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A Resilient Behavior Approach Based on

Non-monotonic Logic

Jos´e Luis Vilchis Medina1and Pierre Siegel1and Vincent Risch1and Andrei

Doncescu2

1Aix-Marseille Univ, Universit´e de Toulon, CNRS, LIS

Marseille, France

{joseluis.vilchismedina,pierre.siegel,vincent.risch}@lis-lab.fr

2LAAS, CNRS

Toulouse, France

andrei.doncescu@laas.fr

Abstract. In this article we present an approach for representing a re-

silient system which has the capability of absorb perturbations and over-

come a disaster. A framework called KOSA is depicted, which is a world

that contains a set of knowledge describing objectives, states and actions,

linked by a set of rules. This link is expressed by a default theory. First,

we deﬁne resilience as a relation among states and objectives. Secondly,

from a given state, extensions are calculated, which provides informa-

tion where to go to the future state. The connection, among two or more

states creates diﬀerent conﬁgurations that we call trajectories. These

connections represent an evolution of the knowledge. Consequently, this

reveals the existence of a resilient trajectory. Examples of piloting an

airplane are concerned through this paper. Eventually, we present a dis-

crete theoretical behavior of the complete model. Finally the notion of

distance among extensions is introduced.

1 Introduction

In every approaches, resilience always concerns the capability for a system to

absorb perturbations and overcome a disaster1 2. For example, nature exhibits

many resilient behaviors: ﬂock of birds, school of ﬁsh, ecological disasters. . . This

behavior allows ﬂocks of birds and schools of ﬁshes to survive in an uncertainty

environment, looking for food in order to preserve the species despite of preda-

tors. Ecological disasters such as seaquakes, tornados and earthquakes involve a

resilient behavior since in all the cases after disasters occurred, elements of the

nature will ﬁnd an equilibrium point. These examples can be translated in an

abstract way. For instance, if we consider two systems, s1and s2, which both

are connected in some manner there exists two types of connections with a view

to keep a balance among both systems. First, a positive feedback is a process in

1https://en.wikipedia.org/wiki/Resilience

2https://dictionary.cambridge.org/search/english/direct/?q=resilience

1

which the eﬀects of small perturbations in s1aﬀects s2, producing disturbances

in s2which in turn will also aﬀect s1. This can lead to collapse of both s1and

s2. A positive feedback causes instability, the output of a system is generally an

exponential growing, chaotic behavior or diverge from an equilibrium point. On

the other hand, a negative feedback produces a reduction at its output, leading

to stability or equilibrium point, reducing the eﬀects of disturbances [1, 2].

1.1 Resilience: State of the Art

Here, we consider Holling’s deﬁnition about resilience. He deﬁned it regarding

four properties: reorganization,exploration,release and conservation [3, 4]. Re-

silience was recently studied from a logical point of view: non-monotonic logic [5]

was used to describe, mainly, two of the four properties deﬁned by Holling: explo-

ration and conservation (mostly, to explore solutions and conserve consistency

when facing of perturbations).

From this view, we discuss some questions that motivated this study: What

is the relation among states and objectives? Is there a resilient behavior? Are

there resilient trajectories?

In this article, we introduce a model which allows to understand the prop-

erty of resilience and the interactions among the elements of a system. Regard-

ing the ﬁrst question about the relation among states and objectives, this is a

non-monotonic relation. Thanks to non-monotonic logic, we can formalize con-

tradictories states, when perturbations occurred, and conserve the consistency

of the knowledge according to the objectives. Regarding the resilient behavior

and resilient trajectories, we need deﬁne some concepts. In general physics, a

trajectory is deﬁned as the successions of the positions of a body in a framework

[6, 7]. In here, a trajectory will be deﬁned as successions of jumps among ﬁxed

points. A ﬁxed-point [8] is a solution of an equation or a system of equations.

These ﬁxed points are obtained using defaults from a default theory. This default

theory is the core of default logic [9] that is a type of non-monotonic logic [10].

To get a resilient behavior, we redeﬁne Minsky model which, according to Mar-

vin Minsky [11, 12], refers to three fundamental parts: a current state in which a

situation develops, a second state on which we want to stay, and ﬁnally, the dif-

ference among both states. Through this study, examples of piloting an airplane

are going to be explained. There are many cases that involve contradictories sit-

uations, e.g. emergency landing, bad human decisions, system failures. . .[13–15].

Recently, fatal accidents have occurred with 737 airplanes operated by Boieng3.

On the one hand, software in MCAS4was not resilient to adapt the physical

modiﬁcations of the engines in order to optimize fuel consummation. On the

other hand, pilots on board also were not resilient to overcome the situation

since they were not trained to solve the bad measures displayed on the cockpit.

