=Paper= {{Paper |id=Vol-1584/paper18 |storemode=property |title=Situations and Evidence for Identity Using Dempster-Shafer Theory |pdfUrl=https://ceur-ws.org/Vol-1584/paper18.pdf |volume=Vol-1584 |authors=William Nick,Yenny Dominguez,Albert Esterline |dblpUrl=https://dblp.org/rec/conf/maics/NickDE16 }} ==Situations and Evidence for Identity Using Dempster-Shafer Theory== https://ceur-ws.org/Vol-1584/paper18.pdf

William Nick et al. MAICS 2016 pp. 81–87

Situations and Evidence for Identity Using Dempster-Shafer Theory

William Nick, Yenny Dominguez and Albert Esterline
Department of Computer Science, North Carolina A&T State University, Greensboro, NC 27411
wmnick@aggies.ncat.edu, ydomingu@aggies.ncat.edu, esterlin@ncat.edu

Abstract The remainder of this paper is organized as follows. The
next section introduces situation theory, and the following
We present a computational framework for identity based on one outlines the Semantic Web standards we use for repre-
Barwise and Devlin’s situation theory. We present an exam-
senting and reasoning about situations and the information
ple with constellations of situations identifying an individual
to create what we call id-situations, where id-actions are per- they contain. There follows a section where we describe
formed, along with supporting situations. We use Semantic how we represent and reason about situations and their in-
Web standards to represent and reason about the situations in formation, drawing on our running example. We then in-
our example. We show how to represent the strength of the troduce the Demptser-Shafer theory of evidence and apply
evidence, within the situations, as a measure of the support it to our running example. The next section outlines how
for judgments reached in the id-situation. To measure evi- we might exploit the structure of a constellation of situa-
dence of an identity from the supporting situations, we use the tions involved in an identification in combining evidence in
Dempster-Shafer theory of evidence. We enhance Dempster- Dempster-Shafer theory. The penultimate section outlines
Shafer theory in two ways to leverage the information avail- another way Dempster-Shafer theory may be applied in situ-
able in a constellation of situations. One way exploits the
ation theory, where a pattern of situations provides the struc-
structure within the situations, and the other way interprets
the information-relationships in terms of argument schemes. ture for an argument scheme. The last section concludes.

Situation Theory
Introduction We follow Devlin’s account of situations and information
We here present our computational framework for identity. (Devlin 1995). Information is represented using infons. An
State of the art in identity is represented by the Superidentity infon is the basic item of information, with the general form
project (Creese et al. 2013)(Hodges, Creese, and Goldsmith << R, a1 , ..., an , l, t, i >>, where R is an n-place relation,
2012), which developed a model in identity that connects el- a1 , ..., an are objects appropriate for the corresponding ar-
ements from both the cyber and physical universes. In their gument places of R, l is a location, t is a temporal location,
terminology, an element of identity has a type, and a charac- and i is the polarity, 0 or 1. A polarity of 1 indicates that
teristic is a multiset of elements of identity of the same type. the objects are thus related in l at t; 0 indicates otherwise.
A superidentity is a set of characteristics. Examples of el- Where s is a situation and an infon, s � is a proposition
ements of identity include real names and email addresses. and may be true or false; if true, s is said to support (
An initial superidentity has a seed identity element and is indeed is information available in s).
enriched by deriving new elements of identity via functions A real situation is a single entity that is part of reality
that transform one or more elements of given types to an el- and supports an indefinite number of infons, while an ab-
ement of another type. For example, an email address may stract situation is a set of infons. An event is essentially a
be transformed to usernames on social network sites. The kind of situation, and an action is a kind of event (involving
enriching continues, creating a directed graph that outlines an agent). We take situations as they relate to identity (id-
the provenance of the elements of identity. situations) to be those that include identity-relevant actions
It became apparent, however, that the elements of iden- (id-actions). We use situation theory to be able to represent
tity and transforms of the Superidentity project do not sup- id-situations and the situations that support them.
port the internal structure we require. For an alternative, we Situation theory arose as part of the development of situ-
turned to situation theory based on Devlins account (Devlin ation semantics by Barwise and his colleagues (Barwise and
1995). When we attribute identity, we want something like Perry 1981). In situation semantics, one identifies an ut-
a legal case. Evidence includes provenance of information, terance situation, in which a speech act is performed, and
records of how procedures were followed, how information a described situation, which the speech act is about. Be-
was communicated, and critical narrative detail. Central to sides supporting information, a situation may carry infor-
our account, a version of Dempster-Shafer theory is used for mation about another situation. This is made possible by
a quantitative account of the impact of evidence. constraints. Some such constraints are natural (as in smoke

