\documentclass{tlp}

\hyphenation{either}

\providecommand\AMSLaTeX{AMS\,\LaTeX}
\newcommand\eg{\emph{e.g.}\ }
\newcommand\etc{\emph{etc.}}
\newcommand\bcmdtab{\noindent\bgroup\tabcolsep=0pt
  \begin{tabular}{@{}p{10pc}@{}p{20pc}@{}}}
\newcommand\ecmdtab{\end{tabular}\egroup}
\newcommand\rch[1]{$\longrightarrow\rlap{$#1$}$\hspace{1em}}
\newcommand\lra{\ensuremath{\quad\longrightarrow\quad}}
\usepackage{times}



\usepackage{color}
\usepackage{xspace}
\usepackage{graphicx} 
\usepackage{amssymb} 
\usepackage{multirow,caption,hhline} 
\usepackage{balance}
\usepackage[noend]{algorithmic}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{proposition}{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem{fact}{Fact}
\usepackage{mathtools}
\def\imp{\mbox{$<\hspace*{-0.5mm}\simx\hspace*{+0.5mm}\simx\hspace*{+1.3mm}$}}

\newcommand{\attacks}[1]{\ensuremath{\overset{#1}{\longrightarrow}}}
\newcommand{\support}[1]{\ensuremath{\overset{#1}{\Longrightarrow}}}

\newcommand{\supportplus}{\ensuremath{\overset{+}{\Rightarrow}}}
\newcommand{\supportone}{\ensuremath{\overset{\beta_1}{\Rightarrow}}}
\newcommand{\supporttwo}{\ensuremath{\overset{\beta_2}{\Rightarrow}}}
\newcommand{\supportn}{\ensuremath{\overset{\beta_n}{\Rightarrow}}}


\newcommand{\wrt}{w.r.t.\xspace}
\renewcommand{\iff}{iff\xspace}
\newcommand{\ie}{\textit{i.e.}\xspace}

\newcommand{\ASAFname}{ASAF\ }

\newcommand{\ASAFex}{\ensuremath{\Delta}\xspace}
\newcommand{\ASAFexN}[1]{\ensuremath{\ASAFex_{#1}}\xspace}

\newcommand{\Set}[1]{\ensuremath{\mathbf{#1}}}

\newcommand{\aspic}{\mbox{\textsc{aspic+}}\xspace}

\newcommand{\Arg}[1]{\ensuremath{\mathcal{#1}}\xspace}
\newcommand{\Args}{\ensuremath{\mathbbm{A}}\xspace}
\newcommand{\ArgN}[2]{\ensuremath{\mathcal{#1}_{#2}}\xspace}
\newcommand{\afarg}{\ensuremath{\hbox{\rm S}}\xspace}
\newcommand{\afneg}{\ensuremath{\hbox{\rm C}}\xspace}

\newcommand{\adfarg}{\ensuremath{\hbox{\rm S}}\xspace}
\newcommand{\adflink}{\ensuremath{\hbox{\rm L}}\xspace}
\newcommand{\adfcond}{\ensuremath{\hbox{\rm C}}\xspace}

\newcommand{\adfsymbol}{\ensuremath{\hbox{\rm D}}\xspace}

\newcommand{\AFargs}{\ensuremath{\Args^{AF}}\xspace}
\newcommand{\AFattRel}{\ensuremath{\mathbbm{R}^{AF}}\xspace}
\newcommand{\AF}{\ensuremath{\langle \AFargs, \AFattRel\rangle}\xspace}

\newcommand{\ArgsNew}{\ensuremath{\mathsf{A}}\xspace}

\newcommand{\AFargsNew}{\ensuremath{A}\xspace}
\newcommand{\AFattRelNew}{\ensuremath{\Omega}\xspace}
\newcommand{\AFNew}{\ensuremath{\langle \AFargsNew, \AFattRelNew\rangle}\xspace}




\newcommand{\AFRA}{\ensuremath{\langle \AFRAargs, \AFRAattRel \rangle}\xspace}
\newcommand{\AFRAargs}{\Args\xspace}
\newcommand{\AFRAattRel}{\ensuremath{\mathbbm{R}}\xspace}

\newcommand{\AFRAn}[1]{\ensuremath{\langle \AFRAargs_{#1}, \AFRAattRel_{#1} \rangle}\xspace}

\newcommand{\AFRAatt}[1]{\ensuremath{\stackrel{#1}{\longrightarrow}}\xspace}

\newcommand{\AFRAdungAF}{\ensuremath{\langle\widetilde{\AFRAargs}, \widetilde{\AFRAattRel}\rangle}\xspace}
\newcommand{\AFN}{\ensuremath{\langle \AFNargs, \AFNattRel, \AFNsuppRel \rangle}\xspace}

