An Automata-Based Method to Formalize Psychological Theories:
The Case Study of Lazarus and Folkman’s Stress Theory
Alain Finkel
a
, Gaspard Fougea
b
and St
´
ephane Le Roux
Universit
´
e Paris-Saclay, CNRS, ENS Paris-Saclay, LMF, 91190, Gif-sur-Yvette, France
Keywords:
Automata-Based Method, Formalization of Psychological Theories, Stress Theory, Compact-Automata,
Refinement.
Abstract:
Formal models are important for theory-building, enhancing the precision of predictions and promoting col-
laboration. Researchers have argued that there is a lack of formal models in psychology. We present an
automata-based method to formalize psychological theories, i.e. to transform verbal theories into formal mod-
els. This approach leverages the tools of theoretical computer science for formal theory development, for
verification, comparison, collaboration, and modularity. We exemplify our method on Lazarus and Folkman’s
theory of stress, showcasing a step-by-step modeling of the theory.
1 INTRODUCTION
Context: For decades, some researchers have been
arguing that the field of psychology is in a state of cri-
sis. (Meehl, 1978) expressed that ”Theories in “soft”
areas of psychology lack the cumulative character of
scientific knowledge”. The study (Open Science Col-
laboration, 2015) showed that about half of the liter-
ature in psychology does not replicate. Several re-
searchers express that these crises stem partly from a
lack of formal models in psychology (Meehl, 1978;
Borsboom et al., 2021; Robinaugh et al., 2021).
Most psychological theories are verbal theories,
i.e. expressed in natural language, with all of its im-
precision (Haslbeck et al., 2022). This imprecision
makes it difficult to make precise predictions (Robin-
augh et al., 2021), which are necessary to validate or
falsify a theory. Furthermore, according to (Robin-
augh et al., 2021), ”verbal theories do not lend them-
selves to collaborative development”, explaining why
theories in psychology are like ”toothbrushes” (“no
self-respecting person wants to use anyone else’s”,
(Mischel, 2008)).
(Smaldino, 2019), (Muthukrishna and Henrich,
2019)) and (Robinaugh et al., 2021) argue that formal
models will address some of the theoretical issues of
psychology: ”We argue that formal theories provide
this much needed set of tools, equipping researchers
a
https://orcid.org/0000-0003-2482-6141
b
https://orcid.org/0009-0004-8357-5340
with tools for thinking, evaluating explanation, en-
hancing measurement, informing theory development,
and promoting the collaborative construction of psy-
chological theories” (Robinaugh et al., 2021).
Expected Properties of a Formal Model for
Psychology: Following (Wing, 2006), we are in-
spired by the computational thinking that involves de-
composing complex theories, identifying patterns, fo-
cusing on essential details, designing step-by-step so-
lutions and evaluating their effectiveness. Let us pro-
pose a list of desirable properties that, we believe, for-
mal method for psychological theories should satisfy.
1. Openness to all psychological theories, both cog-
nitive and behavioral,
2. Modularity (easy to modify, compose, and refine),
3. Having a formal semantics,
4. Formal composition and refinement,
5. Capability to handle large systems,
6. Possibility of step-by-step simulation,
7. Formal verification of properties (psychological
model checking) with the use of automatic tools,
8. Formal (and automatic) comparison of models,
with automatic determination of compatibility be-
tween theories.
Properties Satisfied by Psychological Models:
We identify 13 well-known frameworks used in psy-
204
Finkel, A., Fougea, G. and Roux, S. L.
An Automata-Based Method to Formalize Psychological Theories: The Case Study of Lazarus and Folkman’s Stress Theory.
DOI: 10.5220/0013175400003896
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 13th International Conference on Model-Based Software and Systems Engineering (MODELSWARD 2025), pages 204-215
ISBN: 978-989-758-729-0; ISSN: 2184-4348
Proceedings Copyright © 2025 by SCITEPRESS – Science and Technology Publications, Lda.

chology which we will list below. For each of these
frameworks, we examine which of the eight above-
mentioned properties are satisfied.
Let us recall briefly these 13 models.
ACT-R (Adaptive Control of Thought - Rational) sim-
ulates human cognitive processes (Anderson, 2004)
with formal and non formal models. ACT-R satis-
fies modularity, step-by-step simulation, and open-
ness to theories but lacks formalization, verifica-
tion, and automatic comparison. Friston’s Predic-
tive Coding and Free Energy Principle postulates that
the brain minimizes prediction errors (Friston, 2009),
and it is used to explain perception and decision-
making. Friston’s Free Energy Principle excels in
formal aspects and adaptability to complex systems
but struggles with modularity and automatic verifica-
tion. GWT (Global Workspace Theory), initiated by
Baars, suggests consciousness arises when informa-
tion is shared across the brain (Baars, 1997, 2005).
LIDA (Learning Intelligent Distribution Agent) is
based on global workspace theory, simulating atten-
tion, memory, and decision-making (Stan Franklin,
2007). LIDA is modular and handles large systems
but lacks formal semantics and verification tools. The
Working Memory Model breaks memory into com-
ponents for processing (Baddeley and Hitch, 1974).
Global Workspace Theory is conceptual, suited for
theoretical exploration but lacks formalization and
automatic verification. Dual-Process Theory distin-
guishes between fast, automatic thinking (System 1)
and slow, deliberate thinking (System 2) (Kahneman,
2011). Dual-Process Theory is not formalized, re-
maining a conceptual framework. Bayesian networks
are widely used to model decision-making, proba-
bilistic reasoning, and learning (Pearl and Macken-
zie, 2018). Bayesian networks satisfy most crite-
ria, with formal semantics and modularity. Neural
networks simulate cognitive processes like memory,
learning, and pattern recognition, mimicking brain
function (Goodfellow et al., 2016). Neural networks
manage complex systems but lack formalization and
verification. Dynamic systems are used to model con-
tinuous behaviors over time, such as emotional dy-
namics, motivation, and behavioral regulation (Lake
et al., 2017). Game theory is applied in social psy-
chology to examine how persons make strategic de-
cisions in competitive or cooperative contexts (Sun,
2016). Propositional and modal logic are often used
to formalize human reasoning (Marr, 2015). Percep-
tion, attention, and memory are treated as information
processing systems, applying entropy and redundancy
to explain encoding, storage, and retrieval (Goodfel-
low et al., 2016). Markov decision processes (MDP)
model decision-making in uncertainty, while agent-
based models simulate complex social and ecological
interactions (Sun, 2016). The last 4 models listed pre-
viously (dynamic systems, game theory, propositional
and modal logic, and MDPs) largely meet the criteria,
offering powerful tools for formal modeling and sim-
ulation, though some lack psychological verification
or automatic comparability.
