Decidable Problem

A decidable problem is one that we can come to a yes/no answer given any input. An examle would be given a number determine if it is divisible by 3. We know that the algorithm below would provide the correct ouput every time.

PRODECURE divisbleByThree(num)
    IF (num MOD 3 = 0) 
        RETURN true
    ELSE
        RETURN false

Undecidable Problem

Halting Problem

The halting problem is defined as: Given an arbitrary computer program with given inputs will the program stop or will it run forever?

Undecidable Problems

A problem where an algorithm cannot make a correct yes/no answer every time.

One way would be to test for an ending, but what if that ending is not easily found? What if it takes an unreasonable amount of time to find the ending? Is that because there is an ending or does one not exist?

You see where the problem comes in? This is an undecidable problem – there is no algorithm which can always produce a yes/no answer for every input of the problem.

Halting Problem in Computers Where may we have seen this in computers today? When a website or program takes too long to load it. It may never load, or it may just be taking a long time. Either way, after a certain time the computer decides the program should be stopped.

Popcorn Hack #1

An algorithm can be used to solve an undecidable problem. (True/False)

  • False, because if the problem is undecidable then it cannot be solved by an algorithm

Popcorn Hack #2

If a programmer encounters an undecidable problem, they can just use an algorithm that works most of the time. (True/False)

  • False. Just like explained above, if the problem is undeciable, it is not guranteed that an algorithm will solve it.

Scenarios of Undecidable Problems in Computing

  1. Infinite Loop in Program Execution:
    • When a program enters an infinite loop, it becomes undecidable whether it will eventually terminate or run indefinitely.
  2. Complex Conditional Statements:
    • Programs with intricate conditional statements or complex control flow may pose undecidable scenarios, making it challenging to determine their termination.
  3. Non-Terminating Recursive Functions:
    • Recursive functions that do not have a base case or have poorly defined termination conditions can result in undecidability regarding their halting behavior.
  4. Unknown Input Space Size:
    • In cases where the size of the input space is unknown or unbounded, it becomes difficult to ascertain if a program will halt for all possible inputs.
  5. Multithreading and Concurrency:
    • Undecidability may arise in concurrent programs where multiple threads interact, introducing intricate synchronization and communication challenges.

Popcorn Hack 3

An algorithm exists that can always produce a yes/no answer for the halting problem. (True/False)

  • False, halting problems may continue forever so an algorithm that produces a yes/no answer doesn’t exist.

Homework Question

Research and explain how modern systems or browsers deal with the aspects of the halting problem when a program takes too long to load. Provide examples of mechanisms or strategies implemented in real-world scenarios to manage unresponsive programs or prolonged loading times.

  • Modern systems or browsers deal with the aspects of the halting problems by just telling the user that an error occured. Since halting problems are undeciable and there isn’t a known ending or halt, the browser or system will just tell the user an error has occured so that it doesn’t waste the user’s time.
    • Strategies used by websites or browsers include caching or only asking for data when asked for. For example, if I go to Amazon’s website frequently, it will use caching which will store data that is frequently used so that it takes less time to load. Furthermore, since Amazon has so much categories and products(which people won’t visit), it will only load the data that the user actually wants. This saves the amount of time that it takes for the user to load the website.