Computer Science 101

More Theory of Computing

Another Example: Odd Parity

• A parity bit is added to the right end of a string of bits if it makes the total number of 1’s odd; otherwise, add a 0

• 100011 ->• 100010 ->• 000000 ->

Design Strategy

• Three states– Even parity: if we read a blank here, we have

an even number of 1s, so we write a 1 and goto the halt state

– Odd parity: if we read a blank here, we have an odd number of 1s, so we write a 0 and goto the halt state

– Halt

Design Strategy

• Even parity– We stay here while we keep reading 0s– Otherwise, we goto odd parity

• Odd parity– We stay here while we keep reading 0s– Otherwise, we goto even parity

State Diagram for Odd Parity TM

Even parity Odd parity

Halt

Tuples for Transitions

• 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

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

Emulating an Algorithm on a TM

A Universal TM

• Represent the algorithm as symbols on the tape

• Put the algorithm’s inputs on the tape as well

• Run the universal TM to run the algorithm

• The universal TM is like a stored program computer (von Neumann machine)

Unsolvable Problems

• Problem to investigateDecide, 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

The Proposed Solution

T* = TM being testedt = T’s inputP = The testing TM

A Proposed Counterexample

T* = TM being testedt = T’s inputQ = The testing TM

An Actual Counterexample

S* = TM being testedS*bS* = S’s inputS = The testing TM

Other Unsolvable Problems

• 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