# Abstract Weighted Based Gradual Semantics in Argumentation Theory

Assaf Libman, Nir Oren
University of Aberdeen
Scotland
{a.libman<EMAIL_ADDRESS>
&Bruno Yun
Univ Lyon, UCBL, CNRS, INSA Lyon, LIRIS
UMR5205, F-69622 Villeurbanne
France
<EMAIL_ADDRESS>

###### Abstract

Weighted gradual semantics provide an acceptability degree to each argument
representing the strength of the argument, computed based on factors including
background evidence for the argument, and taking into account interactions
between this argument and others.

We introduce four important problems linking gradual semantics and
acceptability degrees. First, we reexamine the inverse problem, seeking to
identify the argument weights of the argumentation framework which lead to a
specific final acceptability degree. Second, we ask whether the function
mapping between argument weights and acceptability degrees is injective or a
homeomorphism onto its image. Third, we ask whether argument weights can be
found when preferences, rather than acceptability degrees for arguments are
considered. Fourth, we consider the topology of the space of valid
acceptability degrees, asking whether “gaps” exist in this space. While
different gradual semantics have been proposed in the literature, in this
paper, we identify a large family of weighted gradual semantics, called
abstract weighted based gradual semantics. These generalise many of the
existing semantics while maintaining desirable properties such as convergence
to a unique fixed point. We also show that a sub-family of the weighted
gradual semantics, called abstract weighted $(L^{p},\lambda,\mu,A)$-based
gradual semantics and which include well-known semantics, solve all four of
the aforementioned problems.

_K_ eywords Argumentation $\cdot$ Gradual semantics $\cdot$ Inverse problems

##  1 Introduction

In the context of Dung’s abstract argumentation [1], we consider a set of
abstract arguments and a binary attack relation between them, encoding these
as a directed graph. Argumentation semantics then identify which sets of
arguments are justified together by considering inter-argument interactions
[2, 3, 4, 5]. Within the argumentation community, there has been increasing
interest in so-called _ranking-based_ semantics [6, 7, 8, 9]. These aim to
identify a ranking over the arguments, with higher ranked arguments considered
more justified (i.e., “less attacked”) than arguments ranked lower (i.e.,
“more attacked”). One approach to creating such a ranking, called gradual
semantics [10, 11, 12], involves associating a numerical _acceptability
degrees_ to all arguments within the system, with the final ranking computed
according to the numeric ordering. Furthermore, some ranking-based semantics
compute the acceptability degree of an argument based not only on the topology
of the argumentation framework, but also based on some _initial weight_
assigned to each argument. These _weighted gradual semantics_ , exemplified by
the weighted _max-based_ , _card-based_ and _h-categorizer_ semantics, are
widely studied and shown to have various desirable properties [13, 14, 10].

While a specific weighted gradual semantics takes an argumentation system and
initial weights as input, and outputs the arguments’ acceptability degrees,
the _inverse problem_ seeks to identify the initial weights for a given
semantics, argumentation system and acceptability degrees [14]. Previous
research used an iterative numeric technique to identify such initial weights
for the three weighted gradual semantics mentioned above. However, no
guarantee was made that such a solution always exists. This paper continues
investigation into the inverse problem and in doing so obtains some critical
insights. Namely, we ask ourselves the questions: Can we always find a
solution to the inverse problem for the three weighted gradual semantics? If
yes, can we generalise this result to a more general family of gradual
semantics and how is this family caracterised? For any preference ordering on
arguments (rather than simply numerical acceptability degrees), can we always
find some weights on arguments so that the degree obtained will follow the
preference ordering? Our work makes the following contributions.

  * •

We establish a general family of “abstract weighted based gradual semantics”.
In doing so, we generalise the result of Pu et al. [15] to this large, general
family of semantics. This result is used across the remainder of the paper.

  * •

We introduce a sub-family of abstract weighted based gradual semantics, called
abstract weighted $(L^{p},\lambda,\mu,A)$-based gradual semantics, which
includes, among others, the three weighted gradual semantics mentioned above.

  * •

We show that the inverse problem is solved for any semantics in this sub-
family.

  * •

We show that any preference ordering between the arguments is realised by some
acceptability degree in all the semantics in this sub-family.

  * •

We describe some topological properties of the acceptability degree space of
semantics in this sub-family.

The importance of this sub-family of abstract weighted
$(L^{p},\lambda,\mu,A)$-based gradual semantics goes beyond giving a uniform
treatment of the three semantics above. It allows us, among other things, to
define new gradual semantics for which the above inverse-type problems are
solved. In turn, inverse problems can be used — for example — to automatically
elicit and identify users’ preferences based on the acceptability degrees over
arguments, and as part of an argument strategy to help decide which arguments
to advance based on the implicit weights attached to arguments by other
dialogue participants. Such applications are not however the focus of the
current paper, and are left for future work. Instead, we focus on the formal
underpinnings of inverse problems and gradual semantics.

While inverse-type problems for weighted gradual semantics are new, we also
want to highlight that there has been some recent research in this area. Skiba
et al. [16] have studied whether, given a ranking and a ranking-based
semantics, we can find an argumentation framework (without weights) such that
the selected ranking-based semantics induces the ranking when applied to the
framework. Kido and Liao [17] have studied whether, given noisy sets of
acceptable arguments, we can find attack relations that explain the sets well
in terms of acceptability semantics. This is similar to the work of Oren and
Yun [18], where the authors identify the complexity class for the problem of
identifying a set of attacks that will yield the desired acceptability degrees
for a set of weighted arguments with respect to three weighted gradual
semantics. Mailly [19] also studied realization problems for extension
semantics but made used of auxiliary arguments, i.e., given a set of
extensions $S$ and using $k$ auxiliary arguments, can we find $m$
argumentation frameworks such that the union of their extensions is exactly
equal to $S$? The existing body of research demonstrates the growing interest
and diverse approaches in exploring various aspects of inverse problems in the
argumentation domain.

The remainder of this paper is structured as follows. In Section 2, we briefly
recap the background notions and formalism needed in this paper. In Section 3,
we introduce the notion of scoring schemes and rephrase weighted gradual
semantics as collections of functions mapping adjacency matrices to scoring
schemes. Then, we formalise four inverse-type problems in terms of scoring
schemes, define the family of abstract weighted based gradual semantics, and
show that they always converge to a unique fixed point. In Section 4, we
provide conditions for based scoring schemes to solve the four inverse-type
problems. This leads us to define the sub-family of abstract weighted
$(L^{p},\lambda,\mu,A)$-based gradual semantics which always solve the four
problems in Section 5. In Section 6, we show multiple examples of concrete
semantics in this class and introduce new semantics. We summarize and conclude
our work in Section 7.

##  2 Background

This section introduces the necessary argumentation notions used in the rest
of the paper. Our departure point in this paper is the three weighted gradual
semantics described in [13]. As input, these semantics take in a _weighted
argumentation framework_ [20, 21, 22]. A weighted argumentation framework is a
Dung’s abstract framework (with arguments and binary attacks) augmented with a
weighting function which assigns a number between 0 and 1 to each argument.

######  Definition 2.1 (WAF).

A weighted argumentation framework (WAF) is a triple
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$ where $\mathcal{A}$ is a
finite set of arguments; $\mathcal{D}\subseteq\mathcal{A}\times\mathcal{A}$ is
a binary attack relation; and $w:\mathcal{A}\to[0,1]$ is a weighting function
assigning an initial weight to each argument. The underlying argumentation
framework (of $\mathcal{F}$) is
$\mathcal{G}=\langle\mathcal{A},\mathcal{D}\rangle$. The set of attackers of
an argument $a\in\mathcal{A}$ is denoted
$\mathtt{Att}(a)=\\{b\in\mathcal{A}|(b,a)\in\mathcal{D}\\}$.

Upon ordering $\mathcal{A}=\\{a_{1},\dots,a_{n}\\}$, the adjacency matrix of
$\mathcal{G}$ is the $n\times n$-matrix $A$ with $A_{i,j}=1$ if
$(j,i)\in\mathcal{D}$ and $A_{i,j}=0$ otherwise.

A gradual semantics is a function that takes as input a weighted argumentation
framework and produces a scoring function. Note that the latter function is
sometimes called a weighting [13], but we use our terminology to distinguish
it from $w$.

######  Definition 2.2 (Gradual Semantics).

A gradual semantics $\Sigma$ is a function that for each weighted
argumentation framework $\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$,
associates a scoring function $\Sigma^{\mathcal{F}}:\mathcal{A}\to[0,1]$,
whose values are referred to as the acceptability degrees of the arguments.

Due to various desirable properties, the argumentation literature focuses on
three weighted gradual semantics (see for example [13]), which are specific
instantiations of the gradual semantics of Definition 2.2 above. We thus
introduce these three semantics as concrete examples of gradual semantics.

######  Example 2.3.

The weighted max-based gradual semantics $\Sigma_{\mathbb{MB}}$. Given
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$, the acceptability
degree of an argument $a\in\mathcal{A}$ is defined by

$\Sigma_{\mathbb{MB}}^{\mathcal{F}}(a)=\lim_{k\to\infty}\operatorname{MB}_{k}(a)$

for the sequence of functions $\operatorname{MB}_{k}\colon\mathcal{A}\to[0,1]$
defined recursively by setting $\operatorname{MB}_{0}(a)=w(a)$ for all
$a\in\mathcal{A}$ and

$\operatorname{MB}_{k+1}(a)=\frac{w(a)}{1+\max\limits_{b\in\mathtt{Att}(a)}\operatorname{MB}_{k}(b)}$

######  Example 2.4.

The weighted card-based gradual semantics $\Sigma_{\mathbb{CB}}$. Given
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$, the acceptability
degree of an argument $a\in\mathcal{A}$ is

$\Sigma^{\mathcal{F}}_{\mathbb{CB}}(a)=\lim_{k\to\infty}\operatorname{CB}_{k}(a)$

for the sequence of functions $\operatorname{CB}_{k}\colon\mathcal{A}\to[0,1]$
defined recursively by $\operatorname{CB}_{0}(a)=w(a)$ for all
$a\in\mathcal{A}$ and

$\operatorname{CB}_{k+1}(a)=\frac{w(a)}{1+|\mathtt{Att}^{*}(a)|+\tfrac{1}{|\mathtt{Att}^{*}(a)|}\cdot\sum\limits_{b\in\mathtt{Att}^{*}(a)}\operatorname{CB}_{k}(b)}$

Here, $\mathtt{Att}^{*}(a)=\\{b\in\mathtt{Att}(a)\mid w(b)>0\\}$ and if
$\mathtt{Att}^{*}(a)=\emptyset$ the term involving division by
$|\mathtt{Att}^{*}(a)|$ vanishes.

######  Example 2.5.

The weighted h-categorizer gradual semantics $\Sigma_{\mathbb{HC}}$. Given
$\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$, the acceptability
degree of an argument $a\in\mathcal{A}$ is

$\Sigma_{\mathbb{HC}}^{\mathcal{F}}(a)=\lim_{k\to\infty}\operatorname{HC}_{k}(a)$

for the sequence of functions $\operatorname{HC}_{k}\colon\mathcal{A}\to[0,1]$
defined recursively by $\operatorname{HC}_{0}(a)=w(a)$ and

$\operatorname{HC}_{k+1}(a)=\frac{w(a)}{1+\sum\limits_{b\in\mathtt{Att}(a)}\operatorname{HC}_{k}(b)}$

A crucial issue to address with these definitions is the need to prove that
the limits exist and yield scoring functions. The convergence has been shown
to exist (and be unique) for the three semantics mentioned above by Amgoud et
al. (see in particular Theorems 7, 12, and 17 in [13]).

Using gradual semantics, one can obtain a ranking on arguments by using their
acceptability degrees. With the three gradual semantics above, it is the case
that $a$ is at least as preferred as $b$ iff $a$’s acceptability degree is at
least as high as $b$’s. We use the following notation to describe preferences
over arguments, where $a\trianglerighteq b$ denotes that $a$ is at least as
preferred as $b$, $a\simeq b$ iff $a\trianglerighteq b\wedge b\trianglerighteq
a$, $a\triangleright b$ iff $a\trianglerighteq b\wedge a\not\trianglelefteq
b$, and $a\trianglelefteq b$ iff $a\not\triangleright\ b$.

######  Example 2.6.

(taken from [14]). Let $\mathcal{F}=\langle\mathcal{A},\mathcal{D},w\rangle$
be a WAF depicted in Figure 1 with weights
$w(a_{0})=0.43,w(a_{1})=0.39,w(a_{2})=0.92$, and $w(a_{3})=0.3$. Table 1 lists
the acceptability degrees and the associated rankings on arguments for the
semantics $\Sigma_{\mathbb{MB}},\Sigma_{\mathbb{CB}}$ and
$\Sigma_{\mathbb{HC}}$ described in Examples 2.3, 2.4 and 2.5.

$a_{0}$$a_{1}$$a_{2}$$a_{3}$ Figure 1: Graphical representation of a WAF | $a_{0}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ |
---|---|---|---|---|---
$w(a_{i})$ | 0.43 | 0.39 | 0.92 | 0.3 | Argument ranking
$\Sigma^{\mathcal{F}}_{\mathbb{MB}}(a_{i})$ | 0.43 | 0.30 | 0.58 | 0.30 | $a_{1}\simeq a_{3}\triangleleft a_{0}\triangleleft a_{2}$
$\Sigma^{\mathcal{F}}_{\mathbb{HC}}(a_{i})$ | 0.43 | 0.30 | 0.38 | 0.30 | $a_{1}\simeq a_{3}\triangleleft a_{2}\triangleleft a_{0}$
$\Sigma^{\mathcal{F}}_{\mathbb{CB}}(a_{i})$ | 0.43 | 0.18 | 0.17 | 0.30 | $a_{2}\triangleleft a_{1}\triangleleft a_{3}\triangleleft a_{0}$
Table 1: Acceptability degrees of the arguments from Figure 1

We now make a historical note about the problem which inspired this paper. The
inverse problem seeks a set of initial weights which, when applied to a
specific argumentation framework under the chosen semantics will result in a
desired preference ordering, derived from the numeric acceptance degree
computed for each argument.

