=Paper=
{{Paper
|id=Vol-1433/tc_45
|storemode=property
|title=A Logic-Based Approach to Understanding
Lone-Actor Terrorism
|pdfUrl=https://ceur-ws.org/Vol-1433/tc_45.pdf
|volume=Vol-1433
|dblpUrl=https://dblp.org/rec/conf/iclp/AlrajehG15
}}
==A Logic-Based Approach to Understanding
Lone-Actor Terrorism==
https://ceur-ws.org/Vol-1433/tc_45.pdf
Technical Communications of ICLP 2015. Copyright with the Authors. 1
A logic-based approach to understanding
lone-actor terrorism
Dalal Alrajeh
Department of Computing, Imperial College London
London SW7 2RH, United Kingdom
dalal.alrajeh@imperial.ac.uk
Paul Gill
Department of Security and Crime Science, University College London
London WC1H 9EZ, United Kingdom
paul.gill@ucl.ac.uk
submitted 29 April 2015; accepted 5 June 2015
Abstract
The need for systematic research into behavioural factors of individual terrorists has been
highlighted by much recent work on terrorism. Many existing methods follow a hypothesis-
testing approach in which statistical modelling and analysis of existing data is conducted
to either confirm or refute a hypothesis. However, the initial construction of hypotheses
is not trivial, nor is the decision upon which of the variables are to be considered relevant
for the testings. It has been argued that the lack of a methodical approach to represent,
analyse, interpret and infer from existing data presents a pressing challenge to the progress
of lone-actor terrorism research in particular, and the terrorism field more generally.
This paper sets a new agenda for such research. We propose the use of a logic pro-
gramming approach to address the shortcomings of existing methodologies in the study of
lone-actor terrorism. Our method is based on transforming characteristic and behavioural
codes into a logic program and applying inductive logic programming to learn hypothe-
ses about potentially relevant factors associated with terrorist behaviour, as well as the
influence of specific factors on such behaviour. This paper is an exploratory study of 111
lone-actor terrorists’ target selections (civilian vs. high-value targets) and the agency of
their ideological orientation in determining their target choices.
1 Introduction
An emerging consensus within terrorism studies posits analysing what terrorists
do as opposed to merely studying who they are is more instructive (Horgan 2014).
A growth in datasets focused upon individual, as opposed to group, behaviour
has fostered a major new development in our understanding of terrorist behaviour
(LaFree 2013). Rather than employing a single conception of the ‘terrorist’, these
analyses disaggregate the sample and compare the subsets across specific charac-
teristics and behaviours (Gill and Corner 2013). However, a number of problems
remain endemic within the study of the individual terrorist. First, most of these new
analyses rely upon testing hypotheses derived from the study of general criminal
2 Dalal Alrajeh and Paul Gill
offenders whose psychologies and decision-making repertoires may not be gener-
alizable to violent, politically-oriented offenders. In a field as underdeveloped as
terrorism studies, hypothesis generation based purely on studies of terrorist be-
haviour can be onerous. Second, commonly used statistical methods (bivariate and
multivariate analyses) fail to capture causality relations between variables. These
approaches also concentrate on subsets of variables theoretically linked with the
observation being investigated. However, in some cases, this is not sufficient as they
may only be explained by combinations of loosely related variables that potentially
remain untested. Statistical findings do not speak for themselves: moving from ‘fac-
tors’ to policy is not straightforward (Farrington 2000). As a consequence, it has
been argued that a knowledge-base capable of supporting policy must contain more
than a catalogue of factors, however exhaustive. It must include theories which
advance explanations of how these factors are related to the outcome of interest
(Wikstrom 2011). Knowledge is achieved when outcomes are explained, rather than
merely described, or even predicted.
