Temporal Cognitive Maps

Adrian Robert, David Genest and Stéphane Loiseau

LERIA, Université d’Angers, France

Keywords:

Cognitive Map, Ontology, Time Representation, Query Language, Semantic Model, OWL.

Abstract:

A cognitive map is an oriented graph whose nodes are labeled by concepts and edges represent inﬂuences.

It provides a way to model strategies or inﬂuence systems. Cognitive maps do not take into account any

temporal features. This article proposes a solution to this lack: a temporal cognitive map model deﬁned

on a temporal ontology. The temporal ontology is used to represent temporal domain knowledge and to

temporally characterize the concepts of the cognitive map. An extension, named TCMQL, of the cognitive

map query language CMQL, is proposed in order to access the concepts’ temporality and compare them

making inferences.

1 INTRODUCTION

The cognitive map (Axelrod, 1976) model is a se-

mantic model coming from cognitive psychology.

This model is used to represent strategies, or more

generally, inﬂuence systems. Cognitive maps are

close to Bayesian networks (Pearl, 2014). While

Bayesian networks focus on the computation of in-

ﬂuences based on conditional probabilities, cogni-

tive maps focus mainly on the visualisation and are

easier to understand for several types of user. A

cognitive map is an oriented graph whose nodes

are labeled by concepts and edges, called inﬂu-

ences, are labeled by an inﬂuence value. Inﬂu-

ence values belong to a predeﬁned value set which

can contain symbolic values such as {−,+} (Tol-

man, 1948), {none,some,much,alot} (Kosko, 1986),

or numeric values such as [−1,1] (Kosko, 1986),

{−4,−3,−2,−1, 0,1,2, 3, 4} (Le Dorze, 2013). A se-

quence of inﬂuences from a node to another makes

a path. The model can infer a propagated inﬂuence

value from a node to another. A taxonomic cogni-

tive map (Le Dorze et al., 2012) is a cognitive map

deﬁned on a taxonomy. The taxonomy organizes the

concepts with ’kind of’ type relations: the nodes of

the taxonomic cognitive map are labeled by concepts

of the taxonomy. The taxonomy can be used to infer

the taxonomic inﬂuence value from a concept of the

taxonomy to an other one.

Cognitive maps are used in many ﬁelds such as so-

cial sciences (Axelrod, 1976), biology (Martin et al.,

2000) and geography (Çelik et al., 2005). It is typ-

ically the case of the project in geography Kifanlo

This project aims to study the evolution of the ﬁsh-

ing strategies in the Atlantic coast from 1970 to 2016.

About ﬁfty cognitive maps have been designed with

ﬁshermen to model their ﬁshing strategies. Half of

those maps represent ﬁshing strategies in the seven-

ties, the other half represent current ﬁshing strate-

gies, each map contains 25 to 50 nodes. A cognitive

map edition software, VSPCC, has been used and im-

proved

In the Kifanlo project, there is a signiﬁcant num-

ber of concepts that have a temporal semantics. These

concepts usually repeat periodically over time like

seasons, ﬁshing seasons and so on... This periodicity

of the concepts should be taken into account in cog-

nitive maps. Notice that the only articles that study

time in cognitive maps stem not on concepts but on

inﬂuence : (Park and Kim, 1995; Zhong et al., 2008)

consider the delay or duration process of an inﬂuence

in fuzzy cognitive maps.

So, this paper introduces temporal cognitive maps,

which is a new model that extends taxonomic cogni-

tive maps with a temporal ontology for representation

and reasoning.

Because of the periodicity of the concept’s seman-

tics, the temporal ontology aims to represent periodic

intervals (Osmani, 1999). It uses temporal assertions

Kifanlo is a project ﬁnanced by the Fondation de

France. It has been led from 2013 to 2017.

The edition software VSPCC (LeDorze and Robert,

2014) has been implemented after the thesis of Aymeric

LeDorze (Le Dorze, 2013), for the project Kifanlo.

Robert, A., Genest, D. and Loiseau, S.

Temporal Cognitive Maps.

