Qualitative Reasoning and Design Space Exploration

Baptiste Gueuziec

1 a

, Jean-Pierre Gallois

1 b

and Frédéric Boulanger

2 c

Université Paris-Saclay, CEA List, F-91120 Palaiseau, France

Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire Méthodes Formelles, F-91190, Gif-sur-Yvette, France

Keywords:

Design Space Exploration, Qualitative Reasoning, Constraint Solving.

Abstract:

The design of complex systems is a challenging task, combining optimization and testing techniques. Design

space exploration allows the designer to optimize the parameters of the system, but is usually very costly

and induces many inherent difﬁculties that require speciﬁc computation techniques to be solved. Qualitative

reasoning was ﬁrst introduced to describe system structures and causality, and developed to study the behavior

of the system and discretize its state space into qualitative states. It is now used in diagnosis and veriﬁcation

to reduce computation time and precision while still preserving major properties of the behavior. This article

presents how we think that qualitative reasoning can be applied to design space exploration in addition to

state space discretization, i.e. how it may help in the choice of parameters for a system, by reducing the

computational cost of this exploration.

1 INTRODUCTION

Qualitative reasoning is the area of computer science

and engineering that focuses on the representation and

reasoning on models with very little and imprecise in-

formation (Forbus, 2014). Our previous works on this

topic were applied to the study of hybrid systems and

allowed us to study the system’s state space based on

its dynamics, constraints, and discrete controller. We

obtained an entirely qualitative model of the behavior

of a system, discretizing the state space into qualita-

tive states that represent phases of the system’s be-

havior. This discretization allows us to abstract both

the state of the system and its behavior. The computa-

tion of the qualitative behavior generates path condi-

tions requiring to be solved by symbolic solvers such

as Sympy and Z3. In the instantiated systems, we

obtained very satisfying results to study and antici-

pate the possible trajectories of the system. However,

when we try to generalize our approach to the study

of the design space, we meet the limits of these tools

when computing the transitions. The design space

of a system is the space of the possible values of

its constant parameters, and its exploration requires

more generality and abstraction than the exploration

of the state space. Very few computational tools allow

https://orcid.org/0000-0002-4596-9769

https://orcid.org/0000-0001-6982-6247

https://orcid.org/0000-0003-3185-2807

for the symbolic resolution of multivariate inequali-

ties with variables of multiple orders. The difﬁculty

comes from the fact that SMT solvers only ﬁnd one

solution that respects the given constraints, while the

exploration of the design space would require ﬁnding

all the values of the parameters for which there exists

a solution in the variable space that satisﬁes the con-

straint. For a system with parameters P and variables

X, exploring the state space amounts to seeking a val-

uation (A, x) of (P, X ) satisfying a predicate R(A, x).

When exploring the design space, we instead look for

⊂ D

such that ∀A ∈ V

, then ∃x ∈ D

verifying

R(A, x) where D

and D

are the domains of the pa-

rameter P and the variable X. This logic problem can-

not be solved generally by simply using classic SMT

solvers.

Design Space Exploration (DSE) is a critical do-

main of the conception of complex systems. It con-

sists in choosing, testing, and comparing different

conﬁgurations of values for the parameters of the sys-

tem before its construction to verify some property or

optimize some utility function (Aiguier et al., 2010).

High-level DSE can be considered as multi-objective

optimization (Schafer and Wang, 2019) as we may

want to optimize a set of conﬂicting constraints and

utility functions. The solutions of the parameters are

often searched in a ﬁnite space of authorized values

(Lattmann et al., 2014), and the arising problem con-

sists both in the explosion of the number of possible

Gueuziec, B., Gallois, J. and Boulanger, F.

Qualitative Reasoning and Design Space Exploration.