In this paper, we explore a logic programming approach to representing and rea-
soning about lone-actor terrorists’ characteristics and behaviours. We propose an
approach for automatically generating hypotheses about lone actor terrorists with
the aim of gaining better understanding of the links between individuals’ charac-
teristics and behaviour with respect to the outcome of their event planning. In
particular, we investigate the use of inductive logic programming (ILP) to con-
duct two types of analyses: (i) identifying factors that are associated with and can
differentiate between terrorists’ target selections (i.e., learning associations), and
(ii) capturing the influence of specific factors in explaining such differences (i.e.,
learning influences). The approach is applied to a dataset containing antecedent
behaviours and characteristics of 111 lone actors terrorists, originally described in
(Gill et al. 2014). The overall objective of this work is to provide an exploratory ap-
proach that overcomes limitations of the methods standardly applied in this area of
study by: (a) automatically generating explanations that are guaranteed to cover
all the observations; (b) suggesting alternative hypotheses to be tested; and (c)
demonstrating how the presence or absence of a single factor can be associated
with different observed outcomes when combined with others. The approach pre-
sented in this paper forms the initial steps towards developing a logic-based, causal
framework for reasoning about criminal behaviour with wider applicability in crime
science studies.
2 Background
2.1 Lone-actor Terrorism Characteristic and Behavioural Codes
In this paper, terrorism is defined as a violent action, or threat of violent action,
aimed at intimidating and coercing a government or sections of the public, typically
for political, religious or ideological ends. Terrorism can involve violence against a
person, damage to property, endangering a person’s life other than that of the
terrorist, creating a serious risk to the health or safety of the public or a section
of the public. A lone-actor is either a single terrorist or an isolated dyad (a pair
of individuals), operating independently of a group. Antecedent behaviour is the
A logic-based approach to understanding lone-actor terrorism 3
behaviour of the offenders leading up to their planning or conducting a terrorist
act. Demographic characteristics of lone-actor terrorists include gender, education
and socio-economic indicators such as employment status.
For analysis purposes, antecedent behaviour and demographic characteristics are
typically represented as codes (also called variables). Each code c has a domain,
denoted dom(c), of possible values. When C is a set of codes, we write dom(C )
as a shorthand for the set of domains for each c ∈ C . We are interested here
in codes whose domains are non-empty, finite sets of discrete values. A code is
binary if its domain contains two values only, and multi-valued if it contains more
than two. Boolean codes are binary codes with domain {true,false}. A code can be
assigned one or more values from its domain. Given a set of codes C , the function
singlevalue : C → {true, false} returns true if a code takes one value only and false
otherwise. Boolean codes are single valued. Given a set ID containing a unique
identifier for each lone-actor, we define a labelling function θc : ID → 2dom(c)
for each code c ∈ C . Where id ∈ ID is a lone-actor’s identifier, θc (id ) is a set
of values from dom(c), giving the actual values of id ’s c code. We require that
θc (id ) 6= ∅ in all cases and ∀c : C · (singlevalue(c) → ∀id ∈ ID · |θc (id )| = 1).
For example, the terrorist with identifier pg011 targets both civilian and high-value
groups: θtargetgroup (pg011) = {civilian, hvt}. We use θC to denote the set of assignment
functions for each code in C . We sometimes refer to code assignments as factors. The
full set of codes used in this paper is available at www.doc.ic.ac.uk/∼da04/iclp15/.
2.2 Inductive Logic Programming
Inductive logic programming (ILP) (Muggleton 1991) is a logic-based machine
learning technique the aims at automatically generating hypotheses from observed
phenomena and a background theory expressed as logic programs. In this paper, we
are concerned with learning normal logic programs. A normal logic program is one
in which the clauses (or rules) are of the form A ← B1 , . . . , Bn , not C1 , . . . , not Cm
where A is the head atom, Bi are positive body literals, and not Cj are negative
body literals. Normal logic programs may have one, none, or several (minimal)
models. The semantics of our logic programs are based on stable models semantics
(Gelfond and Lifschitz 1988). Given a normal logic program Π, the reduct of Π with
respect to I , denoted ΠI , is the program obtained from the ground instances of Π
by (a) removing all clauses with a negative literal not a in its body where a ∈ I
and (b) removing all negative literals from the bodies of the remaining clauses. If
I is the least Herbrand model of ΠI then I is said to be a stable model of Π. This
along with the notion of entailment are given below.
Definition 1
A model I of Π is a stable model if I is the least Herbrand model of ΠI where ΠI is
the definite program ΠI = {A ← B1 , . . . , Bn | A ← B1 , . . . , Bn , not C1 , . . . , not Cn
is the ground instance of a clause in Π and I does not satisfy any of the Ci }.
Definition 2
A logic program Π entails an expression φ (under the credulous stable model se-
mantics), denoted Π |= φ, iff φ is satisfied in at least one stable model of Π.
