mapping of the solving process with regard to the
steps introduced in figure 1. The third problem is presented to show the need for symbolic reasoning capabilities, such as for example logical inference.
Finally, the fourth and fifth problems are just
sketched (with regard to the solution process), but we
introduce them to stress the needed interplay of multimodal comprehension in deep reasoning: both
problems indeed come with text and diagrams.
Example 1: Three Friends
This first problem belongs to the easiest category,
aimed toward primary school students.
Jacob, Lucy, and Frank are three friends. All together
they are 28 years old. In how many years they will be
together 37 years old?
Step 1: Understanding Text (and Pictures)
Natural language processing techniques allow the
extraction of lexical, syntactical, and semantics infor-
mation contained in the text. Notice, however, that,
in the context of this challenge, understanding a text
means also that a computer has to identify the prob-
lem components like assertions, goals, constraints,
and others. In this specific puzzle, this would mean
discovering at least the explicit knowledge contained
in the text, namely that:
There are three friends.
All together here means the sum operator.
All together refers to the age of each friend.
The sum of their ages now is 28.
Ages and years are natural numbers.
How many refers to a quantity X of years.
In X years the sum of their ages will be 37.
We have to find X.
We highlighted in italic a few words that should have
a semantic link with specific concepts. For example,
the expression “All together” should be related to the
concept of mathematical sum, while “age” and
“years” should be linked with the natural numbers
concept. All the items in the list except the last one
refer to the problem’s assertions. The last item instead
refers to the goal.
Step 2: Identify Modeling and Solving Techniques
The specific problem of the three friends could be
modeled as a system of linear equations. Notice that
once a model has been identified, there is still an
open choice about the best solving technique to be
adopted: one solution might be to adopt some algebraic method for linear equations. Another method
would be to exploit constraint-satisfaction problem
(CSP) techniques for problem solving. In both cases
some sort of metaknowledge for reasoning would be
required, in order to select the specific solving technique.
Step 3: Identify Problem Components and Hidden
The problem components are the following. We have
three variables AJ, AL, and AF that represent the ages
of the three friends. X is the number of years that we
have to find. The three variables as well as X, are integers. Moreover the domain of X could be defined as X
; [0.. 28].
There is (at least) one other piece of hidden information that requires a commonsense knowledge base
about the time and the flowing of time. In particular,
the time will flow with the same speed for all three
friends. Summing up, the information that the three
friends will together be 37 years old can be modeled
as the fact that after X years their ages will be respectively AJ + X, AL + X, and AF + X.
Step 4: Frame the Model
Given the problem components, we can now state
the following model based on linear equations:
Figure 1. Human Intervention in a Problem-Solving Process.
text & figures