DOI: 10.5220/0012417300003645

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 12th International Conference on Model-Based Software and Systems Engineering (MODELSWARD 2024), pages 203-210

ISBN: 978-989-758-682-8; ISSN: 2184-4348

203

solutions when taking into account all the parameters

and in the critical time needed to test the validity of

all of these solutions. To this extent, an effective DSE

requires three elements: a representation, an analy-

sis, and an exploration method (Kang et al., 2011).

The representation of the system must be designed to

optimize the research, the test, and the comparison

of solutions. The analysis requires a metric, a utility

function, or a set of logic predicates to evaluate and

compare the different valuations of the parameters of

the system. It also requires the choice of a veriﬁcation

method. For this veriﬁcation task, various methodolo-

gies are employed, with different levels of reliability,

depending on the complexity of the system and the

number of solutions to test. These methods mainly

consist of simulating the system to verify a property,

measuring an output value, or using some veriﬁcation

process to check the validity of a speciﬁc property. Fi-

nally, the exploration method is necessary to choose

the valuations to test and will strongly depend on the

nature of the design space. For example, some contri-

butions have studied the possibility of searching the

design space using machine learning methods (Blan-

chard et al., 2018; Kim et al., 2018; Mudgal et al.,

2018), others use brute force search in discrete sets,

and others use gradient descent methods in the case

of continuous spaces (Blanchard et al., 2018).

2 CONTEXT AND INITIAL

PROBLEM

2.1 Hybrid Systems and Qualitative

Modeling

We place ourselves in the context of hybrid cyber-

physical systems, represented as:

Q the current mode of the system (discrete),

X a set of n continuous variables,

I a function mapping each mode to

its invariant conditions,

F a function mapping each mode to

its ﬂow conditions,

T a set of discrete transitions, each described

by their starting mode, their condition,

their target mode, and their reset function

P a set of m parameters

Q takes values on D

, the ﬁnite set of modes of the

system, X is valuated on K

with K = R in the general

case, and P takes values on D

a set of dimension m.

can be either continuous or discrete. We make the

realistic hypothesis that D

is at least bounded to fa-

cilitate its exploration. We consider that the ﬂow con-

ditions are represented as ordinary differential equa-

tions (ODE), i.e.,

X = f (X,t) with t the time param-

eter. For example, an autonomous car running on a

straight road on the axis x can be represented as:

= {Stop, Accelerate, StableSpeed, Decelerate,

Reverse},

X = (x, ˙x) with x the coordinate of the car on the

variation axis and ˙x its speed on this axis,

X = K

= R

I = {Stop : X = 0 ; Accelerate : ˙x > 0 ;

StableSpeed : ˙x > 0 ; Decelerate : ˙x > 0 ;

Reverse : ˙x < 0},

F = {Stop :

X = 0 ;

Accelerate :

X = X



0 0

1 0





0 a



;

StableSpeed :

X = X



0 0

1 0





0 0



;

Decelerate :

X = X



0 0

1 0





0 b



;

Reverse :

X = X



0 0

1 0





0 0



}

with a > 0, and b < 0

T = {(Stop, ˙x > ε, Accelerate, X 7→ X),

(Accelerate, ˙x = v

max

, StableSpeed, X 7→ X ),

(StableSpeed, x > x

max

− γ, Decelerate,

X 7→ X),

(Decelerate, ˙x < ε, Stop, X 7→ (x, 0)),

(Stop, ˙x < −ε, Reverse, X 7→ X),

(Reverse, ˙x > −ε, Stop, X 7→ (x, 0))}

with ε > 0 a negligibility criterion under

which a value is considered as null, γ a

security distance from the objective point

max

and v

max

the maximal speed allowed.