In [14] the authors examined how weights could be computed for a given
acceptance degree. The approach described there consists of two phases. In the
first phase, a target acceptance degree is computed for each argument. In the
second phase, a numerical method (the bisection method) is used to find the
initial weights which lead to this acceptance degree. Since the bisection
method is designed to find the zeros of a function with only one variable, and
since changing the initial weight of one argument can affect the acceptance
degree of other arguments, repeated applications of the bisection method are
often necessary. The authors of [14] identify several strategies for selecting
the argument to which the bisection method should be applied, and demonstrate
that selecting the argument whose current acceptance degree is furthest away
from its target acceptance degree works well in practice.

The inverse problem for $\Sigma_{\mathbb{HC}},\Sigma_{\mathbb{MB}}$ and
$\Sigma_{\mathbb{CB}}$ was the focus of previous research [14]. In Theorem 5.6
below we give a complete characterisation of the inverse problem for these
semantics. In fact we give a positive answer to several other related
problems, specifically the _projective preference ordering problem_ , two
variants of the _reflection problem_ , and the _radiality problem_. These are
described in detail next.

##  3 Rephrasing Weighted Gradual Semantics with Scoring schemes

We start by presenting a slightly different approach to weighted gradual
semantics. For any $A$ and $B$, let $\operatorname{Func}(A,B)$ denote the set
of all functions $f\colon A\to B$ and $\operatorname{End}(A)$ the set of
endomorphisms (i.e., self functions) $f\colon A\to A$. Given
$T\in\operatorname{End}(A)$, a sequence $a_{0},a_{1},\dots\in A$ is called a
$\mathbf{T}$-sequence if $a_{k+1}=T(a_{k})$ for all $k\geq 0$. Given a
directed graph $\mathcal{G}=\langle\mathcal{A},\mathcal{D}\rangle$, we may,
and will, assume throughout that $\mathcal{A}=\\{1,\dots,n\\}$ and $n\geq 1$.

A scoring function $s\colon\mathcal{A}\to[0,1]$ is identified with a vector
$(x_{1},\dots,x_{n})$ in $X=[0,1]^{n}$ which we call the space of scores.
Similarly the space of weights, i.e., functions $w\colon\mathcal{A}\to[0,1]$,
can be identified with vectors in $W=[0,1]^{n}$. We also set $K=[0,1]^{n}$
which we call the space of vertices. We regard $X,W$, and $K$ as subspaces of
the Euclidean space $\mathbb{R}^{n}$, which we equip with the partial order
$x\preceq y$ if $x_{i}\leq y_{i}$ for all $1\leq i\leq n$. Thus, they also
inherit this order.

######  Definition 3.1.

A scoring scheme is a function $\sigma\colon W\to X$. The space of all scoring
schemes is $\operatorname{Scheme}=\operatorname{Func}(W,X)$.

The space of acceptability degrees of $\sigma\in\operatorname{Scheme}$ is
$D_{\sigma}=\\{\sigma(w)|\forall w\in W\\}=\sigma(W)$, the image of $W$.

Let $\Sigma$ be a weighted gradual semantics. For any argumentation framework
$\mathcal{G}=\langle\mathcal{A},\mathcal{D}\rangle$, we obtain a scoring
scheme $\sigma^{\mathcal{G}}\colon W\to X$

$\sigma^{\mathcal{G}}(w)=\Sigma^{\langle\mathcal{A},\mathcal{D},w\rangle}.$

This means that a gradual semantics, which takes a weighted argumentation
framework and returns a scoring function, can be seen as a function which
takes an argumentation framework (without weights) and return a particular
scoring scheme. We formalise this in the next explanatory comment.

######  Scholium.

A weighted gradual semantics $\Sigma$ can be viewed as a function

$\Sigma\colon\\{\text{argumentation frameworks $\mathcal{G}$}\\}\xrightarrow{\
\mathcal{G}\mapsto\sigma^{\mathcal{G}}\ }\\{\text{scoring schemes}\\}.$

By partitioning argumentation frameworks according to the number of their
arguments $n=|\mathcal{A}|$ and using the notation above, a weighted gradual
semantics is a collection of functions, one for each $n\geq 1$

$\Sigma\colon\\{\text{argumentation frameworks $\mathcal{G}$ with
$|\mathcal{A}|=n$}\\}\xrightarrow{\mathcal{G}\mapsto\sigma^{\mathcal{G}}}\operatorname{Scheme}.$

The acceptability degree space of the semantics $\Sigma$ for a graph
$\mathcal{G}$ is $D_{\sigma^{\mathcal{G}}}$.

Since every argumentation framework $\mathcal{G}$ is determined by its
adjacency matrix, a weighted gradual semantics is the same as a collection of
functions, one for each $n\geq 1$,

$\Sigma\colon\\{\text{ $n\times n$ adjacency
matrices}\\}\to\operatorname{Scheme}.$

If $A\in\operatorname{Mat}_{n\times n}(\mathbb{R})$ we write $A\geq 0$ if
$a_{i,j}\geq 0$. Denote

$\operatorname{Mat}_{n\times
n}^{+}(\mathbb{R})=\\{A\in\operatorname{Mat}_{n\times n}(\mathbb{R}):A\geq
0\\}.$

######  Definition 3.2.

An abstract weighted gradual semantics is a collection of functions, one for
each $n\geq 1$,

$\Sigma\colon\operatorname{Mat}_{n\times
n}^{+}(\mathbb{R})\to\operatorname{Scheme}.$

Thus, by restriction to $n\times n$ adjacency matrices (those whose entries
are only zeroes and ones) any abstract weighted gradual semantics restricts to
an ordinary weighted gradual semantics.

######  Example 3.3.

Let $\mathcal{G}$ be a complete graph with arguments
$\mathcal{A}=\\{1,2,3\\}$. Then, the acceptability degree space of semantics
$\Sigma_{\mathbb{HC}}$ for $\mathcal{G}$ is $D_{\sigma^{\mathcal{G}}}$, where
$\sigma^{\mathcal{G}}$ is the associated scoring scheme and for every $w\in
W,\sigma^{\mathcal{G}}(w)=\Sigma_{\mathbb{HC}}^{\langle\mathcal{A},\mathcal{D},w\rangle}$.
A representation of $D_{\sigma^{\mathcal{G}}}$ is shown in Figure 2. Note that
the boundaries of the acceptability degree space are not linear.

Figure 2: Representation (in blue) of the acceptability degree space of
$\Sigma_{\mathbb{HC}}$ for a complete argumentation graph with 3 arguments.
Note that the surfaces in the figure are non-linear.

###  3.1 The inverse problems

We generalise the inverse problem introduced in [14] to a family of _inverse-
type problems_. These revolve around moving from acceptability degrees to
weights, but consider different facets in doing so. Thus, we are — for example
— interested in finding conditions on scoring schemes which guarantee that
acceptability degrees satisfying certain properties are realised, and in this
case calculate the weights that realise them. We formulate four inverse-type
problems that are the focus of this paper.

The first problem asks whether, for a given scoring scheme $\sigma$, we can
find an inverse function $\operatorname{inv}$ such that for any element $x\in
X$, we can determine whether $x$ is in the space of acceptability degrees of
$\sigma$ by testing whether $\operatorname{inv}(x)\in W$, and in this case
$w=\operatorname{inv}(x)$ is a weight such that $\sigma(w)=x$.

######  Problem 3.4.

The inverse problem for a scoring scheme $\sigma\in\operatorname{Scheme}$
seeks a (computable) function $\operatorname{inv}\colon X\to\mathbb{R}^{n}$
such that

  1. (i)

$x\in D_{\sigma}\iff\operatorname{inv}(x)\in W$, and in this case

  2. (ii)

$\sigma(\operatorname{inv}(x))=x$.

The second problem (reflection) ask whether the scoring scheme is injective or
a homeomorphism onto its image. This allows us to answer questions such as:
“Given an argumentation framework and a gradual semantics $\Sigma$, is it
possible to find two distinct weighting functions such that the acceptability
degrees will be the same?”

######  Problem 3.5.

The reflection problem for $\sigma\in\operatorname{Scheme}$ asks whether
$\sigma\colon W\to X$ is injective, i.e., every acceptability degree is
obtained by a unique weight $w\in W$.

The closely related topological reflection problem asks whether $\sigma\colon
W\to D_{\sigma}$ is a homeomorphism onto its image.

The third problem asks whether any element $y$ of the space of scores can be
projected into an element $x\in\mathcal{D}_{\sigma}$ by re-scaling it. Note
that the acceptability degrees $x$ have the same arguments’ preference
ordering and ratio between them as the acceptability degrees $y$. This will
allow us to answer questions such as: “Given an argumentation framework and a
gradual semantics $\Sigma$, is it possible to find a weighting function such
that the acceptability degree of argument $a$ is $n$ times as much as the one
of $b$?”.

######  Problem 3.6.

The projective preference ordering problem for a scoring scheme $\sigma$ asks
whether for any $y\in X$ there exists some $t>0$ such that $t\cdot y\in
D_{\sigma}$.

The fourth problem ask whether for any element $x$ of the acceptability degree
space of $\sigma$ ($D_{\sigma})$, all the elements on the line from $0^{n}$ to
$x$ are in $D_{\sigma}$. In other words, a positive answer to this question
means there are no “gaps” in the acceptability degree space along any line
starting at the origin and a point in the acceptability degree space. This is
important as it allows us to scale down any element along the line (via
Problem 3.6).

######  Problem 3.7.

The radiality problem for a scoring scheme $\sigma$ asks whether for any $x\in
D_{\sigma}$, the line segment $[0,x]=\\{tx:0\leq t\leq 1\\}$ in
$\mathbb{R}^{n}$ is contained in $D_{\sigma}$. In this case we call
$D_{\sigma}$ radial.

######  Definition 3.8.

We say that a weighted gradual semantics $\Sigma$ has a solution for the
inverse problem 3.4, the (topological) reflection problem 3.5, the projective
preference ordering problem 3.6 or the radiality problem 3.7 if for any
argumentation framework $\mathcal{G}$ the scoring scheme
$\sigma^{\mathcal{G}}=\Sigma(\mathcal{G})\in\operatorname{Scheme}$ has a
solution for these problems.

One of the achievements of this paper is the identification of a large family
of abstract weighted gradual semantics which includes the semantics
$\Sigma_{\mathbb{HC}},\Sigma_{\mathbb{MB}}$ and $\Sigma_{\mathbb{CB}}$ from
Examples 2.3, 2.4, and 2.5 and for which all four inverse-type problems above
are solved. Moreover, we give explicit and easily computable inverse functions
$\operatorname{inv}_{\mathbb{MB}},\operatorname{inv}_{\mathbb{CB}}$ and
$\operatorname{inv}_{\mathbb{HC}}$ for these three semantics. See Theorem 5.6
below for a detailed statement of the result.

###  3.2 Based scoring schemes

In this section, we introduce the important notion of _based scoring schemes_
as particular scoring schemes which are obtained through a weighted scoring
base (described in Definition 3.18). This formalism is crucial to structure
the following properties and discussion.

We first start with the necessary definitions to introduce the notion of a
scoring base.

######  Definition 3.9.

A function $f\colon\mathbb{R}^{n}\to\mathbb{R}$ is called increasing if for
any $x,y\in\mathbb{R}^{n}$

$x\preceq y\implies f(x)\leq f(y).$

It is called homogeneous if for all $x\in\mathbb{R}^{n}$ and all $t\geq 0$

$f(t\cdot x)=t\cdot f(x).$

A function $f\colon X\to\mathbb{R}^{n}$ is called homogeneous (resp.
increasing) if each component $f_{i}\colon X\to\mathbb{R}$ is homogeneous
(resp. increasing). We say it is _non-negative_ , written $f\geq 0$, if
$f_{i}(x)\geq 0$ for all $1\leq i\leq n$. It is _bounded_ if there is some
$M>0$ such that $f_{i}(x)\leq M$ for all $1\leq i\leq n$. Let

$\mathcal{B^{+}HI}_{n}(X)$

denote the set of all functions $f\colon X\to\mathbb{R}^{n}$ which are
bounded, non-negative, homogeneous and increasing. For $n=1$, we write
$\mathcal{B^{+}HI}(X)$.

The following two propositions show that the collection $\mathcal{B^{+}HI}(X)$
is very rich. Their proofs are straightforward and left to the reader.

######  Proposition 3.10.

$\mathcal{B^{+}HI}(X)$ contains all projections $\pi_{i}\colon
X\xrightarrow{x\mapsto x_{i}}[0,\infty)$.

######  Proposition 3.11.

Let $\psi_{1},\dots,\psi_{k}\in\mathcal{B^{+}HI}(X)$. Then the following
functions obtained from $\psi_{1},\dots,\psi_{k}$ are also in
$\mathcal{B^{+}HI}(X)$.

  1. (i)

$\sum_{j}a_{j}\psi_{j}$ for any $a_{1},\dots,a_{k}\geq 0$.

  2. (ii)

$\max\\{\psi_{1},\dots,\psi_{k}\\}$.

  3. (iii)

$\sqrt[k]{\psi_{1}\cdots\psi_{k}}$ (geometric mean).