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William Nick et al. MAICS 2016 pp. 81–87

means fire), and some are conventional, such as those con-
straints by virtue of which a speech act is about a described
situation.

Running Examples
We present a series of situations involved in identifying an
individual by mugshot and by fingerprint. Our running ex-
ample is shown in Figures 1 and 2 and involves six situa-
tions (within clouds), s1 -s6 . The situations on the right of
each figure (s1 and s2 ) are id-situations that are coordinated
in that they result in identifying the same individual (via
their “name,” actually any identifier unique in the context).
Id-situation s1 has an analyst who matches fingerprints on
file with those on a doorknob. Id-situation s2 has the same
analyst matching the face in a group shot to a face in the
mugshot. The fingerprints on file were produced in s3 , and
the fingerprints on the doorknob were produced in s4 . Situa-
tion s4 is a (spatiotemporal) part of the situation portrayed in
the ellipse in the portrayal of s5 , where a group of people is
Figure 1: Fingerprint Situation
socializing. This situation is in turn part of s5 , where some-
one takes a picture of the group. In situation s6 , a mugshot
of the person of interest is produced. It is used in s2 to pick
out the person in question in the group photo. We thus have are URIs but not vice-versa.)A URI reference (URIref) is it-
two id-cases: the fingerprint case, s1 -s3 -s4 , and the mugshot self a URI with an optional fragment identifier at the end.
case, s2 -s4 -s5 . URIrefs are written typically as qnames, which are in the
The dashed lines between situations shown on the left and form of prefix:lp, where the namespace prefix is a URI.
id-situations connect things produced (left) and used (right). A blank node (bnode) is a resource that is not identified by
In all cases except where the objects produced are them- a URIref.
selves used in the id-situation, there are additional copy- To represent RDF statements in a machine readable way,
ing or rendering situations not shown in in the figures. In the W3C has defined several serializations. One of these se-
a sense, we have one id-situation made of two coordinated rializations is the Notation 3 (N3) serialization. Triples in
id-situations. the N3 serialization are expressed as each of the three com-
We use the empty prefix : for the namespace in which ponents separated by whitespace. When a subject is shared
we define the basic classes and properties. An instance s of amongst triple, we can abbreviate this by having the sub-
class :Situation generally appears as subject in triples iden- ject listed once and separated predicate-object pairs by semi-
tifying the time and location of the situation in terms of sub- colons:
classes of classes defined in the WGS84 Geo Positioning
vocabulary. We thus do not represent time and spatial lo- subject predicate1 object1;
cation in an infon but rather just assume that all infons in a predicate2 object2.
given situation share a common time and place. We move RDFS allows for classes and properties to be defined us-
on to using the Semantic Web standards to implement our ing RDF triples. We state that individual x is an instance
running examples as per situation theory. of class C with the triple x rdf:type C. These individ-
uals could be denoted by a URIref or a bnode. N3 allows
Semantic Web “a” to be used as an abbreviation for “rdf:type”. A class
The Semantic Web is based on two World Wide Web Con- may be a subclasse of other classes, and a property may be
sortium (W3C) standards: 1) the resource description frame- a subproperty of other properties. If p is a subproperty of q,
work (RDF) and 2) RDF schema (RDFS). These standards then x p y implies x q y. If we have x p y, then x is
are enhanced by the much more expressive OWL (Web on- an instance of the class that is the domain of p, and y is an
tology language) standard. RDF is a W3C recommendation instance of its range.
that provides a data model for annotations in the Semantic SPARQL is a SQL-like query language for triple stores
Web. An RDF statement (triple) is of the form subject pred- where a variable is a sequence of alphanumeric characters
icate object. RDF allows users to annotate web resources proceeded by ‘?’, a WHERE clause is a sequence of triples
in terms of named properties. The values of these named each of which might have a variable for its subject, object,
properties can be URIrefs of web resources or literals. Re- or both. SPARQL reports only the variables that appear in
sources that are annotated by RDF are named by uniform the SELECT clause.
resource identifiers (URIs). A URL is a string that identifies SWRL is a rule language for the Semantic Web. SWRL
a resource on the web. A URI has the same structure as a rules are in the form head → body where head is the
URL but need not identify a resource on the web. (URLs antecedent and body is the consequence.