\newcommand{\AFNex}{\ensuremath{\ASAFex^{AFN}}\xspace}

\newcommand{\AFNargs}{\ensuremath{\ASAFargs^{AFN}}}
\newcommand{\AFNattRel}{\ensuremath{\ASAFattRel^{AFN}}\xspace}
\newcommand{\AFNsuppRel}{\ensuremath{\ASAFsuppRel^{AFN}}\xspace}

\newcommand{\AFNatt}{\AFatt}
\newcommand{\AFNsupp}{\ensuremath{\Longrightarrow}\xspace}



\newcommand{\AFatt}{\ensuremath{\longrightarrow}\xspace}


\newcommand{\ASAF}{\ensuremath{\langle \ASAFargs, \ASAFattRel, \ASAFsuppRel \rangle}\xspace}
\newcommand{\ASAFargs}{\Args}
\newcommand{\ASAFattRel}{\ensuremath{\mathbbm{R}}\xspace}
\newcommand{\ASAFsuppRel}{\ensuremath{\mathbbm{S}}\xspace}

\newcommand{\ASAFunivAttRel}{\ensuremath{\mathbbm{\ASAFattRel}_\mathbbm{U}}\xspace}
\newcommand{\ASAFunivSuppRel}{\ensuremath{\mathbbm{\ASAFsuppRel}_\mathbbm{U}}\xspace}


\newcommand{\ASAFatt}[1]{\ensuremath{\stackrel{#1}{\longrightarrow}}\xspace}
\newcommand{\ASAFsupp}[1]{\ensuremath{\stackrel{#1}{\Longrightarrow}}\xspace}

\newcommand{\ASAFargsN}[1]{\ensuremath{\ASAFargs_{\mathit{#1}}}\xspace}
\newcommand{\ASAFsuppRelN}[1]{\ensuremath{\ASAFsuppRel_{\mathit{#1}}}\xspace}
\newcommand{\ASAFattRelN}[1]{\ensuremath{\ASAFattRel_{\mathit{#1}}}\xspace}

\newcommand{\ASAFn}[1]{\ensuremath{\langle \ASAFargsN{#1}, \ASAFattRelN{#1}, \ASAFsuppRelN{#1} \rangle}\xspace}



\newcommand{\DeltaAF}{\ensuremath{\Delta^{\mathit{AF}}}\xspace}
\newcommand{\AFTargs}{\ensuremath{\ASAFargs^{\mathit{AF}}}\xspace}
\newcommand{\AFTattRel}{\ensuremath{\ASAFattRel^{\mathit{AF}}}\xspace}
\newcommand{\AFT}{\ensuremath{\langle \AFTargs, \AFTattRel \rangle}\xspace}

\newcommand{\src}[1]{\ensuremath{\mathsf{src}(#1)}\xspace}
\newcommand{\trg}[1]{\ensuremath{\mathsf{trg}(#1)}\xspace}



\newcommand{\ddef}{\ensuremath{\mathsf{d}$-$\mathsf{def}}\xspace}
\newcommand{\ddefs}[2]{\mbox{\ensuremath{#1} \ddef \ensuremath{#2}}\xspace}

\newcommand{\idef}{\ensuremath{\mathsf{i}$-$\mathsf{def}}\xspace}
\newcommand{\idefs}[2]{\mbox{\ensuremath{#1} \idef \ensuremath{#2}}\xspace}

\newcommand{\extdef}{\ensuremath{\mathsf{e}$-$\mathsf{def}}\xspace}
\newcommand{\extdefs}[3]{\mbox{\ensuremath{#1} \extdef \ensuremath{#2}} \ensuremath{\mathsf{given}} \ensuremath{#3}\xspace}

\newcommand{\eidef}{\ensuremath{\mathsf{ei}$-$\mathsf{def}}\xspace}
\newcommand{\eidefs}[3]{\mbox{\ensuremath{#1} \eidef \ensuremath{#2}} \ensuremath{\mathsf{given}} \ensuremath{#3}\xspace}

\newcommand{\Lab}{\ensuremath{\mathcal{L}ab}\xspace}
\newcommand{\LabellingsASAF}{\ensuremath{\mathfrak{L}}\xspace}
\newcommand{\LabToExt}{\ensuremath{\mathtt{Lab2Ext}}\xspace}
\newcommand{\ExtToLab}{\ensuremath{\mathtt{Ext2Lab}}\xspace}