  4. (iv)

$\sqrt[p]{\psi_{1}^{p}+\cdots+\psi_{k}^{p}}$ for any $p>0$.

We can now introduce the fundamental notion of a scoring base.

######  Definition 3.12.

A scoring base is a pair $(c,f)\in K\times\mathcal{B^{+}HI}_{n}(X)$. That is,

  * •

$c\in K$ and

  * •

$f\colon X\to[0,\infty)^{n}$ is bounded, homogeneous, non-negative, and
increasing.

The set of all scoring bases is denoted

$\operatorname{Base}(X)=K\times\mathcal{B^{+}HI}_{n}(X)\qquad(\subseteq
K\times\operatorname{Func}(X,\mathbb{R}^{n})).$

From a scoring base, one can construct a particular endomorphism from $X$ to
$X$.

######  Definition 3.13.

Let $(c,f)$ be a scoring base, and $i\in\\{1\ldots|n|\\}$, be an index w.r.t
arguments. Define $T_{(c,f)}\in\operatorname{End}(X)$ by

$T_{(c,f)}(x)_{i}=\frac{c_{i}}{1+f(x)_{i}}.$

$T_{(c,f)}$ is well defined because $0\leq c_{i}\leq 1$ and $f(x)_{i}\geq 0$
so $T_{(c,f)}(x)\in X$ for any $x\in X$.

A crucial question is: “Given an arbitrary element $x\in X$, will the
successive application of $T_{(c,f)}$ lead to convergence and will it always
converge to the same fix point?”. To formalise this, we introduce the
following notion of scoring dynamics so that the previous question can thus be
rephrased as: “Is $T_{(c,f)}$ a scoring dynamics?”.

######  Definition 3.14.

A scoring dynamics is a function $T\in\operatorname{End}(X)$ such that

  1. (i)

$T$ has a unique fixed point denoted $\mathrm{fix}(T)\in X$, and

  2. (ii)

Every $T$-sequence $x_{0},x_{1},\dots\in X$ is convergent to
$\mathrm{fix}(T)$.

Denote by $\mathcal{D}{yn}(X)$ the collection of all scoring dynamics. The
assignment $T\mapsto\mathrm{fix}(T)$ gives a function

$\mathrm{fix}\colon\mathcal{D}{yn}(X)\to X.$

To answer the question “Is $T_{(c,f)}$ a scoring dynamics?”, we start by
showing, in Theorem 3.16, that many order reversing functions of
$\operatorname{End}(X)$ are scoring dynamics.

While our proof for Theorem 3.16 is inspired by the work of Pu, Zhang and Luo
([23, Theorems 1 and 2]) who showed that a unique fixed-point exists for the
h-categorizer semantics, i.e., in the case where all arguments have an initial
weight of 1; we are able to prove such a result for a much larger class of
functions.

######  Definition 3.15.

A function $f\colon X\to\mathbb{R}^{n}$ is called order preserving if
$x\preceq x^{\prime}\implies f(x)\preceq f(x^{\prime})$ for any
$x,x^{\prime}\in X$. It is called order reversing if $x\preceq
x^{\prime}\implies f(x^{\prime})\preceq f(x)$ for any $x,x^{\prime}\in X$.

######  Theorem 3.16.

Let $f:X\to X$ be a function such that

  1. (a)

$f$ is order reversing, and

  2. (b)

there is some $0<\alpha\leq 1$ such that for any $x\in X$ and any $0\leq t\leq
1$

$f(tx)\preceq\frac{1}{t+\alpha(1-t)}f(x).$

Then $f$ has a unique fixed point $y\in X$. Furthermore,
$y=\lim_{k\to\infty}x^{(k)}$ where $x^{(k)}$ is any sequence defined
recursively by choosing $x^{(0)}\in X$ arbitrarily and $x^{(k+1)}=f(x^{(k)})$.

The next theorem shows that this generalisation applies to $T_{(c,f)}$
(obtained via a scoring base), and we will show below how this therefore
applies to a very large class of gradual semantics (see Definition 3.19).

######  Theorem 3.17.

Let $(c,f)\in\operatorname{Base}(x)$ be a scoring base. Then
$T_{(c,f)}\in\mathcal{D}{yn}(X)$.

A weighted scoring base is a function which associate a scoring base to each
element of $W$ by making use of two functions which instantiate each of the
elements of the scoring base.

######  Definition 3.18.

A weighted scoring base is a function $b\colon W\to\operatorname{Base}(X)$.
More specifically, $b=(\kappa,\varphi)$ where $\kappa$ and $\varphi$ are
functions

$\displaystyle\kappa\colon W\to K$ $\displaystyle\varphi\colon
W\to\operatorname{Func}(X,\mathbb{R}^{n})$

such that $(\kappa(w),\varphi(w))\in\operatorname{Base}(X)$ for all $w\in W$,
that is $\varphi\colon W\to\mathcal{B^{+}HI}_{n}(X)$. The set of all weighted
scoring bases is denoted $\operatorname{w-Base}(X)$.

The associated scoring scheme $\sigma_{b}\in\operatorname{Func}(W,X)$ is
defined by

$\sigma_{b}(w)=\mathrm{fix}(T_{(\kappa(w),\varphi(w))}).$

That is, $\sigma_{b}$ is the composition from $X$ to $X$ via a scoring base
and scoring dynamics.

$\sigma_{b}\ \colon\
W\xrightarrow{b}\operatorname{Base}(X)\xrightarrow{T}\mathcal{D}{yn}(X)\xrightarrow{\mathrm{fix}}X.$

We will also write $\sigma_{(\kappa,\varphi)}$ for $\sigma_{b}$. We can thus
obtain a function

$\operatorname{w-Base}(X)\xrightarrow{\
(\kappa,\varphi)\mapsto\sigma_{(\kappa,\varphi)}\ }\operatorname{Scheme}.$

A scoring scheme is based if it has the form $\sigma_{(\kappa,\varphi)}$ for
some weighted scoring base $b=(\kappa,\varphi)$. Its acceptability degree
space is denoted $D_{(\kappa,\varphi)}$.

Figure 3 summarises the notions and the results of Section 3.2.

Figure 3: Representation of a based scoring scheme $\sigma_{b}$ and the
contributions of Section 3.2.

######  Definition 3.19.

An abstract weighted based gradual semantics is one which is obtained by
composing a function

$\operatorname{Mat}_{n\times
n}^{+}(\mathbb{R})\xrightarrow{\beta}\operatorname{w-Base}(X)$

with the map
$\operatorname{w-Base}(X)\xrightarrow{b\mapsto\sigma_{b}}\operatorname{Scheme}$
in Definition 3.18.

By construction, if $\Sigma$ is an abstract weighted based gradual semantics
then for any $A$ the scoring scheme $\sigma(A)=\sigma_{\beta(A)}$ (Definition
3.18) is based.

##  4 Solving Inverse Problems

In this section, we give conditions on weighted scoring bases $b$ so that the
associated scoring scheme $\sigma_{b}$ admits a solution for the inverse
problems described in Section 3.1.

We start by introducing the necessary definitions used to describe our
constraints.

The support of $x\in\mathbb{R}^{n}$ is the subset of $\\{1,\dots,n\\}$ defined
as $\mathrm{supp}(x)=\\{1\leq i\leq n:x_{i}\neq 0\\}$.

######  Definition 4.1.

We say that a function $f\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ preserves
supports if for every $x\in\mathbb{R}^{n}$

$\mathrm{supp}(x)=\mathrm{supp}(f(x))$

We say that $f\colon\mathbb{R}^{n}\to Y$, for some set $Y$, is independent of
supports if

$\mathrm{supp}(x)=\mathrm{supp}(y)\implies f(x)=f(y).$

In this case, if $I\in\wp(\\{1,\dots,n\\})$ we write $f^{I}\in Y$ for the
value of $f$ on all $x\in X$ with support $I$.

As we will see in the following theorems, some of the essential constraints
for a weighted scoring base $(\kappa,\varphi)$ to have a solution for the
aforementioned inverse problems are that $\kappa$ preserves supports and that
$\varphi$ is independent of supports.

######  Definition 4.2.

Let $A$ be a subset of $\mathbb{R}^{n}$. A discerning right inverse for
$g\colon A\to X$ is a function $r\colon X\to\mathbb{R}^{n}$ such that $r(x)\in
A\iff x\in g(A)$ and in this case $g(r(x))=x$. That is, $g\circ
r|_{g(A)}=\operatorname{id}_{g(A)}$.

Figure 4: Representation of the discerning right inverse $r$ of $g$ (left) and
a discerning right inverse of a scoring scheme $\sigma$ (right).

######  Remark.

The inverse problem 3.4 for a scoring scheme $\sigma$ seeks a computable
discerning right inverse $\operatorname{inv}$ for $\sigma$ (see Figure 4).

The next theorem gives conditions on a weighted scoring base
$(\kappa,\varphi)\in\operatorname{w-Base}(X)$ so that the associated based
scoring scheme $\sigma_{(\kappa,\varphi)}$ admits a solution to the inverse
and the reflection problems (3.4, 3.5). It also gives conditions for it to
admit a solution to the topological reflection problem.

######  Theorem 4.3.

Let $b=(\kappa,\varphi)$ be a weighted scoring base. Assume that

  1. (i)

$\kappa\colon W\to K$ is the restriction of
$\tilde{\kappa}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ which is bijective and
preserves supports.

  2. (ii)

$\varphi\colon W\to\operatorname{Func}(X,\mathbb{R}^{n})$ is independent of
supports.

Then $\sigma_{(\kappa,\varphi)}\colon W\to X$ solves the inverse and
reflection problems. That is, $\sigma_{(\kappa,\varphi)}$ is injective and
admits a discerning right inverse $\operatorname{inv}\colon
X\to\mathbb{R}^{n}$ which is defined as follows. Recall the notation of
$\mathrm{supp}$ from Definition 4.1 and set

$\displaystyle\overline{\operatorname{inv}}(x)_{i}=x_{i}(1+\varphi^{\mathrm{supp}(x)}(x)_{i})$
$\displaystyle\operatorname{inv}(x)=\tilde{\kappa}^{-1}\circ\overline{\operatorname{inv}}.$

If $\tilde{\kappa}$ is a homeomorphism and $\varphi\colon
W\to\operatorname{Base}(X)$ is constant with value $(c,f)$ such that $f\colon
X\to[0,\infty)^{n}$ is continuous, then $\sigma_{(\kappa,\varphi)}\colon W\to
X$ is a homeomorphism onto its image. That is, $\sigma_{(\kappa,\varphi)}$
solves the topological inverse problem.

Our next goal is to find conditions on a weighted scoring base
$(\kappa,\varphi)$ so that the projective preference ordering problem 3.6
admits a solution. We view $K=[0,1]^{n}$ as a subspace of $\mathbb{R}^{n}$.
Let $0\in K$ denote the origin in $\mathbb{R}^{n}$. Recall that a
neighbourhood $U$ of $0$ in $K$ is the intersection of $K$ with an open subset
of $\mathbb{R}^{n}$ containing $0\in\mathbb{R}^{n}$. Thus, there exists
$\epsilon>0$ such that $U$ contains any $x\in\mathbb{R}^{n}$ such that
$\|x\|<\epsilon$ and $x_{i}\geq 0$ for all $1\leq i\leq n$.

######  Theorem 4.4.

Let $b=(\kappa,\varphi)$ be a weighted scoring base. Suppose that

  1. (i)

$\kappa\colon W\to K$ preserves supports.

  2. (ii)

$\kappa(W)\subseteq K$ contains a neighbourhood $U$ of $0\in K$.

  3. (iii)

$\varphi$ is independent supports.

Then the projective preference ordering problem 3.6 has a solution for the
scoring scheme $\sigma_{(\kappa,\tau)}$.

Finally, we find conditions on a weighted scoring base $(\kappa,\varphi)$
which guarantees that the scoring scheme $\sigma_{(\kappa,\varphi)}$ admits a
solution to the radiality problem (Problem 3.7), i.e $D_{(\kappa,\varphi)}$ is
radial subset of $X$. Recall the partial order $\preceq$ on $\mathbb{R}^{n}$.

######  Definition 4.5.

A subset $A$ of $K=[0,1]^{n}$ is $\preceq$-closed in $K$, respectively
$(\mathrm{supp},\preceq)$-closed in $K$, if for any $a\in A$ and any $x\in K$,

$\displaystyle x\preceq a$ $\displaystyle\implies x\in A$ $\displaystyle
x\preceq a\text{ and }\mathrm{supp}(x)=\mathrm{supp}(a)$
$\displaystyle\implies x\in A.$

######  Theorem 4.6.

Let $b=(\kappa,\varphi)$ be a weighted scoring base. Suppose that

  1. (i)

$\kappa\colon W\to K$ preserves supports.

  2. (ii)

$\kappa(W)$ is $(\mathrm{supp},\preceq)$-closed in $K$.

  3. (iii)

$\varphi\colon W\to[0,\infty)^{n}$ is independent of supports.

Then $\sigma_{b}$ solves the radiality problem, i.e
$D_{(\kappa,\varphi)}=\sigma_{(\kappa,\varphi)}(W)$ is radial.

By Theorems 4.3, 4.4 and 4.6 all four Problems 3.4–3.7 have positive solution
for particular abstract weighted based gradual semantics. Now that we have
defined the necessary constraints on weighted scoring bases for all the
inverse problems, can we find a concrete family of based scoring schemes
(scoring schemes obtained via a weighted scoring base) that have a positive
solution to all the inverse problems?

##  5 $(L^{p},\lambda,\mu,A)$-based scoring schemes

In this section we construct a family of based scoring schemes which answers
our four problems in the positive, and link this family back to the max-based,
h-categoriser and card-based semantics.