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on the doorknob. Situation s3 has one important infon, i3,
an instance of :TakeFpInfon, that a given officer takes
the fingerprint of our suspect.
The photo id-case also involves three situations: s2 (id-
situation), s5 (taking the forensic photo), and s6 (taking
the mug shot). s2 is analogous to s1 but lacks the ana-
logue of the fingerprint on the doorknob. s6 is analogous
to s3 (taking the fingerprint on file). s5 is only roughly
analogous to s4. One of s5’s infons, i5, is an instance of
:ForensicPicInfon and is the subject of triples identi-
fying the photographer, the camera, the photo produced, and
the situation, s5a, caught on camera. One infon that s5a
has is that our suspect is touching the doorknob; it also has
sit:s5a :inSituation group:5342;
This says that this group is in the situation but does not identify
any information associated with the group, yet infon i5 includes
the information that s5a is the situation pictured. We also have
(where insys:201 is our suspect)
group:5342 a foaf:Group;
foaf:member insys:201, insys:563.
There is thereby in i5 the information that insys:201
is pictured in the photo produced; we do not necessar-
ily have the information that insys:201 is a member of
Figure 2: Mugshot Situation group:5342. And we have (where foaf:depicts is
an information relation)
fshot:812 a biom:GroupImage;
RDF/OWL/SWRL Representation of Examples # The group photo (s2, s5)
That a given situation s has an infon i (an instance of class foaf:depicts sit:s5a .
:Infon) is expressed as s :hasInfon i. Infon i it- There is also a part-whole (mereological) relation be-
self has a polarity (property :hasPolarity). The vari- tween s4 and s5: s4, where the suspect touches the door-
ous relations are captured by various subclasses of :Infon. knob, is a proper part of s5a, the situation caught on film in
If R is a relation with roles r1 , r2 , ..., rn , then we define a situation s5 .
subclass :RInfon of :Infon and properties r1, r2, Assuming all our information is available (possibly dis-
..., rn with domain :RInfon. This avoids RDF’s re- tributed) on the Web, we can issue SPARQL queries that
striction of relations to binary relations (“properties”) since navigate across situations connected by, say, shared individ-
any instance of :RInfon may be a subject of any number uals.
of triples with one of r1, r2, ..., rn as the property. We have identified a few important infons for each real
The fingerprint id-case involves three situations: s1 (id- situation s1 -s6 , but each supports an unbounded number of
situation), s3 (taking the fingerprint on file), s4 (taking the infons. We need abstract situations as types to classify real
forensic fingerprint). We discuss only s1 in detail. It has situations and constellations in a way conducive to identi-
three important infons: i1, i1a, and i14. Like all our fication. For classifying, we use SWRL rules. Where C is
infons, they have positive polarity; henceforth we assume a class and x is an individual, C(x) is true iff the triple x
this. We discuss only i1 in detail. It is an instance of rdf:type C holds. Where p is a property, x is a URIref
:AnalystMatchingFpInfon, information that an an- or bnode, and y is a URIref, a bnode, or a literal, p(x,
alyst is matching the forensic fingerprint and the fingerprint y) is true iff the triple x p y holds. If certain conditions
on file (no suggestion of objective similarity). Three prop- hold of a situation ‘?s’ (note that SWRL variable names
erties are recorded for it: :fpObserved, whose value begin with ‘?’), we classify it as some subclass of class
is the URIref of the forensic fingerprint, :fpRecorded, :Situation. Our classifying SWRL rules, then, have the
whose value is the URIref of the fingerprint on file, and form
:fpAnalyst is for the officer making the match. In N3,
this is (i1, like all our infons, is represented by a bnode.) Situation(?s), ... -> SituationSubClass(?s)
_:i1 a :AnalystMatchingFpInfon; The conditions that fill in the ellipsis relate to the infons
:fpAnalyst officer:117; that ?s has, one or more sequences like
:fpObserved forensicfp:822; hasInfon(?s, ?i), ...,
:fpRecorded fpfile:496; hasPolarity(?i, ?po),
:hasPolairty :PositivePolarity. polarityValue(?pol, ?val), equal(?val, 1)
Infon i1a is an instance of :SimilarFpInfon, that The ellipsis here is filled in with specifics on the roles of
the forensic fingerprint and the one on file have a similarity the relation represented by the infon. The sequence of atoms
measure of 0.94 according to a certain procedure. Infon i14 after the ellipsis forces positive infon polarity. All our infons
is an instance of :OnInfon, that the forensic fingerprint is have positive polarity, so we ignore this.