4 Dalal Alrajeh and Paul Gill
In ILP, mode declarations are used as a form of language bias to reduce the
hypotheses search space. They provide a mechanism for specifying which predicates
may appear in the heads and bodies of rules and for controlling the placement and
linking of constants and variables within those clauses. A mode declaration M
is either a head or body declaration, respectively modeh(s) and modeb(s) where
s is called a schema. A schema s is a ground literal containing placemarkers. A
placemarker is either ‘+type’ (input), ‘−type’ (output), ‘#type’ (ground) where
type is a constant. Given the above, an ILP task is defined as follows where |= is
interpreted under brave induction (Sakama and Inoue 2009).
Definition 3
A nonmonotonic ILP task is a tuple hB , E + , E − , M i where E + and E − are sets of
ground literals, called positive examples and negative examples respectively, B is a
normal logic program, called background theory and M is a set of mode declarations
defining a hypothesis space s(M ). An inductive solution (or hypothesis), H ⊆ s(M ),
for E + ∪ E − w.r.t B is a set of clauses such that:
B ∪ H |= e + , ∀e + ∈ E + and B ∪ H 6|= e − , ∀e − ∈ E −
under brave induction (Sakama and Inoue 2009).
We sometimes write B ∪ H |= E where E = E + ∪ E − as a shorthand for the two
conditions above.
In this paper, we focus on the use of a learning technique first introduced in
(Ray 2009) and its implementation XHAIL. The technique is based on a three-
phase Hybrid Abductive Inductive Learning (HAIL) approach (Ray et al. 2004).
The XHAIL language and search bias mechanisms are based upon a compression
heuristic that favours solutions containing the fewest number of literals.
3 Approach
We first introduce an automated mechanism for mapping characteristic and be-
havioural codes into a logic program. We then describe an ILP approach for con-
ducting two types of analyses. The first aims at identifying combinations of factors
that are associated with specific observations. We call this learning associations.
The second is to understand the influence of specific factors when explaining these
observations. We refer to this type as learning influences. Our focus in this paper
will be on explaining terrorists’ target group selection, civilian targets vs. high-value
targets, where the latter includes targets such as government, business, schools etc.
We demonstrate the approach using a dataset of 111 individuals (of which 91 have
known target selections) and 185 codes described in (Gill et al. 2014).
3.1 Modelling Codes in Logic Programs
Our representation uses a sort ID for capturing the domain of unique lone-actor
terrorists. It includes a unique predicate for each code c and type predicate to rep-
resent the domain of c. The number of arguments for this predicate depends on the
size of the code’s domain. If it is Boolean, e.g., mentalill, then, as a simplification,
a predicate with a single argument of sort ID is introduced e.g., mentalill(id). If it
is a non-Boolean variable, then a two argument predicate is introduced in which
A logic-based approach to understanding lone-actor terrorism 5
the first argument is of sort ID and the second is of sort Dom(c). For instance,
the nature of the location at which the attack occurred is encoded using a pred-
icate location nature ⊆ ID × dom(location nature) where dom(location nature) =
{government, business, private citizens, religious, military, other}. The language also
contains a type predicate for representing dom(c). The encoding is given below.
Definition 4
Let ID be a set of unique identifiers for lone-actors, C a set of codes named {c 1 , · ·
·, c n }, D(C ) the set of C domains named {dc 1 , · · ·, dc n } respectively and θC their
assignment functions. The logic program IDLP ∪ CLP ∪ DLP (C) ∪ ALP encoding
of ID, C, D(C ) and θC is constructed such that:
• IDLP contains a fact la iden(id ) for each id ∈ ID;
• DLP (C) contains a fact dc k (j ) for each j ∈ dom(c k );
• for each id ∈ ID and c ∈ C, CLP contains a fact c k (id) where c k is a Boolean
code and θc (id ) = {true};
• for each id ∈ ID and c ∈ C, CLP contains a fact c k (id , j ) where |dom(c k )| ≥ 2,
c k is non-Boolean, j ∈ θc (id ) and j 6∈ {false, unknown};
• for each c ∈ C, if c is non-Boolean and singlevalue(c) = true, ALP contains
the clause ← c(I, D1 ), c(I, D2 ), D1 6= D2 ·
Our encoding deploys a closed world assumption where unknown code values are
treated as false, as assumed in (Gill et al. 2014). An example of the encoding is:
IDLP = {la iden(pg018). la iden(pgpg101)...}
DLP (C) = {ideo(rightwing). ideo(single issue)... loc(government)...
tg(hvt). tg(citizens)...}
CLP = {imprisoned(pg018). ideology(pg018, rightwing). mentalill(pg018).
location nature(pg018, government).