DOI: 10.5220/0008937300580068

In Proceedings of the 12th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2020) - Volume 2, pages 58-68

ISBN: 978-989-758-395-7; ISSN: 2184-433X

that are triples made of two periodic intervals related

by a comparison predicate. The proposed temporal

ontology could be added to other temporal ontologies

of reference like owl-time (Hobbs and Pan, 2006a)

which lacks of such periodic temporal entities.

A temporal cognitive map is deﬁned on a temporal

ontology; it contains a set of temporal assertions that

link the nodes of the cognitive map to the temporal

ontology. The nodes can thus be temporally charac-

terized, meaning that a certain inﬂuence holds with

respect to the temporal assertions of its nodes.

To reason with a temporal cognitive map, this ar-

ticle proposes the ’Temporal Cognitive Map Query

Language’ TCMQL, which is an extension of the

query language for cognitive maps CMQL (Robert

et al., 2018). TCMQL is made with two temporal

primitives: TimeInfo and Compare. TimeInfo lets

the user access the periodic interval associated with

a node. Compare infers new information using tem-

poral assertions of nodes and the temporal ontology.

This extension provides a way to use the temporal in-

formation of the model for a further analysis of cog-

nitive maps.

TCMQL, as well as VSPCC extended to temporal

cognitive maps, have been delivered to the researchers

in geography that work in the Kifanlo project for fur-

ther analysis

This article is composed of three parts. The ﬁrst

recalls the taxonomic cognitive map model. The sec-

ond introduces the temporal cognitive map model.

The third one presents the TCMQL language.

2 TAXONOMIC COGNITIVE

MAP

A taxonomic cognitive map is a graph whose nodes

and edges are respectively labeled by a concept of a

taxonomy and by an inﬂuence value; the taxonomy

aims to organize the concepts. The taxonomy is even

more useful when using a set of cognitive maps, to

make sure that different cognitive maps use the same

concepts (Chauvin et al., 2009).

2.1 Taxonomic Cognitive Map Model

The taxonomy organizes the concepts by specifying

a specialization relation between them.

This work is being led in the project Analyse Cognitive

de Savoirs granted by the french region Pays de la Loire

from 2017 to 2020.

Deﬁnition 1 (Taxonomy). Let C be a concept set.

A taxonomy T = (C, ≤) is a set of rooted trees of

concepts that represents a partial order relation ≤

whose meaning is ’kind of’.

Example 1. T

is the taxonomy of the ﬁgure 1. Some

concepts are ordered by a relation of specialization.

For instance, the relation MultiPurposeShip

≤ Ship, meaning that MultiPurposeShip is a

kind of Ship, is represented by an arrow in the ﬁgure.

The most specialized concepts of the taxonomy

are said elementary.

Deﬁnition 2 (Elementary concepts). Let T = (C,≤)

be a taxonomy. The elementary concepts of T are:

elem

= {c ∈ C/ ∀c

∈ C,c

≤ c =⇒ c

= c}.

Example 2. In T

(ﬁgure 1), the elementary

concepts are elem

={MultiPurposeShip,

RemoteArea...}; only the concepts Ship,

FishingActivity and Pleasure are not

elementary.

A taxonomic cognitive map is a graph whose

nodes and edges are respectively labeled by an

elementary concept of a taxonomy and an inﬂuence

value. The inﬂuence value represents the strength

of the inﬂuence and belongs to a deﬁned value set

which can be qualitative or quantitative, discrete or

continuous.

Deﬁnition 3 (Taxonomic cognitive map). A taxo-

nomic cognitive map deﬁned on a taxonomy T =

(C,≤) and a value set I is an oriented labeled graph

CM=(N,E,labelN,labelE) such that:

• N : the nodes of the graph.

• E ⊂ N × N: the edges are called inﬂuences.

• labelN :N → elem

is a label function on the

nodes.

• labelE :E → I is a label function on the edges.