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The case-type with the photos involves three situation Available evidence (e.g., similarity measures) provides
types, identified with the classes :Mug (a mug shot is some degree of support (“mass”), from 0.0 to 1.0, for sub-
taken), :Pic (a forensic picture is taken), and :PicId (id- sets of W ; those subsets with non-zero mass are called focal
situation type). A situation is of type :Mug if it has an in- elements. The sum of the mass for all subsets of W is 1.0.
fon of type :TakeMugshotInfon involving a recorded Where U ⊆ W , the belief that U holds, Bel(U ), is the sum
mugshot, a subject, and an administrative officer responsi- of the support (mass) on subsets of U , a number in [0,1].
ble for the mugshot. A situation ?s is of type :Pic if it Where m(.) is the mass function, m(U ) is the probability
has a :ForensicPicInfon involving an officer taking of observing U , so the definition of the belief function in
the photo, a group image that is the photo, and a situation terms of the mass function is Belm (U ) = ∑U ∗ ⊆U m(U ∗ ).
captured by the photo and that includes the group depicted For our example, suppose that the similarity measures for
in the photo. This describes a situation that references an- the singletons {Fred}, {Bill}, {Sue}, and {Mary} to the fin-
other. A :Pic situation, then, is like an utterance situation, gerprint on the doorknob are, respectively, 0.4, 0.075, 0.075,
best compared to a situation where the “uttering” is writing, and 0.0. In addition, there is some evidence, mass 0.05, of
although speech and writing abstract away information. the fingerprint belonging to {Sue, Bill} (i.e., to Sue or Bill
The case-type with fingerprints also involves three situa- without distinction). And perhaps someone other than the
tion types, identified with the classes :FpFile (a finger- people mentioned left the fingerprint on the doorknob. The
print is recorded), :Touch (a forensic fingerprint is left), evidence for this chance has about half the strength as the
and :FpId (id-situation type). evidence for {Fred}; as a singleton set, it receives mass 0.2.
Recall that an id-situation together with its supporting sit- We suppose that there is some interest in whether the person
uations is an id-case. We form id-case types, abstract ver- is either male or female. Since there is no reason to imply the
sions of id-cases. Generally, an id-case type glues together unknown fingerprint belongs to a male rather than a female
several situation types, which requires (for connections) ex- or vice versa, we split this mass between a fictional female,
posing more information in the situations than is exposed for Nulla, and a fictional male, Nullus. The sum of the masses
the situation types. We define SWRL rules to classify cases so far is 0.8. The remaining 0.2 covers all ways the fin-
as subclasses of a generic :Case class. gerprint could have got on the doorknob, not only by those
In the envisioned scenario, the two id-cases are coordi- mentioned, but perhaps left before or after the situation con-