We conclude that no existing model, tool or
method totally satisfy all the 8 previous properties.
More specifically, most models satisfy neither formal
verification of properties (property 7) nor decidable
comparison between theories (property 8). Surpris-
ingly, almost none of these models use the automaton
model, which is the basis of computer science.
Formalization with Systems of Automata: Al-
though the notion of algorithms is used in many fields,
few researchers are familiar with the classical models
of computability. This is the case in neuroscience and
psychology. Yet, the conceptual foundations of com-
puter science would be quite useful for psychology,
which also deals with notions of states, actions, be-
haviors, simulation, and process equivalence, among
others. Even computational psychiatry (Montague
et al., 2012), which emerged in the 2010s, does not
use computability but rather game theory, probabilis-
tic models, statistics, and machine learning. Further-
more, finite automata diagrams are intuitive and un-
derstandable for researchers without a formal training
in mathematics or computer science.
Our Contributions:
• We present a new automata-based method to for-
malize psychological theories. In the spirit of
(Fodor, 1983), we build our model by defining and
composing modules. Our methodology is based
on the principle of modeling different modules
with different finite automata which will interact
in a very specific way. These modules can be
easily modified and refined without changing the
whole model.
• We provide a method that satisfies the eight prop-
erties listed previously.
• Our method is demonstrated using the example of
stress theory.
• We propose a list of new open questions.
Our (first) modeling, based on finite automata, is only
a first stage, and we will continue by adding time
(with timed automata), probabilities (with Markov
chains), and differential equations on continuous vari-
ables.
An Automata-Based Method to Formalize Psychological Theories: The Case Study of Lazarus and Folkman’s Stress Theory
205

Table 1: Models and properties.
Model 1 2 3 4 5 6 7 8
ACT-R Yes Yes Partially Partially Partially Yes Partially Partially
Friston’s Free Energy
Principle
Yes Partially Yes Partially Yes Partially No No
GWT (Global
Workspace Theory)
Partially Partially No No Partially Partially No No
LIDA Yes Yes Partially Partially Yes Yes No No
Working Memory
Model
No Yes No No Partially No No No
Dual-Process Theory Partially No No No No No No No
Bayesian Networks Yes Yes Yes Yes Yes Yes Partially Yes
Neural Networks Yes Partially No Partially Yes No No No
Dynamic Systems Yes Partially Yes Partially Yes Yes Partially No
Game Theory Yes Yes Yes Yes Yes Yes Partially Yes
Propositional/Modal
Logic
Partially Yes Yes Yes Partially Yes Yes Yes
Information Processing
Models
Partially Yes Yes Yes Partially Yes Partially Yes
MDPs Yes Yes Yes Yes Partially Yes Partially Yes
Structure of the Paper: Section 2 introduces
compact-automata and refinements to formalize the
development of a theory into successive formal sys-
tems. Section 3 briefly presents the Lazarus and Folk-
man’s (verbal) theory of stress. Section 4 unfolds a
step by step refinement process to translate the verbal
theory into a formal model: we successively add cog-
nitive appraisal, stress, environment, coping, primary
and secondary appraisal and finally commitments.
2 COMPACT AUTOMATA AND
REFINEMENT
Throughout this paper, Σ will be a non-empty finite
set which we call the alphabet. Let us recall that a
finite automaton (without accepting states) on Σ is a
tuple (Σ, Q,δ,I), where Q is the finite set of states,
δ ⊆ Q × Σ × Q is the transition relation, and I ⊆ Q is
the finite set of initial states.
2.1 Synchronized Automata
There are various systems of (extended) finite au-
tomata that synchronise by communication: via
rendez-vous (Milner, 1989; Balasubramanian et al.,
2023), blocking or non-blocking (Guillou et al.,
2023), broadcast, FIFO queues, or shared memory
(Atiya et al., 1991; Moiseenko et al., 2021).
For the remainder of this article, let τ /∈ Σ and Σ
τ
=
Σ ∪ {τ} and [n] = {1,2,...,n}. Let us now introduce a
generalized rendezvous.
Definition 1. A system of n automata synchro-
nized via generalized rendezvous (short: system)
is a n-tuple S = (A
1
,A
2
,...,A
n
) where every A
i
=
(Σ
τ
,Q
i
,∆
i
,I
i
) is a finite automaton.
The operational semantics of a system S =
(A
1
,A
2
,...,A
n
) is the finite automaton A(S) =
(Σ
τ
,Q,∆, I) defined as follows. The set of (global)
states is Q = Q
1
× ... × Q
n
. For all q ∈ Q, let us write
q
i
for the i
th
element of a (global) state q = (q
1
,...,q
n
).
The set of (global) initial states is I = I
1
×...×I
n
. The
(global) transition relation, ∆ ⊆ Q × Σ
τ
×Q, is defined
by the two following rules:
1. τ-transitions: Let i ∈ [n], q,q
′
∈ Q such that
(q
i
,τ,q
′
i
) ∈ ∆
i
, then if for all j ∈ [n]−{i}, q
j
= q
′
j
,
there is a τ-transition q
τ
−→ q
′
, which stands for
(q,τ,q
′
) ∈ ∆.
2. Synchronised Transitions: Let q,q
′
∈ Q,a ∈
Σ. Let us note J
a
= {i ∈ [n] | ∃p
i
,r
i
∈ Q
i
s.t.
(p
i
,a,r
i
) ∈ ∆
i
} the set of indices of the automata
that contain at least one a-transition. If J
a
̸= ∅,
and for all i ∈ J
a
, (q
i
,a,q
′
i
) ∈ ∆
i
and for all i /∈ J
a
,
q
i
= q
′
i
, then there is a synchronised transition
q
a
−→ q
′
, which stands for (q,a,q
′
) ∈ ∆.