We fix

  * •

$1\leq p\leq\infty$,

  * •

$\mu\colon\mathbb{R}^{n}\to(0,1]^{n}$ independent of supports,

  * •

$\lambda\colon\mathbb{R}^{n}\to\operatorname{End}([0,\infty)^{n})$ independent
of supports,

  * •

$n\times n$ matrix $A=(a_{i,j})$ such that $A\geq 0$, i.e $a_{i,j}\geq 0$.

Recall that the $L^{p}$-norm on $\mathbb{R}^{n}$ is defined by

$\|(x_{1},\dots,x_{n})\|_{p}=\left\\{\begin{array}[]{ll}(x_{1}^{p}+\cdots
x_{n}^{p})^{1/p}&\text{if $1\leq p<\infty$}\\\
\max\\{x_{1},\dots,x_{n}\\}&\text{if $p=\infty$}\end{array}\right.$ (1)

Given vectors $x,y\in\mathbb{R}^{n}$ we denote

$x\wedge y=(x_{1}y_{1},\dots,x_{n}y_{n}).$ (2)

We denote

$a_{i,*}\ =\ \text{the $i^{\text{th}}$ row of $A$.}$

With this data and notation, define $\kappa\colon W\to K$ and $\varphi\colon
W\to\operatorname{Func}(X,[0,\infty)^{n})$ as follows.

$\displaystyle\kappa(w)_{i}=\mu(w)_{i}\cdot w_{i}$ (3)
$\displaystyle\varphi(w)(x)_{i}=\|\lambda(w)(a_{i,*})\wedge x\|_{p}.$

######  Proposition 5.1.

The pair $(\kappa,\varphi)$ defined in (3) is a weighted scoring base.

######  Definition 5.2.

The associated scoring scheme $\sigma_{(\kappa,\varphi)}$ defined in (3) is
called the $(L^{p},\lambda,\mu,A)$-based scoring scheme
$\sigma_{(L^{p},\lambda,\mu,A)}$.

######  Theorem 5.3.

Any $(L^{p},\lambda,\mu,A)$-based scoring scheme has positive solution to the
inverse problem, the reflection problem, the projective preference ordering
problem and the radiality problem.

The discerning right inverse function $\operatorname{inv}\colon
X\to\mathbb{R}^{n}$ has the form

$\operatorname{inv}(x)_{i}=\tfrac{1}{\mu(x)_{i}}x_{i}(1+\|\lambda(x)(a_{i*})\wedge
x\|_{p}).$

If $\mu$ and $\lambda$ are constant functions then the topological reflection
problem is answered positively.

######  Theorem 5.4.

Suppose that $p\leq q$ then

$D_{(L^{p},\lambda,\mu,A)}\subseteq D_{(L^{q},\lambda,\mu,A)}.$

The weighted gradual semantics $\Sigma_{\mathbb{HC}},\Sigma_{\mathbb{MB}}$ and
$\Sigma_{\mathbb{CB}}$ in Examples 2.3–2.5, which are our prime source of
interest, are examples of the framework we have developed.

######  Proposition 5.5.

The weighted gradual semantics $\Sigma_{\mathbb{HC}},\Sigma_{\mathbb{MB}}$ and
$\Sigma_{\mathbb{CB}}$ are abstract weighted $(L^{p},\lambda,\mu,A)$-based
gradual semantics.

As a result we can deduce the following theorem about these important gradual
semantics.

######  Theorem 5.6.

The gradual semantics $\Sigma_{\mathbb{HC}},\Sigma_{\mathbb{MB}}$ and
$\Sigma_{\mathbb{CB}}$ solve the inverse, reflection, projective preference
ordering, and the radiality problems. Moreover
$\Sigma_{\mathbb{HC}},\Sigma_{\mathbb{MB}}$ solve the topological reflection
problem.

It is easy to see that the discerning right inverses $\operatorname{inv}\colon
X\to\mathbb{R}^{n}$ of these gradual semantics are given by the following
formulae. We let $A$ denote the adjacency matrix an argumentation framework
$\mathcal{G}=\langle\mathcal{A},\mathcal{D}\rangle$, whose individual elements
are denoted $a_{i,j}$. Then

$\displaystyle\operatorname{inv}^{\mathcal{G}}_{\mathbb{HC}}(x)_{i}$
$\displaystyle=x_{i}(1+\sum_{j}a_{i,j}x_{j}).$
$\displaystyle\operatorname{inv}^{\mathcal{G}}_{\mathbb{MB}}(x)_{i}$
$\displaystyle=x_{i}(1+\max_{j}\\{a_{i,j}x_{j}\\})$
$\displaystyle\operatorname{inv}^{\mathcal{G}}_{\mathbb{CB}}(x)_{i}$
$\displaystyle=x_{i}\Big{(}1+S(x)_{i}+\tfrac{1}{S(x)_{i}}(1+\sum_{j\in\mathrm{supp}(x)}a_{i,j}x_{j})\Big{)}.$

In the last formula $S(x)_{i}=\sum_{j\in\mathrm{supp}(x)}a_{i,j}$ and if
$S(x)_{i}=0$ then by convention $\tfrac{1}{S(x)_{i}}=0$.

######  Remark 5.7.

It is useful to identify $\wp(n)$ with $\\{0,1\\}^{n}$ as a subset of
$\mathbb{R}^{n}$, thus we view $\mathrm{supp}(x)$ as a vector of zeros and
ones. If $v\in\mathbb{R}^{n}$ we write $\operatorname{diag}(v)$ for the
diagonal matrix with $\operatorname{diag}(v)_{ii}=v_{i}$. In matrix form, one
can write $\operatorname{inv}_{\mathbb{HC}},\operatorname{inv}_{\mathbb{CB}}$
and $\operatorname{inv}_{\mathbb{MB}}$ as follows. Set

$S(x)=A\cdot\mathrm{supp}(x)$

and write $S(x)^{-1}:=(\tfrac{1}{S(x)_{1}},\dots,\tfrac{1}{S(x)_{n}})$ where
$\tfrac{1}{S(x)_{i}}=0$ whenever $S(x)_{i}=0$. Also let $max_{c}(M)$ return a
column-wise operation which retains the maximum element in a column, but sets
all other entries in the column to 0 (breaking ties at random).

Then

$\displaystyle\operatorname{inv}_{\mathbb{HC}}^{A}(x)=(I+\operatorname{diag}(x)\cdot
A)\cdot x$
$\displaystyle\operatorname{inv}_{\mathbb{CB}}^{A}(x)=\Big{(}I+\operatorname{diag}(S(x))+\operatorname{diag}(S(x)^{-1})\cdot
A\cdot\operatorname{diag}(\mathrm{supp}(x))\Big{)}\cdot x$
$\displaystyle\operatorname{inv}_{\mathbb{MB}}^{A}(x)=\Big{(}I+max_{c}(A\cdot
diag(x))\Big{)}\cdot x$

This result allows us to easily and precisely compute the initial weights for
an argumentation framework and associated acceptability degrees, without
resorting to numerical methods as was done in [14].

##  6 New gradual semantics

With the machinery we have developed, there is a large collection of desirable
weighted gradual semantics that we can define inspired by the content of
Proposition 5.5. In this section we provide two new example semantics built
using our work.

###  6.1 The Weighted Euclidean-based gradual semantics

The weighted Euclidean-based gradual semantics $\Sigma_{\mathbb{EB}}$ is the
$(L^{p},\lambda,\mu,A)$-based semantics with $p=2$ and $\mu=1$ and $\lambda$
constant with value $\operatorname{id}\in\operatorname{End}([0,\infty)^{n})$.
Thus, $\Sigma_{\mathbb{EB}}^{\mathcal{G}}(w)$ is obtained as the limit of the
sequence of vectors $\operatorname{EB}_{k}\in X$ where $\operatorname{EB}_{0}$
is chosen arbitrarily and

$\operatorname{EB}_{k+1}(x)=\frac{w_{i}}{1+\sqrt{\sum_{j\in\mathtt{Att}(i)}x_{j}^{2}}}.$

From our results, this semantics positively answers the inverse problem,
topological reflection problem, the projective preference order problem and
the radiality problem. The discerning right inverse function
$\operatorname{inv}_{\mathbb{EB}}\colon X\to\mathbb{R}^{n}$ is given by

$\operatorname{inv}_{\mathbb{EB}}^{\mathcal{G}}(x)_{i}=x_{i}\left(1+\sqrt{\sum_{j\in\mathtt{Att}(i)}x_{j}^{2}}\right).$

As one can observe, it is easy to create basic variations of existing weighted
gradual semantics while preserving important properties such as convergence or
satisfactions to inverse problems.

###  6.2 Gradual semantics with remote attacks

In most weighted gradual semantics considered to date (including
$\Sigma_{\mathbb{HC}},$ $\Sigma_{\mathbb{MB}}$, and $\Sigma_{\mathbb{CB}}$),
only the acceptability degree of direct attackers on arguments in the graph
$\mathcal{G}$ were considered. The intuition given by this is that the
acceptability degrees of direct attackers already “compile” the acceptability
degree of indirect attackers. However, there may be cases where it is useful
to access the degree of these indirect attackers.

That is, given $i\in\mathcal{A}$, attacks on $i$ are considered only by
arguments $j\in\mathcal{A}$ such that $(j,i)\in\mathcal{D}$, i.e., $j$ is a
neighbour at distance $1$ from $i$ within the underlying argumentation
framework. The “amount” of attack on $i$ is formulated by means of a function
$\alpha_{i}$ which only depends on $x_{j}$ with $j\in\mathtt{Att}(i)$. For
example, in $\Sigma_{\mathbb{HC}}$ we get
$\alpha_{i}(x)=\sum_{j\in\mathtt{Att}(j)}x_{j}$ and in $\Sigma_{\mathbb{MB}}$
it is $\alpha_{i}(x)=\max_{j\in\mathtt{Att}(i)}x_{j}$. More generally, if
$\Sigma$ is associated to a weighted scoring base $(\kappa,\varphi)$ then the
attack is formulated by means of the functions $\varphi_{i}\colon
X\to[0,\infty)$.

Instead, it is conceivable to consider “secondary level” attacks on $i$ by
arguments $j$ at distance greater than $1$. Such attacks are loosely analogous
to indirect defeaters in bipolar argumentation frameworks [24], but will have
“weaker” strength which is achieved by multiplying the function $\varphi$ by a
discount factor $\delta<1$.

Recall that if $A$ is the adjacency matrix of $A$ then the $(i,j)^{\text{th}}$
entry of $A^{k}$ is equal to the number of (directed) paths of length $k$ from
$j$ to $i$, thus the number of “secondary attacks” of $j$ on $i$ via a
sequence of $k+1$ arguments (including $i$ and $j$). In order to encode such
“remote attacks” one should consider the matrix

$A^{\prime}=A+\delta_{2}A^{2}+\delta_{3}A^{3}+\cdots$

where $0\leq\delta_{k}\leq 1$ indicate the multiplicative strength factors of
attacks from distance $k$.

As a concrete example, let us show how we modify $\Sigma_{\mathbb{HC}}$ to a
new weighted gradual semantics $\Sigma_{\mathbb{HC},r}$ which takes into
account remote attacks from distance $2$. Choose some $0<\delta<1$. Suppose
that $\mathcal{G}$ is an argumentation framework and let $A$ be its adjacency
matrix. Set

$A^{\prime}=A+\delta\cdot A^{2}.$

Consider the abstract $(L^{p},\lambda,\mu,A^{\prime})$-based scoring scheme
with $p=1$ and $\lambda=\mu=1$. Then attacks on $i$ are made using the formula

$\frac{w_{i}}{1+(A^{\prime}x)_{i}}=\frac{w_{i}}{1+(Ax)_{i}+\delta\cdot(A^{2}x)_{i}}$

and observe that $Ax_{i}=\sum_{j\in\mathtt{Att}(i)}x_{j}$ and
$(A^{2}x)_{i}=\sum_{\gamma\in\mathtt{Att}_{2}(i)}x_{j}$ where
$\mathtt{Att}_{2}(i)$ is the set of all paths $\gamma$ of length $2$ in
$\mathcal{G}$ ending at $i$, and the term $x_{j}$ in the sum refers to the
argument starting the path $\gamma$. We highlight that here, the degree of
indirect attackers may be counted multiple times if there are multiple paths.

To generalise this result to remote attacks further away, it is reasonable to
use the matrix

$A^{\prime}=\sum_{k=1}^{\infty}\delta^{k-1}A^{k}.$

This second example shows the richness of the family of abstract weighted
$(L^{p},\lambda,\mu,A)$-based gradual semantics.

##  7 Conclusions

We began this paper by rephrasing the notion of a weighted gradual semantics
in terms of scoring schemes. This allowed us to formulate four problems,
including the inverse problem of [14] around such scoring schemes. These four
problems revolve around the uniqueness of solutions, and continuity of
solutions, for weighted gradual semantics, and thus encode highly desirable
properties of any semantics.

We introduced a very general class of semantics, the
$(L^{p},\gamma,\mu,A)-$based scoring schemes. Any semantics which can be
encoded as such as scoring scheme (including the weighted h-categoriser, max-
based and card-based semantics) answer the four problems we describe in the
positive. We also demonstrated that an analytical solution to the original
inverse problem can be derived, in contrast to the numeric approach used in
[14].

Finally, building on the generality of our semantics and the desirable
properties of our problems, we described two new semantics.

Our paper addresses several important theoretical gaps in argumentation
theory, and facilitates computationally efficient solutions to the inverse
problem. In the context of future work, the new semantics we propose in
Section 6 are very interesting. One obvious avenue of such future work
involves determining which of the properties described in [13] this family of
new semantics comply with. In addition, the $\sigma$-acceptability-maximal
semantics and ratio semantics differ significantly from existing semantics,
and offer the possibility of making elicitation from expert knowledge easier,
reducing knowledge engineering effort. Thus, exploring the properties of these
semantics, and how closely they mirror human reasoning, is an avenue of work
we intend to pursue.

