How can I prove algorithm correctness? [closed]

How can I prove algorithm correctness ? when i face a problem and come up with a solution the only way to know if this a valid solution or not is by trying some test cases. if they pass through the algorithm and produce the expected output then my algorithm most properly true. but obviously this is not hold all the time because i may forget some corner cases or it is hard to figure out all the test cases. So how can I prove mathematically if my algorithm produce the expected output or not ?

For example, consider the program below.

You’re given a read only array of $n$ integers. Find out if any integer occurs more than $n/3$ times in the array in linear time and constant additional space.

Algorithm: We will use an array of size 3 to count occurrence of numbers let it be count adding numbers to our count array with its proper count. If we reach the size of count array we decrement one from count of each number. If number count becomes zero it can be safely eliminated from the count array.

Here is an example:

• Input: 4 3 3 7 2 3 4 5
• count arr (4,1) 4 as first element and 1 is its count till now
• count arr (4,1)(3,1)
• count arr (4,1)(3,2)
• count arr (4,1)(3,2)(7,1). Here we reach the max allowed size for count then we need to decrement count by one and if it reaches zero its item will be removed from our count array so count arr becomes count arr (3,1). We will proceed with next element in the array which is 2.
• count arr (3,1)(2,1)
• count arr (3,2)(2,1)
• and so on. At the end count arr will be (3,1)(5,1).

We will make another loop to the input array to count occurrence of 3 and 5 and if any one exceeds $n/3$ it will be printed out.

• You will need to tell us what algorithm you have in mind, otherwise your question is way too general. Jun 20, 2015 at 3:40
• You use mathematical proof. Jun 20, 2015 at 4:47
• I believe that Yuval meant that the way to prove depends much on the specific algorithm. There are plenty of methods: by induction, case analysis, etc. etc. Try to search this site within the algorithm tag questions that have "correctness" in their title, you will find several different examples Jun 20, 2015 at 4:48
• This is far too broad to answer. It's at essentially the level of "How do I prove a mathematical theorem?" Jun 20, 2015 at 7:54
• @RenatoSanhueza When someone tries to understand a concept, it is more important to try help his intuition rather than telling him first that it is hopelessly useless. Can we prove things about programs: Yes we can, and we do have methods for doing it in many cases. And we find new ones. The cosmological fact that there is no unique method to do it is second order for most people What is the point of telling it is undecidable when he does not really know what proving correctness means to begin with. And how many people bother to define what it means before saying it is undecidable. Jun 20, 2015 at 17:06

In practice, to prove an algorithm you should search a good invariant property for each loop. For example, if you compute in a given order the sum of $n$ integers with a for loop (indexed by $i$), the invariant could be : "At the end of an iteration, the $sum$ variable contains the sum of the $i$ first values". It's invariant over iterations number and always true, it's easy to prove this by recurrence on the iterations number. Afterwards you can easily conclude that, at the end of the loop, $i=n$ and thus that $sum$ is the expected sum.