P = {a, b, c, ε, γ, x

max

, v

max

}

Qualitative modeling consists in the creation of

a model adapted to support qualitative reasoning

(Sugeno and Yasukawa, 1993). As this reasoning

paradigm has the ambition to avoid numerical ele-

ments, the stake of qualitative modeling is to de-

sign models of Cyber-Phyical-Systems that integrate

enough information from the system to perform ef-

ﬁcient reasoning without using numerical knowledge

or computation. We call qualitative model the model

resulting from this design. A qualitative model must

not consider variables as numerical valuations but as

sets of values. For example, the autonomous car rep-

resented in subsection 2.1 can be qualitatively repre-

sented by imposing:

X = {−, 0, +};

F = {Stop :

X = 0 ;

Accelerate :

X =





;

StableSpeed :

X =





;

MODELSWARD 2024 - 12th International Conference on Model-Based Software and Systems Engineering

204

Decelerate :

X =



+−



;

Reverse :

X =



−0



}

In our previous works (Gueuziec et al., 2023),

we studied the suitability of qualitative reasoning

(Kuipers, 1986; Tiwari and Khanna, 2002) to repre-

sent, study, and reﬁne the state space of hybrid sys-

tems. We used polynomial solving to create a qual-

itative map of this state space and SMT solvers to

determine all the possible transitions between the ob-

tained qualitative states. This computation allowed us

to generate a complete behavioral tree of the system

between its modes and qualitative states. However,

the limit of this approach is that it requires all the pa-

rameters of P to be known. Considering that one of

the main causes of interest in qualitative reasoning is

its high level of abstraction, its use in a context where

all the values of P are not entirely deﬁned seems ap-

propriate. However, since the computation of the

transitions in the state space of the system requires

solving symbolic inequalities with a SMT solver (we

use Z3 in our model), the presence of symbolic con-

stants in the equations of the system creates signiﬁ-

cant problems. These symbolic constants should not

be treated on the same logic level, and using a sim-

ple SMT solver as previously done is not sufﬁcient

anymore because we are now searching for a subset

⊂ D

of parameter values such that for any set p

of parameters in V

, there exists x ∈ K

that satisﬁes a

predicate R(p, x). For example, if we consider a Van

der Pol oscillator system whose dynamics is given by

the equation system :

(

˙x = 10(y + x −

)

˙y = p − x −

(1)

with p a positive parameter, we can look for a value of

p such that the system is convergent. To this extent,

the expression of the predicate R(p, X) would be: ˙x =

˙y = 0, i.e. 10(y + x −

) = p − x −

= 0. The DSE

implies to look for a value of p > 0 such that ∃(x, y)

such that 10(y + x −

) = p − x −

= 0.

SMT solvers require well-designed predicates on

the qualitative states and frontiers and allow us to

check composed properties containing conjunctions

and disjunctions, meaning that it is either possible to

operate on the state space by giving the satisﬁability

for a valuation (complete or partial) of P, or on the

design space to evaluate the pertinence of a choice of

P. Our motivation in this work was to apply qualita-

tive reasoning and analysis to the simulation of cyber-

physical systems. Therefore, our motivation is to de-

velop our methodology to be able to use qualitative

reasoning for DSE to choose or optimize the values

of the parameters P to satisfy some constraints.

2.2 Design Space Exploration

Design Space Exploration aims to ﬁnd a design sat-

isfying a set of constraints Const or optimizing some

utility function f . Depending on the situation, the for-

mulation will have the form of a satisﬁability prob-

lem or an optimization problem. Since we want to

explore the possibility of using qualitative reasoning,

we mainly focused on the satisﬁability case. There-

fore, the problem takes the form:



max(V

= (V

, ...,V

) ⊂ D

)

s.t. Const(V

) = True

where Const(V

) represents the complete set of con-

straints that the parameters must satisfy. As qualita-

tive reasoning is based on the loss of numerical val-

ues, optimization problems are not adapted for this

modeling paradigm.

The sought maximum of the subspace V

is a

maximum regarding inclusion. Getting the maximum

valid set regarding the design constraints is essential

to compare the allowed values on other criteria after-

ward. This comparison is at the base of the design op-

timization (Fuchs and Neumaier, 2010) and requires

further analysis steps. They will not be treated in this

article.