##  Appendix A Appendix

This appendix contains the proofs of the results presented in this paper. We
kept the order in which we prove our results in the same order in which we
present them. All definitions, lemmas, theorems, and propositions in this
appendix are intermediary results which are used in the proofs.

We will use the following notation: Given $a,b\in X$ such that $a\preceq b$,
the interval between $a$ and $b$ in $X$ is
$[a,b]_{X}\overset{\text{def}}{=}\\{x\in X\,:\,a\preceq x\preceq b\\}.$

###### Proof of Theorem 3.16.

Define a sequence $u^{(n)}$ in $X$ by recursion by $u^{(0)}=0$ and
$u^{(n+1)}=f(u^{(n)})$ for every $n\geq 0$. Then $u^{(0)}\preceq u^{(1)}$
since $0\in X$ is the minimum element in $(X,\preceq)$. Since $f$ is order
reversing, we get $u^{(0)}\preceq u^{(2)}\preceq u^{(1)}$. By induction one
easily shows

  1. (i)

$u^{(2k)}\preceq u^{(2k-1)}$ for all $k\geq 1$,

  2. (ii)

$u^{(2k)}\preceq u^{(2k+2)}$ for all $k\geq 0$,

  3. (iii)

$u^{(2k+1)}\preceq u^{(2k-1)}$ for all $k\geq 1$.

Thus, the sequence $u^{(2k)}$ is increasing in the sense that $u^{(2k)}_{i}$
is an increasing sequence in $\mathbb{R}$ for every $i=1,\dots,n$, and bounded
above by $1\in X$. Similarly, $u^{(2k-1)}$ is decreasing and bounded below by
$0\in X$. We may therefore define

$\displaystyle u^{(\operatorname{ev})}$
$\displaystyle=\sup_{k}u^{(2k)}=\lim_{k}u^{(2k)}$ $\displaystyle
u^{(\operatorname{odd})}$
$\displaystyle=\inf_{k}u^{(2k-1)}=\lim_{k}u^{(2k-1)}.$

It follows from (i) that $u^{(\operatorname{ev})}\preceq
u^{(\operatorname{odd})}$.

Our next goal is to show that
$u^{(\operatorname{ev})}=u^{(\operatorname{odd})}$. For every $k\geq 1$, set:

$\pi_{k}=\sup\ \\{0\leq t\leq 1\ :\ t\cdot u^{(2k-1)}\preceq u^{(2k)}\\}.$

The set on the right is not empty since $0\cdot u^{(2k-1)}=0\preceq u^{(2k)}$,
so:

$0\leq\pi_{k}\leq 1.$

It is clear from the construction that for every $k\geq 1$

$\pi_{k}\cdot u^{(2k-1)}\preceq u^{(2k)}.$

By applying hypotheses (a) and (b) to this inequality and then using (ii), for
every $k\geq 1$

$\displaystyle u^{(2k+1)}$ $\displaystyle=f(u^{(2k)})$ $\displaystyle\preceq
f(\pi_{k}\cdot u^{(2k-1)})$
$\displaystyle\preceq\tfrac{1}{\pi_{k}+\alpha(1-\pi_{k})}\cdot f(u^{(2k-1)})$
$\displaystyle=\tfrac{1}{\pi_{k}+\alpha(1-\pi_{k})}\cdot u^{(2k)}$
$\displaystyle\preceq\tfrac{1}{\pi_{k}+\alpha(1-\pi_{k})}\cdot u^{(2k+2)}.$

Hence $(\pi_{k}+\alpha(1-\pi_{k}))\cdot u^{(2k+1)}\preceq u^{(2k+2)}$, so by
the definition of $\pi_{k+1}$

$\pi_{k}+\alpha(1-\pi_{k})\leq\pi_{k+1}.$

It follows that $1-\pi_{k+1}\leq(1-\alpha)(1-\pi_{k})$ for all $k\geq 1$.
Therefore

$1-\pi_{k+1}\leq(1-\pi_{1})(1-\alpha)^{k}\xrightarrow{\ k\to\infty\ }0.$

But $\pi_{k}\leq 1$ for all $k$, so $\lim_{k}\pi_{k}=1$.

Consider some $\epsilon>0$. Then $\pi_{k}>1-\epsilon$ for all $k\gg 0$.
Equation (i) implies

$(1-\epsilon)\cdot u^{(2k-1)}\preceq\pi_{k}\cdot u^{(2k-1)}\preceq
u^{(2k)}\preceq u^{(2k-1)}.$

Letting $k\to\infty$, this implies
$(1-\epsilon)u^{(\operatorname{odd})}\preceq u^{(\operatorname{ev})}\preceq
u^{(\operatorname{odd)}}$. Since $\epsilon>0$ was arbitrary,

$u^{(\operatorname{ev)}}=u^{(\operatorname{odd})}$

as needed. We will denote
$u^{*}:=u^{(\operatorname{ev)}}=u^{(\operatorname{odd})}$.

Write $f^{n}\colon X\to X$ for the $n$-fold composition of $f$ with itself,
i.e $f^{n}=f\circ\cdots\circ f$. Since $0\in X$ is minimal, $u^{(0)}\preceq x$
for any $x\in X$. Since $f$ is order reversing, $u^{(0)}\preceq f(x)\preceq
u^{(1)}$. It then follows by induction that for all $k\geq 1$

$\displaystyle f^{2k-1}(X)\subseteq[u^{(2k-2)},u^{(2k-1)}]_{X}$ $\displaystyle
f^{2k}(X)\subseteq[u^{(2k)},u^{(2k-1)}]_{X}.$

It follows from Equation (ii) that

$\displaystyle\bigcap_{n\geq 1}f^{n}(X)$ $\displaystyle=\bigcap_{k\geq
1}f^{2k-1}(X)\cap f^{2k}(X)$ $\displaystyle\subseteq\bigcap_{k\geq
1}[u^{(2k-2)},u^{(2k-1)}]_{X}\cap[u^{(2k)},u^{(2k-1)}]_{X}$
$\displaystyle=\bigcap_{k\geq 1}[u^{(2k)},u^{(2k-1)}]_{X}$
$\displaystyle=[u^{(\operatorname{ev})},u^{(\operatorname{odd})}]_{X}=\\{u^{*}\\}.$

Notice that $u^{*}\in[u^{(2k)},u^{(2k-1)}]_{X}$ for all $k\geq 1$. Since $f$
is order reversing $f(u^{*})\in[u^{(2k)},u^{(2k+1)}]_{X}$ for all $k\geq 1$.
Therefore $f(u^{*})\in\\{u^{*}\\}$, so $u^{*}$ is a fixed point of $f$.

If $y\in X$ is a fixed point of $f$ then by induction $y\in f^{n}(X)$ for all
$n$, so $y\in\cap_{n}f^{n}(X)=\\{u^{*}\\}$ and it follows that $y=u^{*}$.
Therefore $u^{*}$ is the unique fixed point of $f$.

Finally, choose some $x\in X$ and define a sequence by recursion $x^{(0)}=x$
and $x^{(n+1)}=f(x^{(n)})$. Then $u^{(0)}=0\preceq x$ and since $f$ is order
reversing, $u^{(0)}\preceq x^{(1)}\preceq u^{(1)}$. Then one proves by
induction that $u^{(2k-2)}\preceq x^{(2k-1)}\preceq u^{(2k-1)}$ and
$u^{(2k)}\preceq x^{(2k)}\preceq u^{(2k-1)}$ for all $k\geq 1$. By the
sandwich rule $\lim_{n}x^{(n)}=u^{*}$. ∎

###### Proof of Theorem 3.17.

Let $(c,f)\in\operatorname{Base}(X)$. Thus $c\in K$ and $f\colon
X\to[0,\infty)^{n}$ is bounded, homogeneous and increasing. Set $F=T_{(c,f)}$,
see Definition 3.13. Recall that $F\in\operatorname{End}(X)$.

Claim 1: $F$ is order reversing.

Proof of Claim 1: Suppose that $x\preceq y$, i.e $x_{i}\leq y_{i}$ for all
$i=1,\dots,n$. Since $f_{i}\colon X\to[0,\infty)$ is increasing

$F(y)_{i}=\frac{c_{i}}{1+f(y)_{i}}\leq\frac{c_{i}}{1+f(x)_{i}}=F(x)_{i}.$

It follows that $F(y)\preceq F(x)$. This proves Claim 1.

Claim 2: There exists $0<\alpha\leq 1$ such that for any $x\in X$ and any
$0\leq t\leq 1$

$F(tx)\preceq\frac{1}{t+\alpha\cdot(1-t)}\cdot F(x).$

Proof of Claim 2: Say that the functions $f_{i}\colon X\to[0,\infty)$ are
bounded above by $M>0$. Set $\alpha=\tfrac{1}{1+M}$. Then for any $x\in X$

$\frac{1}{1+f(x)_{i}}\geq\frac{1}{1+M}=\alpha.$

Since $f_{i}$ is homogeneous, for any $x\in X$ and any $0\leq t\leq 1$,

$\displaystyle F(t\cdot x)_{i}$ $\displaystyle=\frac{c_{i}}{1+f_{i}(t\cdot
x)}$ $\displaystyle=\frac{c_{i}}{1+t\cdot f_{i}(x)}$
$\displaystyle=\frac{c_{i}}{(1-t)+t(1+f_{i}(x))}$
$\displaystyle=\frac{c_{i}}{1+f_{i}(x)}\cdot\frac{1+f_{i}(x)}{(1-t)+t(1+f_{i}(x))}$
$\displaystyle=F(x)_{i}\cdot\frac{1}{(1-t)\tfrac{1}{1+f_{i}(x)}+t}$
$\displaystyle\leq\frac{1}{(1-t)\alpha+t}\cdot F(x)_{i}.$

The theorem follows by applying Theorem 3.16 to $F$. ∎

######  Definition A.1.

Let $Y$ be a set. Let $U$ be a subset of $W$. A function $f\colon W\to Y$ is
called independent of weights on $U$ if $f|_{U}\colon U\to Y$ is a constant
function. The constant value of $f|_{U}$, i.e the element $y\in Y$ such that
$f|_{U}\equiv y$, is called the $U$-value of $f$.

We say that $f$ is independent of weights if it is independent of weights on
$W$, i.e it is constant.

Recall that a topological space $X$ is called locally compact if every $x\in
X$ has a compact neighbourhood, i.e a neighbourhood whose closure is compact.

######  Lemma A.2.

Let $X,Y$ be metric spaces, $X$ compact. Let $D\subseteq X$ be a subspace and
$C\subseteq Y$ locally compact subspace. Let $k\colon X\to Y$ be a continuous
function such that

  1. (a)

$k(D)=C$ and the restriction $k|_{D}\colon D\to C$ is a bijective function
with inverse $f\colon C\to D$.

  2. (b)

For every $x\in X$,

$k(x)\in C\iff x\in D.$

Then $k|_{D}$ and $f$ are homeomorphisms.

###### Proof.

Since $k|_{D}$ is continuous, it remains to show that its inverse $f$ is
continuous. To do this it suffices to show that $f$ is continuous at any $c\in
C$.

Let $c\in C$. Since $C$ is locally compact, choose a neighbourhood $c\in
U\subseteq C$ whose closure $\overline{U}$ in $C$ is compact. Set

$E:=f(\overline{U})\subseteq D.$

We will next show that $E$ is a closed subset of $X$. Let $(e_{n})$ be a
convergent sequence in $X$ contained in $E$ and set
$x=\lim_{n\to\infty}e_{n}$. We must show that $x\in E$.

By construction $e_{n}=f(u_{n})$ for some $u_{n}\in\overline{U}$. Since $k$ is
continuous, the sequence $k(e_{n})$ is a convergent sequence in $Y$ and

$\lim_{n\to\infty}k(e_{n})=k(x).$

By the assumption (b) $k(e_{n})\in C$ (because $e_{n}\in E\subseteq D$), and
assumption (a) implies that

$f(k(e_{n}))=e_{n}=f(u_{n}).$

Since $f$ is injective, $k(e_{n})=u_{n}\in\overline{U}$. Since $\overline{U}$
is compact it is closed in $Y$, hence $k(x)\in\overline{U}\subseteq C$.
Assumption (b) implies that $x\in D$, and by assumption (a)

$x=f(k(x))\in f(\overline{U})=E.$

This completes the proof that $E$ is a closed subset of $X$.

Since $X$ is compact, so is $E$. Assumption (a) now implies that $k|_{E}\colon
E\to\overline{U}$ is a bijective continuous function between compact metric
spaces, hence it is a homeomorphism. Therefore its inverse $f|_{\overline{U}}$
is continuous and in particular, since $U$ is open, $f$ is continuous at $c$.
This completes the proof. ∎

######  Theorem A.3.

Let $b\colon W\to\operatorname{Base}(X)$ be a weighted scoring base. Write
$b=(\kappa,\varphi)$ as in Definition 3.18. Assume that $\varphi$ is
independent of weights on some $U\subseteq W$ with $U$-value $\psi\colon
X\to[0,\infty)^{n}$. Then

  1. (a)

There exists a bijective function

$\overline{\sigma_{b,U}}\colon\kappa(U)\xrightarrow{\ \cong\
}\sigma_{b}(U)\subseteq X$

which renders the following triangle commutative

---
$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{b}}$$\scriptstyle{\kappa}$$\textstyle{X}$$\textstyle{\kappa(U)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\sigma_{b,U}}}$

  2. (b)

The inverse of $\overline{\sigma_{b,U}}$ is given by the restriction to
$\sigma_{b}(U)\subseteq X$ of the function

$\overline{\operatorname{inv}_{b,U}}\colon X\to\mathbb{R}^{n}$

defined on each components by

$\overline{\operatorname{inv}_{b,U}}(x)_{i}=x_{i}(1+\psi(x)_{i}).$

  3. (c)

For any $x\in X$,

$\overline{\operatorname{inv}_{b,U}}(x)\in\kappa(U)\iff x\in\sigma_{b}(U).$

  4. (d)

If $\kappa(U)$ is locally compact then $\overline{\sigma_{b,U}}$ is a
homeomorphism.

