I have a problem with my current research that I have come across repeatedly over my research career in various different fields. It goes like this.
You are trying to characterise instances of some particular category that you study. This can be trees, buildings, chemical substances or English sentences. You determine a few discrete variables that you want to measure for each particular instance. For example, if we are talking about buildings, you may decide to focus on the main construction material (stone, wood, brick, etc.), use (residence, office space, public institution, etc.), construction decade (1940s, 1950s, 1960s, etc.), height bracket (under 5 m, between 5 and 100 m, between 100 and 300 m, over 300 m), plus others. Each individual instance that you observe (each building in our example) can be characterised by giving it a value for each variable. For example, building A may be made of wood, used for residence, constructed in the 1960s, and having a height in the 5-100 m bracket.
In real-world situations, you may have many variables (ten or twenty is not uncommon), and each variable may be able to take multiple values (many take 5-10 values; some may take up to 40 or 50). This creates a "solution space" of as many dimensions as variables, and as many individual points as distinct combinations of values. In my current research, I am working with 6 variables, some of which may take up to 12 different values, adding up to a solution space having over 15.000 points.
Now, I want to verify if every possible combination of values is possible. In other words, I want to check if there might be instances of the things I'm studying at every point in the solution space. In the example above, there aren't any buildings constructed in the 1880s and having a height of over 300 m. I have a suspicion that many areas in the solution space are similarly "impossible" areas, in the sense that there are combinations of values that are incompatible amongst each other and can't be found in the real world.
Of course, I can create a multi-dimensional matrix with thousands of cells, and start evaluating the possibility of finding an instance for each one. But this would take ages and is difficult to visualise and process without making mistakes. So, my question is, is there any particular technique to tackle a problem like this?