Download Lecture 27 - Theory of Computation

Computer Science 101
Theory of Computing
Computer Science is . . .
• The study of algorithms, with respect to
– their formal properties
– their linguistic realizations (in programming
languages, from Python down to machine code)
– their hardware realizations (machine
architecture, logic circuits, transistors)
Things to Desire in an Algorithm
• Correctness
• Maintainability
• Efficiency
Wait, can it be solved?
• There are problems that do not have any
algorithmic solution!
• Algorithms are carried out by computing
agents
• What can these agents do, and what can’t
they do?
What is a model?
• Models
– 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
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 algorithmic instructions
– Produce output
The Turing Machine
• Developed by Alan Turing to demonstrate
theoretical properties of computation
• 1935, before real computers
• Still the theoretical basis of computing
The Turing Machine
• Developed by Alan Turing to demonstrate
theoretical properties of computation
• 1935, before real computers
• Still the theoretical basis of computing
Components of a TM
• Memory – a paper tape to which symbols can be written
or from which they can be read
• States – a set of descriptions of actions to be performed
(reading, writing, and moving the read/write head on the
tape)
• Alphabet – a set of symbols to be input or output
The TM Tape
Consists of squares in which symbols are written
Extends an infinite distance in both directions (can always
buy more tape)
Can read or write a symbol from/to one square at a time
The Read/Write Head and States
At startup, the read/write head (the arrow) is placed next to a
square on the tape
State 1 says “if the tape symbol is a 0, write a 1, move one
square to the right, and go into state 2”
The Read/Write Head and a State
The change from one state to the next one is called a
transition
On each transition, an input symbol is read, an output
symbol is written, the read/right head moves left or right,
and the machine goes into the next state
The Read/Write Head and a State
Sometimes, the current state and the next state are the same
(like a loop)
Sometimes, the input and output symbols are the same
The alphabet can contain any symbols
What Can a TM Do?
• Can execute a whole sequence of
instructions
• Can accept input
• Can store information in and retrieve it from
memory
• Can take actions according to algorithm
instructions
• Can produce output
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
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
Bit Inverter
• State diagram
– Visual representation of a Turing machine
algorithm
– Circles represent states, and arrows represent
transitions from one state to another
Invert the Bits
This TM needs another transition, which reads a blank,
writes a blank, moves left, and goes into State 2
State 2 then has no transitions (no actions), so it’s the halt
state
Tuple Representation of States
A state can also be represented as a set of tuples
Each tuple has the form IN OUT NEXT DIR
States for the bit inverter TM:
State 1
IN OUT NEXT DIR
0
1
1
R
1
0
1
R
b
b
2
L
State 2
(halt state)