Undecidable Problems • 10 min read

Description

:D

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

Popcorn Hack #2

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

  • False

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

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.

Ways/strategies modern systems or browsers deal with the halting problem:

  • Timeouts:
    • Uused to limit the amount of time a system or browser waits for a response from a program/operation.
    • Ex. If a webpage takes too long to load, the browser usually displays an error message or offer to stop loading/exit the page
  • User Interface Feedback:
    • Providing feedback to the user is crucial when dealing with potentially lengthy operations. If a certain process takes more time, the system could inform the user that it takes time rather than the user just assuming it is unresponsive
    • Ex. The spinning loading thing that replaces the cursor
  • Caching:
    • Hardware or software component that stores data so that future requests for that data can be served faster
    • Ex. Web browsers store images, stylesheets, and scripts to speed up page loading. When a user revisits a page, the browser can retrieve these resources from the cache rather than downloading them again.