Example 3. CC1 and CC2 are the two taxonomic

cognitive maps of the ﬁgure 2. They are deﬁned

on the taxonomy T

of the ﬁgure 1 and the value

set I = [−1,1]. Note that, in the ﬁgure, each node

has a unique identiﬁer per map n

... that is dis-

played only for clarity in this paper. An inﬂuence

labeled by 1 (resp. -0.25) means that the source

node inﬂuences strongly (resp. weakly) and posi-

tively (resp. negatively) the destination node. In

Temporal Cognitive Maps

Figure 1: A taxonomy T

Figure 2: Two taxonomic cognitive maps, CC1 (top) and CC2 (bottom).

our application, each ﬁsherman designs a cogni-

tive map: CC1 has been designed by ﬁsherman1,

and CC2 by ﬁsherman2. In CC2, the node n

(CrabFishing) inﬂuences strongly and negatively

(-1) the node n

(BossPleasure); which means

that the boss does not like ﬁshing crab.

2.2 Taxonomic Cognitive Map Inference

A path is a sequence of inﬂuence which represents a

way a node of the map inﬂuences another. A path is

said minimal if it does not contain any cycle. Notice

that between two nodes, there can be more than one

minimal path.

Deﬁnition 4 (Path). Let CM =(N,E,labelN,labelE) be

a taxonomic cognitive map deﬁned on T = (C,≤)

and I. Let a,b ∈ N be two nodes of CM.

• A path P from a to b is a sequence of length

length

≥ 1 of inﬂuences (u

i+1

) ∈ E (with i ∈

[0;length

−1]) such that a = u

is the source of P

and b = u

length

is the destination of P. This path

is denoted by a → u

→ ··· → b.

• A path P is said minimal if ∀i, j ∈ [0; length

],i 6=

j ⇒ u

6= u

• The set of all minimal paths on CM is denoted by

Paths

Example 4. This example is based on CC2 (ﬁgure 2).

= n

(CrabFishing) → n

(TimeAtSea) →

(Rentability) is a minimal path of length=2,

from the source node n

to the destination node n

= n

(CrabFishing) → n

(Rentability) is

a minimal path of length=1.

One of the main features of cognitive maps is

their ability to infer the propagated inﬂuence from any

node to any other one, which denotes a value of inﬂu-

ence. To do that, every inﬂuence path from the node

to the other is involved.

The propagated inﬂuence from a node to another

can be calculated differently depending on the map’s

semantics and on the value set on which it is deﬁned.

In all cases, the computation of the propagated inﬂu-

ence ﬁrst assigns a path value for each path with a

function, then secondly aggregates those values with

an other function.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

Deﬁnition 5 (Propagated Inﬂuence). Let

CM=(N,E,labelN,labelE) be a taxonomic cogni-

tive map deﬁned on T = (C,≤) and I.

• The path value is a function PV

path

: Paths

→ I

which infers the propagated inﬂuence of a path.

• The propagated inﬂuence value is a function

PV : N × N → I which infers the propagated

inﬂuence from a node to another one, aggregating

the path values of each path between the two

nodes.

In this paper, we will use the value set I = [−1,1].

A product function will be used as path value and a

mean function for the propagated inﬂuence value as it

is often done in cognitive maps (Genest and Loiseau,

2007).

Example 5. This example is based on CC2 (ﬁgure 2).

The paths p

and p

come from the example 4.

Let’s infer the propagated inﬂuence value between

and n

, respectively labeled by CrabFishing

and Rentability. The set of all minimal paths

between those two nodes is Paths

= {p

, p

}, it

contains two paths. To infer the propagated inﬂu-

ence value between n

and n

we need PV

Path

)

and PV

Path

). From the chosen product func-

tion, we have PV

Path

) = 0.5 ∗ 0.25 = 0.125 and

Path

) = 0.75. Then, aggregating the path

values, PV (n

) =

(0.125+0.75)

= 0.44. So the

propagated inﬂuence value from n

to n

is 0.44.

The taxonomic cognitive map model can also infer

a taxonomic inﬂuence value which is used to infer the

inﬂuence value between any pair of concepts of the

taxonomy. Note that the propagated inﬂuence value

is a particular case of the taxonomic inﬂuence value

where the concepts are elementary. The taxonomic

inﬂuence value is not presented in this article, but is

described in (Chauvin et al., 2009).