An a-transition (p,a,r) ∈ ∆ is enabled when all the
automata A
i
such that i ∈ J
a
(i.e., A
i
have at least a
local a-transition) are ready to enable an a-transition.
Every τ-transition (p, τ,r) is local and do not depend
on other automata for its execution. There is no block-
ing condition on τ-transitions.
Example 1. Consider the system (A
1
,A
2
,A
3
) below:
We have J
a
= {1, 2}. Since 3 /∈ J
a
and there are
enabled (local) a-transitions in (local) states q
1
and
MODELSWARD 2025 - 13th International Conference on Model-Based Software and Systems Engineering
206

q
1
q
2
a
τ
τ
A
1
p
a
A
2
r
b
τ
A
3
Figure 1: Synchronised transitions and τ-transitions.
p, we have that (q
1
, p,r)
a
−→ (q
2
, p,r) is a transition in
∆ (we write (q
1
, p,r)
a
−→ (q
2
, p,r) ∈ ∆).
However, (q
2
, p,r)
a
−→ (q
2
, p,r) /∈ ∆, because 1 ∈
J
a
and there are no (local) a-transition is enabled in
(local) state q
2
.
Since there is no blocking-condition on τ-
transitions, we have that (q
2
, p,r)
τ
−→ (q
1
, p,r) ∈ ∆.
In writing (q
2
, p,r)
τ
−→ (q
2
, p,r) ∈ ∆, it is not pre-
cised whether the τ−transition is executed in A
1
or
in A
3
. This level of precision is enough for what we
model.
Here are two executions of our generalised ren-
dezvous system A(A
1
,A
2
,A
3
) :
(q
1
, p,r)
a
−→ (q
2
, p,r)
b
−→ (q
2
, p,r)
τ
−→ (q
2
, p,r)
τ
−→
(q
1
, p,r)
τ
−→ (q
1
, p,r)
(q
1
, p,r)
b
−→ (q
1
, p,r)
a
−→ (q
2
, p,r)
b
−→ (q
2
, p,r)
τ
−→
(q
2
, p,r)
2.2 Compact-Automata
Some of the automata that we will consider in Sec-
tion 4 contain a large number of states and transitions.
We will thus use compact-automata as a way of rep-
resenting these automata. Compact-automata do not
have a behaviour of their own, and they are useful
since they are compact versions of finite automata,
able to represent large and complex automata in a
compact and simple way. Below we define compact-
automata and the (deterministic) unfolding process.
Later we will explain the (non-deterministic) fold-
ing process. Let us fix notations. As previously,
Σ
τ
= Σ ∪ {τ} is an alphabet. Let also V = {v
1
,v
2
,v
3
}
be a set of three variables, and FO(V ,=,P
G
) denotes
the set of first order logic formula with relation =,
variables in V and with predicate P
G
meaning that
(v
1
,v
2
,v
3
) is an edge of G (for a graph G).
Definition 2. A compact-automaton is a tuple
˜
A =
(Σ
τ
,
˜
Q,X, f ,
˜
δ,I), where
˜
Q is the set of (compact)
states, X is the non-empty set, f :
˜
Q −→ P (X) is the
unfolding function,
˜
δ ⊆
˜
Q × FO(V ,=,P
G
) ×
˜
Q is the
(compact) transition relation, and I ⊆ ∪
˜q∈
˜
Q
f ( ˜q).
The unfolding of such
˜
A is A = (Σ
τ
,Q,δ, I) with
Q = ∪
˜q∈
˜
Q
f ( ˜q) and where the transition relation δ of
A is defined by δ = δ
1
∪ δ
2
, where:
δ
1
=
[
( ˜q,φ, ˜p)∈
˜
δ, ˜q̸= ˜p,a∈Σ
τ
{(x,a,y) | x ∈ f ( ˜q),y ∈ f ( ˜p),φ(x,a,y)} (1)
δ
2
=
[
( ˜q,φ, ˜q)∈
˜
δ,a∈Σ
τ
{(x,a,x) | x ∈ f ( ˜q),φ(x,a,x)} (2)
In Definition 2, term (1) refers to (compact) tran-
sitions where the origin and target (compact) states
are different. In this case, the unfolding can gener-
ate (unfolded) transitions for all pairs of origin and
target (unfolded) states. Term (2) refers to (compact)
loops. In this latter case, the unfolding may only gen-
erate (unfolded) loops: (with previous notations) if
( ˜q,φ, ˜q) ∈
˜
δ,x,y ∈ f ( ˜q),a ∈ Σ
τ
s.t. x ̸= y ∧ φ(x,a,y),
then (x,a,y) is not added to δ from term (2).
˜q
˜p
(v
2
= a)
(v
2
= b)
compact-automaton
˜
A
1
I = {x
0
}, f ( ˜q) = f ( ˜p) = X
⇝
x
0
x
1
x
2
a a
a
a
a
a
(finite) automaton A
1
unfolding
of
˜
A
1
a,b
a,b
a,b
Figure 2: Unfolding of a compact-automaton.
Example 2. Figure 2 is the unfolding of the compact-
automaton
˜
A
1
for X = {x
0
,x
1
,x
2
}.
Graphical Representation of Compact-Automata.
Let
˜
A = (Σ
τ
,
˜
Q,X, f ,
˜
δ,I) be a compact-automaton.
We use three types of graphical notations in compact-
automata diagrams:
1. Sets and Singletons: Let ˜q ∈
˜
Q. If | f ( ˜q) |> 1, we
represent the (compact) state ˜q as a double circle
around ˜q. If f ( ˜q) = {q}, we represent the (com-
pact) state ˜q as a simple circle around ˜q (same rep-
resentation as for finite automata).
˜q
| f ( ˜q) |> 1
˜q
f ( ˜q) = {q}
Figure 3: Graphical representations of sets and singletons.
2. Transitions with a Letter or a Variable: when
an element of FO(V ,=,P
G
) is of the form (v
2
=
a) for a letter a ∈ Σ
τ
, we will simply label the tran-
sition as a. When an element of FO(V ,=, P
G
) is
of the form [v
1
] = v
2
, and the origin state is written
[ ˜q], we can simply label the transition with ˜q.