nated since the filed fingerprint and mugshot of a single sus- sidered.
pect are used to establish his presence in a gathering. We Corresponding to the belief function is the plausibility
introduce symmetric property :coordinatedIdCase function. The plausibility that U holds, P laus(U ), is the
whose domain and range are :IdCase. We have a SWRL sum of the probabilities of the evidence compatible with the
rule for determining that an instance of MugIdCase and an world being in U : P lausm (U ) = ∑U ∗ s.t.U ∗ ∩U ≠� m(U ∗ ).
instance of FingerpIdCase are coordinated by checking For U ⊆ W , Bel(U ) ≤ P laus(U ). Note that, where Ū
not only that the label on the mugshot is the same as that is the complement of U , P laus(U ) = 1 − Bel(Ū ) and
on the fingerprint on file but also that we have one and the Bel(U ) = 1 − P laus(Ū ).
same id-situation. The criterion for identity of situations is Table 1 shows the values of the Dempster-Shafer func-
beyond the scope of this paper. tions for each focal element for s1 . All represents the entire
frame of discernment; its mass was not assigned elsewhere.
We show the values of the belief and plausibility functions
Dempster-Shafer Theory of Evidence only for focal elements; there are other subsets of W that
We want a measure of how the evidence supports the judg- have non-zero belief and plausibility.
ment in an id-situation. It should reflect the structure of an
id-case and fuse belief constraints from different sources. In Focal element Mass Belief Plausibility
our example, s1 and s2 are essentially a single utterance sit- {Fred} 0.400 0.400 0.600
uation (the identity judgment), and the situation in the photo All 0.200 1.000 1.000
in s5 is the described situation. Imagine that, in s1 , the an- {Sue} 0.075 0.075 0.325
alyst has access to fingerprints for several likely suspects, {Mary} 0.000 0.000 0.200
each associated with a supporting situation in which a finger- {Bill} 0.075 0.075 0.325
print was recorded. The RDF for s1 includes a measure of {Nullus} 0.100 0.100 0.300
how similar the fingerprint on file is to the fingerprint from {Bill,Sue} 0.050 0.200 0.400
the scene; in the expanded view, it includes such measures {Nulla} 0.100 0.100 0.300
for all available fingerprints.
We adapt the Dempster-Shafer theory of evidence Table 1: Fingerprint Mass, Belief, and Plausibility
(Halpern 2003). The frame of discernment (the set of pos-
sible values), W , includes here people who might have left For s2 , focal elements are subsets of W with non-zero
the fingerprint or have their mugshot considered. In s1 , we probability of containing the person whose mugshot matches
have a measure of how well the fingerprint on file matches the picture of the culprit in the forensic picture. Suppose
the fingerprint on the scene. We also have similarity mea- that the mass for the focal elements are Fred: 0.35; All:
sures for other people who might have left the fingerprint on 0.2; {Sue}: 0.0; {Mary}: 0.05; {Bill}: 0.05; {Mary,Nulla}:
the door in Figure 1, say, Fred, Bill, Sue, and Mary. 0.15; {Nullus}: 0.100; and {Nulla}: 0.100. Given mass