In summary, these problems were the result of two non-resilient systems, because

of a system failure and incomplete information.

3https://www.boeing.com/commercial/737max/737-max-software-updates.page

4Maneuvering Characteristics Augmentation System

2

1.2 Classical Logic

Logic is a particular way of thinking, it studies the formal principles of inference.

This is logical consequences from given axioms. Formal systems, e.g., proposi-

tional, predicate, modal. . . are symbolic constructions in a particular language

which allows to study inference [10]. Propositional logic is deﬁned as least set of

expressions satisfying: >(true) and ⊥(false) are formulas, and if A and B are

formulas: ¬A(not A), A∧B(A and B), A∨B(A or B) and A→B(A implies

B). A proposition can be any sentences, e.g., “It’s a sunny day”, “Robert can

pilot his airplane”. Propositional variables are denoted by a, b, c.. .The sentence

“It’s a sunny day” could be represented by a, and the other sentence “Robert

can pilot his airplane” by b. It can be composed to create complex sentences,

e.g., “It’s a sunny day and Robert can pilot his airplane”, resulting: a∧b. First-

Order Logic (FOL) or predicate logic is an extension of propositional logic that

includes universal and existential quantiﬁers, ∀and ∃, respectively. Predicates

are denoted by P, Q, R. . .FOL is very expressive, so it is very natural to formal-

ize sentences. For instance, We can formalize the next sentence: “all airplanes

land on wheels”, with the following rule:

∀y, Airplane(y)→Land on wheels(y) (1)

But we also know that some ﬂoatplanes are airplanes that do not land on wheels

and some airplanes use skis to land on ice or snow. So, we have the following

rules:

∀y, Ski airplane(y)→Airplane(y) (2)

∀y, Ski airplane(y)→ ¬Land on wheels(y) (3)

We can see that formalizations (1) and (3) are contradictory. This is because clas-

sical logic, such as FOL, is monotonic. This property is very important in the

world of mathematics, because it allows to describe lemmas previously demon-

strated. But this property cannot be applied to uncertain, incomplete informa-

tion or exceptions. In such situations, we would expect adding new information

or set of formulas to a model, the set of consequences of this model to be reduced.

Formally the property of monotony is: A`wthen A∪B`w. The problem leads

directly to the general representation of common sense knowledge. By moving

to non-monotonic framework, we can carry out the principle of explosion and

nevertheless reach a conclusion.

1.3 Default Logic

It is one of the best known formalization for default reasoning, founded by Ray-

mond Reiter. This kind of formalization allows to infer arguments based on

partial and/or contradictory information as premises [9]. A default theory is a

pair ∆= (D, W ), where Dis a set of defaults and Wis a set of formulas in

FOL. A default dis: A(X):B(X)

C(X), where A(X), B(X), C (X) are well-formed for-

mulas. A(X) are the prerequisites,B(X) are the justiﬁcations and C(X) are the

3

consequences. Where X= (x1, x2, x3, . . . , xn) is a vector of free variables (non-

quantiﬁed). Intuitively a default means,“if A(X) is true, and there is no evidence

that B(X) might be false, then C(X) can be true”. The use of defaults implies

the generation of sets containing the consequences of these defaults, called ex-

tensions. An extension can be seen as a set of beliefs of acceptable alternatives.

Formally, an extension of a default theory ∆is the smallest set Eof logical

formulas for which the following property holds: If dis a default of D, whose the

prerequisite is in E, and the negation of its justiﬁcation is not in E, then the

consequent of dis in E[9].

Deﬁnition 1. Let ∆= (D, W ), an extension Eof ∆is deﬁne:

–E=S∞

i=0 Eiwith:

–E0=Wand,

–for i > 0: Ei+1 =T h(Ei)∪{C(X)|A(X):B(X)

C(X)∈D,A(X)∈Ei,¬B(X)6∈ E}

Where as T h(Ei) is the set of formulas have derived from Ei. A default is said to

be normal when defaults have the form: A(X):C(X)

C(X). The main result regarding

normal defaults theories is that at least one extension is always guaranteed.

The original version of the deﬁnition of an extension is diﬃcult to compute in

practice. Since it involves checking that ¬B6∈ Ewhile Eis not yet calculated.

In the case of normal defaults, Eis an extension of ∆if and only if: we replace

¬B(X)6∈ Eby ¬C(X)6∈ Ei. Regarding the rules (1) and (3), we can reﬁne the

sentence “all airplanes land on wheels” by “generally, airplanes land on wheels”.