The contribution presented here aims to study the

possibility of linking the DSE problem with quali-

tative reasoning, as already mentioned in (Lattmann

et al., 2014). Our goal is to open the capabilities

of qualitative reasoning to broader perspectives and

to present a hint at its possible use for more com-

plex applications than state space partitioning. More-

over, since DSE as presented in (Kang et al., 2011)

is mainly limited to structural considerations, the ad-

dition of qualitative reasoning to the DSE tools may

increase the ﬁeld of resolution with dynamic and be-

havioral considerations.

3 QUALITATIVE REASONING

As explained in the introduction, the ﬁrst signiﬁ-

cant element required to process the exploration of

the design space is an adapted representation of the

model. Using a discretization process, we get a qual-

itative representation of the system corresponding to

QM = hQ, X, B, F, I, T, Pi with B a mapping from each

mode q in Q to a set of qualitative borders deﬁning the

qualitative state they separate.

Deﬁnition: We call qualitative state of a system

the pair (m, qs) associated with a list of polynomials P

with m ∈ D

the current mode and qs ∈ {+, 0, −}

|P|

vector such that ∀i ∈ J1, |P| K, qs[i] = − if P[i](X) <

0, 0 if P[i](X) = 0, and + otherwise.

Qualitative Reasoning and Design Space Exploration

205

The qualitative states are then represented as an

array of {−, 0, +}, showing the position of the corre-

sponding state with respect to each frontier in B. This

representation can be applied to fully designed sys-

tems or more abstract ones with non-valuated parame-

ters of P. The second case implies the impossibility of

computing the transitions between qualitative states,

as this knowledge requires the computation of the sign

of the Lie derivative (Tiwari and Khanna, 2002) to de-

duce the allowed transition directions.

The Lie derivative (Yano, 2020) consists in a

vector ﬁeld differentiation method enabling the di-

rectional derivation of a function. For a polyno-

mial p whose zeros deﬁne borders of the system, the

Lie derivative corresponding to p can be written as

(p) =

∑

∈X

∂p

∂X

∂t

where the X

are the compo-

nents of X the set of continuous variables and t is the

time parameter. Determining the sign of this expres-

sion requires the ability to fully evaluate p and the

derivatives of all the variables X

This computation allows the creation of a quali-

tative model, where the states and evolution are not

represented with numerical values but with variable

signs (+, 0, −) and variation directions (correspond-

ing to the sign of the associated derivative).

As the DSE requires optimization criteria or nec-

essary constraints, it is possible to integrate these el-

ements in the construction of the frontiers during the

abstraction process. If R(P, X) ∈ K

n+m

is a predicate

on X and P that should be satisﬁed by the system,

then the equations composing R(P, X) can be consid-

ered individually to deﬁne new borders.

We anticipate that qualitative reasoning may im-

prove the DSE efﬁciency and applicability to com-

plex systems as qualitative reasoning applied on early

stages of the design of a system can link the found so-

lutions to non valuated variables. It can apply purely

symbolic solving to estimate adapted values of P to

different use cases of a same system depending on

the anticipated variations in the state space evolution.

Moreover, the abstraction of the state space may al-

low to propose local solutions to adapt the design of

the required qualitative behaviors.

4 PARAMETERS EVALUATION

The second important aspect that is necessary for

DSE is an analysis method. When a value or a subset

of values is available for P, it is essential to test its per-

tinence using a solver adapted for the equation struc-

tures. Here, as we made the hypothesis of systems

deﬁned with polynomial equations and predicates, we

use a symbolic polynomial solver. Many behavioral

properties of a dynamic system can be conveniently

represented using temporal logic (Han and Sanfelice,

2020). However, predicate satisﬁability for tempo-

ral logic is more complex and costly than ﬁrst-order

logic (FOL) evaluation. Moreover, qualitative reason-

ing gives us an adapted support to reason on ﬁrst-

order logic predicates. We will limit ourselves to the

properties expressible as FOL predicates to apply the

possibilities of qualitative reasoning. This choice im-

plies representing the behavioral constraint we want

to express as predicates.