An Automata-Based Method to Formalize Psychological Theories: The Case Study of Lazarus and Folkman’s Stress Theory
207

3. Graph Transitions: if ( ˜q,φ, ˜p) ∈
˜
δ, and there is a
directed labeled graph G, with labels of G belong-
ing to L ⊆ Σ
τ
, with f ( ˜q) ̸= f ( ˜p), f ( ˜q) = f ( ˜p) = X,
and the vertices of G are X. If φ(v
1
,v
2
,v
3
) =
((v
1
,v
2
,v
3
) is an edge of G), then we label the
(compact) transition G/L.
x
y
G/L
Representation for
φ(v
1
,v
2
,v
3
) = ((v
1
,v
2
,v
3
) is an edge of G)
compact-automaton
˜
A
2
+
x
0
x
0
⇝
x
1
x
1
x
2
x
2
a
b
b
x
0
x
1
x
2
Unfolding of
˜
A
2
for X = {x
0
,x
1
,x
2
},
G and L = {a, b}
a
b
b
G
Figure 4: Graphical representation of graph transitions.
Folding. The folding process is non-deterministic.
From a large finite automaton, first we identify sets of
states for which the structure of transitions is similar;
second, we create a small number of (compact) states
and (compact) transitions that represent all the states
and transitions of the initial automaton.
2.3 Refinements of Automata
The notion of refinement helps us understand how
successive systems relate to one another: a more re-
fined system is modeled more precisely with regards
to the theory. Refinement between systems is ex-
pressed as a binary relation. In order to define it, we
used two intermediary definitions which correspond
to refinements over labels and transitions, and over
automata.
Let us consider two systems S = (A
1
,A
2
,...,A
n
)
and S
′
= (A
′
1
,A
′
2
,...,A
′
m
) with their asso-
ciated automata A(S) = (Σ
τ
,Q,∆, I) and
A(S
′
) = (Σ
′
τ
,Q
′
,∆
′
,I
′
). As usual, we note
Σ
τ
= Σ ∪ {τ} and Σ
′
τ
= Σ
′
∪ {τ}.
Definition 3. Refinement of Labels and Transitions:
We define the following relation ≤
1
on Σ
τ
∪ Σ
′
τ
as the
smallest relation satisfying the following condition:
For all labels λ ∈ Σ
τ
∪ Σ
′
τ
, τ ≤
1
λ and λ ≤
1
λ.
We extend this relation to elements of ∆ ∪∆
′
in the
following way : for all q
1
,q
2
∈ Q, p
1
, p
2
,∈ Q
′
,λ ∈
Σ
τ
,λ
′
∈ Σ
′
τ
we have (q
1
,λ, p
1
) ≤
1
(q
2
,λ
′
, p
2
) if and
only if λ ≤
1
λ
′
.
Definition 4. Refinement of Automata: let us
consider i ∈ [n], j ∈ [m]. We say that A
i
=
(Σ
τ
,Q
i
,∆
i
,I
i
) ≤
2
A
′
j
= (Σ
′
τ
,Q
′
j
,∆
′
j
,I
′
j
) iff there exists a
partition (Q
q
)
q∈Q
i
of Q
′
j
for which the following con-
ditions hold:
• Σ
τ
⊆ Σ
′
τ
• Transition Refinement: for all t
′
= (p
′
,λ
′
,q
′
) ∈ ∆
′
j
,
for p,q ∈ Q
i
such that p
′
∈ Q
p
and q
′
∈ Q
q
, there
exists a transition t = (p,λ,q) such that t ≤
1
t
′
• Refinement of initial states: for all q
′
∈ I
′
j
, ∃q ∈
I
i
,q
′
∈ Q
q
Example 3. Figure 5 explicits all possible ways of
refining transitions. Since there are no loops on q
2
there cannot be any transition between p
1
, p
2
, and
p
3
.
q
1
q
2
Automaton A
i
τ
a
τ
p
1
p
2
p
3
p
4
p
5
Automaton A
′
j
refinement of A
i
τ
b
a
c
Figure 5: Refinement of an automaton.
Let us recall that for all a ∈ Σ
τ
, J
a
= {i ∈ [n] |
∃p
i
,r
i
∈ Q
i
s.t. (p
i
,a,r
i
) ∈ ∆
i
} is the set of indices
of the automata that contain at least one a-transition.
For system S
′
and a ∈ Σ
′
we note it J
′
a
.
Definition 5. Refinement of Systems: For S =
(A
1
,A
2
,...,A
n
) and S
′
= (A
′
1
,A
′
2
,...,A
′
m
), we say that
S ≤
3
S
′
if and only if:
• n ≤ m
• For all 1 ≤ i ≤ n, A
i
≤
2
A
′
i
• For all a ∈ Σ,J
a
⊆ J
′
a
Remark 1. The three relations ≤
1
,≤
2
, and ≤
3
are
quasi-orders.
3 LAZARUS AND FOLKMAN’S
THEORY OF STRESS
This section explains the elements of Lazarus and
Folkman’s theory of stress that we have identified as
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208

both important and suitable for modeling, see sec-
tion 4. These elements come from the book Stress,
appraisal and coping (Lazarus and Folkman, 1984),
and hence, every reference to pages in this section
points to pages of it. This book
1
is arguably one
of the most representative of this theory. We detail
these elements below, namely the relational definition
of stress, cognitive appraisal, primary appraisal, sec-
ondary appraisal, coping, emotional-focused coping,
problem-focused coping, and commitments.
Relational Meaning of Stress. Until the 1960s,
stress was defined either as a stimulus (stressor)
(p.12), or as a physiological response (stress re-
sponse) (p.14). These definitions fail to explain
the discrepancies in response between persons to
the same stimulus (p.19). This motivated Lazarus
and Folkman to introduce a relational definition of
stress, taking both aspects into account: ”Psychologi-
cal stress is a particular relationship between the per-
son and the environment that is appraised by the per-
son as taxing or exceeding his or her resources and
endangering his or her well-being” (p.19, l.26).
Cognitive Appraisal: is the process by which a per-
son determines whether a relationship is stressful or
not (p.19). This process is continuous during wak-
ing life (p.31) and contains two main processes: pri-
mary appraisal and secondary appraisal(p.31), which
”cannot be considered as separate” (p.43, l.8).
