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Let me show you how this works on an example.

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Here is a square root function.

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It takes x and returns the square root.

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Let's assume we invoke this with the values 2, 4, and 16.

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When we invoke the square root with the value of 2,

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we could infer that x has a value of 2 and eps has a value of 10^-7.

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However, these patterns would also be instantiated:

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x is smaller or equal than two, and x is greater or equal than 2,

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because these patterns also hold for the values that we observed.

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In the next iteration, we invoke the square root with the number of four,

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and now, the invariant of x always being 2 is eliminated.

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What we get now, however, is that x being less or equal than 4 still holds.

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We can do so by merging the earlier invariant with a new one.

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And x greater or equal than 2 still holds for the new value.

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When we invoke the square root of 16, we now retain the invariant

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that x is less or equal than 16 and greater or equal than 2,

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and this is what we get in the end:

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x is between 2 and 16, and eps is always 10^-7.

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For the postcondition, we get similar ranges for the return value.

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The return value is between the square root of 2 and 4,

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which is the square root of 16.

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However, what we also get is that the return value squared is equal to x,

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and we get this because Daikon has an appropriate pattern for that,

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namely a pattern where the multiplication of any two variables equals a third variable,

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and this is instantiated with a return value, again with a return value and with x

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and this pattern then holds for all runs--

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at least for all runs with integer numbers.

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If we put in floating point numbers, then eps also comes into play,

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because of rounding errors, and then this pattern would no longer be discovered.

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So whatever Daikon can produce is constrained to the pattern library it has,

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but if you add more patterns, then you'll be able to discover more properties,

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which will take Daikon a bit longer, though, to discover them.

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Still, even with a perfect set of patterns,

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approaches like these are dependent on the actual numbers

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that are being fed in there.

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What Daikon produces is relevant for all the runs observed,

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but we all know that the real precondition for the square root

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does not have specific range constraints on x

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except that x should be greater or equal than 0.

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Likewise, the return value of square root is not necessarily between the square root of 2

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and the square root of 16,

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but it can actually be anything that's, again, greater than 0.

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So, tools for dynamic inference of invariants can work well

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if they do have a good test suite in the beginning.

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How can we get the correct ranges for x and the return value?

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By invoking square root with a value of 0?

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By invoking square root with a value of 1?

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By invoking square root with a value of maxint, where maxint is the highest available integer?

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Or by invoking square root with a negative value?

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Hint: you need multiple invocations.

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Check those which you need to get the correct ranges. Over to you.