-
1. Introduction
Bipolar Argumentation Frameworks (BAFs) [
1
] extend Abstract Argumentation
frameworks (AFs) [
2
], consisting of arguments and attacks between them, to include supports
between arguments. Formally a BAF is a triple X , R−, R+ consisting of a set X of
arguments, a binary (attack) relation R− on X and a binary (support) relation R+ on X .
In recent years, the standard, qualitative acceptance semantics first proposed for
AFs in [
2
] and for BAFs in [
1
] have been supplemented with quantitative semantics,
e.g. as in [
3,4,5
]. These measure arguments on a gradual scale (usually I = [
0, 1
], as in
this paper) so that arguments can be ranked against one another. Some of these
quantitative semantics are defined for variants of BAFs, e.g. for the Quantitative
Argumentation Debate (QuAD) frameworks of [
7
], corresponding to restricted forms of BAFs,
namely (acyclic) trees, extended with base scores (i.e. intrinsic strengths) for arguments,
the Discontinuity-Free QuAD (DF-QuAD) algorithm of [
6
] is defined. Similarly, the
social models of [
8
] are defined for the Social Abstract Argumentation Frameworks (SAFs)
of [
8
]. SAFs are cyclic or acyclic AFs (without supports) extended with positive and
negative votes for arguments used to determine their social support (i.e., again, intrinsic
strength). These methods define an overall strength for each argument, based on their
intrinsic strength and the strengths of its attackers and, where present, supporters.
In this paper we define a novel quantitative semantics for evaluating the strength of
arguments in BAFs by adapting the DF-QuAD algorithm originally defined for QuAD
frameworks. We also relate the new semantics and other methods in the literature, namely
the gradual valuation semantics [
9
] for BAFs and Social Models for SAFs [
8
].
2. Adapting DF-QuAD
In this section, unless specified otherwise, we will assume as given a generic BAF
X , R−, R+ . With an abuse of terminology, given any a ∈ X , we will use R−(a) to
denote {b ∈ X (b, a) ∈ R−}, referred to as the set of all attackers against a, and R+(a) to
denote {b ∈ X (b, a) ∈ R+}, referred to as the set of all supporters for a. Also, we will
use I∗ i≥0 Ii to denote the set of all possible sequences of elements of I.
We now present our adaptation of the DF-QuAD semantics for BAFs. This
determines the strength of an argument based on the aggregated strength of its attackers
and supporters. The semantics differs from the original DF-QuAD semantics for QuAD
frameworks in that it does not require a base score for arguments. Like the original
DFQuAD semantics for QuAD frameworks though it relies upon a strength aggregation
function, defined exactly as in [
6
] as follows:
Definition 1. The strength aggregation function is defined as s ∶ I∗ → I, where for S =
(v1, . . . , vn) ∈ I∗:
if n = 0 ∶ s (S) = 0
if n = 1 ∶ s (S) = v1
if n = 2 ∶ s (S) = f (v1, v2)
if n > 2 ∶ s (S) = f (s (v1, . . . , vn−1), vn)
with the base function f ∶ I×I → I defined, for v1,v2 ∈ I, as:
f (v1, v2) = v1 + (1 − v1) ⋅ v2 = v1 + v2 − v1 ⋅ v2
Thus, the base function is the building block to handle sequences of strengths of
attackers or supporters by s , proportionally increasing the attacking or supporting
arguments’ strength towards 1, as in [
6
].
Once the strengths of attacking (resp. supporting) arguments against (resp. for) an
argument have been aggregated separately using s , the following mediating function is
used to obtain the overall strength of arguments, using the distance between the strengths
of attacking and supporting arguments:
Definition 2. The mediating function is defined as µ ∶ I × I → I, where for va, vs ∈ I:
µ (va, vs) = 0.5 + 0.5 ⋅ (vs − va)
(1)
The intuition behind this function is that if the strength of the attacking arguments
is larger (resp. smaller) we obtain a score that is below (resp. above) 0.5, using the
difference between the aggregated supporting and attacking strengths to obtain a score that
is proportionally closer to 0 (resp. 1).
The argument’s overall strength is then determined by the following score function:
Definition 3. The score function is defined as SF ∶ X → I where, for any a ∈ X :
SF (a) = µ (s (SEQSF (R−(a))), s (SEQSF (R+(a))))
where if (a1, . . . , an) is an arbitrary permutation of the (n ≥ 0) attackers in R−(a),
SEQSF (R−(a)) = (SF (a1), . . . , SF (an)) (similarly for supporters).