For example, let us suppose we want a determin-

istic behavior for the system modeled through its dis-

cretization in qualitative states. In that case, we may

wish to have a direct transition from the qualitative

state (m

, s

) to (m

, s

). In temporal logic and con-

sidering a temporal structure T, <, we could write

the predicate corresponding to a necessary transition

= (m

, s

) U q

= (m

, s

) with s

and s

quali-

tative states of the system in mode m

, i.e., abstrac-

tions of values of X following an abstraction function

adapted to the system. In FOL, we could write it us-

ing the Lie derivative such that (m

, s

) =⇒ ∀b ∈

B(m

, s

)\ B(m

, s

), L

)∗ p

) > 0 ∧ for b

∈

B(m

, s

) ∩ B(m

, s

), L

) ∗ p

) < 0 where

B(m

, s

) is the set of borders deﬁning the qualitative

state s

and p

the polynomial function whose zero

corresponds to b. This predicate corresponds to the

evaluation of the sign of each L

) and its compar-

ison to the evolution of p

after crossing the corre-

sponding qualitative frontier. A qualitative descrip-

tion of the system gives us a convenient toolbox to

express the dynamic logic of the system as ﬁrst-order

logic predicates.

In our case, we keep using Z3 as it perfectly han-

dles property veriﬁcation on parameter values. We

use it as a Python library and create a solver s to

which we add the constraints on the system’s design,

as will be shown later in the algorithms we use for

design space exploration. The command s.check() re-

turns the satisﬁability of the created problem under

the given constraints. Compared to constraint solving

on the state space, the main difﬁculty here is that there

must be many calls to the solver to ﬁx a value or a set

of values of P that verify the constraints.

5 DESIGN SPACE EXPLORATION

The last element of an efﬁcient DSE is an optimized

research method to choose the values or the sets of

values of P. As already mentioned, we assume that

the design space is bounded. To develop a research

method, we have to separate two prominent cases de-

MODELSWARD 2024 - 12th International Conference on Model-Based Software and Systems Engineering

206

pending on the nature of the Design space.