Having a default theory that is composed of D={Airplanes(y):Land on wheels(y)

Land on wheels(y)},

and a knowledge about airplanes:

W={Floatplane(y)→Airplane(y),Floatplane(y)→ ¬Land on wheels(y)}.

Using D, we can note that the prerequisite Airplane(y) is true and the jus-

tiﬁcation Land on wheels(y) is inconsistent with W, because of:

Floatplane(y)→ ¬Land on wheels(y),

then we cannot conclude that ﬂoatplanes land on wheels. But we know that some

ﬂoatplanes have wheels, formally, W∪ {Floatplane wheels(y)→Airplane(y)}.

With this a new information, the prerequisite of Dis true and the justiﬁcation

is consistent, hence we can conclude that there are ﬂoatplanes that have wheels

and land on wheels.

2 KOSA

KOSA is an acronym for Knowledge-Objectives-States-Actions framework. It is

a formalization in default logic which allows to study the property of resilience.

KOSA is a theory which use non-temporal logic to describe its evolution. This

theory is used to describe a resilience system.

4

Deﬁnition 2. AKOSA theory is a default theory ∆= (D, W ), where W=

(R∪I):

–D is a set of defaults which represents uncertain rules. It contains actions:

A(x):B(x)

C(x), and perturbations: :C(x)

C(x),

–R is a set of formulas in FOL which represents certain rules,

–I is a set of grounded literals which represents the state of a system, thereafter

we will say that Iis a state.

D={d1, d2, d3,· · · }, R ={r1, r2, r3,· · · }, I ={i1, i2, i3,· · · }

Therefore a KOSA theory,∆= (D, (R∪I)), has two types of knowledge:

static and dynamic.Dand Rare static, when KOSA evolves these rules do not

change. On the other hand, Iis the dynamic system.

Deﬁnition 3. Atransition is a change of state, ∆= (D, (R∪I)) ∆0=

(D, (R∪I0)). Considering that Dand Rare static, the evolution occurs when I

changes. Hence a transition amounts to I I0.

Example 1. In the context of piloting an airplane, a transition I I0can be as

follows:

D={emergency :Land()

Land() ,¬obstacle :Land()

Land() ,:Y oke()

Y oke() ,:¬M otor()

¬Motor() · · · }

R={Aircraf t() →F light(),· · · }

We have Dand Rthat are ﬁxed rules, and Iwill change, for instance:

I={Altitude(50), Compass(north), AirS peed(80),· · · }

I0={Altitude(80), Compass(west), AirS peed(70),· · · }

Deﬁnition 4. Aperturbation is a modiﬁcation of some values of I(Fig.1 rep-

resents a perturbation that can trigger a transition).

In practice, perturbations can have many causes, e.g. when the pilot pulls the

yoke (yoke’s position changes), the wind changes (airspeed changes), instructions

are given by the control tower (state of ﬂight changes). . . In a real system, pertur-

bations occur very often, e.g., airplane’s position changes even if all parameters

are stables 5.

Deﬁnition 5. Atrajectory T={I=i0, i1, i2,· · · , in−1, in=I0}is a sequence

of states with W= (R∪I)consistent (Fig.2 is a trajectory Twith some per-

turbations). A long-term objective I0is the last element of a sequence T, and

intermediates objectives are ikwith 0≤k < n.

5In fact, there are two types of disturbances, internal (pilot pulls the yoke) and

external (changes in the environment). We just mention them but we are not going

to detail them because of place unavailable.

5

I I0

Fig. 1: A vertical arrow represents a perturbation that can trigger a transition.

T={I=i0, i1, i2,· · · , in−1, in=I0}

Fig. 2: A trajectory Twith some perturbations.

For the moment we can consider that short-terms objectives are intermediates

objectives with a reduced number of states. Further on, we will detail the formal

deﬁnition of short-term objectives.

Example 2. During a take-oﬀ, an airplane should have above stall speed as a

short-term objective. Once take-oﬀ is done, he should climb to increase in al-

titude, which is another short-term objective. He will maintain this objective

until he reaches a speciﬁc altitude and keep on, a long-term objective. To sum-

marize, piloting an airplane is following diﬀerent objectives and changing them

depending of the perturbations.

Deﬁnition 6. Let ∆aKOSA theory, T is a tra jectory of ∆and I0an objective

of T (that means I0is the last element of T).

–K= (∆, T )is resilient, if for all perturbations on T there exists K0=

(∆, T 0), such that I0is the objective of T0.

–∆is resilient, if for all trajectories T, K= (∆, T )is resilient.

