In Need of Methods to Solve Imprecisely Posed Problems
Andreas Geiger, Andreas Kroll and Hanns J. Sommer
Faculty of Mechanical Engineering, University of Kassel, M
¨
onchebergstrasse 7, 34121 Kassel, Germany
Keywords:
Artificial General Intelligence, Computational Intelligence, Imprecisely Posed Problems.
Abstract:
Problem solving and problem understanding are interwoven. Often at the beginning of a project, engineers’
knowledge is insufficient with regard to the posed problems. The correct specification of a task may depend
on its solution, which is why the conventional sequence of specification and solution cannot be maintained.
Methods are needed to deal with ’imprecisely posed problems’, by which the solution process will be adapted
to the essential problem structures only. We present procedures to detect these essential structures and to
acquire problem solving skills from the solution process itself.
1 EXPOSITION OF THE THESIS
Albert Einstein stated in his Herbert Spencer lec-
ture of 1933 (Kragh, 2013) that “we can discover by
means of pure mathematical considerations the con-
cepts and the laws ..., which furnish the key to the
understanding of natural phenomena”.
Following Einstein’s dictum, many scientists be-
lieve that also engineering is restricted to the solutions
of mathematically precisely posed problems. But is
this attitude really the most fruitful? Einstein declared
in subsequent lectures: “experience alone can decide
on truth”, and our experience from successful engi-
neering projects tells us that a thinking that restricts
himself exclusively to well posed problems will never
be open for the needs of the problems emerging from
practice. The openness for new ideas is a capacity
which is indispensable for every engineer involved in
practical work. Tools are needed to support this ca-
pacity:
Thesis 1. Instead of solving engineering problems by
first transforming them into presisely posed problems,
it is more effective to solve them with methods suitable
for solving imprecisely posed problems.
To bring this thesis into a precise form, we
suggest the following definitions:
Definition 1. By an imprecisely posed problem we
understand a problem whose parameters, boundary
conditions or other descriptions are not completely
fixed.
Definition 2. A method to solve imprecisely posed
problems is a solution method which accepts detailed
definitions and specifications with the benefit of hind-
sight.
The requested methods should not only be able to
transform imprecisely known data into a mathemat-
ically well-defined fuzzy-knowledge (K
¨
opcke, 2004)
but also to prepare and to elaborate solution methods
whilst essential requirements for a specification of the
solution are missing. Soft constraints are in terms of
Definition 1 mathematically precisely specified. It is
our intention, to select from an imprecisely specified
problem the essential structures by which the general
features of solution method can be determined.
To justify Thesis 1, we demonstrate first that im-
portant objectives of engineering design strategies are
in contradiction with an early specification of the fi-
nal solution. In the third section, three methods are
presented that correspond to the different types of in-
accuracies. Section 3.1 discusses a concept-oriented
approach, by which the essential structure of a prob-
lem will be detected and used for the selection of an
adequate solution method. The approach presented in
section 3.2 is suitable when no structures are cognis-
able and, as every determinate approach implies some
bias, in section 3.3 a method is discussed to acquire
problem solving skills from the solution process it-
self. Finally we summarise by emphasising the im-
portance and universality of the demand for methods
to solve imprecisely posed problems. Our considera-
tion shows that Artificial General Intelligence (AGI)
provides a theoretical foundation for the design of
strategies to solve imprecisely posed problems.
294
Geiger A., Kroll A. and J. Sommer H..
In Need of Methods to Solve Imprecisely Posed Problems.
DOI: 10.5220/0004592702940299
In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 294-299
ISBN: 978-989-8565-70-9
Copyright
c
 2013 SCITEPRESS (Science and Technology Publications, Lda.)

2 ENGINEERS NEED FLEXIBLE
DESIGN METHODS
The following examples demonstrate that engineers
in design, development and construction should and
very often must avoid an early determination of the
final outcome of their task. Some principal reasons
for this demand are presented.