Primary Appraisal: is the process through which
a person answers the question of whether a particular
stimulus is beneficial, irrelevant or stressful (p.32).
Secondary Appraisal: is the process through
which a person evaluates the coping strategies avail-
able to deal with a stimulus appraised as stressful
(p.35).
Coping: is defined as ”constantly changing cogni-
tive and behavioral efforts to manage specific exter-
nal and/or internal demands that are appraised as tax-
ing or exceeding the resources of the person” (p.141,
l.10). Furthermore, it is said that the dynamic of
coping are ”a function of the continuous appraisals
(...) of the evolving person-environment relationship”
(p.142, l.32).
1
Which has gathered more than 95000 quotes according
to google scholar as of now
Emotion-Focused Coping and Problem-Focused
Coping. Coping strategies fall into two categories:
emotion-focused coping efforts are meant to regu-
late emotions, whereas problem-focused coping ef-
forts are meant to resolve the situation itself (p.150).
Commitments: ”express what is important to the
person, what has meaning for him or her.”(p.56, l.4)
They play a key role in the appraisal process, since
the stress experienced by a person is directly related
to how much meaning is tied to its commitments.
4 A MODEL FOR LAZARUS AND
FOLKMAN’S THEORY OF
STRESS
This section unfolds a step by step refinement process
to translate a verbal theory into a formal model. This
methodology consists in starting from a trivial initial
system, and at each step, either adding a new con-
cept from the theory, or refining a previously mod-
eled concept. At each step, new concepts or develop-
ments must be based on previously modeled concepts.
From this condition, next concepts or developments to
model are chosen with regards to their importance and
their simplicity. Every new system is a refinement of
the previous one. In this paper, the whole methodol-
ogy is not formally defined.
In the remainder of this paper, we will refer to
compact-automata as automata, referring to the un-
folding of a compact-automaton when it is mentioned.
Subsequent systems are noted S
1
,S
2
,S
3
,..., and for all
natural number i, and (compact or normal) automata
of system S
i
will be noted A
i.1
,A
i.2
,A
i.3
,...
All explanations from Section 3 were based on
the book Stress, appraisal and coping (Lazarus and
Folkman, 1984). Likewise, we will base our model-
ing solely on this book, and hence, every reference to
pages in this section points to pages of the book.
4.1 The Initial System
We begin with a (trivial) initial system S
1
= (A
1.1
),
which is meant to represent a system where nothing
yet has been specified. The universe is represented as
having an unique state, with some observed activity.
When we model a dynamic taking place which we do
not specify, we add a τ-transition. This is the role of
the τ-loop on state uni.
A
1.1
= (Σ
τ,1
,Q
1.1
,∆
1.1
,I
1.1
) with Σ
τ,1
= {τ},
Q
1.1
= {uni}, ∆
1.1
= {(uni,τ, uni)} and I
1.1
= {uni}.
An Automata-Based Method to Formalize Psychological Theories: The Case Study of Lazarus and Folkman’s Stress Theory
209

uni
τ
A
1.1
: initial automaton
Figure 6: System S
1
.
4.2 Adding Cognitive Appraisal
Here we model the key notion of cognitive appraisal
2
defined as follows: ”Cognitive appraisal can be most
readily understood as the process of categorizing an
encounter, and its various facets, with respect to its
significance for well-being(...) it is largely evaluative,
focused on meaning or significance, and takes place
continuously during waking life” (p.31, l.18)
Since what happens outside of waking life does
not seem to be explicited in the book, for the sake of
simplicity, we decide that appraisal happens only dur-
ing waking life. Since we know from the theory that
some activity takes place during appraisal, we add a
τ-loop on the ”appraisal” state. The fact that the loop
is labeled by τ means that we do not specify the na-
ture of this activity. Nothing is mentioned in the the-
ory about activity during the ”non-awake” state, and
hence we do not add a τ-loop on top of this state. This
leads to the following system S
2
= (A
2.1
) with one au-
tomaton with two states.
A
2.1
: Cognitive appraisal
non-awake
appraisal
τ
τ
τ
Figure 7: System S
2
.
Formally, the system S
2
consists only of au-
tomaton A
2.1
, where A
2.1
= (Σ
τ,2
,Q
2.1
,∆
2.1
,I
2.1
) is
defined by Σ
τ
= {τ}, Q
2.1
= {non-awake,awake},
∆
2.1
= {(non-awake,τ, appraisal),
(appraisal,τ,appraisal), (appraisal,τ,non-awake)},
and I
2.1
= {non-awake}.
System S
2
= (A
2.1
) is a refinement of system
S
1
= (A
1.1
): 1 ≤ 1, A
1.1
≤ A
2.1
, and since Σ
τ,1
=
{τ}, the third condition is always true. A
1.1
≤ A
2.1
because: if Q
uni
= {non − awake,appraisal}, then
(Q
uni
) is clearly a partition of Q
2.1
. For all transi-
2
blue-colored text points to other sections of the paper
tions (q,τ, p) of A
2.1
the transition (uni,τ,uni) satis-
fies τ ≤ τ and p,q ∈ Q
uni
. Finally, I
2.1
= {non-awake}
and non-awake ∈ Q
uni
.
4.3 Adding Stress
In Section 3 we discussed the relational definition of
stress. For now we will only distinguish between two
consequences of cognitive appraisal : ”no-stress”, and
”stress”.
To our previous system S
2
, we add a second au-
tomaton that calculates stress. The calculation of
stress can only happen during the appraisal state of
the first automaton. This way of modeling these sen-
tences is not unique, but will be coherent with the fur-
ther development of the theory.
Automata A
3.1
and A
3.2
synchronise on two let-
ters: ”stress” and ”no-stress”. The set of automata
that possess a (local) stress-transition is J
stress
=
{(3.1),(3.2)}. Hence, both automata must have an
enabled (local) stress-transition for a global stress-
transition to take place. This is why in (global) state
(non − awake, f ), a (global) stress-transition cannot
occur, whereas in (global) state (appraisal, f ), the
(global) stress-transition is enabled. Here is an ex-
ecution of the system:
(non − awake, f )
τ
−→ (appraisal, f )
stress
−−−→
(appraisal, f )
no−stress
−−−−−→ (appraisal, f )
τ
−→
(non − awake, f )
This execution described a person who wakes
up, appraises the person-environment relationship as
stressful, appraises again this relationship as non-
stressful, then goes back to sleep.
non-awake
appraisal
A
3.1
: Cognitive appraisal
τ
τ
stress
no − stress
τ
f
A
3.2
: Calculation of stress
stress
no − stress
τ
Figure 8: System S
3
.