3 TIME REPRESENTATION

This section introduces the periodic intervals, then

proposes a temporal ontology deﬁned on those peri-

odic intervals and temporal assertions that compare

pairs of them. So, the temporal cognitive map can be

introduced, it is a taxonomic cognitive map deﬁned

on a temporal ontology.

3.1 Periodic Intervals

A periodic interval (Ermolayev et al., 2014; Ermo-

layev et al., 2008; Poveda-Villalón et al., 2014; Os-

mani, 1999) is a type of non-convex interval (Lad-

kin, 1986), which is an interval composed of several

unconnected convex subintervals. Periodic intervals

have the particularity to be composed of subintervals

that have the same length and are equally spaced. For

instance ’winter’ is a periodic interval.

The periodic intervals of Osmani and Balbiani (Os-

mani, 1999; Balbiani and Osmani, 2000) that also

considers qualitative relations between them are cho-

sen. This approach is relevant to the Kifanlo project

and, in general, seems suited for cognitive maps as it

offers more ﬂexibility and handles the lack of precise

information.

Deﬁnition 6 (Periodic Interval). A periodic interval

is a non-convex interval whose subintervals are

equally spaced and have equal length.

Example 6. January is a periodic interval since

all its subintervals last one month and occur every

year. Summer is also a periodic interval, with subin-

tervals lasting three months and occurring every year.

This paper proposes to specify those periodic

intervals with qualitative relations between two inter-

vals using a comparison predicate. Those predicates

are the 16 relations of Osmani (Osmani, 1999) plus

5 relations. The relations of Osmani are very similar

to the 13 relations of the Allen’s intervals, except

that the precedence and its inverse are replaced by

5 relations which consider the periodicity. This

paper also considers two relations (Inside/Disjoint)

that combine some of Osmani’s relations and three

relations (<,>,=) that compare duration of intervals,

which can not be done with Osmani’s intervals.

Deﬁnition 7 (Comparison predicate). A com-

parison predicate is a binary relation whose

domain and range are periodic intervals. P

is the set of the 21 comparison predicates:

{m,mi,s,si,d,di,f,ﬁ,o,oi,eq,ppi,mmi,moi,omi,ooi,

in,dis,<,=,>}

The table below shows the 16 relations of Os-

mani & Balbiani, the column meaning explains

the relations through an ordering of the boundaries

(A1,A2,B1,B2) of the periodic intervals A and B. This

ordering comes from the CYCORD theory (Röhrig,

1994). Two added relations are: ’in’(Inside) which

Temporal Cognitive Maps

Figure 3: Two cyclic representations of relations between

periodic intervals.

is the disjunction of ’s’,’d’,’f’,’eq’ and ’dis’(Disjoint)

which is the disjunction of ’m’,’mi’,’mmi’,’ppi’.

To these relations are also added three relations to

compare the duration of periodic intervals : ’<’,’>’

and ’=’.

Periodic intervals and comparison predicates

deﬁned above are used to represent temporal knowl-

edge through temporal assertions. A temporal

assertion is an assertion which represents a relation

between two periodic intervals. It is a triple (inter-

val,predicate,interval).

Deﬁnition 8 (Temporal assertion). P is the set of

the 21 comparison predicates. A temporal assertion

is an assertion which constitute a triple (e

, p, e

)

such that p∈ P and e

and e

are periodic intervals.

Example 7. The relations between periodic intervals

are often represented on a circle (ﬁgure 3) which

is to be read clockwise. The ﬁrst circle represents

the temporal assertion (Spring, meets, Summer) and

it matches the ordering (’SpringBegins’, ’Sprin-

gEnds’=’SummerStarts’, ’SummerEnds’) of the

second line of the table. Its inverse relation is

mi (is met by), so we have (Summer, is met by,

Spring). The second circle illustrates the temporal

assertion (CrabSeason, meets&ismet, Summer).