\newcommand{\AFtoCAF}{\ensuremath{\mathtt{AFtoCAF}}\xspace}
\newcommand{\CAFtoAF}{\ensuremath{\mathtt{CAFtoAF}}\xspace}

\newcommand{\ASAFtoCAF}{\ensuremath{\mathtt{ASAFtoCAF}}\xspace}
\newcommand{\CAFtoASAF}{\ensuremath{\mathtt{CAFtoASAF}}\xspace}

\def\imply{{\text{implies}}}

\def\arg{{\mathcal{A}}}
\def\aaf{{\mathcal{A}}}
\def\ebaf{{\mathcal{E\!A}}}
\def\maaf{{\mathcal{M}}}
\def\cmaf{{\mathcal{CM}}}

\def\big{{\mathcal{G}}}
\def\S{\ensuremath{\sigma}}
\def\ssup{\ensuremath{\textit{sup}}}
\def\pssup{\ensuremath{\textit{psup}}}
\def\E{\ensuremath{\mathcal{E}}}
\def\M{{\mbox\cal{M}}}
\def\D{{{D}}}
\def\W{{\mathcal{W\!}}}
\def\CW{{\mathcal{CW\!}}}
\def\DA{{\mathcal{R}}}
\def\SA{{\mathcal{S\!\!\!A}}}
\def\I{{\mathcal{I}}}
\def\NI{{\overline{\mathcal{I}}}}
\def\Ig{{\mathcal{I}_g}}
\def\Iv{{\mathcal{I}_v}}
\def\Ie{{\mathcal{I}_e}}
\def\IAF{\I\!\!\!\aaf}
\def\IFP{\it IFP}
\def\reachable{\it Reach}
\def\reachA{\it Reach\mbox{$\!\!_{_\aaf}\!$}}
\def\reach0{\it Reach\mbox{$\!\!_{{_\aaf}_0}\!$}}
\def\Project{\mbox{$\Pi$}}

\def\gr{ \mbox{$\tt gr$}}
\def\pr{ \mbox{$\tt pr$}}
\def\co{ \mbox{$\tt co$}}
\def\sta{ \mbox{$\tt st$}}
\def\sst{\mbox{$\tt sst$}}
\def\id{\mbox{$\tt id$}}
\usepackage{soul}
\def\simx{\sim\!\!}
\def\C{{\textbf{C}}}
\def\R{{\textbf{R}}}
\def\AR{{Arg}}
\def\tern{{\langle \AR,\sqsubseteq ,\Sigma\rangle}}
\def\ternp{{\langle \AR^{'},\sqsubseteq^{'} ,\Sigma{'}\rangle}}
\def\ternstar{{\langle \AR,\sqsubseteq,\Sigma^{*}\rangle}}
\def\ternd{{\langle \AR_{D},\Sigma_{D}\rangle}}
\def\quartet{{\langle \AR,\sqsubseteq ,\C,\R\rangle}}

\def\+{\mbox{+}}
\def\-{\mbox{-}}
\def\<{\langle}
\def\>{\rangle}
\def\styleex{\rm}
\def\styledef{\rm}
\def\axiom{\textbf{Axiom\ 1 }}
\def\L{{\mathcal{L}}}
\def\i{\textsc{in}}
\def\o{\textsc{out}}
\def\u{\textsc{undec}}

\def\styledef{\rm}
\def\styleproof{\rm}
\def\styleth{\it}

\def\real{{\texttt{REAL}}}
\def\syn{{\texttt{SYN}}}

\def\maf{\Delta^m}
\newcommand{\ldot}{\,{\bf .}\,}
\def\nrDef{\ensuremath{\textit{def}}}
\def\nrAcc{\ensuremath{\textit{acc}}}

\newcommand{\Def}{\textsc{Def}}
\newcommand{\Acc}{\textsc{Acc}}

\def\={\mbox{=}}
\def\IN{\mox{$\!\in\!$}}
\def\true{\mbox{$\tt true$}}
\def\false{\mbox{$\tt false$}}
\def\undec{\mbox{$\tt undec$}}
\def\a{\mbox{$\tt a$}}
\def\b{\mbox{$\tt b$}}
\def\c{\mbox{$\tt c$}}
\def\d{\mbox{$\tt d$}}
\def\Sb{\Set{S}}
\def\sb{\Set{s}}
\def\tb{\Set{t}}

\newcommand{\blue}[1]{{\color{blue}{#1}}}
\newcommand{\Deltan}{\Delta_n}
\newcommand{\Deltad}{\Delta_d}
\newcommand{\Omegan}{\Omega_n}
\newcommand{\Omegad}{\Omega_d}
\newcommand{\rebaf}{Rec-BAF}
\newcommand{\reaf}{Rec-AF}


