Invitation to Computer Science
5th Edition
Chapter 12
Models of Computation
Objectives
In this chapter, you will learn about:
• What a model is
• A model of a computing agent
• A model of an algorithm
• Turing machine examples
• The Church-Turing thesis
• Unsolvable problems
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Introduction
• There are problems that do not have any
algorithmic solution!
• Algorithms
– Carried out by computing agents
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What Is a Model?
• Models
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–
–
–
Capture the essence of the real thing
Probably differ in scale from the real thing
Omit some of the details of the real thing
Lack the full functionality of the real thing
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A Model of a Computing Agent
• Computing agent should be able to:
–
–
–
–
Accept input
Store information in and retrieve it from memory
Take actions according to algorithm instructions
Produce output
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The Turing Machine
• Turing machine
– Includes a (conceptual) tape that extends infinitely in
both directions
• Alphabet for a given Turing machine
– Always contains a special symbol b (for “blank”)
– Usually contains the symbols 0 and 1 and
sometimes a limited number of other symbols
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Figure 12.1 A Turing Machine Tape
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Figure 12.2 A Turing Machine Configuration
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Figure 12.3 The Next Turing Machine Configuration after
Executing One Instruction
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The Turing Machine (continued)
• Turing machine
– Can execute a whole sequence of instructions
• Clock governs the action of the machine
–
–
–
–
Can accept input
Can store information in and retrieve it from memory
Can take actions according to algorithm instructions
Can produce output
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A Model of an Algorithm
• An algorithm must:
– Be a well-ordered collection
– Consist of unambiguous and effectively computable
operations
– Halt in a finite amount of time
– Produce a result
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Turing Machine Examples
• Turing machine
– Must begin in state 1 on the leftmost nonblank cell
– Machine state 1 must be a state in which 0s are
changed to 1s and 1s are changed to 0s
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A Bit Inverter
• State diagram
– Visual representation of a Turing machine algorithm
– Circles represent states, and arrows represent
transitions from one state to another
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Figure 12.4 State Diagram for the Bit
Inverter Machine
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A Parity Bit Machine
• Odd parity bit
– Extra bit
– Can be attached to the end of a string of bits
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A Parity Bit Machine (continued)
• The Turing machine program
– (1,1,1,2,R) Even parity state reading 1, change state
– (1,0,0,1,R) Even parity state reading 0, don’t
change state
– (2,1,1,1,R) Odd parity state reading 1, change state
– (2,0,0,2,R) Odd parity state reading 0, don’t change
state
– (1,b,1,3,R) End of string in even parity state, write 1
and go to state 3
– (2,b,0,3,R) End of string in odd parity state, write 0
and go to state 3
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Figure 12.5 State Diagram for the Parity Bit Machine
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Machines for Unary Incrementing
• Unary representation of numbers
– Any unsigned whole number n is encoded by a
sequence of n + 1 1s
• Incrementer
– Turing machine that adds 1 to any number
• Turing machine for the incrementer
(1,1,1,1,R) Pass to the right over 1s
(1,b,1,2,R) Add a single 1 at the righthand end of
the string
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Figure 12.6 State Diagram for Incrementer
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Machines for Unary Incrementing
(continued)
• The Turing machine program for:
– (1,1,1,1,L) Pass to the left over 1s
– (1,b,1,2,L) Add a single 1 at the lefthand end of the
string
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Figure 12.7 Time Efficiency for Two Turing
Machine Algorithms for Incrementing
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A Unary Addition Machine
• Turing machine
– Can be written to perform the addition of two
numbers
• Turing machine program
–
–
–
–
(1,1,b,2,R)
(2,1,b,3,R)
(3,1,1,3,R)
(3,b,1,4,R)
Erase the leftmost 1 and move right
Erase the second 1 and move right
Pass over any 1s until a blank is found
Write a 1 over the blank and halt
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Figure 12.8 State Diagram for the Addition Machine
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The Church-Turing Thesis
• Church-Turing Thesis
– If there exists an algorithm to do a symbol
manipulation task, then there exists a Turing
machine to do that task
• Thesis
– Statement advanced for consideration and
maintained by argument
• Theorem
– Ideas that can be proved in a formal, mathematical
way
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Figure 12.9 Emulating an Algorithm by a Turing Machine
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Unsolvable Problems
• Problem to investigate
Decide, given any collection of Turing machine
instructions together with any initial tape contents,
whether that Turing machine will ever halt if started
on that tape
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Figure 12.10 Hypothetical Turing Machine P Running on T* and t
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Figure 12.11 Hypothetical Turing Machine Q Running
on T* and t
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Figure 12.12 Hypothetical Turing Machine S Running on S*
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Unsolvable Problems (continued)
• Unsolvable problems, related to the halting
problem, have the following consequences
– No program can be written to decide whether any
given program always stops eventually, no matter
what the input
– No program can be written to decide whether any
two programs are equivalent
– No program can be written to decide whether any
given program run on any given input will ever
produce some specific output
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Summary
• Models are an important way of studying physical
and social phenomena
• Church-Turing thesis
– If there exists an algorithm to do a symbol
manipulation task, then there exists a Turing
machine to do that task
• Turing machine
– Can be accepted as an ultimate model of a
computing agent
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Summary (continued)
• Turing machine program
– Can be accepted as an ultimate model of an
algorithm
• Turing machines
– Define the limits of computability
• Uncomputable or unsolvable problem
– We can prove that no Turing machine exists to solve
the problem
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