CrabSeason is related to Summer by the relation

’meets&ismet’ which means that the crab season

starts when summer ends and ends when summer

starts. Some comparison predicates are used to

compare duration, for instance in the temporal

assertion (Day, <, Month) the comparison predicate

’<’ is used to compare the duration of Day and

Month.

3.2 Time Ontology

Many temporal ontologies exist, amongst those

owl-time ontology (Hobbs and Pan, 2006b) is a W3C

reference and one of the most used. It turns out that

time ontologies do not take into account periodic

intervals and certainly not the qualitative relations to

compare them. That is why this paper introduces a

new temporal ontology that considers periodic inter-

vals and could be added to existing heavier temporal

ontologies like owl-time. Our light-weight temporal

ontology is composed of the class PeriodicInterval,

the 21 comparison predicates as object properties,

a set of instances of PeriodicInterval and a set of

temporal assertions on these individuals.

Deﬁnition 9 (Temporal Ontology). A temporal ontol-

ogy O = (P, E ,A ) is an ontology such that :

• P is the set of the comparison predicates.

• E is a set of periodic intervals.

• A is a set of temporal assertions of the ontology.

Example 8. The ﬁgure 4 represents the temporal on-

tology O

. The periodic intervals of this ontology are

E ={Spring, CrabSeason, Year ...} and the temporal

assertions are A = {(Season,<,Year), (CrabSeason,

meets&ismet, Summer)...}.

3.3 Temporal Cognitive Map

A temporal cognitive map is a taxonomic cognitive

map deﬁned on a temporal ontology. Each node of

the map is labeled by a periodic interval and a set

of temporal assertions links those periodic intervals

to the ontology. This way, nodes may be temporally

characterized.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

Figure 4: A partial representation of the temporal ontology O

Deﬁnition 10 (Temporal cognitive map). Let

O = (P, E ,A ) be a temporal ontology. Let

CM=(N,E,labelN,labelE) be a taxonomic cogni-

tive map deﬁned on T (C,≤) and I. A temporal

cognitive map TCM deﬁned on O is a triple

(CM,labelT,A

TCM

) such that :

• labelT : is a label function on the nodes of the map

which attaches a unique periodic interval e

to a

node n.

• A

TCM

is a set of temporal assertions (e

, p, e

)

where labelT

−1

) ∈ N and e

∈ E .

In the next part, a set of temporal cognitive maps

based on the same taxonomy, value set and temporal

ontology is considered. So to specify an associated

periodic interval, the following notation is used.

Notation 1 (Associated Periodic interval). The

periodic interval associated with a node labeled by a

concept ’c’ of a map ’m’ is noted ’m_c’.

Example 9. This example describes the two tempo-

ral cognitive maps of the ﬁgure 5: TCM1 and TCM2.

A temporal assertion (in yellow) of a temporal cog-

nitive map is visually represented below the node

(in blue) that it characterizes. The periodic inter-

val attached to the node is visually omitted, that is

why temporal assertions are written as couples and

not triples. For instance in TCM1, the node la-

beled by OffSeason is characterized by the tempo-

ral assertion (TCM1_OffSeason, si, Summer) where

TCM1_OffSeason is the omitted periodic interval at-

tached to this node and ’si’ is the comparison pred-

icate ’isStartedBy’. Notice that several temporal as-

sertions can be attached to the same node, as it is

the case for the node labeled by CrabFishing in

TCM1. This node is characterized by a periodic in-

terval that lasts one month (=,Month) at the end of

the CrabSeason (f,CrabSeason). The ﬁsherman 1

ﬁshes crab for one month at the end of the crab sea-

son.

4 TCMQL

CMQL is a query language whose syntax is close to

the one of SQL and whose semantics is similar to

the one of the relational domain calculus (Louis and

Pirotte, 1982; Robert et al., 2019). CMQL’s particu-

larity resides in the use of many primitives that allow

to access the various features of a taxonomic cognitive

map set. TCMQL is the extension of CMQL that in-

tegrates two temporal primitives, TimeInfo and Com-

pare, which allow to access the concepts’ temporal

assertions and compare them. TCMQL is designed to

query a set of temporal cognitive maps deﬁned on the

same temporal ontology.