... dryruns(pg101). ideology(pg101, single issue). f2f(pg101)...}
ALP = {← ideology(I, D1 ), ideology(I, D2 ), D1 6= D2 ·, · · ·}
The expressiveness of the formalism allows us to capture relationships between
codes. For example, a code value both may be introduced for capturing individuals
who have targeted both civilian and high-value groups, in which case dom(targetgroup)
is extended with the value both and ALP is amended with the following:
{targetgroup(I, both) ← targetgroup(I, hvt), targetgroup(I, civilian)·,
← targetgroup(I, both), not targetgroup(I, hvt)·,
← targetgroup(I, both), not targetgroup(I, civilian)·}
3.2 Learning Characteristic and Behavioural Associations
In the context of characteristic and behavioural analyses, a code c1 is said to be
associated with another code c2 if c1 forms part of at least one explanation of the
observed behaviour represented by c2. In ILP terms, identifying codes associated
with an observation amounts to finding an inductive solution in which the literals
corresponding to these codes appear in the body of at least one rule in the solution.
In the presented approach, we do not distinguish between codes that are causes and
those that are correlates. Future work will clarify the distinction.
Definition 5
Let ID be a set of unique lone-actor identifiers, C a set of codes, and D(C ) a set of
C domains. Let ct ∈ C be a code representing the observed terrorist behaviour to
6 Dalal Alrajeh and Paul Gill
be explained, and R = {cr |cr ∈ C, cr 6= ct } be the rest of variables in the language.
The ILP task is defined with:
• B = IDLP ∪ RLP ∪ DLP (R) ∪ DLP ({ct }) ∪ ALP ;
• E + includes a fact ct (id ) for each id ∈ ID where ct is Boolean and θct (id ) =
{true}, or a fact ct (id , j ) where ct is non-Boolean and j ∈ θct (id );
• E − includes a fact ct (id ) for each id ∈ ID where ct is Boolean and θct (id ) =
{false}, or a fact ct (id , j ) where ct is non-Boolean, for every j ∈ dom(ct ) such
that j 6∈ θct (id );
• M includes modeh(ct (+la iden)) (or modeh(ct (+la iden, #dct ))), and a pair of
body declarations modeh(cr (+la iden)) (or modeh(cr (+la iden, #dcr ))) and
modeb(not cr (+la iden)) (or modeb(not cr (+la iden, #dcr ))) for each cr ∈ R.
Note that in this type of the analysis, we do not impose any restrictions on the
number of rules, within an inductive solution, in which a body literal appears to be
said associated with the observation, nor on the number of observations explained
by the rule in which it appear. However, to quantify the relevance of that association
for a given dataset, our algorithm calculates a measure (relative significance value)
for each hypothesis in an inductive solution as defined below.
Definition 6
Let B be a background theory, E = E + ∪ E − the set of examples, M the mode
declaration and H = {h1 , · · ·hn } ⊆ s(M ) an inductive solution to E w.r.t B . Let
Eh+i ⊆ E + be the set of positive examples explained by the hypothesis hi ∈ H such
that B ∧ hi |= eh+i , for each eh+i ∈ Eh+i . Then the relative significance σ of hi , denoted
|E + |
σ(hi ), is calculated as: σ(hi ) = |Eh+i |
In addition to the above, we also calculate for each hypothesis a measure, we
call the predictive value of a hypothesis, which is based on the number of target
selections the hypothesis infers for individuals with unknown target selections.