###### Proof.

(a) and (b). Define $\overline{\sigma_{b,U}}\colon\kappa(U)\to\sigma_{b}(U)$
as follows. Given $c\in\kappa(U)$ choose $u\in U$ such that $\kappa(u)=c$ and
set

$\overline{\sigma_{b,U}}(c)=\sigma_{b}(u)$

We need to show that this is independent of the choices. If $u^{\prime}\in U$
is another preimage of $c$ then by the definition of $\sigma_{b}$ in
Definition 3.18 and since $\varphi$ is independent of weights on $U$ and
$\kappa(u)=c=\kappa(u^{\prime})$

$\sigma_{b}(u^{\prime})=\mathrm{fix}(T_{(\kappa(u^{\prime}),\varphi(u^{\prime}))})=\mathrm{fix}(T_{(\kappa(u),\varphi(u)}))=\sigma_{b}(u).$

Hence $\overline{\sigma_{b,U}}(c)$ is independent of the choice of the
preimages of $c$ in $U$. By construction,
$\overline{\sigma_{b,U}}(c)\in\sigma_{b}(U)$ and the diagram is commutative,
that is

$\overline{\sigma_{b,U}}\circ\kappa|_{U}=\sigma_{b}|_{U}.$

In particular $\overline{\sigma_{b,U}}$ is onto $\sigma_{b}(U)$ which we
henceforce denote $D_{b,U}$.

Look at $\overline{\operatorname{inv}_{b,U}}\colon X\to\mathbb{R}^{n}$ defined
in the statement of the theorem and recall that
$\psi\in\mathcal{B^{+}HI}_{n}(X)$ (Definition 3.9). Consider some
$c\in\kappa(U)$ and set $y=\overline{\sigma_{b,U}}(c)$. Then $c=\kappa(u)$ for
some $u\in U$ and by construction

$y=\overline{\sigma_{b,U}}(c)=\sigma_{b}(u)\in D_{b,U}.$

By the definition of $\sigma_{b}$,

$y=\mathrm{fix}(T_{b(u)})$

and by Definition 3.13, $T_{b(u)}=T_{(c,\psi)}$ and

$y_{i}=\frac{c_{i}}{1+\psi(y)_{i}}\qquad(1\leq i\leq n).$

Then $c_{i}=y_{i}(1+\psi(y)_{i})$ and notice that $0\leq y_{i}\leq 1$ so $y\in
X$. It follows that $c=\overline{\operatorname{inv}_{b,U}}(y)$. We have shown
that

$\overline{\operatorname{inv}_{b,U}}|_{D_{b,U}}\,\circ\,\overline{\sigma_{b,U}}\,=\,\operatorname{id}_{\kappa(U)}.$

It follows that $\overline{\sigma_{b,U}}$ is injective, hence bijective onto
its image $D_{b,U}$ and $\overline{\operatorname{inv}_{b,U}}|_{D_{b,U}}$ is
its inverse.

(c). Let $x\in X$. Suppose that
$\overline{\operatorname{inv}_{b,U}}(x)\in\kappa(U)$. Then
$\overline{\operatorname{inv}_{b,U}}(x)=\kappa(u)$ for some $u\in U$. Thus,
$\kappa(u)_{i}=x_{i}(1+\psi(x)_{i})$ for all $i=1,\dots,n$. Definition 3.13

$x_{i}=\frac{\kappa(u)_{i}}{1+\psi(x)_{i}}=T_{(\kappa(u),\psi)}(x)_{i}.$

It follows that $T_{(\kappa(u),\psi)}(x)=x$. Since $\psi=\varphi(u)$ and
$b(u)=(\kappa(u),\varphi(u))$ is a scoring base, it follows that
$\mathrm{fix}(T_{b(u)}(x)=x$ so $\sigma_{b}(u)=x$, see Definition 3.18. Hence
$x\in\sigma_{b}(U)=D_{b,U}$.

Conversely, suppose that $x\in D_{b,U}$. By construction there exists $u\in U$
such that $x=\sigma_{b}(u)=\overline{\sigma_{b,U}}(\kappa(u))$. By definition,
$x=\mathrm{fix}(T_{\kappa(u),\varphi(u)})$, hence

$x_{i}=T_{(\kappa(u),\varphi(u))}(x)_{i}=T_{(\kappa(u),\psi)}(x)_{i}=\frac{\kappa(u)_{i}}{1+\psi(x)_{i}}.$

It follows that

$\kappa(u)_{i}=x_{i}(1+\psi(x)_{i})=\overline{\operatorname{inv}_{b,U}}(x)_{i}\qquad(1\leq
i\leq n).$

Thus, $\overline{\operatorname{inv}_{b,U}}(x)=\kappa(u)\in\kappa(U)$.

(d). Apply Lemma A.2 with the spaces $X=[0,1]^{n}=K$, $C=\kappa(U)\subseteq
K$, $Y=\mathbb{R}^{n}$ and $D=D_{b,U}$, and with
$k=\overline{\operatorname{inv}_{b,U}}$ and $f=\overline{\sigma_{b,U}}$.
Condition (a) of that lemma is guaranteed by parts (a) and (b) of this theorem
which we have proven above, and condition (b) of Lemma A.2 is part (c). ∎

Recall the concept of preservation of support and independence of supports
from Definition 4.1.

######  Proposition A.4.

The function $\overline{\operatorname{inv}_{b,U}}$ defined in Theorem A.3
preserves supports.

###### Proof.

$x_{i}=0\iff
x_{i}(1+\psi^{I}(x)_{i})=0\iff\overline{\operatorname{inv}_{b,U}}=0$. ∎

######  Lemma A.5.

Let $(c,f)$ be a scoring base (Definition 3.12). Consider $T_{(c,f)}\colon
X\to X$ (Definition 3.13 and Theorem 3.17). Then

$\mathrm{fix}(T_{c,f})_{i}=0\iff c_{i}=0.$

###### Proof.

Set $x=\mathrm{fix}(T_{(c,f)})$. By construction of $T_{(c,f)}$ we get
$c_{i}=x_{i}(1+f(x)_{i})$ and the result follows since $f\geq 0$ by
hypothesis. ∎

######  Theorem A.6.

Let $b=(\kappa,\varphi)$ be a weighted scoring base. Then $\sigma_{b}\colon
W\to X$ preserves supports if and only if $\kappa\colon W\to K$ preserves
supports.

###### Proof.

Consider some $w\in W$. By Lemma A.5 and the definition of $\sigma_{b}$

$\sigma_{b}(w)_{i}=0\iff\mathrm{fix}(T_{(\kappa(u),\varphi(u)})_{i}=0\iff\kappa(w)_{i}=0.$

We have shown that $\mathrm{supp}(\sigma_{b}(w))=\mathrm{supp}(\kappa(w))$ for
all $w\in W$. So $\sigma_{b}$ preserves supports (i.e
$\mathrm{supp}(\sigma_{b}(w))=\mathrm{supp}(w)$ for all $w$) if and only if
$\kappa$ preserves supports (i.e $\mathrm{supp}(\kappa(w))=\mathrm{supp}(w)$
for all $w$). ∎

Let $\wp(n)$ be the power set of $\\{1,\dots,n\\}$. The support gives a
function

$\mathrm{supp}\colon\mathbb{R}^{n}\to\wp(n)$

Given $I\in\wp(n)$ write

$\mathbb{R}^{n}(I)=\mathrm{supp}^{-1}(I)=\\{v\in\mathbb{R}^{n}:\mathrm{supp}(v)=I\\}.$

More generally, if $A\subseteq\mathbb{R}^{n}$ we write
$A(I):=A\cap\mathbb{R}^{n}(I)$, the set of all $x\in A$ with support $I$.

With the terminology of Definition 4.1, the statement that a function $f\colon
W\to Y$, where $Y$ is some set, is independent of supports is equivalent to
the assertion that $f$ is independent of weights on $W(I)$, see Definition
A.1, for all $I\in\wp(n)$.

###### Proof of Theorem 4.3.

Write $b=(\kappa,\varphi)$. Clearly $\\{W(I)\\}_{I\in\wp(n)}$ forms a
partition of $W$. Since $\kappa$ preserves supports, it easily follows that
$\kappa(W(I))=\kappa(W)\cap\mathbb{R}^{n}(I)$ for every $I\in\wp(n)$. In
particular, $\\{\kappa(W(I))\\}_{I\in\wp(n)}$ forms a partition of
$\kappa(W)$.

Set $D_{b}=\sigma_{b}(W)$. By Theorem A.6 $\sigma_{b}\colon W\to X$ preserves
supports, hence $\sigma_{b}(W(I))=D_{b}(I)$. Therefore
$\\{\sigma_{b}(W(I))\\}_{I\in\wp(n)}$ forms a partition of $D_{b}$.

By Theorem A.3 for every $I\in\wp(n)$ there is an injective function

$\overline{\sigma_{b,W(I)}}\colon\kappa(W(I))\to X$

with image $\sigma_{b}(W(I))$ such that
$\overline{\sigma_{b,W(I)}}\circ\kappa|_{\kappa(W(I))}=\sigma_{b}|_{W(I)}$.
There is also a discerning right inverse for $\overline{\sigma_{b,W(I)}}$,

$\overline{\operatorname{inv}_{b,W(I)}}\colon X\to\mathbb{R}^{n}.$

Since $\\{\kappa(W(I))\\}_{I}$ form a partition of $\kappa(W)$, we define
$\overline{\sigma_{b}}\colon\kappa(W)\to X$ by the requirement

$\overline{\sigma_{b}}|_{\kappa(W(I))}=\overline{\sigma_{b,W(I)}}.$

Since the image of $\overline{\sigma_{b,W(I)}}$ is $D_{b}(I)$ and since
$\\{D_{b}(I)\\}_{I}$ forms a partition of $D_{b}$, it follows that
$\overline{\sigma_{b}}$ is injective with image $D_{b}=\sigma_{b}(W)$. For any
$w\in W$ set $I=\mathrm{supp}(\kappa(w))$. Since $\kappa$ preserves supports
$\mathrm{supp}(w)=I$ and we get
$\overline{\sigma_{b}}\circ\kappa(w)=\overline{\sigma_{b,W(I)}}\circ\kappa(w)=\sigma_{b}|_{W(I)}(w)$.
We deduce that

$\overline{\sigma_{b}}\circ\kappa=\sigma_{b}.$

Since $\\{X(I)\\}_{I\in\wp(n)}$ is clearly a partition of $X$, we define
$\overline{\operatorname{inv}_{b}}\colon X\to\kappa(W)$ by the requirement

$\overline{\operatorname{inv}_{b}}|_{X(I)}=\overline{\operatorname{inv}_{b,W(I)}}|_{X(I)}.$

By the defining formula for $\overline{\operatorname{inv}_{b,W(I)}}|_{X(I)}$
in Theorem A.3(b), for any $x\in X$, if we set $I=\mathrm{supp}(x)$

$\overline{\operatorname{inv}_{b,W(I)}}(x)_{i}=x_{i}(1+\psi^{I}(x)_{i}).$

Thus, $\overline{\operatorname{inv}_{b}}$ we have just defined coincides with
$\overline{\operatorname{inv}}$ in the statement of the theorem. Notice that
by this formula it is clear that $\overline{\operatorname{inv}_{b}}$ preserves
supports since $\overline{\operatorname{inv}_{b}}_{i}=0\iff x_{i}=0$.

Next we claim that
$x\in\sigma_{b}(W)\iff\overline{\operatorname{inv}_{b}}(x)\in\kappa(W)$ and
that in this case
$\overline{\sigma_{b}}\circ\overline{\operatorname{inv}_{b}}(x)=x$. Indeed,
suppose that $x\in\sigma_{b}(W)$ and set $I=\mathrm{supp}(x)$. Since
$\sigma_{b}$ preserves supports, $x=\sigma_{b}(w)$ for some $w\in W(I)$. Thus,
$x\in\sigma_{b}(W(I))$. Since $\overline{\sigma_{b,W(I)}}$ is a discerning
right inverse for $\overline{\sigma_{b,{W(I)}}}$ it follows that
$\overline{\operatorname{inv}_{b}}(x)=\overline{\operatorname{inv}_{b,W(I)}}(x)\in\kappa(W(I))$
as needed. Furthermore, by the definition of $\overline{\sigma_{b}}$

$\overline{\sigma_{b}}(\overline{\operatorname{inv}_{b}}(x))=\overline{\sigma_{b}}(\overline{\operatorname{inv}_{b,W(I)}}(x))=\overline{\sigma_{b,W(I)}}(\overline{\operatorname{inv}_{b,W(I)}}(x))=x.$

Conversely, suppose that $\overline{\operatorname{inv}_{b}}(x)\in\kappa(W)$.
Set $I=\mathrm{supp}(x)$. Since $\overline{\operatorname{inv}_{b}}$ preserves
supports,
$\overline{\operatorname{inv}_{b}}(x)\in\kappa(W)\cap\mathbb{R}^{n}(I)=\kappa(W(I))$.
By definition,
$\overline{\operatorname{inv}_{b}}(x)=\overline{\operatorname{inv}_{b,W(I)}}(x)$
so $\overline{\operatorname{inv}_{b,W(I)}}(x)\in\kappa(W(I))$. Since
$\overline{\operatorname{inv}_{b,W(I)}}$ is a discerning right inverse for
$\overline{\sigma_{b,W(I)}}$, it follows that
$x\in\sigma_{b}(W(I))\subseteq\sigma_{b}(W)$. We have thus shown that
$\overline{\operatorname{inv}_{b}}$ is a discerning right inverse for
$\overline{\sigma_{b}}$.