In the remainder of this article, we will note na
for ”non-awake”, a for ”appraisal”, s for ”stress” and
s for ”no-stress”. Formally, S
3
= (A
3.1
,A
3.2
), Σ
τ
=
{s,s,τ}, and:
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210

A
3.2
= (Σ
τ,3
,Q
3.2
,∆
3.2
,I
3.2
),Q
3.2
= { f }
∆
3.2
= {( f , s, f ), ( f ,s, f ),( f ,τ, f )}
I
3.2
= { f }.
A
3.1
= (Σ
τ,3.1
,Q
3.1
,∆
3.1
,I
3.1
),Q
3.1
= {na,a}
∆
3.1
= {(na,τ, a),(a,s, a),(a,s, a),(a,τ,na), (a,τ,a)}
I
3.1
= {r}.
System S
3
= (A
3.1
,A
3.2
) is a refinement of sys-
tem S
2
= (A
2.1
): 1 ≤ 2, A
2.1
≤ A
3.1
, and since Σ
τ,2
=
{τ}, the third condition is always true. A
2.1
≤ A
3.1
because if Q
non-awake
= {na} and Q
appraisal
= {a}
then (Q
non-awake
,Q
appraisal
) is a partition of Q
3.1
=
{na,a}. The two added transitions in ∆
3.2
from ∆
3.2
are (a,s, a) and (a, s,a), the other ones remaining ex-
actly the same. (appraisal,τ,appraisal) satisfies τ ≤ s,
τ ≤ s, and a ∈ Q
appraisal
. Finally, initial states are not
modified. In the remainder of this paper, we will not
detail the proof that each new system is a refinement
of the previous one.
4.4 Adding the Environment
We consider that the environment is a compact-
automaton A
4.3
= (Σ
τ
,
˜
Q
3
,X, f ,
˜
δ
3
,I
3
), on parameters
(X,G
τ
), where X is a finite set of states and G
τ
is a
directed labeled graph whose vertices are X and la-
bels are τ. Since this is the first compact-automaton
of our modeling, we detail it here. In the remain-
der of this paper, the details will only appear on
the graphical representations. We have
˜
Q = {x, y},
and f (x) = f (y) = X. We have that
˜
δ = {(x, (v
1
=
[v
2
]),y),(x, ((v
1
,v
2
,v
3
) is an edge of G
τ
),y)}. The
second transition of
˜
δ is graphically labeled by
G
τ
/{τ}, see Subsection 2.2.
Appraisal happens ”continuously during waking
life”(p.31, l.23) (see Section 3). One way of mod-
eling the continuity of the appraisal process during
waking life is to have the ”Environment” automaton
”send” periodic updates of its state to the ”Cogni-
tive appraisal” automaton while it is in the ”appraisal”
state.
This is done via synchronised transitions: au-
tomata A
4.1
and A
4.3
synchronise over letters of X .
On automaton A
4.1
, the transition labeled by X means
that this transition exists for all x ∈ X. Let’s fix
x ∈ X. The name of the state associated with x in
the unfolded automaton of A
4.3
is [x], to avoid confu-
sion between states and transitions of the automaton.
Since J
x
= {(4.1),(4.3)}, the (global) x-transition
(a, f ,[x])
x
−→ (a, f ,[x]) is enabled. Automaton A
4.3
in
state [x] synchronises on letter x with A
4.1
when it is
in state a.
Here is an execution, when we have x
1
∈ X such
that (x
0
,τ,x
1
) is an edge of G
τ
:
(na, f ,[x
0
])
τ
−→ (a, f ,[x
0
])
x
0
−→ (a, f ,[x
0
])
s
−→
(a, f ,[x
0
])
τ
−→ (a, f , [x
1
])
x
1
−→ (a, f , [x
1
])
s
−→ (a, f , [x
1
])
This execution represents a person who wakes
up, perceives its environment, appraises the person-
environment relationship as stressful, the environ-
ment changes by itself, the person perceives it, and
appraises the new person-environment relationship as
non-stressful.
na a
A
4.1
: Cognitive appraisal
τ
τ
s,s
X
τ
f
A
4.2
: Calculation of stress
s
s
τ
[x]
[y]
A
4.3
: Environment
I
4.3
= {[x
0
]}, f ([x]) = f ([y]) = X
G
τ
/{τ}
x
Figure 9: System S
4
.
4.5 Adding Coping
We introduce a ”Coping”(see explanations in Section
3) automaton that models the decision of engaging in
coping efforts. Once a ”stress” appraisal has been cal-
culated in A
5.2
as a (global) s-transition, the ”Coping”
automaton gets to a state from which it can ”instruct”
the ”Environment” automaton to engage with coping
efforts, via new synchronised transitions.
As the environment, the modeling of coping has
parameters (C, G
C
), where C is a finite set of coping
strategies, and G
C
is a directed labeled graph whose
vertices are X and whose edges are labeled by ele-
ments of C. The ”Coping” automaton A
5.4
has two
states: ρ for ”rest” and [C] for the state where coping
efforts can be engaged. The ”Environment” automa-
ton was modified by adding transitions corresponding
to the edges of G
C
: for all x,z ∈ X ,c ∈ C such that
(x,c,z) is an edge of G
C
, ([x],c, [z]) is a transition of
automaton A
5.3
. Each of these (local) transitions can
synchronise with automaton A
5.4
when it is in state
[C]. Although it is clear that humans can act when
they are not stressed, we only consider here coping
efforts which occur during psychological stress.