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William Nick et al. MAICS 2016 pp. 81–87

functions m1 (e.g., for the fingerprints) and m2 (e.g., for adding information relevant to the acceptability of the fin-
the mugshots) defined on some frame W , we use Demp- gerprint file. This might reduce the belief due to matching
ster’s Rule of Combination to construct a new mass func- the fingerprint. Instead of something like Mary as a frame
tion m1 ⊕ m2 that fuses the belief constraints of m1 and m2 element, we have things like (Mary, off23, 11/25/2007) for
(e.g., combining the evidence from both the fingerprints and a fingerprint purportedly of Mary; the frame is effectively
mugshots): a product space, Suspects × AdministeringOf f icers ×
(m1 ⊕ m2 )(h) = ∑U1 ,U2 s.t.U1 ∩U2 =U m1 (U )m2 (U )�c Dates. We effectively expand the id-situation to include the
supporting situations.
where normalizing constant c is the sum of the products Issues arise with respect to the structure of this product
m1 (U1 ) ⊕ m2 (U2 ) of all overlapping pairs U1 , U2 : space and how the mass is aggregated to contribute to ev-
c = ∑U1 ,U2 s.t.U1 ∩U2 ≠� m1 (U1 )m2 (U2 ) idence in the id-situation (where a judgment is made). A
Table 2 shows the shows the values of the Dempster-Shafer focal element is a subset of this product space that is as-
functions for each focal element of m1 ⊕ m2 . signed a non-zero mass. We can consider something like
marginal distributions: for a given (suspect, administering-
Focal element Mass Belief Plausibility officer) pair, we add up the mass across all the dates for that
{Fred} 0.536 0.536 0.610 pair. Going further, for a given suspect, we add up all the
mass for the triples that involve that suspect.
All 0.074 1.000 1.000
The certainty on the constraint from the fingerprint-
{Sue} 0.028 0.028 0.120
producing situation to the existence of the fingerprint file in
{Mary} 0.018 0.018 0.148 the id-situation is a measure of the general acceptability of
{Bill} 0.058 0.058 0.150 introducing a fingerprint file into an investigation. What the
{Mary,Nulla} 0.055 0.194 0.268 mass of the supporting situation is taken to be depends on
{Nullus} 0.092 0.092 0.166 how the evidence is being used. If the investigation engen-
{Bill,Sue} 0.018 0.104 0.178 ders suspicion of a given administering officer, then, within
{Nulla} 0.120 0.120 0.249 the supporting situation, the mass for identifying the suspect
would be reduced. These considerations revolve around the
Table 2: Combined for fingerprint and mugshot relation between the id-situation and a supporting situation
as well as the nature of the supporting situation. These are
essentially ontological considerations.
Dempster-Shafer Theory & Situation Theory
To reflect the structure of an id-case in our account of evi- Dempster-Shafer Argument Schemes
dence, we consider the work by Lalmas et al. (Lalmas and How evidence regarding supporting situations is incorpo-
Van Rijsbergen 1994), who combine situation theory and rated into an overall evaluation can perhaps be answered in
Dempster-Shafer theory for an account of information re- a nonontological manner following the work by Tang et al.
trieval. They consider constraints as conditionals, → , in combining argumentation with an explicit representation
where and are types, with a measure of certainty, of evidence (Tang et al. 2013) (see also (Tang et al. 2012)).
cert( → ). If cert( → ) < 1, then → leads They introduce a logical language L with the usual truth-
from one situation s (say, where there is smoke) to another, functional connectives. Atomic propositions are constructed
s′ (where there is fire), which may be just an extension of from a finite set of predicate symbols and a finite set of indi-
s in that it supports all the infons supported by s. They vidual constants, with no function symbols, so there are only
require that, for type , where C is the set of constraints, finitely many possible ground terms. Individual variables
∑ → ∈C cert( → ) = 1 occur only in rules (for generality, with uniform substitution
One of our constraints is that there must be an appropriate across premises and conclusion). The frame of discernment,
supporting situation in which the fingerprint file was pro- , is the set of possible truth assignments to all the (ground)
duced. We read this, as it were, backwards or teleologically: atomic propositions: if there are n such propositions, there
if there is a situation in which a fingerprint file is produced, are 2n elements of ( i.e., rows in the truth table). The in-
then there is a situation in which it is used. Our frame of terpretation of proposition (atomic or not), I( ), is a sub-
discernment W is a finite number of fingerprint files. The set of . Propositions , ∈ L are logically equivalent iff
masses in the singletons are now on the constraints (or sets I( ) = I( ). Where true and f alse are the obvious con-
of constraints). Where → leads from situation s to s′ , stants, I(true) = and I(f alse) = �. An inference rule
Lalmas et al. define the mass of s′ in terms of the mass of s for L is of the form:
and the certainty of → : mi+1 (s′ ) = cert( → )mi (s).
s′ itself may actually be a set of alternative situations the = p1 ,...,p
c
m