4.1 Primitive ’TimeInfo’

The extraction primitive TimeInfo links a cognitive

map, a concept of this map, and the periodic interval

associated with the node labeled by this concept in

this map.

Deﬁnition 11 (Primitive: TimeInfo). Let S be a set

of temporal cognitive maps deﬁned on the same tem-

poral ontology O = (P,E , A ), taxonomy T = (C,≤)

Temporal Cognitive Maps

Figure 5: Two temporal cognitive maps, TCM1 (top) and TCM2 (bottom).

and I. Let E

be the set of all periodic intervals asso-

ciated with the nodes of the maps in S. The primitive

TimeInfo(map:S, concept:C, interval:E

) is a relation

made of the set of the triples (map,concept,interval)

such that ∃n ∈ N

map

,LabelT

map

(n) =interval and

labelN

map

(n) =concept.

Example 10. In TCMQL, a variable is a syntactic ex-

pression prefaced with "?" like in SPARQL (Harris

et al., 2013). The following examples uses the maps

TCM1 and TCM2 (ﬁgure 5).

TimeInfo(?map,TimeAtSea,?interval) is a primitive

formula, which is the syntactic expression of a prim-

itive. Its meaning is a binary relation whose value

is the set of tuples (?map,?interval) in which ?inter-

val is associated with the node labeled by the concept

TimeAtSea in ?map :

?map ?interval

TCM1 TCM1_TimeAtSea

TCM2 TCM2_TimeAtSea

The primitive TimeInfo is used here to link

concepts and maps to associated periodic intervals,

TimeAtSea is then used in TCM1 and TCM2.

Used alone the usefulness of this primitive is lim-

ited, it is often used in conjunction with the primitive

Compare deﬁned below.

4.2 Primitive ’Compare’

When the designer of a temporal cognitive map adds

his domain knowledge, he adds the least amount of

temporal assertions and expects the implicit ones to

be taken into account: an inference is thus necessary.

151 inference rules are used for these inferences, they

are OWL2(Hitzler et al., 2009) rules of two types de-

scribed in the next example. The comprehensive list

of rules is not given in the paper as it is too long but

available online (Robert, 2019). The rules about the

16 Balbiani’s relations can be found also in the refer-

ences (Balbiani and Osmani, 2000), a few other rules

about the new predicates are added.

Example 11. Here are some inference rules:

• SubObjectPropertyOf(during inside) which

means (e

, during, e

)→(e

, inside, e

)

• SubObjectPropertyOf(starts <) which means (e

starts, e

)→(e

, <, e

)

• SubObjectPropertyOf(

ObjectPropertyChain(meets startedBy) meets)

which means (e

, meets, e

) ∧ (e

, startedby,

) → (e

, meets, e

)

Using the ontology of the ﬁgure 4 and the cognitive

maps ﬁgure 5, the inference produces assertions like :

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

• (CrabSeason, disjoint, Summer) which means that

the crab season is outside the summer. This asser-

tion comes from the assertion (CrabSeason, mmi,

Summer) of O

and the rule (e

, mmi, e

)→(e

disjoint, e

• (TunaSeason, >, Season) which means that the

season of tuna is longer than a calendar sea-

son. This assertion comes from the assertions (Tu-

naSeason, ﬁ, Summer), (Season, =, Summer) of

and the rules (e

, ﬁ, e

)→(e

, <, e

) and (e

>, e

) ∧ (e

,=, e

) → (e

, >, e

• (TCM1_CrabFishing, meets, Summer) which

means that Summer starts when the crab season

ends. This assertion comes from the assertions

(TCM1_CrabFishing, f, CrabSeason) of TCM1,

(CrabSeason, mmi, Summer) of O

and the rule

, f, e

) ∧ (e

, mmi, e

) → (e

, meets, e

Inferences can be carried out on a set that contains

the temporal assertions of the ontology and the

temporal assertions of each temporal cognitive map.

The saturated set is the set of temporal assertions that

can be deduced from all these temporal assertions

and the inference rules.