2.1 Plant Engineering
In this subsection we present experiences of five years
of collaboration in the construction of broadcasting
plants. Plants have a quite long lifetime (30 years
in the case of broadcasting plants). This means that
when new plants are designed and constructed, the
experiences of the operators of the old plants as well
as new technological advances should be considered
(Aztiria et al., 2010). The operators of the older
plants are usually not familiar with new technologies
and can appreciate them only after some testing time
whilst they gain real experience. The design of a plant
must be adapted to the operator’s feedback, but this
feedback will only be disposable after the main com-
ponents of the plant and its control system have been
realised.
2.2 Evolvability
Other industries, like automobile industry, face the
problem of plant modernisation every two or three
years. The knowledge obtained in the production pro-
cess must be transferred from the actual plant to the
future designs. The concept of evolvability plays an
important role in all engineering tasks (van Beek and
Tomiyama, 2012). For factories, it is not only im-
portant to have good products but just as much nec-
essary that these products and including their pro-
duction processes offer the possibility to be contin-
uously adapted to new, today unknown requests of fu-
ture clients. Examples from the history of industry
demonstrate that is is often better to have no solution
than to have the second best solution. Already more
than hundred years ago, Thomas Edington made this
sad experience. He lost a lot of money because his so-
lution for an electrification of the United States (using
DC technology) was only the second best compared
with Westinghouse’s solution using AC technology.
2.3 Chemical Engineering
Process design in chemical engineering (Gross, 1999)
confronts the problem that exact models are not avail-
able. Exact knowledge has always to be comple-
mented with heuristic rules whose adequacy cannot
be guarantied or tested in advance. As model evalu-
ation is impossible without fixed parameters and pa-
rameter estimation is only possible for well defined
model classes, in the early stages, plant design can
never be transposed into a mathematically well de-
fined optimisation problem. Flexibility is therefore
one of the main exigencies a design process of chem-
ical plants should satisfy. Even in its use, the task to
control the unknown dynamics of many components
corresponds to the class of imprecisely posed prob-
lems. Many chemical processes had to be optimised
during their operation time.
3 METHODS TO SOLVE
IMPRECISELY POSED
PROBLEMS
In this section we present and discuss strategies which
we consider suitable for solving imprecisely posed
problems. Each of these strategies corresponds to a
specific problem of common designs.
3.1 Concept - Orientation
The worst case for system engineers occurs, when
complete components of a project must be replaced.
For example in software technology and system en-
gineering optimised algorithms are often strongly
adapted to a special problem and not flexible enough
to accept small changes without significant redesign.
Algorithms are highly specialised entities, which are
very costly and it is a priori never an obvious task and
often even impossible to adapt them. To prove the
conformity of an algorithm with a problem, the al-
gorithm must be completely accomplished. But does
an alternative exist to a mathematical theory of algo-
rithms?
Computational Intelligence (CI) offers (compared
to the huge and intractable set of algorithms) a better
manageable set of concepts which have the property
of being adaptable by a suitable realisation to special
problems. Whereas algorithms yield most efficiently
the solutions to well-defined problems, CI-procedures
are universally applicable (Kroll, 2013).
Concepts can be characterised by principles de-
scribing their main idea. In Table 1 some CI methods
with their underlying operational principles are listed.
A further option, to obtain concepts for problem
solving consists in generalising and representing the
methods of mathematical procedures in a general lan-
guage. This method has been used by Kroll and Som-
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295

Table 1: CI-methods and their corresponding principles.
Neural Networks Let many groups of
agents try to solve a
problem and select the
most effective one.
Evolutionary
Computing (cross
over-strategy)
Combine good parts of
several entities to obtain
a better new entity.
Computed Swarm Learn from the best.
Tabu Search Realise a greedy search
but avoid to test twice
the same element of the
search space.
Diversity Search Distribute the search
over all parts of the
search space.
Fuzzy Logic Information granulation.
mer (Kroll and Sommer, 2013) to deduce principles
for solving coordination problems from a list of meth-
ods for constraint satisfaction problems that was given
by a theorem of Schaefer.
The concept based approach presented in this sub-
section has its root in the spirit of Computational In-
telligence. To solve a problem using a concept based
approach, the following steps must be realised (com-
pare Figure 1):
Steps of a Concept-oriented approach:
(I) Choose a list of CI-methods with it’s correspond-
ing concepts.