sum of whose masses equals cert( → )mi (s). So we where p1 , ..., pm ; c ∈ L. The pi are the set of premises of the
invoke the notion of a frame of discernment W ′ being the rule, and c is its conclusion.
refinement of a frame W ; essentially W ′ is a finer partition It is straightforward to go from a frame of discern-
of the universe of possibilities than W . ment where elements are structures on individuals to a
When we impose a constraint that leads from a frame of discernment where elements are logical propo-
fingerprint-producing situation, the frame is refined by sitions over a finite set of predicate symbols. To take

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William Nick et al. MAICS 2016 pp. 81–87

our example (M ary, of f 23, 11�25�2007) ∈ Suspects × Person Fred Bill Sue Nulla
AdministeringOf f icers × Dates, assume we have one- Thief? Yes 0.32 0.06 0.06 0.08
place predicates suspect(x), adminOf f icer(x), and No 0.08 0.015 0.015 0.04
date(x), meaning, respectively, that x is the suspect
(whose fingerprint is taken), that x is the administrat- Person Nullus Bill or Sue All
ing officer, and that x is the date (when the finger- Thief? Yes 0.08 0.04 0.16
print was taken). Assume also that we have individual No 0.02 0.01 0.04
constants Mary, off23, and 11/25/2007 with the obvious
denotations. Then our example triple translates to the Table 3: Result of Applying our Rule with evidence E5
conjunction suspect(M ary) ∧ adminOf f icer(of f 23) ∧
date(11�25�2007). Set-theoretical operators correspond in
obvious ways to truth-functional operators, which again re- P laus(f print(Bill) ∧ thief (Bill)) = 0.26. This example
late to set-theoretical operators on the interpretations of is particularly simple, and we intentionally avoided combin-
propositions. ing evidence to indicate how the rules are applied.
We have a set of formulae �h, E� where �h, E� is an Tang et al. (Tang et al. 2013) consider the several ways of
evidence argument that has h ∈ L associated with support- combining evidence that have been suggested in the context
ing evidence E for which there is a mass function, E = {e1 ∶ of Dempster-Shafer theory, considering them all to fit into
m1 , ..., en ∶ mn } such that ∑ni=0 mi = 1.0. (Note that here the general pattern of:
� A rule pattern in = p1 ,...,p
a mass function is being associated with a single proposi-
: m
tion.) To write a mass function value in isolation, we write c
m(E, ei ) . Given evidence argument �h, e�, the belief b(h), � A Dempster-Shafer argument scheme specifying
disbelief d(h), and uncertainty u(h) of h are defined as � the pattern of the evidence of the premises:
• b(h) = ∑I(ei )⊆I(h) m(E, ei ) = the sum of the mass of all �h1 , E1 �...�hn , En �
the focal elements in E that are part of the evidence for h. � optional evidence for rule applicability E
• d(h) = ∑I(ei )∩I(h)=� m(E, ei ) = the sum of all the mass � an an associated conclusion evidence derivation pro-
for all the focal elements that are evidence for ¬h. cess: we compute the evidence for the conclusion from
• u(h) = 1 − b(h) − d(h) = the sum of the mass of the the evidence for the premises possibly including the
formulae that imply neither h nor ¬h. rule evidence
We can define the plausibility of h as 1 − d(h). We also When we go to apply an argument scheme, we ask certain
have a set of rules { , E} where rule is associated with critical questions. Only if the answers to all these questions
evidence E. are affirmative are we entitle to apply the scheme. Each
For an example of the use of a rule, suppose that scheme is associated with a particular rule for combining