Note that whenever X is finite with X = n, the output of the score function SF for
a ∈ X can be seen as the (appropriate) projection of the output of a function SF ∶ In → In
which takes a ‘valuation’ (v1, . . . , vn) ∈ In of all the arguments in X and maps it to another
‘valuation’. In particular, the ith coordinate vi of (v1, . . . , vn) ∈ In is mapped thus:
If I is a closed interval [a, b] ⊆ R on the real line, then a generalised version of Brouwer’s
fixed-point theorem ensures that SF has a fixed-point, provided SF is continuous [
10
].
Clearly, SF is continuous, because it is obtained by composition of continuous
functions, namely sum, product and projections. Consequently, if X is finite and I is a closed
interval on the real line, in particular if I = [
0, 1
], then SF ∶ X → I has at least one
solution, i.e. the arguments’ overall strengths are well-defined.
To show that there is a unique solution, one could utilize Banach’s fixed-point
theorem, by showing that SF is a contraction mapping, e.g. as in [
8
]. To this end, note that
the Jacobian J of SF is continuous and differentiable, and its derivative’s J ′ matrix
norm J ′ given by the maximum absolute value of its entries (since matrix norms on
Rn are equivalent, we can use the most suitable one) satisfies J ′(v1, . . . , vn) 1 for
any (v1, . . . , vn) ∈ In. The latter inequality needs to be strict in order to apply Banach’s
fixed-point theorem, but we conjecture that this is attainable and that with finite X and
I = [
0, 1
], the arguments’ strength is guaranteed to be unique, i.e. equations defined by
SF have a unique solution. Proving this conjecture is left for future work.
For illustration, Figure 1 shows some BAFs with arguments’ strengths as indicated.
Most of the properties of the DF-QuAD algorithm for QuAD frameworks, given
and proven in [
6
], also hold for our adaptation of the DF-QuAD algorithm for BAFs. In
particular, the following properties hold, mirroring the corresponding properties in [
6
]
(i.e. Proposition 7 and Proposition 9, resp.)
Proposition 1. For any S = (v1, . . . , vn) ∈ I∗ (for n ≥ 0), s (S
(1)) = 1.
In other words, adding an attacker/supporter with maximum strength to the list of
attakers/supporters (resp.) saturates the strength aggregation function.
Proposition 2. For any a ∈ X with s (SEQSF (R−(a))) = va, s (SEQSF (R+(a))) = vs:
SF (a) = 0 iff va = 1 ∧ vs = 0
SF (a) = 1 iff va = 0 ∧ vs = 1
In other words, the extreme values, 0 and 1, can only be achieved as the strength of
an argument if they are somehow already present.
3. Comparison with Other Approaches
QuAD frameworks [
7
] where all arguments have a base score of 0.5, under the DF-QuAD
algorithm of [
6
], can be mapped to BAF frameworks, under the new semantics given in
Section 2. The mapping is defined as follows:
Definition 4. Let Q = A, C, P , R, BS0.5 be a QuAD framework with BS0.5 (A ×
C ∪ P ) × {0.5}.2 The corresponding BAF is defined as B = X , R−, R+ such that X =
A ∪ C ∪ P , R− = R ∩ (C × X ) and R+ = R ∩ (P × X ).
Since R is acyclic, by definition of QuAD framework, the BAF corresponding to a
QuAD framework is guaranteed to be acyclic. Thus, QuAD frameworks can be seen as
restricted types of BAFs.
We now show that the strength of arguments in a QuAD framework by the
DFQuAD semantics in [
6
] is the same as the strength of arguments in the corresponding
BAF obtained by Definition 3. We have already stated that the strength aggregation
function in Definition 1 is borrowed from the DF-QuAD semantics in [
6
]. The notion of
strength in [
6
] is however defined in terms of a combination function (rather than a
mediating function as in our case) defined in [
6
] as c ∶ I × I × I → I, where for v0, va, vs ∈ I:
c(v0, va, vs) = v0 − v0 ⋅ vs − va
c(v0, va, vs) = v0 + (1 − v0) ⋅ vs − va
if va ≥ vs
if va < vs
(2)
(3)
The following proposition shows that the mediating function is equivalent to the
combination function when the base score (v0) is set to 0.5:
Proposition 3. For any va, vs ∈ I, c(0.5, va, vs) = µ (va, vs).