Define $\operatorname{inv}_{b}\colon X\to\mathbb{R}^{n}$ by
$\operatorname{inv}_{b}=\tilde{\kappa}^{-1}\circ\overline{\operatorname{inv}_{b}}$.
Then $\operatorname{inv}_{b}$ coincides with $\operatorname{inv}$ in the
statement of the theorem. Since $\sigma_{b}=\overline{\sigma_{b}}\circ\kappa$
we get

$x\in\sigma_{b}(W)\iff
x\in\overline{\sigma_{b}}(\kappa(W))\iff\overline{\operatorname{inv}_{b}}(x)\in\sigma_{b}(W).$

Furthermore, since $\overline{\operatorname{inv}_{b}}$ is a discerning right
inverse of $\overline{\sigma_{b}}$

$\sigma_{b}\circ\operatorname{inv}_{b}|_{D_{b}}=(\overline{\sigma_{b}}\circ\kappa)\circ(\tilde{\kappa}^{-1}\circ\overline{\operatorname{inv}_{b}}|_{D_{b}})=\overline{\sigma_{b}}\circ\overline{\operatorname{inv}_{b}}|_{D_{b}}=\operatorname{id}_{D_{b}}.$

Thus, we have shown that $\operatorname{inv}_{b}$ is a discerning right
inverse for $\sigma_{b}$ as needed.

Finally, suppose that $\varphi$ is independent of weights with values
$\psi\colon X\to[0,\infty)^{n}$ which is continuous. Since $\tilde{\kappa}$ is
a homeomorphism and $W$ is compact, $\kappa(W)=\tilde{\kappa}(W)$ is compact.
Theorem A.3(d) shows that $\sigma_{b}=\overline{\sigma_{b}}\circ\kappa$ is a
homemorphism onto its image. ∎

###### Proof of Theorem 4.4.

Set $I=\mathrm{supp}(y)$. By the hypothesis $\varphi$ is independent of
weights on $W(I)$. Let $\psi\colon X\to[0,\infty)^{n}$ be the $W(I)$-value of
$\varphi$.

Let $\overline{\sigma_{b,W(I)}}\colon\kappa(W(I))\to X$ and
$\overline{\operatorname{inv}_{b,W(I)}}\colon X\to\mathbb{R}^{n}$ be as in
Theorem A.3. Thus, $\overline{\sigma_{b,W(I)}}$ is bijective onto
$\sigma_{b}(W(I))$. By Theorem A.6 $\sigma_{b}$ preserves supports, and it
easily follows that $\sigma_{b}(W(I))=D_{b}(I)$ where $D_{b}=\sigma_{b}(W)$.
Similarly, since $\kappa$ preserves supports,

$\kappa(W(I))=\mathbb{R}^{n}(I)\cap\kappa(W).$

Clearly, $t\cdot y\in X(I)$ for any $0<t\leq 1$. By definition of
$\overline{\operatorname{inv}_{b,W(I)}}$ in Theorem A.3 and since by
hypothesis $\psi_{i}$ are homogeneous

$\overline{\operatorname{inv}_{b,W(I)}}(t\cdot
y)_{i}=ty_{i}(1+\psi(ty)_{i})=ty_{i}(1+t\cdot\psi(y)_{i})\xrightarrow{\ t\to
0\ }0.$

It follows that there exists $s>0$ such that
$\overline{\operatorname{inv}_{b,W(I)}}(ty)\in U$ for every $0<t\leq s$.
Moreover, for any $t>0$ it is clear that
$\overline{\operatorname{inv}_{b,W(I)}}(ty)_{i}=0$ if and only if $y_{i}=0$,
hence
$\mathrm{supp}(\overline{\operatorname{inv}_{b,W(I)}}(ty))=\mathrm{supp}(y)=I$
for any $t>0$. Since $\kappa$ preserves supports, for any $0<t\leq s$.

$\overline{\operatorname{inv}_{b,W(I)}}(t\cdot y)\in\mathbb{R}^{n}(I)\cap
U\subseteq\mathbb{R}^{n}(I)\cap\kappa(W)=\kappa(W(I)).$

Part (c) of Theorem A.3 implies that $ty\in\sigma_{b}(W(\alpha))\subseteq
D_{(\kappa,\varphi)}$. ∎

###### Proof of Theorem 4.6.

Consider some $x\in D_{b}=\sigma_{b}(W)$. We need to prove that the interval
$[0,x]\subseteq D_{b}$, i.e $tx\in D_{b}$ for any $0\leq t\leq 1$. By Theorem
A.6 $\sigma_{b}$ preserves supports. In particular, $\sigma_{b}(0)=0$ (where
$0\in W$ and $0\in X$). Thus, for the rest of the proof we assume that
$0<t\leq 1$.

Set $I=\mathrm{supp}(x)$. Since $x=\sigma_{b}(w)$ for some $w\in W$ and Since
$\sigma_{b}$ preserves supports, it follows that $w\in W(I)$. By hypothesis
$\varphi$ is independent of supports on $W(I)$. Let $\psi\colon
X\to[0,\infty)^{n}$ denote the $W(I)$-value of $\varphi$. Then
$\psi\in\mathcal{B^{+}HI}_{n}(X)$.

Consider $\overline{\sigma_{b},W(I)}\colon\kappa(W(I))\to X$ and
$\overline{\operatorname{inv}_{b,W(I)}}\colon X\to\mathbb{R}^{n}$ from Theorem
A.3. Recall that the image of $\overline{\sigma_{b}}$ is $\sigma_{b}(W(I))$.
Since $x=\sigma_{b}(w)=\overline{\sigma_{b,W(I)}}(\kappa(w))$, it follows from
Theorem A.3 that $\overline{\operatorname{inv}_{b,W(I)}}(x)=\kappa(w)$. By the
construction of $\overline{\operatorname{inv}_{b,W(I)}}$ and since $\psi$ is
homogeneous

$\displaystyle\overline{\operatorname{inv}_{b,W(I)}}(tx)_{i}$
$\displaystyle=tx_{i}(1+\psi(tx)_{i})$ $\displaystyle=tx_{i}(1+t\psi(x)_{i})$
$\displaystyle\leq x_{i}(1+\psi(x)_{i})$
$\displaystyle=\overline{\operatorname{inv}_{b,W(I)}}(x)_{i}$
$\displaystyle=\kappa(x)_{i}.$

Thus,
$\overline{\operatorname{inv}_{b,W(I)}}(tx)\preceq\kappa(w)\in\kappa(W(I))$.
It is clear from the defining formula of
$\overline{\operatorname{inv}_{b,W(I)}}$ that it preserves supports, so
$\mathrm{supp}(\overline{\operatorname{inv}_{b,W(I)}}(tx))=\mathrm{supp}(tx)=\mathrm{supp}(x)=I$.
Since $\kappa(W)$ is $\mathrm{supp}\operatorname{-}\preceq$ closed in $K$ and
since $\kappa(W(I))=\kappa(W)\cap\mathbb{R}^{n}(I)$, we deduce that
$\overline{\operatorname{inv}_{b,W(I)}}(tx)\in\kappa(W(I))$. By Theorem A.3(c)
implies that $tx\in\sigma_{b}(W(I))\subseteq D_{b}$. ∎

Since $\lambda$ and $\mu$ are independent of weights, given $I\in\wp(n)$ we
write $\lambda^{I},\mu^{I}\in\mathbb{R}^{n}$ for the value of these functions
on $\mathbb{R}^{n}(I)$.

###### Proof of Proposition 5.1.

Since $0\leq\mu(w)_{i}\leq 1$ it is clear that $\kappa(w)\in K$ for all $w\in
W$ so $\kappa\colon W\to K$ is well defined.

Choose some $w\in W$ and set $\psi=\varphi(w)$. We need to show that
$\psi\in\mathcal{B^{+}HI}_{n}(X)$, see Definition 3.9. It is clear that
$\psi\geq 0$ since $\|\ \|_{p}\geq 0$. It is also clear that $\psi$ is
continuous since $\|\ \|_{p}$ is continuous as well as $x\mapsto v\wedge x$
for a fixed $v\in\mathbb{R}^{n}$. Then $\psi$ is bounded since Since $X$ is
compact, $\psi$ is bounded. For any $t\geq 0$ it is clear that

$\psi(tx)_{i}=\|\lambda(w)(a_{i*})\wedge tx\|_{p}=t\|\lambda(w)(a_{i*})\wedge
x\|_{p}=t\cdot\psi(x)_{i}.$

So $\psi$ is homogeneous. Suppose that $x\preceq x^{\prime}$ in $X$, i.e
$x_{i}\leq x^{\prime}_{i}$. Since $\lambda(w)(a_{i,*})\geq 0$ it follows that
$0\leq\lambda(w)(a_{i,*})_{j}x_{j}\leq\lambda(w)(a_{i,*})_{j}x_{j}$. By the
definition of the $L^{p}$-norms (1) it is easy to check that
$\|\lambda(w)(a_{i,*})(a_{i,*})\wedge x\|_{p}\leq\|\lambda(w)(a_{i,*})\wedge
x^{\prime}\|_{p}$, so $\psi(x)\preceq\psi(x^{\prime})$. Hence, $\psi$ is
increasing. ∎

######  Proposition A.7.

Let $(\kappa,\varphi)$ be the weighted scoring base defined in (3). Then

  1. (i)

$\kappa$ is the restriction of a bijective support-preserving function
$\tilde{\kappa}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$. If $\mu$ is independent
of weights then $\tilde{\kappa}$ is a homeomorphism.

  2. (ii)

$\kappa(W)$ contains a neighbourhood of $0$ in $K$.

  3. (iii)

$\kappa(W)$ is $(\mathrm{supp},\preceq)$-closed in $K$.

  4. (iv)

$\varphi$ is independent of supports.

  5. (v)

$\varphi(w)\colon X\to[0,\infty)^{n}$ is continuous for all $w\in W$.

###### Proof.

Define $\tilde{\kappa}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ by

$\tilde{\kappa}(u)_{i}=\mu(u)_{i}u_{i}.$

Then $\tilde{\kappa}$ preserves supports because $\mu(u)_{i}\neq 0$ so
$\tilde{\kappa}(u)_{i}=\iff u_{i}=0$. If $\mu$ is independent of weights, i.e
constant with value $d\in(0,1)^{n}$, then $\tilde{\kappa}$ is the linear
transformation given by the diagonal matrix
$D=\operatorname{diag}(d_{1},\dots,d_{n})$. In particular, it is a
homeomorphism. This proves (i).

Let $\mu^{I}\in(0,1)^{n}$ denote the constant value of $\mu$ on $W(I)$. Then
$\tilde{\kappa}|_{\mathbb{R}^{n}(I)}$ is multiplication by the matrix
$D^{I}=\operatorname{diag}(\mu^{I}_{1},\dots,\mu^{I}_{n})$ which preserves
supports. It follows that

$\kappa(W)=\coprod_{I\in\wp(n)}\left(\prod_{j=1}^{n}[0,\mu^{I}_{j}]\right).$

It is now clear that $\kappa(W)$ is $(\mathrm{supp},\preceq)$-closed and that
it contains the cube $[0,m]$ where $m=\min\\{\mu^{I}_{j}\\}$. This proves (ii)
and (iii).

Since $\lambda$ is independent of supports it is clear that $\varphi$ is
independent of supports with
$\varphi^{\mathbb{R}^{n}(I)}(x)_{i}=\|\lambda^{\mathbb{R}^{n}(I)}(a_{i,*})\wedge
x\|_{p}$. It also follows that $\varphi^{\mathbb{R}^{n}(I)}$ is continuous.
Thus, (iv) and (v) follow. ∎

###### Proof of Theorem 5.3.

By Proposition A.7(i) and (iv), all the conditions of Theorem 4.3 are
satisfied. We deduce that $\sigma_{\kappa,\varphi}$ admits a solution to the
inverse and reflection problems 3.4, 3.5. By that theorem, the inverse
function is computed as follows. Define $\overline{\operatorname{inv}}\colon
X\to\mathbb{R}^{n}$ by

$\overline{\operatorname{inv}}(x)_{i}=x_{i}(1+\varphi^{\mathrm{supp}(x)}(x)_{i})=x_{i}(1+\|\lambda(x)(a_{i,*})\wedge
x\|_{p}).$

Set $I=\mathrm{supp}(x)$. Since $\tilde{\kappa}$ is given on
${\mathbb{R}^{n}(I)}$ by the matrix
$\operatorname{diag}(\mu^{I}_{1},\dots,\mu^{I}_{n})$, and since
$\mu^{I}=\mu(x)$ and $\lambda^{I}=\lambda(x)$,

$\operatorname{inv}(x)_{i}=\tilde{\kappa}^{-1}(\overline{\operatorname{inv}(x)})_{i}=\tfrac{1}{\mu(x)_{i}}x_{i}(1+\|\lambda(x)(a_{i,*})\wedge
x\|_{p}).$

If $\lambda,\mu$ are independent of weights, i.e they are constant, then by
Proposition A.7(i) and (v) the conditions for a solution of the topological
reflections problem hold.