Definition 7
Given an ILP task hB , E + , E − , M i constructed using Def. 5, let H = {h1 , · · ·hn } ⊆
s(M ) be an inductive solution to E + ∪E − , w.r.t B . Let ct be the predicate appearing
in E . Let |P | be the set of ct (id ) atoms where ct is Boolean, or ct (id,j ) atoms where
ct is non-Boolean, with id ∈ ID, j ∈ dom(ct ) and θct (id) = {unknown}, entailed by
H ∪ B , such that H ∪ B |= p for every p ∈ P . Let Phi ⊆ P be the set of ct (id ), or
ct (id,j ), atoms entailed by hi ∪ B where hi ∈ H , such that B ∪ hi |= phi , for every
|Phi |
phi ∈ Phi . Then the predictive ρ of hi , denoted ρ(hi ), is calculated as: ρ(hi ) = |P| ·
Our algorithm uses the Answer Set solver clingo (Gebser et al. 2007) to find
models of B ∪ hi from which the relative significance and predictive values for
each hypothesis are calculated. In summary, a logic program is constructed for
each hypothesis hi ∈ H in conjunction with B . From this, the number of atoms
representing target selections for each individual in the dataset in the answer set of
the B ∪ hi is computed. We distinguish between the number derived for those with
A logic-based approach to understanding lone-actor terrorism 7
known and unknown target selections by comparing the individuals’ id’s appearing
in the answer set with those provided in the original dataset.
In our case, to learn characteristics and antecedent behaviour that are associated
with specific target selections, B includes the clauses representing code assign-
ments and relationships for the dataset of 111 lone-actor terrorists. The positive
examples include facts about the observed target selection for each individual, e.g.,
{targetgroup(pg018,hvt), targetgroup(pg101,civilian)} and the negatives includes
facts about the groups that were not targeted, e.g., {targetgroup(pg018,civilian),
targetgroup(pg101,hvt)}. The total number of positive examples is 95, and the total
number of negative examples is 87.
The mode declarations M includes modeh(targetgroup(+la iden, #tg)). It also
comprises a total of 38 modeb declarations for codes such as crimcon, verbfam,
dryruns and mentalill amongst others.
Having defined the ILP task, we use the ILP system XHAIL (Ray 2009), to com-
pute the hypotheses. Table 1 shows an extract of the inductive solution. A full list of
the mode declarations and hypotheses can be found at www.doc.ic.ac.uk/∼da04/iclp15/.
i hi |eh+ | |phi |
i
1 targetgroup(I, civilian) ← not crimcon(I ), not history(I ), not f2f (I ), 4 2
otherknowledge(I ), not children(I )).
2 targetgroup(I,civilian,) ← dryruns(I ), not warning(I ), not imprisoned(I ), 3 1
not training(I ), mentalill(I ), not virtualinteract(I ).
3 targetgroup(I,civilian) ← not imprisoned(I ), not milexp(I ), not livealone(I ), 8 4
not training(I ), not virtuallearn(I ), not otherknowledge(I ).
4 targetgroup(I,civilian) ← not milexp(I ), verbfam(I ), livealone(I ), 4 4
not f2f (I ).
5 targetgroup(I,civilian) ← uniexp(I ), not warning(I ), not milexp(I ), 12 1
not virtuallearn(I ).
... ... ... ... ...
52 targetgroup(I,hvt) ← not warning(I ), mentalill(I ), children(I ). 2 2
53 targetgroup(I,hvt) ← not imprisoned(I ), virtuallearn(I ), f2f (I ), 1 0
otherknowledge(I ), recruit(I ) , not children(ˇ).
54 targetgroup(I,hvt) ← not dryruns(I ), not warning(I ), not religcon(I ), 5 2
verbfam(I ), not virtuallearn(I ), not f2f (I ).
55 targetgroup(I,hvt) ← not univexp(I ), not religcon(I ), mentalill(I ), 10 3
not recruit(I ).
56 targetgroup(I,hvt) ← not warning(I ), not imprisoned(I ), crimcon(I ), 1 0
livealone(I ), not training(I ), not mentalill(I ),
not history(I ), not recruit(I ).
Unique total 95 29
Table 1: Hypothesis for target selection for the full sample
The solution shown above is not the minimal one. We have redefined the algo-
rithm to terminate once it has found an optimal solution within a specified time
bound. From the table above, we see that h5 has a higher relative significance value
than h1 since σ(h5 ) = 0 · 12 > σ(h4 ) = 0 · 04, but a lower predictive value with
ρ(h5 ) = 0 · 03 < ρ(h4 ) = 0 · 13. From a criminological perspective, the solution
demonstrates that civilian targeting is associated with individuals with a history
8 Dalal Alrajeh and Paul Gill
of mental illness (h2 ) who engage in dry runs and have not been imprisoned, un-
dergone training, amongst others. At the same time, we observe that mental illness
alone cannot determine the target selection outcome as shown by (h55 ) where men-
tal illness exhibited with other characteristics, including no university experience
or religious conversion prior to the attack, explains high-value target selections.