2By definition, A, C and P are pairwise disjoint and R ⊆ (C ∪ P) × (A × C ∪ P) is acyclic, see [
7
].
The next proposition follows immediately, given SF Q defined in [
6
] as SF Q ∶
A ∪ C ∪ P → I where, for any a ∈ A ∪ C ∪ P , SF Q(a) = c(BS(a), s (SEQSFQ (R−(a))),
s (SEQSFQ (R+(a)))) where if (a1, . . . , an) is an arbitrary permutation of the (n ≥ 0)
attackers in R−(a), SEQSFQ (R−(a)) = (SF Q(a1), . . . , SF Q(an)) (similarly for
supporters).
Proposition 4. Given a QuAD framework Q = A, C, P , R, BS0.5 and the corresponding
BAF B = X , R−, R+ , for a ∈ X , let sQ(a) be SF Q(a) in Q and sB(a) be SF (a) in B.
Then for all a ∈ X : sQ(a)=sB(a).
Thus, our new semantics for BAFs can be seen as a specialisation (setting all base
scores to 0.5) as well as a generalisation (to any BAFs, rather than those corresponding
to QuAD frameworks) of the original DF-QuAD semantics in [
6
].
Since BAFs are an extension of AFs [
1
], social models of SAFs [
8
] give the same
output as our score function when the social support resulting from votes amounts to 0.5
(this equates to an approximately equal number of positive and negative votes, depending
on parameter selection). We omit the details of this correspondence for lack of space.
The gradual valuation semantics for BAFs, given in [
9
], is also restricted to acyclic
BAFs. This semantics can be presented as follows, for direct comparison with our
semantics. Let the equivalent strength aggregation function be defined as s ′ ∶ [
−1, 1
]∗ → [0, ∞),
where for S = (v1, . . . , vn) ∈ [
−1, 1
]∗:
s ′(S) =
n vi + 1
i=1
2
1 1
µ ′(va, vs) = 1 + va − 1 + vs
Let the equivalent mediating function be defined as µ ′ ∶ [0, ∞) × [0, ∞) → [
−1, 1
], where
for va, vs ∈ [0, ∞):
Finally, let the equivalent score function be defined as our score function, but using the
equivalent strength aggregation and mediating function, namely as SF ′ ∶ X → I where,
for any a ∈ X :
SF ′(a) = µ ′(s ′(SEQSF′ (R−(a))), s ′(SEQSF′ (R+(a))))
where if (a1, . . . , an) is an arbitrary permutation of the (n ≥ 0) attackers in R−(a),
SEQSF′ (R−(a)) = (SF ′(a1), . . . , SF ′(an)) (similarly for supporters).
For lack of space we omit to prove that this formulation of gradual valuation is
indeed equivalent to the original formulation in [
9
]. Here, we note however that our
semantics differs from this formulation of gradual valuation. Indeed, the equivalent of
Proposition 1 is not held by s ′, because each argument with maximum strength incrementally
increases the aggregated attacking/supporting strength towards infinity, and the
equivalent of Proposition 2 is not held by SF ′, since, then minimum (resp. maximum) value, -1
(resp. 1), can only be achieved if the attacking (resp. supporting) component is ∞ and the
supporting (resp. attacking) component is 0, requiring an infinite sequence of attackers
(resp. supporters) and no supporters (resp. attackers).
4. Conclusion
We have defined a quantitative semantics for evaluating the strength of arguments in
(possibly non-acyclic) BAFs (including both attacks and supports). The semantics is
defined as an adaptation of the DF-QuAD algorithm [
6
] assuming that all arguments
have a base score of 0.5 and, differently from the original algorithm, it can deal with any
BAFs rather than restricted (acyclic) BAFs corresponding to QuAD frameworks [
7
]. The
semantics generalises, by dealing with supports too, the social model semantics of [
8
]
when arguments’ social support is 0.5. The semantics differs from other semantics for
BAFs, and in particular the gradual valuation of [
9
].
We have sketched and conjectured, resp., existence and uniqueness of strength of
arguments obtained by our semantics. We leave formal proofs to future work, alongside
proving other properties of the semantics and relationship to other existing semantics.
The semantics defines the strength of arguments, within the [
0, 1
] interval of the real
numbers, as solutions of non-linear equations, and can be computed, algorithmically, by
any solver for systems of non-linear equations, e.g. via Newton’s method. The definition
of a system supporting the computation is left as future work.