Proposition A.7(i),(ii),(iii) and (iv) show that all the conditions of
Theorems 4.4 and 4.6 are satisfied so $\sigma_{\kappa,\varphi}$ solve the
projective preference ordering problem 3.6 and the radiality problem 3.7. ∎

######  Proposition A.8.

Let $b=(\kappa,\varphi)$ and $b^{\prime}=(\kappa^{\prime},\varphi^{\prime})$
be weighted scoring bases. Suppose that

  1. (i)

$\kappa$ and $\kappa^{\prime}$ preserve supports.

  2. (ii)

$\kappa(W)$ and $\kappa^{\prime}(W)$ are $(\mathrm{supp},\preceq)$-closed in
$K$.

  3. (iii)

$\varphi$ and $\varphi^{\prime}$ are independent of supports. Let $\psi^{I}$
and $\psi^{\prime}{}^{I}$ denote their $W(I)$-value where $I\in\wp(n)$.

  4. (iv)

For every $I\in\wp(n)$ there exists $0<\alpha^{I}\leq 1$ such that that

     * •

$\psi^{\prime}{}^{I}\leq\alpha^{I}\cdot\psi^{I}$

     * •

$\kappa|_{W(I)}\leq\alpha^{I}\cdot\kappa^{\prime}|_{W(I)}$.

Then $D_{(\kappa,\varphi)}\subseteq D_{(\kappa^{\prime},\varphi^{\prime})}$.

###### Proof.

Consider some $x\in D_{(\kappa,\varphi)}$ and set $I=\mathrm{supp}(x)$. Then
$x=\sigma_{b}(w)$ for some $w\in W$. By Theorem A.6 $\sigma_{b}$ preserve
supports. Therefore, $w\in W(I)$ and
$x\in\sigma_{b}(W(I))=\sigma_{b}(W)\cap\mathbb{R}^{n}(I)$. In particular
$\kappa(W(I))\subseteq K(I)$.

Apply Theorem A.3 to obtain functions

$\displaystyle\overline{\sigma_{b,W(I)}}\colon\kappa(W(I))\to X$
$\displaystyle\overline{\operatorname{inv}_{b,W(I)}}\colon X\to\mathbb{R}^{n}$

where $\overline{\sigma_{b,W(I)}}$ is bijective onto $\sigma_{b}(W(I))$ and
$\overline{\operatorname{inv}_{b,W(I)}}$ is a discerning right inverse. Then

$\overline{\operatorname{inv}_{b,W(I)}}(x)=\overline{\operatorname{inv}_{b,W(I)}}(\sigma_{b}(w))=\overline{\operatorname{inv}_{b,W(I)}}(\overline{\sigma_{b,W(I)}}\kappa(w))=\kappa(w)\in\kappa(W(I)).$

By Proposition A.4 $\overline{\operatorname{inv}_{b,W(I)}}$ preserves
supports. Since $0<\alpha^{I}\leq 1$ we get $\alpha^{I}x\in X(I)$, so

$\overline{\operatorname{inv}_{b,W(I)}}(\alpha^{I}x)\in\mathbb{R}^{n}(I).$

Clearly $\alpha^{I}x\preceq x$ and since $\psi^{I}$ is increasing

$0\leq\overline{\operatorname{inv}_{b,W(I)}}(\alpha^{I}x)=\alpha^{I}x_{i}(1+\psi^{I}(\alpha^{I}x)_{i})\leq\\\
x_{i}(1+\psi^{I}(x)_{i})=\overline{\operatorname{inv}_{b,W(I)}}(x)_{i}=\kappa(w)_{i}\leq
1.$

It follows that $\overline{\operatorname{inv}_{b,W(I)}}(\alpha^{I}x)\in
K\cap\mathbb{R}^{n}(I)=K(I)$ and
$\overline{\operatorname{inv}_{b,W(I)}}(\alpha^{I}x)\preceq\kappa(w)\in\kappa(W(I))$.

By assumption $\kappa(W)$ is $(\mathrm{supp},\preceq)$-closed in $K$, so
$\overline{\operatorname{inv}_{b,W(I)}}(\alpha^{I}x)\in\kappa(W)\cap\mathbb{R}^{n}(I)=\kappa(W(I))$.
Therefore $\overline{\operatorname{inv}_{b,W(I)}}(\alpha^{I}x)=\kappa(U)$ for
some $u\in W(I)$. It follows that

$\displaystyle\overline{\operatorname{inv}_{b^{\prime},W(I)}}(x)_{i}$
$\displaystyle=x_{i}(1+\psi^{\prime}{}^{I}(x)_{i})$ $\displaystyle\leq
x_{i}(1+\alpha^{I}\psi^{I}(x)_{i})$
$\displaystyle=\tfrac{1}{\alpha^{I}}(\alpha^{I}x_{i})(1+\psi^{I}(\alpha^{I}x)_{i})$
$\displaystyle=\tfrac{1}{\alpha^{I}}\overline{\operatorname{inv}_{b,W(I)}}(\alpha^{I}x)_{i}$
$\displaystyle=\tfrac{1}{\alpha^{I}}\kappa(u)$
$\displaystyle\leq\tfrac{1}{\alpha^{I}}\alpha^{I}\kappa^{\prime}(u)_{i}$
$\displaystyle=\kappa^{\prime}(u)_{i}.$

Thus, we have shown that

$\overline{\operatorname{inv}_{b^{\prime},W(I)}}(x)\preceq\kappa^{\prime}(u).$

By Proposition A.4
$\mathrm{supp}(\overline{\operatorname{inv}_{b^{\prime},W(I)}}(x))=\mathrm{supp}(x)=I$
and also $\mathrm{supp}(\kappa^{\prime}(u))=\mathrm{supp}(u)=I$ and
$0\leq\overline{\operatorname{inv}_{b^{\prime},W(I)}}(x)_{i}\leq\kappa^{\prime}(u)_{i}\leq
1$. Since $\kappa^{\prime}(W)$ is $(\mathrm{supp},\preceq)$-closed in $K$ we
get
$\overline{\operatorname{inv}_{b,W(I)}}(x)\in\kappa^{\prime}(W)\cap\mathbb{R}^{n}(I)=\kappa^{\prime}(W(I))$.
By Theorem A.3(c)
$x\in\sigma_{b^{\prime}}(W(I))\subseteq\sigma_{b^{\prime}}(W)=D_{(\kappa^{\prime},\varphi^{\prime})}$.
∎

###### Proof of Theorem 5.4.

Let $b=(\kappa,\varphi)$ be the $(L^{p},\lambda,\mu,A)$ weighted scoring base
and $b^{\prime}=(\kappa^{\prime},\varphi^{\prime})$ be the
$(L^{q},\lambda,\mu,A)$ weighted scoring base. By construction
$\kappa=\kappa^{\prime}$.

Conditions (i), (ii) and (iii) of Proposition A.8 hold by Proposition A.7.
Condition (iv) also holds by setting $\alpha^{I}=1$ since $\|\ \|_{q}\leq\|\
\|_{p}$ and $\kappa\leq\kappa^{\prime}$ (in fact, equality). ∎

###### Proof of Proposition 5.5.

Let $\bar{1}\in\mathbb{R}^{n}$ denote the vector $(1,\dots,1)$. We first
highlight the following three items (and their proofs).

  1. 1.

The gradual semantics $\Sigma_{\mathbb{HC}}$ is an abstract
$(L^{p},\lambda,\mu)$-based gradual semantics with $p=1$ and
$\mu\colon\mathbb{R}^{n}\to(0,1]^{n}$ and
$\lambda\colon\mathbb{R}^{n}\to\operatorname{End}([0,\infty)^{n})$ given by

$\displaystyle\mu(v)=\bar{1}\qquad(v\in\mathbb{R}^{n})$
$\displaystyle\lambda(v)=\operatorname{id}\in\operatorname{End}([0,\infty)^{n})\qquad(v\in\mathbb{R}^{n}).$

###### Proof.

By construction $\tilde{\kappa}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the
identity because $\mu(v)_{i}=1$ for all $v\in\mathbb{R}^{n}$. Hence
$\kappa\colon W\to K$ is the identity of $[0,1]^{n}$. The function
$\varphi\colon W\to\operatorname{Func}(X,[0,\infty)^{n})$ is

$\varphi(w)(x)_{i}=\|a_{i,*}\wedge x\|_{1}=\sum_{j}a_{i,j}x_{j}.$

By definition,
$\sigma_{(L^{p},\lambda,\mu,A)}(w)=\mathrm{fix}(T_{(\kappa(w),\varphi(w))})$
and it is the limit of any $T_{(\kappa(w),\varphi(w))}$-sequence in $X$
(Definition 3.14 and Theorem 3.17). Now,

$T_{(\kappa(w),\varphi(w))}(x)_{i}=\frac{\kappa(w)_{i}}{1+\varphi(w)(x)_{i}}=\frac{w_{i}}{1+\|a_{i,*}\wedge
x\|_{1}}.$

Since $A=(a_{i,j})$ is the adjacency matrix of a graph $\mathcal{G}$,

$\|a_{i,*}\wedge
x\|_{1}=\sum_{j}a_{i,j}x_{j}=\sum_{j\in\mathtt{Att}(i)}x_{j}.$

From the construction of $\Sigma_{\mathbb{HC}}$,
$\Sigma_{\mathbb{HC}}^{A,w}=\mathrm{fix}(T_{(\kappa(w),\varphi(w))})=\sigma_{(L^{p},\lambda,\mu,A)}(w)$.
Thus, $\Sigma_{\mathbb{HC}}=\Sigma_{(L^{p},\lambda,\mu)}$. ∎

  2. 2.

The gradual semantics $\Sigma_{\mathbb{MB}}$ is an abstract
$(L^{p},\lambda,\mu)$-based gradual semantics with $p=\infty$ and
$\mu\colon\mathbb{R}^{n}\to(0,1]^{n}$ and
$\lambda\colon\mathbb{R}^{n}\to\operatorname{End}([0,\infty)^{n})$ given by

$\displaystyle\mu(v)=\bar{1}\qquad(v\in\mathbb{R}^{n})$
$\displaystyle\lambda(v)=\operatorname{id}\in\operatorname{End}([0,\infty)^{n})\qquad(v\in\mathbb{R}^{n}).$

###### Proof.

The proof of item 1 can be read verbatim upon replacing $p=1$ with $p=\infty$
and $\Sigma_{\mathbb{HC}}$ with $\Sigma_{\mathbb{MB}}$ in Example 2.3 and
observing that

$\|a_{i,*}\wedge
x\|_{\infty}=\max_{j}\\{a_{i,j}x_{j}\\}=\max_{j\in\mathtt{Att}(i)}x_{j}.$

The details are left to the reader. ∎

  3. 3.

The gradual semantics $\Sigma_{\mathbb{CB}}$ is an abstract
$(L^{p},\lambda,\mu)$-based gradual semantics with $p=1$ and
$\mu\colon\mathbb{R}^{n}\to(0,1]^{n}$ and
$\lambda\colon\mathbb{R}^{n}\to\operatorname{End}([0,\infty)^{n})$ given as
follows. First, define $\theta\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ by

$\theta(v)_{i}=\|a_{i,*}\wedge\mathrm{supp}(v)\|_{1}$

Then define $\mu$ and $\lambda$ by

$\displaystyle\mu(v)_{i}=\frac{1}{1+\theta(v)_{i}}$
$\displaystyle\lambda(v)_{i}=\left\\{\begin{array}[]{ll}\tfrac{\mu(v)_{i}}{\theta(v)_{i}}\cdot(a_{i,*}\wedge\mathrm{supp}(v))&\text{if
$\theta(v)_{i}\neq 0$}\\\ 0&\text{if $\theta(v)_{i}=0$}\end{array}\right.$

###### Proof.

Choose some $w\in W$. By the definition of $\kappa$ and $\varphi$ in
Subsection 5

$\displaystyle T_{(\kappa(w),\varphi(w))}(x)_{i}$
$\displaystyle=\frac{w_{i}/(1+\theta(w)_{i})}{1+\tfrac{\mu(v)_{i}}{\theta(v)_{i}}\|\mathrm{supp}(w)\wedge
a_{i,*}\wedge x\|_{1}}$
$\displaystyle=\frac{w_{i}}{1+\theta(w)_{i}+\tfrac{1}{\theta(w)_{i}}\|\mathrm{supp}(w)\wedge
a_{i,*}\wedge x\|_{1}}$

with the convention that $\tfrac{1}{\theta(w)_{i}}=0$ if $\theta(w)_{i}=0$.
Now,
$\sigma_{(L^{p},\lambda,\mu,A)}(w)=\mathrm{fix}(T_{(\kappa(w),\varphi(w))})$
is the limit of any $T_{(\kappa(w),\varphi(w))}$-sequence in $X$. We observe
that since $A=(a_{i,j})$ is the adjacency matrix of $\mathcal{G}$,

$\displaystyle\theta(w)_{i}=|\mathtt{Att}^{*}(i)|$
$\displaystyle\|\mathrm{supp}(w)\wedge a_{i,*}\wedge
x\|_{1}=\sum_{j\in\mathtt{Att}^{*}(i)}a_{i,j}x_{j}=\sum_{j\in\mathtt{Att}^{*}(i)}x_{j}.$

It follows from the construction of $\Sigma_{\mathbb{CB}}$ in Example 2.4 that
$\Sigma_{\mathbb{CB}}^{\mathcal{G},w}=\sigma_{(L^{p},\lambda,\mu,A)}$. Thus,
$\Sigma_{\mathbb{CB}}=\Sigma_{(L^{p},\lambda,\mu)}$. ∎

We can conclude from items 1, 2 and 3 and Theorem 5.3. ∎

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