(II) Select from the list those methods whose princi-
ples match the characteristics of the problem.
For an imprecisely posed problem, those methods
are selected whose specification parameters en-
able a later fixation of a priori unknown problem
specifications.
(III) The experiences obtained from tests with realisa-
tions of the selected methods will be used to fix
the unknown problem specifications.
(IV) Finally the more precisely stated problem will be
solved with the algorithm that has been obtained
from the selected methods.
The concept oriented approach has been success-
fully applied to solve a
Coordination Problem for Plant Control:
Find control sequences for the agents controlling
a plant such that the mutual influences will be
minimised which the actions of one agent produce
in the parts which are in the responsibility of others
(D
¨
urrbaum et al., 2012).
selection of
methods
parameter
adaptation
test results
imprecisely
and
CI−methods
concepts
posed
problem
problem
specification obtained algorithm
solution
Figure 1: A Concept-oriented Approach.
For this problem it is a priori unknown:
• Which rules must be applied by the agents to guar-
antee absence of mutual interference?
• Are these rules acceptable for the plant operators?
• Do additional constraints exist that may be satis-
fied? (E.g. smoothness of actions, time restric-
tions, etc.)
The selection of problem adapted methods had pro-
vided for Tabu Search and Diversity Search. A flexi-
ble procedure was realised with these methods which
made it possible to find the unknown data with the
benefit of hindsight after a period of extensive tests
with the system.
3.2 Belief-based Modeling
In the case when no physical model exists, the
most often applied approach to obtain a mathemati-
cal frame is system identification (Ljung, 1987). But
even this approach may be impossible in cases where
no model class can be selected or where no error
model is available. An example of such a problematic
task was presented by the requirement to develop an
information system to forecast drill fracture in chip-
ping processes (D
¨
urrbaum et al., 2008).
Due to the variety in design and material, it was
absolutely unrealistic to develop a physical model for
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the chipping processes. Even more, the temporal dis-
tribution of these events was not available, as it was
impossible to precisely specify a definition of fracture
events. To solve the problem only the following data
could be utilised:
• During the chipping process measurements of ax-
ial force F(t) and moment of torque M(t) were
available.
• The result of each particular chipping process was
categorised respective to two classes: ’good’ or
’bad’.
As no other information was available, the fracture
forecasting system had to be based on this data
exclusively.
3.2.1 Idea of the Solution Approach and its
Mathematical Background
Each observation made during a test beginning at time
t
start
and ending at time t
end
can be described by a
fuzzy-sentence of the form:
Example 1. In the first time interval I
begin
⊂
[t
start
,t
end
], F(t) is strongly increasing, in the middle
time interval I
mid
⊂ [t
start
,t
end
], F(t) and M(t) are os-
cillating, in the final time interval I
f in
, F(t) and M(t)
are zero and the result is bad.
Let the letter S refers to a particular sentence
describing measurements m = {(F(t), M(t), result) |
t ∈ [t
start
,t
end
]} obtained from one chipping process.
A formal representation of the propositions into a
sentence S (e.g. ’In the first time interval I
begin
⊂
[t
start
,t
end
] F(t) is strongly increasing.’) using fuzzy-
membership-functions, transforms this sentence itself
into a fuzzy-set with a membership-functions µ
S
(m)
for measurements m.
Each measurement m
i
= {(F
i
(t), M
i
(t), result
i
) |
t ∈ [t
start
,t
end
]} for i = 1, . . . , N,that had been obtained
during a test phase, provides a confirmation µ
S
(m
i
) for
a sentence S. The belief-degree bel(S) of a sentence
S is given by the aggregation of all these confirma-
tions µ
S
(m
1
), µ
S
(m
2
), . . . , µ
S
(m
N
) with an aggregation
operator Agg (Jager, 1994). The following Lemma 1
demonstrates that the conditions claimed for an aggre-
gation determine the operator Agg up to a rescaling.