Let’s consider x
1
∈ X,c ∈ C such that (x
0
,c,x
1
) is
an edge of G
C
. Here is a possible execution of our
new system:
An Automata-Based Method to Formalize Psychological Theories: The Case Study of Lazarus and Folkman’s Stress Theory
211

(na, f ,[x
0
],ρ)
τ
−→ (a, f ,[x
0
],ρ)
x
0
−→ (a, f ,[x
0
],ρ)
s
−→
(a, f ,[x
0
],[C])
c
−→ (a, f ,[x
1
],ρ)
x
1
−→ (a, f ,[x
1
],ρ)
s
−→
(a, f ,[x
1
],ρ)
This execution represents a person who wakes
up, perceives its environment, appraises the person-
environment relationship as stressful, engages in cop-
ing efforts with coping strategy c, perceives the
changed person-environment relationship, appraises
it as non-stressful.
na a
A
5.1
: Cognitive appraisal
τ
τ
s,s
X
τ
[x]
[y]
A
5.3
: Environment
G
τ
/τ
G
C
/C
x
I
5.3
= {[x
0
]}, f ([x]) = f ([y]) = X
f
A
5.2
: Calculation of stress
s
s
τ
ρ
[C]
A
5.4
: Coping
s
C
s
Figure 10: System S
5
.
The notation C on a transition means that this tran-
sition exists for any letter c ∈ C.
4.6 Refining Appraisal into Primary
Appraisal and Secondary Appraisal
Primary and secondary appraisal are the two main
processes of cognitive appraisal.
To model these two processes, we refine the state
a of automaton A
5.1
into two states pa and sa. The
function of primary appraisal is the one conducted
by automaton A
5.2
, which remains unchanged as au-
tomaton A
6.2
which we rename ”Primary appraisal”.
Since secondary appraisal happens once a stimulus is
appraised as stressful, and that coping efforts are a
function of previous appraisals, it seems fitting to in-
clude secondary appraisal into the ”Coping” automa-
ton. The system loops between states pa and sa for a
while as a way of determining ”the degree of stress
and the quality (or content) of the emotional reac-
tion” (p.35, l.29).
Secondary appraisal happens after ”stress” has
been calculated by automaton A
6.2
((global) s-
transition) and before engaging in coping efforts. We
refine automaton A
5.4
by refining state [C] into 2|C|
states: [c] and [[c]] for all c ∈ C. We note C
′
= {[c],c ∈
C} and C
′′
= {[[c]],c ∈ C}. A (global) s-transition
leads non-deterministically to one state [c] ∈ C
′
which
will be evaluated. Two results are possible, coming as
two synchronised transitions between A
6.1
and A
6.4
:
”good” (letter g), or ”bad” (letter b). For a c ∈ C, a
”good” evaluation from state [c] brings A
6.1
back to
state pa, and A
6.4
to state [[c]] ∈ C
′′
, from which cop-
ing efforts are engaged for coping strategy c. A ”bad”
evaluation from state [c] will lead A
6.1
back to state pa
and A
6.4
back to state ρ. From this global state, an-
other primary appraisal can occur, and another strat-
egy can be evaluated. This is the loop between the
states pa and sa mentioned earlier.
Let us describe how the model S
6
=
(A
6.1
,A
6.2
,A
6.3
,A
6.4
) with c
1
,c
2
∈ C, x
1
∈ X
such that (x
0
,c
2
,x
1
) is an edge of G
C
, formalizes
the following human sequence: a person wakes up,
perceives its environment, appraises the person-
environment relationship as stressful, wonders if
coping strategy c
1
would be beneficial, perceives
c
1
as a bad strategy, wonders if coping strategy
c
2
would be beneficial, perceives c
2
to be a good
strategy, engages in coping efforts with strategy
c
2
, the environment changes, the person perceives
the new environment, the person appraises the new
person-environment relationship as non-stressful.
Here is the corresponding execution :
(na, f ,[x
0
],ρ)
τ
−→ (pa, f , [x
0
],ρ)
x
0
−→
(pa, f , [x
0
],ρ)
s
−→ (sa, f ,[x
0
],[c
1
])
b
−→ (pa, f ,[x
0
],ρ)
s
−→
(sa, f ,[x
0
],[c
2
])
g
−→ (sa, f ,[x
0
],[[c
2
]])
c
2
−→
(pa, f , [x
1
],ρ)
x
1
−→ (pa, f ,[x
1
],ρ)
s
−→ (pa, f ,[x
1
],ρ).
Remark 2. To our knowledge, the decision-making
mechanism between secondary appraisal and coping
is not detailed in (Lazarus and Folkman, 1984). We
have chosen to model it in a very simple way: cop-
ing efforts are engaged when a coping strategy has
been evaluated as ”good”. Let’s note that any de-
cision making theory could be incorporated here in-
stead.
4.7 Adding Commitments
As seen in Section 3, commitments play an important
role in the appraisal process.
We model a commitment via a functions ϕ : X −→
{0,1}. The set of all such functions is noted Φ. At
this point, the set of internal variables is Φ and the
set of global states of the person-environment whole
is X ×Φ.
We interpret the functions ϕ as follows: in global
state (x,ϕ), the individual is stressed if and only if
ϕ(x) = 0.
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212

na
pa
sa
A
6.1
: Cognitive appraisal
τ
τ
s
g,b
s
X
[x]
[z]
I
6.3
= {[x
0
]}, f ([x]) = f ([y]) = X
A
6.3
: Environment
G
τ
/{τ}
G
X
/C
x
f
A
6.2
: Calculation of stress
s
s
τ
ρ
[c]
[[c]]
s
b
c
([v
1
] = v
3
)
∧(v
2
= g)
s
f ([c]) = C
′
, f ([[c]]) = C
′′
A
6.4
: Secondary appraisal
and coping
Figure 11: System S
6
.
Since Φ is finite, we add a new ”Internal param-
eters” automaton A
6.5
. Transitions between states of
this automaton are coping efforts intending to affect
internal parameters (Emotion-focused forms of cop-
ing, see Section 3). The graph G
C
introduced in Sub-
section 4.5 is now renamed G
X
. The modeling of
coping affecting internal parameters has parameters
(Φ,G
Φ
), where G
Φ
is a directed labeled graph whose
vertices are Φ and whose edges are labeled by ele-
ments of C. For all c ∈ C, the (global) c-transition is
extended to automaton A
6.5
which then synchronises
with A
6.3
and A
6.4
.
Here is a simple example to show how this formal-
ism can be implemented.