Considering that all the parameters of the states can be modiﬁed. We have

that ∆is resilient if K= (∆, T ) can reach an objective I0of T from a perturbed

state Ip, passing from Ipto I0with W= (R∪I) consistent. We give a method

to ﬁnd a trajectory T using default logic. Consider both K= (∆, T ) and an

objective I0of T. Given a perturbation in the current state I(perturbed state Ip),

this will trigger a calculation of extension E. Selecting the best extension Eit will

be possible to reach I0. For that, we consider that each default has a ponderation

with diﬀerent criteria (these could be importance, security, legislation, . . . ), e.g.

dx= [C1, C2, . . . , Cm] with Cm∈[0,1,2,...,100]. Then, E={d1, d2. . . }=

{[d1C1, d1C2, . . .],[d2C1, d2C2, . . .]. . . }, that means a default d1has more than two

ponderations criteria d1C1and d1C2, a default d2has more than two ponderations

criteria d2C1and d2C2, and so on. We can see that each extension has multiple

criteria, we need to separate each criterion for each extension. For this we are

inspired by previous research [16,5]. For a given Ewith two default d1, d2and

6

two criteria C1, C2, a normalization can be as follows:

|d1C1|=d1C1

d1C1

+d1C2

d1C1

+· · · +d1Cm

d1C1

|d2C2|=d2C1

d1C2

+d2C2

d1C2

+· · · +d2Cm

d1C2

.

.

..

.

..

.

..

.

.

|dxCm|=dxC1

d1Cm

+dxC2

d1Cm

+· · · +dxCm

d1Cm

If we continue the normalization for all Ecalculated, we will have an array of

extensions Eand normalization criteria. Applying an opportunistic principle [17,

18] which in decision theory is a minimax function, we obtain a solution En. This

Enis the best solution to reach I0. In this way we have the transition from Ip

to I0. However, if there are more perturbations there will be intermediate objec-

tives, resulting a trajectory T={Ip=i0, i1, i2,· · · , in−1, in=I0}. To be more

realistic, we can considerate a system with an long-term objective I0interacting

through an uncertainty environment, a perturbed state Iptriggers a computation

of extensions Eat moment Sp,E={I0, I1, I3, I5}, then an extension I1is chosen

using the same principle as before in this section. At some moment, perturbation

ζ1occurs and extensions are computed one more time: E={I1, I4, I5, I6}, and

I6is selected. This process occurs every moment a perturbation ζoccurs. In this

sense, an objective I0is the concatenation among states Ikand perturbations ζ,

a trajectory can be as follows: T?={I1·ζ0·I6·ζ1·I3·ζ2·I6·ζ3·I5·ζ4·I4·ζ5·I1· · · }.

Depending on the force of ζdiﬀerent trajectories can be generate, for instance:

TM={I5·ζ0·I4·ζ1·I3·ζ2·I6, ζ3·I5·ζ4·I4·ζ5·I6· · · } (Fig.3 represents the

evolution of trajectories).

The diﬀerent between T?and TMis the magnitudes of the forces ζ, that’s

means if a trajectory is longer then ζhas a great impact and vice-versa. In

practice, grounded states Iare made at each interval of time. This depends of

sampling time of the system.

Deﬁnition 7. Let K= (∆, T )resilient where ∆is a KOSA theory and Tis a

trajectory of ∆. There is a strong or safe resilience on T.

–strong resilience is the ability of K= (∆, T )to reach an objective I0(an

objective I0is the last element of T) regardless of the perturbations it suﬀers,

–safe resilience is the ability of K= (∆, T )to transform a ﬁnal objective I0

to an intermediate objective I00, in order to maintain in good conditions the

elements of a physical system.

Example 3. In aviation, pilots in a twin-engine aircraft can land with a single

one because the other suﬀered damage in mid-ﬂight6, this can be considerate as

astrong resilience.

6https://www.cbsnews.com/news/small-plane-makes-emergency-landing-on-new-

jersey-beach-today-2019-06-01/

7

Sp

I0

I1

I2

I3

I4

I5

I6

.

.

.ζ0ζ1ζ2ζ3ζ4ζ5

· · ·

I0

Fig. 3: Evolution of trajectories T?and TM.

Example 4. When an electric motor rotates certain revolutions per minute and

at some point it has a fault. The motor could demand more current to maintain

the revolutions. Thanks to new technology this kind of electrical systems include

protection systems. In case of a fault, it should enter a safe resilience so as not

to damage resistors and transistors due to this excess current.

T={I=i0, i1, i2,· · · , in−1, in=I0}

{i0

2,· · · , i0

n−1, i0

n=I00 =I0}

Fig. 4: A trajectory Twith a perturbation on i1and the transformation of it

with the same objective I0=I00 .