Targeting high-value groups is also associated with criminal convictions and living
alone which again may speak towards capability (in terms of both criminal ingenu-
ity and having the space to develop a bigger plot). When this criminal ingenuity
is not present, it may necessitate other behaviour like virtual learning, face to face
interactions with co-ideologies and the attempt to recruit others (as witnessed in
h52 ).
3.3 Learning Characteristic and Behavioural Influences
The previous section is concerned with finding explanations using any subset of
possible codes. In this section, we are interested in learning whether an observed
behaviour can be explained with respect to particular characteristics and behaviour.
We refer to this type of learning as learning influences.
A characteristic or behaviour is said to influence the code c2 if every outcome of
the observed behaviour represented by c2 can be explained in terms of the presence
or absence of c1. In an ILP setting, the problem of learning influences is expressed in
terms of finding inductive solutions in which every rule within that solution contains
a body literal representing that characteristic or behaviour. For instance, in the
case of exploring the influence of ideological orientation on target group selection,
all hypotheses in an inductive solution must include a body literal corresponding
to the individuals’ ideological orientation.
Our algorithm for learning influences comprises four steps. First the dataset is
split into subgroups, according to the value assigned to the code whose influence
is being studied. Thus for a code cf , we have |dom(cf )| ≥ 2 subgroups. Note that
these subgroups are not required to be mutually exclusive. For ease of reference, we
use the notation df to denote the subgroup containing data for individuals whose cf
value is df where the reference is obvious form the context. The second step involves
applying the mapping in Def. 4 to each of the created subgroups separately. Then,
a learning task hBdf , Ed+f , Ed−f , M i is defined for each subgroup df upon which the
learning system XHAIL is executed. Once Hdf is generated for each subgroup, the
resulting inductive solutions undergo a post-processing procedure to generate the
final set of hypotheses. This is done by applying a transformation function defined
below to each rules in the inductive solutions.
Definition 8
Let Π be a normal logic program and b a literal. A transformation τ is defined such
that Π0 = τ (Π, b) and Π0 is obtained from Π by adding a condition b to the body
of every rule in Π.
Given the function τ , we have, in the case of a Boolean code cf , the final solution
0 0 0 0
H = Htrue ∪ Hfalse where Htrue = τ (Htrue , cf (I)), and Hfalse = τ (Hfalse , not cf (I)). In
the case of a non-Boolean code cf , the final solution H = {Hd0f } where Hd0f =
i i
A logic-based approach to understanding lone-actor terrorism 9
τ (Hdfi , cf (I, dfi )). Note that the correctness of solutions with respect to the union
of the example sets is only guaranteed when the code values are independent.
In the case of learning the influence of ideological orientation on target selection,
the dataset of 111 individuals is split into three subgroups based on whether the
individual’s ideological orientation is rightwing, single issue or religious. In our ex-
ample, the subgroups contain data for 43, 30, 38 individuals respectively. In our
dataset, these groups are mutually exclusive since singlevalue(ideology)=true. The
XHAIL system is then run using three independent learning tasks, one on each of
the subgroups. The final solutions are shown in Table 2. The relative significance
and predictive values are calculated with respect to each subgroup.
Religious Ideology
i hi |eh+ | |phi |
i
1 targetgroup(I,civilian) ← ideo(I,religious), not virtuallearn(I ), not mentalill(I ). 10 2
2 targetgroup(I,civilian) ← ideo(I,religious), univexp(I ), verbfam(I ), 7 0
not mentalill(I ).
3 targetgroup(I,civilian) ← not univexp(I ), f2f (I ). 3 0
4 targetgroup(I,civilian) ← ideo(I,religious),dryruns(I ), otherknowledge(I ). 5 1
5 targetgroup(I,hvt) ← ideo(I,religious),univexp(I ), mentalill(I ). 6 3
6 targetgroup(I,hvt) ← ideo(I,religious), not univexp(I ), not warning(I ), 9 2
crimcon(I ).
7 targetgroup(I,hvt) ← ideo(I,religious),not verbfam(I ), virtuallearn(I ). 9 2
Unique total 40 9
Single-issue Ideology
i hi |eh+ | |phi |
i
8 targetgroup(I,civilian) ← ideo(I,single issue), not dryruns(I ), 4 0
not crimcon(I ),not livealone(I ).
