By dropping the property of -closure Doignon and Falmagne (1985) arrived at the definition of a so-called `surmise system'.

Let be a set of problems, and a so-called surmise function which assigns to each element a non-empty family of subsets of , the so-called clauses for . Then, together with is called a surmise system on problems.

A clause of contains a set of prerequisites of minimal with respect to in the sense that if is also a clause for and , then =.

Generalization of Birkhoff's (1937) theorem by Doignon and Falmagne (1985): Every surmise system uniquely determines a set of knowledge states which is closed under union and which contains and .
Such a set of states is called a knowledge space.
Clauses of a surmise system are elements of such a knowledge space. Any state of a knowledge space can be obtained by a union of specific clauses.

• Surmise function for the above introduced 4-problem set:
  
• Interpretation: Each participant who can solve problem is also able to solve problem or ; each participant who can solve problem is also able to solve problem .
  
• Problem sets or are supposed to be `prerequisites' for problem . is the prerequisite for .
• Surmise systems can be visualized by so-called `AND-OR graphs'.
• Or-nodes are marked (sometimes) by a curved line and a V symbol.
• And-nodes don't have a special mark.

  Figure 2: AND-OR graph for the example problem set .

The knowledge space resulting from the surmise system is

- no longer -closed.

Example:

• Figure 3 shows an example of an and-or-graph of a domain with five items.
• Knowing a person masters item c, it can be surmised that this person will also master items d and e.
• Which problem/problems is/are a prerequisite for other items?
• To master item a it is sufficient to know about item b or item c (or both of them).
• A person has to understand about both items d and e in order to master item c.
• In this sense, items d and e are prerequisites for c.

Figure 3: Example of an and-or-graph for 5 items.

 

• What are the clauses?

• Interpretation of Q according to Baumunk (1995):

Items:
a Subtraction of uncommon fractions.
b Addition of mixed numbers.
c Addition of uncommon fractions.
d Addition of common fractions.
e Finding the least common multiple.

• The corresponding knowledge space is the set of all subsets of the domain set which conform to the relationships mentioned before:

• Subset {b} is not a member of the knowledge space, since item d is a prerequisite for item b, which yields as the smallest (with regard to the ordering given by set inclusion) member of the knowledge space which contains item b.

• A knowledge space on a set of items has (by definition) the following three properties:

  

• means a knowledge space is closed under union.