Deﬁnition 12 (Saturated set). Let R a set of rules and

S = {(CM

,labelT

),... ,(CM

,labelT

))}

be a set of k temporal cognitive maps deﬁned on

O = (P,E ,A ), I

is the saturated set of temporal

assertions resulting from the inference of the rules of

R on the set A ∪

i=1

The primitive Compare uses the saturated set

of temporal assertions, it is a relation between two

periodic intervals and a comparison predicate which

is a valid comparison between these intervals.

Deﬁnition 13 (Primitive: Compare). Let S be a

set of temporal cognitive maps deﬁned on the same

ontology O = (P, E ,A ) where P is the set of

comparison predicates. Let E

be the set all periodic

intervals associated with the nodes of the maps in S.

Let I

the saturated set of all temporal assertions.

The primitive Compare(e1: E

∪E ,p: P,e2: E

∪E )

is a relation made of the triples (e1, p, e2) such that

(e1, p, e2) ∈ I

Example 12. The following examples use the ontol-

ogy O

(ﬁgure 4) and the temporal cognitive maps

TCM1 and TCM2 (ﬁgure 5).

• Compare(TunaSeason, ?pred,

?interval) is a primitive formula which

aims to compare the periodic interval TunaSea-

son (which is from Spring to Summer according

to O

) to any other periodic interval. There are

many result tuples like (>,Summer) since the

summer ﬁnishes the TunaSeason:

?pred ?interval

isStarted Spring

isFinished Summer

> Summer

> Week

isFinished TCM2_O f f Season

... ...

• Compare(TCM1_TimeAtSea, ?pred,

Month)

is a primitive formula. There is no

answer since we can not evaluate

the comparison of two durations both

greater than a week (TCM1_TimeAtSea,

>, Week) and (Month, >, Week):

?pred

• Compare(Winter,?pred,CrabSeason)

is a primitive formula. This primitive formula

asks the relations between the Winter and the

CrabSeason. Since the CrabSeason starts at the

end of the summer and ends at its beginning, the

winter is during the CrabSeason and thus shorter.

We obtain the three following tuples:

?pred

during

Inside

Although the complexity of the inferences is high

(at least EXPTIME), it has not been a problem in our

system for two reasons. Firstly, a cognitive map is

hand designed and it is a visual model so it is usually

quite a small graph, for instance in the Kifanlo project

a thirty nodes map is a big one. Secondly, the satu-

rated set is precomputed and queries give an answer

in an acceptable time in our application. Neverthe-

less, to go one step further, a study should be done to

evaluate the theoretical complexity and how to face it

depending on maps structure.

4.3 Query Examples

Some primitives of CMQL are recalled here for the

following examples. The primitive IsInMap is one

of them: it is a binary relation whose ﬁrst attribute

is a map and the second a concept. This primitive is

veriﬁed if the concept appears in this map. Path is

an other primitive : it is a relation of four attributes,

Temporal Cognitive Maps

a map, two concepts of this map and a path whose

source and destination nodes are labeled by the two

concepts. KindOf is a binary primitive with two

concepts of the taxonomy where the ﬁrst is a kind

of the second. Value is a relation of four attributes

which links a map, two concepts and the taxonomic

inﬂuence value of the ﬁrst second to the second in

the map. TCMQL’s syntax is close to SQL’s syntax :

SELECT selects variables ?x. . . , FROM indicates the

maps to query and WHERE describes the conditions.

Four examples are given here along with their results

and comments, they are based on T

(ﬁg. 1), O

(ﬁg. 4) and TCM1,TCM2 (ﬁg. 5).

Example 13. The primitives IsInMap and KindOf

are used in this example. In plain English this query

means : ’In which maps are used the concepts types

of Pleasure?’.

SELECT ?map,?concept FROM TCM1,TCM2

WHERE{

KindOf(?concept, Pleasure)

AND IsInMap(?map,?concept)

}

The ﬁrst condition allows to obtain the concepts

that are types of Pleasure in the taxonomy. The

second condition gets the couples (map, concept)

such that the concept belongs to the map. The re-

sult of this query is the list of the following tuples

(?map,?concept):

?map ?concept

TCM1 BossPleasure

TCM1 CrewPleasure

TCM2 BossPleasure

The result shows what are the types of the concept

pleasure and in which maps they appear.