2.1 Minsky’s Model

This is a model that was created by Marvin Minsky [11, 12]. The principle of this

model lies on the fact of having three fundamental parts. First, a current state in

which a situation develops, second at state on which we want to be. Finally, the

diﬀerence between both states. The diﬀerence are the necessary stages to reach

the desired state (Fig. 5 represents Minsky model). The principle of Minsky

model is introduced in K= (∆, T , I0). This will allow to have a measure of

distance among intermediates objectives. That is, for a given state Iand a long-

term objective I0, this gives a distance to another nearby objective Ik.

Deﬁnition 8. For a given state I, a short-term objective is the closest state Ik

where there are fewer disturbances.

8

Want

Now

Diﬀ

Fig. 5: Minsky model.

The purpose of a distance is to know about the shape of the trajectory Tin

∆. To carry out this hypothesis, we include an axis that represents the current

states Iand another axis for the long-term objectives,I0.

Deﬁnition 9. Vertical axis (want-axis) contains the objectives I0. Horizontal

axis (now-axis) is composed of the states I(Fig. 6 represents the axis).

Remark 1. A point on want-axis is an objective that is accessible through a tra-

jectory T.

Proposition 1. The radius pof an extension Eis the sum of its ponderations,

considering the intersection of now–want axis as the origin.

Proof. From a given K= (∆, T, I 0) where ∆is a KOSA theory ∆= (D, W =

(R∪I)) each default dxin Dhas criteria deﬁned by dx= [C1, C2, . . . , Cm] with

Cm∈[0,1,2,...,100].

Then, defaults E={d1, d2. . . }={[d1C1, d1C2, . . .],[d2C1, d2C2, . . .]. . . }[9]. To

obtain the radius pof a Ewe sum the values of each poderation. For n > 0, we

have the radius for all defaults Encomputed:

E0=X{d1, d2. . . dx}=p

E1=X{d0

1, d0

2. . . d0

x}=p0

.

.

..

.

..

.

..

.

.

En=X{dn

1, dn

2. . . dn

x}=pn

The representation of the radius pncan be seen in Fig. 6 the representation of

the radius of E0,1,2and a ﬁxed objective I0.

3 Discussion and Conclusion

The importance of using a non-monotonic logic, particulary default logic, is to

be able to ﬁnd consistent solutions. We presented an approach for representing a

resilient system which has the capability of absorb perturbations and overcome

a disaster. A KOSA theory ∆= (D, W = (R∪I)) is deﬁned with the purpose of

9

now (I)

want (I’)

Objective

p

•

E0

p0•E1

p00

•

E2

Fig. 6: Radius pof defaults E0,1,2and a ﬁxed objective I0(black rectangle).

study a resilient behavior. It is a default theory which use not temporal logic to

describe its evolution. We proved that it exists a resilient trajectory T, for any

perturbation (incomplete, partial and contradictory informations). Considering

that a perturbed state Ipcan be inconsistent. This trajectory is a sequence of

states, T={I=i0, i1, i2,· · · , in−1, in=I0}with W= (R∪I) consistent. A

long-term objective I0is the last element of a sequence T, and intermediates

objectives are ikwith 0 ≤k < n. The introduction of Minsky model to our

KOSA theory is presented, thanks to this we could have a ﬁrst step to study

the shape of the trajectories T. Also the notion of distance among extensions is

introduced.

To answer the questions that motivated this investigation: What is the re-

lation among states and objectives? We can say that the relation among states

and objectives is non-monotonic. Are there resilient trajectories? We demon-

strated that all ∆is resilient, if K= (∆, T ) can reach an objective I0of T from

a perturbed state Ip, passing from Ipto I0with W= (R∪I) consistent. Is

there a resilient behavior? We demonstrated that exists a resilient behavior, if

all trajectories of ∆are resilient. We presented a discrete theoretical behavior

of the trajectories.

The main objective of this research was to conduct a purely logical study.

KOSA theory does not use learning techniques to infer conclusions which will

be interesting for the future, e.g. it could be the use of this type of method

to learn the necessary rules to achieve an objective. This study provides the

basis for generalizing Deﬁnition 6. In which one could consider ﬁnding the

universe of resilient trajectories for any perturbation. That is, no matter what

the perturbation is, we could ﬁnd the universe of trajectories to achieve the

desired objective.

10

A practical application [5] without resilience property was performed on an

embedded computer which calculates the extensions for the stabilization of a

motorized glider.

Acknowledgments

I would like to extend my thanks to the people who contributed their criticisms

and comments in the development of this article, either directly or indirectly.

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