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Pushdown Automata and Turing Machine - Lecture Notes | CMSC 330, Study notes of Programming Languages

University of MarylandProgramming Languages

Material Type: Notes; Class: ORGNZTN PROGM LANG; Subject: Computer Science; University: University of Maryland; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 02/13/2009

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CMSC 330: Organization of
Programming Languages
Pushdown Automata and
Turing Machines
CMSC 330 2
Last Lecture
Parsing
•
Recursive descent
•
FIRST sets
Rewriting grammars
•
Left factoring
•
Eliminating left recursion
Question
•
Where do regular expressions and context free
grammars stand in the world of complexity?
CMSC 330 3
This Lecture
Regular grammars
Pushdown automata
Turing machines
Decidability
Turing tests
CMSC 330 4
Regular Expressions and CFGs
Regular grammar
•
Subset of context free grammar
•
All productions are of the form
¾
A →aB // 1 terminal followed by 1 nonterminal
¾
A →a // 1 terminal
Regular grammars are just as powerful as
regular expressions
•
For any regular expression R, there is a regular
grammar G that accepts the exact same language
CMSC 330 5
Equivalence of DFA and Regular Grammar
Reducing DFA to regular grammar
CMSC 330 6
Equivalence of DFA and Regular Grammar
Reducing regular grammar to DFA
pf3
pf4

Partial preview of the text

Download Pushdown Automata and Turing Machine - Lecture Notes | CMSC 330 and more Study notes Programming Languages in PDF only on Docsity!

CMSC 330: Organization of

Programming Languages

Pushdown Automata and

Turing Machines

CMSC 330 2

Last Lecture

Parsing

  • Recursive descent
  • FIRST sets

Rewriting grammars

  • Left factoring
  • Eliminating left recursion

Question

  • Where do regular expressions and context free grammars stand in the world of complexity?

CMSC 330 3

This Lecture

Regular grammars

Pushdown automata

Turing machines

Decidability

Turing tests

CMSC 330 4

Regular Expressions and CFGs

Regular grammar

  • Subset of context free grammar
  • All productions are of the form ¾ A → aB // 1 terminal followed by 1 nonterminal ¾ A → a // 1 terminal

Regular grammars are just as powerful as

regular expressions

  • For any regular expression R, there is a regular grammar G that accepts the exact same language

CMSC 330 5

Equivalence of DFA and Regular Grammar

Reducing DFA to regular grammar

CMSC 330 6

Equivalence of DFA and Regular Grammar

Reducing regular grammar to DFA

CMSC 330 7

Regular Expressions and CFGs

CFGs are strictly more powerful than REs

  • Since CFGs are superset of regular grammar

This is good, since programming languages are

not regular

  • No regular expression for ( n^ ) n

But programming languages are usually

(almost) context-free

  • With a few language constructs being exceptions

CMSC 330 8

Regular Expressions and CFGs

We can implement REs with DFAs

Question

  • How can we implement CFGs?

Description Machine

context-free grammars

context-free languages

regular DFAs, NFAs expressions

regular languages

pushdown automata (PDAs)

CMSC 330 9

Pushdown Automaton (PDA)

A PDA is an abstract machine similar to a DFA

  • Has a finite set of states
  • But also has a stack 1 0 1 1 q 1 q (^2) 0

(^0 ) 1

CMSC 330 10

Moves of a PDA

Reads

  • An input symbol is read
  • Top symbol on the stack

Based on both inputs, the machine

  • Enters a new state, and
  • Writes zero or more symbols onto the pushdown stack
  • Or pops zero or more symbols from the stack

String accepted if

  • The stack is empty, AND
  • The string has ended

CMSC 330 11

Power of PDAs

PDAs are more powerful than DFAs

  • a n^ b n^ , which cannot be recognized by a DFA, can easily be recognized by the PDA ¾ Stack all a symbols and, for each b, pop an a off the stack. ¾ If the end of input is reached at the same time that the stack becomes empty, the string is accepted

PDAs can recognize some L(CFG)

  • Table-driven predictive parser ¾ Table = Finite automaton ¾ Stack = Stack
  • Limited to LR(1) grammars in general

CMSC 330 12

Power of PDAs and NDPDA

There are also nondeterministic PDA (NDPDA)

  • NDPDA are more powerful than DPDA
  • NDPDA can recognize even length palindromes over {0,1}, but a DPDA cannot. Why? ¾ Hint: Consider palindromes over {0,1}2{0,1}* vs {0,1}* {0,1}*

NDPDAs can recognize all L(CFG)

  • Equivalent in power to context-free languages

CMSC 330 19

“Enhanced” Turing Machines

Attempts to increase power of Turing machine

  • Additional tapes
  • Additional tape heads
  • 2-dimensional tape
  • BlueGene/L @ Livermore Lab ¾ World’s fastest computer in 2007 ¾ 106K IBM PowerPC nodes (596 x10 12 Flops)

Turns out any computation on enhanced machine

  • Can be computed on standard Turing machine
  • Though may require more time

CMSC 330 20

Computability

Computability

  • A language is computable if it can be recognized by some algorithm with finite number of steps

Church-Turing thesis

  • Turing machine can recognize any language computable on any machine

Intuition

  • Turing machine captures essence of computing ¾ Both in a formal sense, and in an informal practical sense

CMSC 330 21

Computability – Turing Test

Turing Test

  • Test machine’s capability to demonstrate intelligence
  • Proposed by Alan Turing in 1950
  • Practical test for “Can machines think?”

Test procedure

  • Judge converses via text (e.g., instant messaging)
  • Pass if can’t reliably tell whether machine or human

CMSC 330 22

More Turing Tests

“On the Internet,

nobody knows

you’re a dog.”

CMSC 330 23

Computability – Reverse Turing Test

Reverse Turing test

  • Computer determines whether human or machine

CAPTCHA

  • Completely Automated Public Turing test to tell Computers and Humans Apart
  • Challenge-response asking for word identification ¾ Useful for foiling scripts

CMSC 330 24

More Reverse Turing Tests