Lemma 1. (Benferhat, 2010), (Sommer and
Schreiber, 2012)
Let Agg be an operator that assigns to val-
ues d
1
, d
2
, . . . , d
N
∈ [0, 1] an aggregated value
Agg(d
1
, . . . , d
N
) ∈ [0, 1] such that the following
conditions hold for all values d
i
∈ [0, 1]:
Agg(d
1
, . . . , d
i
, d
j
, . . . , d
N
)= Agg(d
1
, . . . , d
j
, d
i
, . . . , d
N
),
Agg(d
1
, . . . , d
i
, . . . , d
N
) ≤ Agg(d
1
, . . . ,
¯
d
i
, . . . , d
N
),
for d
i
≤
¯
d
i
,
Agg(d
1
, . . . , d
k
, Agg(d
k+1
, . . . , d
N
)) = Agg(d
1
, . . . , d
N
),
(1)
There exists a value e ∈ (0,1) such that:
Agg(d
1
, . . . , d
N
) = Agg(d
1
, . . . , d
N
, e),
Agg(d
1
, . . . , d
N
) < Agg(d
1
, . . . , d
N
, d
max
)
for d
max
= max{d
1
, . . . , d
N
} < e
(2)
The operator Agg can be represented with the fuzzy-
operators ∧ and ∨,
(with a ∧ b = a · b; a ∨ b = 1 − (1 − a)(1 − b)),
if the values d
i
have been re-scaled by functions
Φ
∧
: [0, e] → [0, 1] and Φ
∧
: [e, 1] → [0, 1]
with the equations:
Agg(d
1
, d
2
) = Φ
−1
∧
(Φ
∧
(d
1
) ∧ Φ
∧
(d
2
))
for d
1
, d
2
∈ [0, e]
Agg(d
1
, d
2
) = Φ
−1
∨
(Φ
∨
(d
1
) ∨ Φ
∨
(d
2
))
for d
1
, d
2
∈ [e, 1]
(3)
For d
1
∈ [0, e] and d
2
∈ [e, 1] various possibilities ex-
ist:
• Remove any value d
2
∈ [e, 1] from the aggregation,
• Remove any value d
1
∈ [0, e] from the aggregation,
• Use the following equations to calculate the ag-
gregation:
Agg(d
1
, d
2
) = Φ
−1
∨

1 −
1 − Φ
∨
(d
1
)
Φ
∧
(d
2
)

for (1 − Φ
∨
(d
1
)) < Φ
∧
(d
2
)
Agg(d
1
, d
2
) = Φ
−1
∧

1 −
1 − Φ
∧
(d
2
)
Φ
∨
(d
1
)

for (1 − Φ
∨
(d
1
)) ≥ Φ
∧
(d
2
)
(4)
For a balanced decision, the last of the possibilities to
aggregate values from different zones d
1
∈ [0, e] and
d
2
∈ [e, 1] will be used.
Lemma 1 has been presented in different formu-
lations by various authors. A prove can be found in
(Sommer, 1995).
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3.2.2 Realisation of the Solution Approach
The method presented in section 3.2.1 had been used
to construct a fracture-forecast system by the fol-
lowing steps (D
¨
urrbaum et al., 2008):
• Construct randomly fuzzy-sentences S describing
the measurements that had been obtained during a
chipping process.
• Compare the constructed sentences S with real
measurements {m
1
, . . . , m
N
}, using their fuzzy-
membershipfunction µ
S
to calculate the values
µ
S
(m
i
).
• Highly believable sentences, (satisfying
bel(S) ≈ 1) which assert bad results are selected.
(Nearness between bel(S) and 1 is the selection
criterion. Lemma 1 shows that a different choice
of the aggregation operator can be compensated
by a modification of this criterion.)
• The selected sentences are included into an alarm
system, forecasting breaks in the work process.
During the use of the fracture-forecast system in the
work process, the following rule will be used:
• If a measurement taken from an individual work
process has a high membership degree into some
of the selected sentences, an alarm will be given
and the drill must be changed.
The method used in this project may be transferred
to many other problems. Belief-Based Modeling is
a search technique for a mathematical description
which can be applied for problems whose mathemat-
ical precision is questionable.