9 targetgroup(I,civilian) ← ideo(I,single issue), crimcon(I ), livealone(I ). 6 0
10 targetgroup(I,civilian) ← ideo(I,single issue), f2f (I ), not otherknowledge(I ). 11 0
11 targetgroup(I,civilian) ← ideo(I,single issue),mentalill(I ), not children(I ). 9 1
12 targetgroup(I,hvt) ← ideo(I,single issue), not livealone(I ), not history(I ), 4 0
children(I ).
13 targetgroup(I,hvt) ← ideo(I,single issue),imprisoned(I ), not mentalill(I ), 2 0
not recruit(I ),not children(I ).
14 targetgroup(I,hvt) ← ideo(I,single issue),livealone(I ), not mentalill(I ), 1 0
not history(I ), not f2f (I ).
15 targetgroup(I,hvt) ← ideo(I,single issue), not crimcon(I ), training(I ). 2 0
Unique total 31 1
Rightwing Ideology
i hi |eh+ | |phi |
i
16 targetgroup(I,civilian) ← ideo(I,rightwing), not dryruns(I ), livealone(I ). 14 4
17 targetgroup(I,civilian) ← ideo(I,rightwing),verbfam(I ), not mentalill(I ), 7 1
not children(I ).
10 Dalal Alrajeh and Paul Gill
18 targetgroup(I,civilian) ← ideo(I,rightwing),not training(I ), 16 4
not otherknowledge(I ), not children(I ).
19 targetgroup(I,civilian) ← ideo(I,rightwing),not warning(I ), mentalill(I ), 10 2
not recruit(I ).
20 targetgroup(I,hvt) ← ideo(I,rightwing), not livealone(I ), children(I ). 5 3
21 targetgroup(I,hvt) ← ideo(I,rightwing), mentalill(I ), 3 0
not virtualinteract(I ), otherknowledge(I ).
22 targetgroup(I,hvt) ← ideo(I,rightwing), not livealone(I ), training(I ), 1 0
not f2f (I ).
23 targetgroup(I,hvt) ← ideo(I,rightwing), warning(I ), mentalill(I ). 3 0
Unique total 46 12
Table 2: Hypothesis for target selection for three ideological orientation groups.
The first part of Table 2 highlights a number of interesting facets related to
a religious-inspired individual’s choice. The presence (or lack thereof) of mental
health problems helps shape target choice toward civilians or high-value targets
respectively depending on whether the individual has university experience or not
(see solution h2 and solution h5 .) The lack of university experience (and perhaps
the skills associated with overcoming complex tasks) can be mitigating for when
targeting high-value targets by the presence of a criminal past and the nous that
may develop through prior antecedent offending; see solution h2 . The second part
in the table that refers to single-issue ideology indicates the need to disaggregate
across ideological domains. Whereas the confluence of criminal histories and living
alone appeared in Table 1 to suggest a close relationship with high-value targeting,
the opposite is true for those individuals inspired by single issues (animal rights, en-
vironmentalists) and may be a direct reflection of different targeting norms within
these movements (see solution h9 ). In the absence of criminal histories, gaining
training from a wider group appears to be a relevant substitute (see solution h15 ).
The last part of the table referring to rightwing ideologies confounds some expecta-
tions in the wider literature as it highlights the presence of mental health problems
(solution h21 ) in terms of attacking high-value targets compared to civilian targets
(the latter of which are presumably easier to plan).
4 Discussion
In our approach, we focused on learning rules that capture associations between
behaviour and influence of specific behaviour when explaining terrorists’ target se-
lection. The performance of the learning algorithm used depended on the size of M
amongst other factors such as the size of the examples. Experiments showed that it
decreased when a larger M was considered. The performance of the algorithm for
same M size was higher in the case of learning influences than it was when learn-
ing associations, as expected given the examples size was smaller. Furthermore, we
found that we were able to find a more optimal solution for learning influences
than we did for associations for the fixed time-frame we gave (which was set to
A logic-based approach to understanding lone-actor terrorism 11
360 minutes) for the same observation (56 rules in the case of learning associa-
tions compared to 23 when learning influences). The choice of codes to use in M
in the presented work was influenced by variables that are commonly investigated
in existing literature. Furthermore, the optimization was driven by the total num-
ber of literals appearing in an inductive solution in some cases resulting in more
hypotheses with small potential significance values as opposed to fewer hypotheses
with higher potential significance. The learning association approach is similar in
essence to task discovery in data-mining, e.g.,(Dehaspe and Toivonen 1999), the
aim here is to provide a reasoning platform capable of handling default negation
which better suits the incremental hypothesis generation and refinement nature of
the problem domain, and allows the integration of domain knowledge both in the
background and in the heuristics defined over the search space.