Example 14. In plain English this query means

:’When does ﬁsherman1(TCM1) ﬁsh crabs in com-

parison to ﬁsherman2(TCM2)?’.

SELECT ?pred FROM TCM1,TCM2 WHERE{

TimeInfo(TCM1, CrabFishing, ?e1)

AND TimeInfo(TCM2, CrabFishing,

?e2) AND Compare(?e1,?pred,?e2)

}

The ﬁrst two conditions allow to get the temporal

entities of the concept CrabFishing in TCM1 and

TCM2. The third condition allows to get all compar-

ison predicates between those two temporal entities

that are characterized by "ﬁnishes CrabSeason" and

" = Month" for the one in TCM1 and by "equals

CrabSeason" for the other. The result is made of the

tuples (?pred):

?pred

f inishes

The result shows that the ﬁsherman1 ﬁshes at the

end of the ﬁsherman2’s ﬁshing period, for a shorter

period.

Example 15. In plain English this query means

:’Which duration of FishingSets inﬂuences BossPlea-

sure?’.

SELECT ?p, ?e2 FROM TCM1,TCM2

WHERE{

Path(?map,FishingSet,BossPleasure,?path)

AND TimeInfo(?map,FishingSet,?e1)

AND Compare(?e1, ?p, ?e2))}

The ﬁrst condition allows to get the maps in which

FishingSet inﬂuences BossPleasure (TCM2). The two

following conditions allow to get the temporal infor-

mation about FishingSet in the right map.

?p ?e2 ?m

= ShortFishingSet TCM2

< Day TCM2

... ... ...

The result shows that according to the ﬁsherman2

(TCM2), the BossPleasure is inﬂuenced by a short

period of ﬁshingset.

Example 16. This query asks the concepts in summer

which inﬂuence a concept kind of Pleasure.

SELECT ?map,?c1,?i,?c2

FROM TCM1,TCM2

WHERE{KindOf(?c2,Pleasure) AND

Value(?map,?c1,?c2,?i) AND ?i != 0

AND

TimeInfo(?map,?c1,?e1) AND

Compare(?e1,in,Summer)}

The ﬁrst condition allows to get all concepts ?c2

kind of Pleasure. The second and third ones allow

to get the concepts ?c1 that inﬂuences ?c2 with their

inﬂuence value. The two last conditions ﬁlter only the

concepts ?c1 in summer.

?map ?c1 ?i ?c2

TCM1 O f f Season −0.5 BossPleasure

TCM1 O f f Season 0.5 CrewPleasure

The results show that, according to the ﬁsher-

man1, the OffSeason which is in Summer inﬂuences

negatively the pleasure of the boss and positively the

pleasure of the crew.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

5 CONCLUSION

This paper introduces an extension of the cognitive

map model called temporal cognitive map which al-

lows to temporally characterize concepts of the map.

To do this the temporal cognitive map is deﬁned on a

temporal ontology which uses periodic intervals. This

paper proposes also an extension of CMQL, named

TCMQL, which allows to query a set of temporal cog-

nitive map and its new temporal features.

The temporal cognitive map model has been

implemented and tested into the VSPCC soft-

ware which provides tools to edit and use cog-

nitive maps. This software can also execute

TCMQL queries, it is available online (LeDorze

and Robert, 2014). The implementation uses the

temporal ontology owl-time to which is added a

class PeriodicInterval as a subclass of the main

class http://www.w3.org/2006/time#TemporalEntity

and comparison predicates as properties. owl-time

contains other temporal entities, such as instants or

Allen’s intervals. They could also be used once ade-

quate inference rules are added.

The temporal features introduced in this paper

come from real application needs for a better mod-

elling of the ﬁshermen’s strategies and for a more

in-depth analysis in the Kifanlo project. The ACS

project that succeeds the Kifanlo project is currently

in progress, using these new tools.

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