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As far as I know, a problem representation is the formulation of the problem in a way that it can be programmed and therefore solved (for example, you can represent the $N$-queens problem by using an array of $N \times N$).

What does problem modelling mean? What is the difference between a problem representation and problem modelling?

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I would say these terms are often used interchangeably in AI. When they differ, I would say that problem modeling means finding a mathematical description of the problem, while problem representation means finding a particular way to represent that mathematical formalism.

For example, a list of numbers can be stored (represented) with a linked list, and array list, a hash table, or a self-balancing tree. All of them can produce a faithful model of the list, but if what you want to do is find the order that elements were entered in, the array list or linked list is far faster and more natural. If what you want to do is determine whether certain pieces of information are present in the list, the hash table is fastest. If what you want to do is find ranges of similar elements, the tree is fastest. Essentially, representational choices are engineering problems, while modelling choices are scientific problems.

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  • $\begingroup$ What do you mean by "mathematical description". If I formulate the problem as a CSP, I have to use mathematical operations to express the constraints. How else could I formulate the problem in a mathematical way? @JohnDoucette $\endgroup$
    – Jay Critch
    Mar 8 '19 at 20:59
  • $\begingroup$ @JayCritch To model a CSP, you write down the constraints. To represent a CSP, you might use a list of strings that state logical conditions; a function that checks consistency of some representation of an assignment by using loops over known indices; or even a compact representation in terms of binary digits and a sequence of bitwise operators. These different instantiations correspond to different representational choices. You can actually write all of these down mathematically, but usually this isn't worthwhile. Where it pays off is when writing down the program that solves the CSP. $\endgroup$
    – John Doucette
    Mar 9 '19 at 0:06

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