Example 4. Here is a person’s description of his re-
lationship with money:
”It’s important for me to have enough money. I
want to feel like I’m safe financially. Enough money
for me is having more than 1000 euros in my bank
account. Right now I have enough money. Sometimes
people steal money from my bank account and I have
no money left. As a way to make myself feel better,
I try to save a lot of money each month, and I try to
think that money is not so important”
There are two states of the world
X = {≥ 1000,< 1000}, where ≥ 1000 (resp. <
1000) is the state where there is more (resp. strictly
less) than 1000 euros in the person’s bank account.
Let’s define ϕ : X −→ {0,1} such that ϕ(≥ 1000) =
1 and ϕ(< 1000) = 0 which corresponds to the com-
mitment expressed by the person.
Φ = {ϕ, 1 − ϕ,1,0}, where 1 (resp. 0) is the func-
na
pa
sa
A
7.1
: Cognitive appraisal
τ
τ
s
g,b
s
X
[x]
[z]
I
7.3
= {[x
0
]}, f ([x]) = f ([y]) = X
A
7.3
: Environment
G
τ
/{τ}
G
X
/C
x
f
A
7.2
: Calculation of stress
s
s
τ
ρ
[c]
[[c]]
s
b
c
([v
1
] = v
3
)
∧(v
2
= g)
s
f ([c]) = C
′
, f ([[c]]) = C
′′
A
7.4
: Secondary appraisal
and coping
ϕ
ϕ
′
G
Φ
/C
I
7.5
= {ϕ
0
}, f (ϕ) = f (ϕ
′
) = Φ
A
7.5
: Internal parameters
Figure 12: System S
7
.
tion always equal to 1 (resp. to 0).
The only τ-transitions in the environmnent au-
tomaton A
6.3
correspond to ”Sometimes people steal
money from my bank account and I have no money
left”. Hence the two edges of G
τ
are (≥ 1000,τ, <
1000) and (< 1000,τ, < 1000).
C = {c
1
,c
2
}
The coping strategy c
1
corresponds to ”saving
money”. This strategy only affects the environment.
The coping strategy c
2
corresponds to ”trying to think
money is not so important”. This strategy only affects
internal parameters. Hence, the edges of G
X
are :
(< 1000,c
1
,< 1000), (< 1000,c
1
,≥ 1000),
(≥ 1000,c
1
,≥ 1000), (< 1000,c
2
,< 1000),
(≥ 1000,c
2
,≥ 1000).
The edges of G
Φ
are:
(ϕ,c
2
,ϕ),(ϕ, c
2
,1),(1, c
2
,1),(ϕ, c
1
,ϕ),(1, c
1
,1).
A transition (ϕ,c
2
,1) corresponds to a shift in the
person’s commitment with money.
With this example, the diagrams of automata A
7.3
and A
7.5
are represented on Figure 13.
4.8 Further Developments
It is possible to refine the concept of primary appraisal
by calculating a result (”stress” or ”non-stress”) based
on the state of A
7.3
× A
7.5
.
The calculation of secondary appraisal can also be
modeled by adding an automaton which corresponds
to the inner representations of the person-environment
An Automata-Based Method to Formalize Psychological Theories: The Case Study of Lazarus and Folkman’s Stress Theory
213

[≥ 1000] [< 1000]
c
1
τ
≥ 1000
c
1
,c
2
c
1
,c
2
,τ
< 1000
A
7.3
: Environment
ϕ
1
1 − ϕ
0
c
1
,c
2
c
1
,c
2
c
2
A
7.5
: Internal parameters
Figure 13: Example of modeled environment and internal
parameters.
relationship, with an imagination process.
It is also possible to add other results of primary
appraisal, namely beneficial, irrelevant, harm/loss,
threat or challenge (p.32-33). For this, we can sim-
ply add other letters instead of s and s.
Several concepts or mechanism can be modelled
by adding internal parameters to automaton A
7.5
and
by changing the commitment functions to take the
new internal parameters into account. This can be
done for the interaction between primary and sec-
ondary appraisal (p.35), for beliefs about personal
control(p.65), as well as the concept of novelty (p.83).
5 CONCLUSION AND
PERSPECTIVES
Conclusion: We presented a new automata-based
method to formalize psychological theories, with an
example applied to stress theory. We demonstrated
how to create increasingly precise systems. Our
modelization allows for a large number of modules
that is not possible with a verbal theory.
Perspectives: We open the way to many directions of
research:
• Since we have represented psychological con-
cepts by both states and transitions, it is natural to
define the language of a theory T modelized by a
system S
n
either as the set of accepted words in Σ
∗
τ
(all states are final) of the system S
n
or as the set
of accepted sequences of transitions in ∆
∗
(where
∆ ⊆ Q × Σ
τ
× Q). We now have a canonical for-
mal object, i.e., a finite automaton, allowing us to
leverage the full range of conceptual and practical
tools from theoretical computer science. We are
now able to compare our theory with other theo-
ries also expressed by automata.
• Let S
i+1
be a system obtained by refinement of
system S
i
; prove (under realistic hypothesis) some
formal properties of refinement like L(S
i+1
) ⊆
L(S
i
) or S
i
simulates S
i+1
(for a simulation order-
ing to choose).
• We plan to apply our methodology and to build
formal models of many other theories like cogni-
tive theory of emotions, properties of short-term
memory, long-term memory, perception, repre-
sentations of the world, imagination, rational
thinking,...
• We will try to formalize into the computability
framework two theories of the mind and con-
sciousness like the Friston theory and the GWT.
• We will also extend our class of models by adding
probabilities, time and continuous variables.
• We will implement and automatically verify
some psychological theories with existing tools
like state-chart (Harel, 1987), event-b (Schnei-
der et al., 2014), NuSMV (Cimatti et al.,
2002) (A symbolic model checker), SPIN (Holz-
mann, 1997) (A tool for the formal verification
of distributed software systems) and UPPAAL
(Behrmann et al., 2004) (A tool for modeling,
simulation, and verification of real-time systems)
(Holzmann, 1997).
• We will host seminars and workshops to train re-
searchers in psychology to model theories with
this tool. This will be the opportunity to validate
the usefulness and usability of our tool.
ACKNOWLEDGEMENTS
We thank the reviewers of the Modelsward confer-
ence for their insightful comments.
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