5.1 Finite Design Space

In the literature, DSE methods are mainly adapted

to optimize the search for solutions in ﬁnite design

spaces. Many practices have emerged, including ge-

netic algorithms (Palesi and Givargis, 2002), ma-

chine learning methods (Motamedi et al., 2016), and

other discrete optimization methods. We chose to ex-

ploit the permissiveness of the SMT solver Z3 to use

branch-and-bound to specify the different parameters

of P sequentially without dealing with every possi-

bility. In branch-and-bound, we compute for each

parameter p

of P the satisﬁability of the constraints

set for every proposed value v

i, j

associated with ev-

ery satisﬁable combination of v

k,l

for k < i. Here, v

i, j

corresponds to the j

authorised value of the i

ele-

ment of P. By cutting the branch of the non-satisﬁable

partial combinations of P

1...i

, we save the computa-

tion time that could be lost to test trivially impossible

solutions. This algorithm is detailed in algorithm 1,

where possibleCon f ig retains the authorized partial

evaluation allowing the constraints Constr to be sat-

isﬁable. The function toConstr transforms the instan-

tiation of the current parameter p

to a constraint ex-

pressed as an equality. This new constraint is noted

newConstr, and its conjunction with Constr is noted

compConstr. D

] contains all the authorized val-

ues v

i, j

of the component p

of P. The operation

pre + v corresponds to the association of the autho-

rized partial conﬁguration pre of the p

, k < i with

the considered value v of p

5.2 Inﬁnite Design Space

Although the exploration of discrete design space is

often practiced on ﬁnite discrete sets, developing the

same techniques for continuous or inﬁnite research

spaces can be crucial. We develop an approach based

on dichotomy, i.e., a successive partition of the search

space in two distinct parts. For a single parame-

ter p, we deﬁne a minimum width of the remain-

ing deﬁnition set, and we subdivide it until we ﬁnd

that there is no solution in the set or that the cur-

rent width of the set is smaller than the minimum

width. The algorithm is shown in algorithm 2, where

DichotomialCut(S

) proceeds to a split of the set S

In the general case, if S

is an interval on R, noted

[a, b], the separation is completed by computing the

center of the interval c =

a+b

. Otherwise, if S

composed of many convex non-contiguous sets, the

separation is done by splitting S

between two non-

contiguous components. minSize corresponds to the

Data: P, D

, Constr

Result: Possible values of the parameters in P

possibleCon f ig = [[]] ∗ length(P)

for i ∈ (0, length(P) − 1) do

p = P[i]

for pre ∈ possibleCon f ig do

for v ∈ D

[p] do

newConstrs = toConstr(pre + v)

compConstr = newConstr ∧Constr

s = SMT solver

s.addConstraints(compConstr)

an = s.Check()

if an == SAT then

possibleCon f ig[i + 1].append(

pre +v)

)

end

if isEmpty(possibleCon f ig[i +1])

∧ i < length(P) − 1 then

Stop

end

return possibleConﬁg

Algorithm 1: Branch-and-Bound.

criteria on max(ka − bk)

a,b∈set

to stop the splitting

process. As in algorithm 1, Constr represents the set

of constraints on the design parameters. allowedSets

contains all the explored sets and Boolean values cor-

responding to the satisﬁability of the design constraint

on each of these sets.

To deal with multi-dimensional P, we can com-

bine algorithm 2 and algorithm 1 to use dichotomy on

every p

∈ P sequentially and to divide the different

combinations with a disjunction of cases.

Interestingly, we process once again to a dis-

cretization of a continuous space just as in the state

space partitioning. The obtained partition of D

based on the respect of the design constraints. Using

an optimization function could also give more tools

to reﬁne such a partition based on its magnitude and

variation. Moreover, the computation of its deriva-

tive would allow us to compute its critical points and,

therefore, its maximums.

5.3 Non-Uniform Probabilistic

Distribution

In the main part of the cases, D

is a subset of R and

often takes the form of an interval. However, in some

situations, it may appear that the design space does

Qualitative Reasoning and Design Space Exploration

207

Data: p, S

, Constr, minSize

Result: Domain of possible values for p

allowedSets = []

, S

= DichotomialCut(S

)

for set ∈ {S

, S

} do

newConstr = toConstr(p ∈ set)

ConstrTot = newConstr ∧Constr

s = SMT solver

s.addConstraint(ConstrTot)

an = s.Check()

if an == U nsat then

allowedSets.append((set, False))

else

if max(abs(a − b))

a,b∈set

< minSize

then

allowedSets.append((set, True))

else

allowedSets.concatenate(

Dichotomial search(

p, set,Constr, minSize

)

end

return allowedSets

Algorithm 2: Dichotomial search.

not take this form. For example, if prior knowledge

is available regarding the probability of the different

possible values, the set can no longer be represented

using a simple interval. Some works, such as (Blan-

chard et al., 2018), proposed solutions using proba-

bility density functions to represent this irregular dis-

tribution. This can be seen as a generalization of the

non-probabilistic case considering that ∀(a, b) ∈ R

such that a < b, the interval [a, b] can be represented

by a probability distribution following the uniform

probability density law U(a, b) on the same interval.

One can also represent discrete sets using discrete

probability laws. Therefore, it is possible to model

many design spaces using probability functions. Us-

ing the expression of the probability density function

for each p ∈ P, it is possible to pick values of P ac-

cording to these density functions and combine this

biased selection with the previously presented meth-

ods. However, this way of exploring the design space

better ﬁts the classical numerical simulation veriﬁca-

tion as it also applies gradient descent to orient the

choice of new values for the next simulation. It con-

strains the differentiability of the utility function ac-

cording to the researched element. This also better

corresponds to the selection of the variable’s initial

values than to parameters valuation.