To evaluate our approach, we compared our results against those produced us-
ing standard methods deployed in terrorism studies, rather than performed cross
validation over our small sample set. In particular, we conducted a Smallest Space
Analysis (SSA), shown in Fig. 1, of the antecedent behaviour and their relationship
to one another for the full dataset of 111 individuals. Such analyses focus upon vari-
able co-occurrence. Prominent examples include (Canter and Heritage 1990) work
on serial rapists, and (Canter 2004) work on serial murder. The lone-actor terrorist
typologies presented below utilises this specific method. It provides geometric rep-
resentations of the level of association between variables. In other words, the Multi
Dimensional Scaling outputs represent a matrix wherein variables that regularly
co-occur are plotted closer together in a Euclidean space. The utility of such a rep-
resentation is that the variable configuration is based upon variables’ relationships
with each other rather than their relationships with pre-determined dimensions
(Davis 2009). SSA is based upon the assumption that the underlying structure of
complex systems is most readily appreciated if the relationship between each and
every other variable is examined, but that such examination is much clearer if the
relationships are represented visually not only in terms of numbers’ (Canter 2004).
The Jaccard co-efficient (which represents the level of association between two vari-
ables) was calculated for each pair-wise set of variables. The closer two variables
appear within the matrix, the higher their co-occurrence across observations. For
example, virtual learning (VirtualLearn) and virtual interaction (VirtualInteract)
are extremely close and therefore occur very regularly together.
The results visually illustrate some of the key findings produced in Table 1. The
SSA output also helps demonstrate which of the “not” behaviour rarely co-occur
with the present behaviour and which are specific to that combination of factors.
As per h1 , other knowledge is situated very far from criminal convictions, history
of violence, children indicated that this particular behaviour rarely co-occurs with
these other factors. However, the SSA output demonstrates a relatively close rela-
tionship between other knowledge and face-to-face interactions. h1 however shows
these rarely co-occur when these other factors are also absent. The SSA output also
helps illustrate the degree to which the “not” behaviour co-occur. Returning to h1 ,
children rarely co-occur with a history of violence. The types of hypotheses that the
SSA struggles with are those that are purely made up of “not” occurrences because
12 Dalal Alrajeh and Paul Gill
what underpins the SSA is the co-occurrence of two variables. To illustrate, h3 is
very difficult to comprehend using the SSA.
Fig. 1. Smallest Space Analysis results for 111 lone actor terrorists.
5 Conclusion and Future Work
The aim of this paper is to examine the applicability of ILP for the purpose of
generating relevant hypothesis about terrorists’ behaviour. The findings reported
in the previous sections collectively show how ILP not only provides the ability
to derive new insights but it is also clearly beneficial in outlining the explanatory
power of rules. Whilst the SSA approach outlines the many diffuse relationships
between a sizeable number of variables, it is very difficult to focus on the most
relevant ones. The clusters that tend to emerge through identification by research
teams tend to be quite subjective - more art than science - and are therefore subject
to potential bias. The SSAs also tend to drag commonly occurring variables into
the centre of the model whilst non-common variables are pushed to the extremes.
The ILP approach, we believe, has the power to delineate which relationships are
highly relevant and more immediately usable.
Our ongoing and future work includes distinguishing between causal and non-
causal factors when generating solutions, prioritizing hypotheses with causal ex-
planations and higher relative significance values, and applications to other crimi-
nological problems such as serial crimes. We plan to conduct further investigation
into prioritizing and optimizing the selection of the body literals when constructing
hypotheses, one possibility is by using the results from SSA or information gain the-
ory . We also intended to investigate the use of probabilistic learning and methods
capable of handling noise and uncertainty in this setting.
A logic-based approach to understanding lone-actor terrorism 13
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