006Costas Busch - RPI22 A string is accepted if there is a computation such that: All the input is consumed AND The last state is an accepting state we do not care about the stack contents at the end of the accepting computation
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Fall 2006Costas Busch - RPI23 Rejection Example: Input current state Time 0 Stack
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Fall 2006Costas Busch - RPI24 Rejection Example: Input current state Time 1 Stack
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Fall 2006Costas Busch - RPI25 Rejection Example: Input current state Time 2 Stack
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Fall 2006Costas Busch - RPI26 Rejection Example: Input current state Time 3 Stack
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Fall 2006Costas Busch - RPI27 Rejection Example: Input current state Time 4 Stack
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Fall 2006Costas Busch - RPI28 Rejection Example: Input current state Time 4 Stack reject
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Fall 2006Costas Busch - RPI29 The string is rejected by the PDA There is no accepting computation for
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Fall 2006Costas Busch - RPI30 Another PDA example PDA :
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Fall 2006Costas Busch - RPI31 Basic Idea: 1.Push v on stack 2. Guess middle of input 3. Match on input with v on stack 4. Match found
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Fall 2006Costas Busch - RPI32 Execution Example: Input Time 0 Stack
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Fall 2006Costas Busch - RPI33 Input Time 1 Stack
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Fall 2006Costas Busch - RPI34 Input Time 2 Stack
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Fall 2006Costas Busch - RPI35 Input Time 3 Stack Guess the middle of string
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Fall 2006Costas Busch - RPI36 Input Time 4 Stack
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Fall 2006Costas Busch - RPI37 Input Time 5 Stack
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Fall 2006Costas Busch - RPI38 Input Time 6 Stack accept
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Fall 2006Costas Busch - RPI39 Rejection Example: Input Time 0 Stack
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Fall 2006Costas Busch - RPI40 Input Time 1 Stack
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Fall 2006Costas Busch - RPI41 Input Time 2 Stack
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Fall 2006Costas Busch - RPI42 Input Time 3 Stack Guess the middle of string
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Fall 2006Costas Busch - RPI43 Input Time 4 Stack
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Fall 2006Costas Busch - RPI44 Input Time 5 Stack There is no possible transition. Input is not consumed
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Fall 2006Costas Busch - RPI45 Another computation on same string: Input Time 0 Stack
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Fall 2006Costas Busch - RPI46 Input Time 1 Stack
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Fall 2006Costas Busch - RPI47 Input Time 2 Stack
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Fall 2006Costas Busch - RPI48 Input Time 3 Stack
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Fall 2006Costas Busch - RPI49 Input Time 4 Stack
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Fall 2006Costas Busch - RPI50 Input Time 5 Stack No accept state is reached
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Fall 2006Costas Busch - RPI51 There is no computation that accepts string
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Fall 2006Costas Busch - RPI52 Pushing & Popping Strings Input symbol Push string Pop string
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Fall 2006Costas Busch - RPI53 top input stack Replace push string Example: pop string top
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Fall 2006Costas Busch - RPI54 pop push Equivalent transitions
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Fall 2006Costas Busch - RPI55 Another PDA example PDA
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Fall 2006Costas Busch - RPI56 Time 0 Input current state Stack Execution Example:
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Fall 2006Costas Busch - RPI57 Time 1 Input Stack
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Fall 2006Costas Busch - RPI58 Time 3 Input Stack
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Fall 2006Costas Busch - RPI59 Time 4 Input Stack
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Fall 2006Costas Busch - RPI60 Time 5 Input Stack
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Fall 2006Costas Busch - RPI61 Time 6 Input Stack
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Fall 2006Costas Busch - RPI62 Time 7 Input Stack
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Fall 2006Costas Busch - RPI63 Time 8 Input Stack accept
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Fall 2006Costas Busch - RPI64 Formalities for PDAs
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Fall 2006Costas Busch - RPI65 Transition function:
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Fall 2006Costas Busch - RPI66 Transition function:
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Fall 2006Costas Busch - RPI67 Formal Definition Pushdown Automaton (PDA) States Input alphabet Stack alphabet Transition function Accept states Stack start symbol Initial state
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Fall 2006Costas Busch - RPI68 Instantaneous Description Current state Remaining input Current stack contents
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Fall 2006Costas Busch - RPI69 Input Stack Time 4: Example:Instantaneous Description
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Fall 2006Costas Busch - RPI70 Input Stack Time 5: Example:Instantaneous Description
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Fall 2006Costas Busch - RPI71 We write: Time 4 Time 5
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Fall 2006Costas Busch - RPI72 A computation:
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Fall 2006Costas Busch - RPI73 For convenience we write:
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Fall 2006Costas Busch - RPI74 Language of PDA Language accepted by PDA : Initial state Accept state
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Fall 2006Costas Busch - RPI75 Example: PDA :
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Fall 2006Costas Busch - RPI76 PDA :
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Fall 2006Costas Busch - RPI77 PDA : Therefore:
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