6 EXPERIMENTATION

At this stage of our research, we have not been able to

apply the presented algorithm to concrete cases, and

we did not set up a test benchmark either. We in-

tend to implement this in the following weeks to put

our propositions to the test. Speciﬁcally, we plan to

test these algorithms on both simple systems such as

a Brusselator system, with positive parameters a and

b and state variables x and y, whose dynamics is given

by Equation 2, and on more complex ones such as the

Lorenz system, with positive parameters σ, ρ and β

and state variables x, y, and z, whose dynamics are

given by Equation 1.



˙x = 1 − (b + 1)x + ax

˙y = bx − ax

(2)







˙x = σ(y − x)

˙y = x(ρ − z) − y

˙z = xy − βz

(3)

Both systems exhibit different behaviors depend-

ing on the value of their parameters. The convergence

or periodicity of such systems being an essential ele-

ment in the trace of a system, we will try to express

the sets of validity of the convergence of each one

for ﬁnite and continuous sets of allowed values. If

our method allows to obtain sufﬁciently precise re-

sults corresponding to theoretical calculations, it will

be possible to consider further analysis and compu-

tation to improve the efﬁciency and the generality of

the process. However, to consider the result interest-

ing, it will be necessary to measure the computation

time of our tests to ensure that there is either a gain in

complexity, permissiveness, or generalization.

7 PERSPECTIVES AND FUTURE

WORKS

The next step to perfectly combine our different sub-

jects would be to express the properties of the state

space and the solution of the various constraints on

the variable X depending on the chosen value of the

parameters P. The validity of a predicate would be ex-

pressed as X = f (P) for the subset of D

that allows

these predicates to have a solution. To achieve this

goal, we searched many different methods and looked

at other tools, such as modal solvers or Wolfram Al-

pha, which give encouraging solutions in that direc-

tion. However, it does not generalize well for systems

with too many variables. Other studied tools allowed

this type of resolution but only for low-degree poly-

nomial functions. There is no possible generalization

for now.

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8 RELATED WORKS

Some works have studied the different possibilities

offered by qualitative reasoning. (Medimegh et al.,

2018) focused on the description and the prediction

of the behavior of systems at a high level of abstrac-

tion, while (Zaatiti et al., 2018) developed the diagno-

sis aspect of qualitative reasoning using more numeri-

cal methods. (Gueuziec et al., 2023) proposed a tech-

nique to automatize system abstraction and the cre-

ation of hybrid automata. All these projects focused

on the state space study and did not delve into state

space partitioning but showed the advantages and

the limits of the different approaches. Many works

proposed very different approaches regarding design

space exploration, from the classic search in a ﬁnite

design space (Lattmann et al., 2014) to works present-

ing deep-learning-based strategies (Motamedi et al.,

2016). Methods also exist to deal with continuous sets

using probability density functions (Blanchard et al.,

2018) to represent the design space based on previous

knowledge and more qualitative constraints regarding

the expected value of parameters. However, this ap-

proach is more appropriate to deal with the choice

of the initial value of variables and is, therefore, at

the edge between state and design space exploration.

Also, many works regarding the solving of tempo-

ral logic constraints and its use for optimization have

been led (Wolff et al., 2014) and may allow signiﬁ-

cant progress in the resolution of temporal and modal

predicates. DSE, in its aspect of design optimiza-

tion, has been more signiﬁcantly treated in (Fuchs and

Neumaier, 2010; Wang and Shan, 2006).

9 CONCLUSION

Theoretically, qualitative reasoning should have an in-

teresting relation with DSE, and we can take advan-

tage of it to optimize performances and computation

time. Qualitative abstraction allows a permissive rep-

resentation of models and a satisfying logical expres-

sion of essential system properties. Using an SMT

solver on these ﬁrst-order logic predicates is a relevant

method to test the constraints’ validity for a speciﬁc

valuation of the parameter list P.

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