3.3 Co-evolution-Method
As mentioned in section 2, over and above the prob-
lem description, solving imprecisely posed problems
calls for information obtained in the whole solution
process. This implies a different view. It is now
necessary to consider the whole context of a prob-
lem. The theory which tries to extend Artificial In-
telligence and to understand this view is called Arti-
ficial General Intelligence (AGI) (Goertzel and Pen-
nachin, 2007),(Sun, 2009). AGI offers a theoretical
background for Computational Intelligence by exam-
ining the relations between phenomena that had been
detected in cognition and their reasons in knowledge
processing. To stress this relation, we use the defini-
tion:
Definition 3. A solution system for an imprecisely
posed problem that has been designed with CI-
methods will be called an AGI-system.
The phenomenon of cognitive synergy implies the
empirically well confirmed experience that to obtain
a powerful AGI-system multiple subsystems focused
on learning regarding different sorts of information
must interact in such a way as to actively aid each
other in overcoming combinatorial explosion. Due
to an effect called “trickiness” by Goertzel, Ihle and
Wigmore (Goertzel et al., 2012), cognitive synergy is
responsible for the difficulties to evaluate and opti-
mise AGI-systems:
”’Trickiness’ means the effect that, in each
case of a practical test [to evaluate an AGI-
system] it seems likely that there is some way
to “game” the test via designing a system
specifically oriented towards passing that test,
and which doesn’t constitute progress towards
a general power [of the tested AGI-system].”
Because of cognitive synergy, methods for search-
ing solutions to imprecisely posed problems should
be composed of various procedures, but due to the
trickiness-effect, the influence of the particular proce-
dures on the overall method is a priori unknown and
only detectable from the final solution. As a conse-
quence, the overall method should be selected from
a co-evolutionary-competitive system in which var-
ious optimisation processes influence each other.
Co-evolution-Method:
When solving a problem, at the very beginning we
dispose of a huge set of possible search processes
(searching for a solution of the problem) which are
differently composed of the basic procedures and
which may use different assumptions with regard to
the missing specifications. Some of these collections
are selected into a competitive system and adapted
with the following steps:
Evaluation: Evaluate the solutions offered by the
individual processes and learn from them their
recommendations of a final problem specification.
Information Exchange: The currently best solu-
tions with their corresponding problem specifica-
tion will be placed at the disposal of the others.
Elimination: Eliminate ineffective processes.
Configuration of new Processes: Insert new pro-
cesses using the information obtained from the
competition.
Co-evolution is an adequate method to solve impre-
cisely posed problems because it uses besides the
problem information the information that will be pro-
vided by the search process itself. In this way, in-
formation can be exploited which is not available for
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a conventional search, using an evolutionary algo-
rithm(Geiger et al., 2010).
4 CONCLUSIONS
The motivation of Thesis 1 reveals the relations be-
tween Computational Intelligence (CI) and Artificial
General Intelligence (AGI). The understanding pro-
vided by these scientific disciplines is indispensable
for every approach to solve imprecisely posed prob-
lems.
In the same way as problem solving is the objec-
tive of Artificial Intelligence (AI), imprecisely posed
problem solving can be understood as the objective
of AGI. Two well known experts of AGI, Richard
Loosemore and Ben Goertzel wrote in a recent paper
(Loosemore and Goertzel, 2012):
“There are two possible types of intelligence
speedup: one due to faster operation of an intelli-
gent system (clock speed increase) and one due to
an improvement in the type of mechanismus that im-
plement the thought process (’depth of thought’ in-
crease).”
But as speedup may be limited by quantum me-
chanical limitations for the hardware and by the in-
tractability of many problems, the second type of in-
telligence amplification seems to be more promis-
ing. Really intelligent systems must be “self-
understanding”. To solve a problem means first, to
understand this problem, but secondly to understand
also the search process and context for its solution.
Many industrial systems like plants are not only tech-
nical systems but also social systems, and as such sys-
tems they enable emergent events. The complexity
residing in those systems requires a parallel design of
it’s parts. For that reason, plant engineering always
confronts imprecisely posed problems. The “depth of
thought” in engineering reasoning is therefor strongly
related to the ability for solving imprecisely posed
problems. The methods of CI and AGI are suitable to
create strategies for that task. The presented strategies
correspond to the strongest confinements of common
designs.
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