the proposition in question is that the fingerprint is evidence. We have seen the oldest and most common rule:
Bill’s, f print(Bill), and suppose that the associated ev- Dempster’s rule. Another common rule is Yager’s rule (see
idence is as follows (which duplicates the mass func- (Curley 2007) for an intuitive comparison with Dempster’s
tion used in the example in the above example) E1 = rule), which treats conflicting evidence as uncertainty. See
{f print(F red) ∶ 0.4, f print(Sue) ∶ 0.075, f print(Bill) ∶ (Sentz and Ferson 2002) for a systematic presentation of var-
0.075, f print(N ullus) ∶ 0.1, f print(N ulla) ∶ 0.1, ious rules for combining evidence.
f print(Bill) ∨ f print(Sue) ∶ 0.05, f print(All) ∶ 0.2} What we are interested in, however, is how to go from in-
Suppose also that we have the following rule with evi- formation produced in supporting situations to its use as ev-
dence idence in an id-situation. (In contrast, the rule above, with-
E5 = {f print(X) ∧ thief (X) ∶ 0.8, f print(X) ∧ out combination, used the results of id-actions as evidence
¬thief (X) ∶ 0.2} for various actors being thieves.) This is usually a combina-
Note that the proposition constituting the premise is carried tion problem. A closer look, however, reveals that the lan-
down to be conjoined with the stated conclusion; this is to guage used in supporting situation s3 is different from the
specialize the conclusion to the individual to which variable language used in the id-situation, s1 . One way they differ is
X is bound. that s1 is an utterance situation while s3 is a described situa-
It is more informative to combine the evidence E1 as a tion. In terms of vocabulary, s3 talks about an administering
whole and the evidence E5 , where we take products, in- officer, the time and place the fingerprint is taken, and the
stantiating X to the individual constant in the correspond- method used. And s1 talks about the fingerprint from the
ing element of E1 (treating f print(Bill)f print(Sue) scene, matching the two fingerprints, and the time and place
as f print(Bill � Sue), where Bill � Sue is a com- the matching is done. Both situations talk about the finger-
posite object). The result is shown in Table 3, where print produced in s3 and used in s1 and the person thereby
a number in a cell is the mass value of f print(X) ∧ identified. Considering the questions Tang et al. pose, the
thief (X) or f print(X) ∧ ¬thief (X), depending on the appropriate rule here is Zhang’s center combination rule,
row, where the value of X is indicated in the column. which is based on two frames of discernment S and T from
Note that the sum of all values is 1.0. We have, for two disjoint sublanguage LS and LT of L. It assumes that
example Bel(f print(Bill) ∧ thief (Bill)) = 0.06 and we are concerned with the truth of sentences in LT but we

86
William Nick et al. MAICS 2016 pp. 81–87

only have evidence expressed in LS and in LS ∈ LT . For work, then, besides including enhancements to the imple-
AT ∪ LT , we are given two pieces of evidence, E1 in LS mentation, will attempt to reconcile these two approaches to
and E2 in LS ∪ LT . This scheme can be used where ques- how supporting situations contribute to the evidence for a
tion 1 for Demster’s rule is answered “no” since the evidence judgment.
does not directly support the conclusion of interest because
of the change in language. Zhang’s rule is especially useful References
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On the other hand, capturing the constraints and refining
the frame of discernment would have the benefit of associat-
ing evidence with the inherent structure of the case. Future