\newtheorem{lemma}{Lemma}[section]




\begin{document}
%
%\lefttitle{G. Alfano et al.}
%
%\jnlPage{1}{18}
%\jnlDoiYr{2024}
%\doival{10.1017/xxxxx}
%
%\title[Cyclic Supports in Recursive Bipolar AFs: Semantics and LP Mapping]{Cyclic Supports in Recursive Bipolar Argumentation Frameworks: Semantics and LP Mapping\thanks{We acknowledge the support from project Tech4You (ECS0000009), and PNRR MUR projects FAIR (PE0000013) and SERICS (PE00000014).}} 
%
%
%\begin{authgrp}
%\author{\sn{Gianvincenzo} \gn{Alfano}, \sn{Sergio} \gn{Greco}\\ \sn{Francesco} \gn{Parisi},   and \sn{Irina} \gn{Trubitsyna}}
%\affiliation{DIMES Department, University of Calabria, Rende, Italy\\
%(e-mail: $\{$g.alfano,\ greco,\ fparisi,\ i.trubitsyna$\}$@dimes.unical.it)}
%\end{authgrp}
%
%\history{\sub{xx xx xxxx;} \rev{xx xx xxxx;} \acc{xx xx xxxx}}
%
%\maketitle


\section*{Appendix A - Partial Stable Models}
The semantics of a logic program is given by the set of its partial stable models (PSMs) (corresponding to complete extensions of AFs \citep{CaminadaSAD15}).
We summarize the basic concepts underlying the notion of PSMs \citep{Sacca97}. 

A (normal) logic program (LP) is a set of rules of the form
$A \leftarrow B_1 \wedge \cdots \wedge B_n$, with $n \geq 0$, where $A$ is an atom, called head, and $B_1\wedge \cdots \wedge B_n$ is a conjunction of literals, called body. 
We consider programs without function symbols.
Given a program $P$, $ground(P)$ denotes the set of all ground instances of the rules in $P$. 
The Herbrand Base of a program $P$, i.e., the set of all ground
atoms which can be constructed using predicate and constant symbols occurring in $P$, is denoted by $B_P$, whereas $\neg B_P$ denotes the set $\{ \neg A \mid A \in B_P \}$. 
Analogously, for any set $S \subseteq B_P \cup \neg B_P$, $\neg S$ denotes the set $\{ \neg A \mid A \in S \}$, where $\neg \neg A = A$.
Given $I \subseteq B_P \cup \neg B_P$, $pos(I)$ (resp., $neg(I)$) stands for $I \cap B_P$ (resp., $\neg I \cap B_P$).
$I$ is \emph{consistent} if $pos(I) \cap \neg neg(I) = \emptyset$, otherwise $I$ is \emph{inconsistent}.

Given a program $P$, $I \subseteq B_P \cup \neg B_P$
is an \emph{interpretation}  of $P$ if $I$ is consistent. 
Also, $I$ is \emph{total} if 
$pos(I) \cup neg(I) = B_P$, 
\emph{partial} otherwise.
A partial interpretation $M$ of a program $P$ is a
\emph{partial model} of $P$ if for each $\neg A \in M$ every rule in $ground(P)$ having as head $A$ contains at least one body literal $B$ such that $\neg B \in M$.
Given a  program $P$ and a partial model $M$, the positive instantiation of $P$ w.r.t. $M$, denoted by $P^M$, is obtained from $ground(P)$ by deleting:\
$(a)$
each rule containing a negative literal $\neg A$ such that $A \in pos(M)$;\
$(b)$\
each rule containing a literal $B$ such that neither $B$ nor $\neg B$ is in $M$;\
$(c)$
all the negative literals in the remaining rules.
$M$ is a partial stable model of $P$ iff $M$ is the minimal model of $P^M$.
Alternatively, $P_M$ could be built by replacing every negated body literal in $ground(P)$ by its truth value.

The set of partial stable models of a logic program $P$, denoted by ${\cal PS}(P)$, define a meet semi-lattice. 
The \textit{well-founded} model (denoted by ${\cal WF}(P)$) and the \textit{maximal-stable} models ${\cal MS}(P)$, are defined by considering $\subseteq$-minimal and $\subseteq$-maximal elements. 
The set of (total) \textit{stable} models (denoted by ${\cal TS}(P)$) is obtained by considering the maximal-stable models which are total.

The semantics of a logic program is given by the set of its partial stable  models or by one of the restricted sets above recalled.


\bibliographystyle{tlplike}
\bibliography{refs}

\end{document}



