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\FXRegisterAuthor{sg}{asg}{SG}  % Sergey
\FXRegisterAuthor{fg}{afg}{FG}  % Florian

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\renewcommand{\xto}[1]{\mathrel{\raisebox{-.15pt}{$\xrightarrow{\;\smash{\raisebox{-.5pt}{\makebox(3,0)[b]{\scriptsize $#1$}}\;}}$}}}

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\newcommand{\Fst}{\Pi_1} %{\oname{Fst}} 
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\title{Soundness and Completeness of Symmetric Relators} %TODO Please add
%Or: It is expected from a symmetric simulation to be a bisimulation
\titlerunning{From Abstract Higher-Order GSOS to Abstract Big-Step Semantics, Abstractly}

\author{Sergey Goncharov}{University of Birmingham, UK}{s.goncharov@bham.ac.uk}{https://orcid.org/0000-0001-6924-8766}{}
\author{Pouya Partow}{University of Birmingham, UK}{p.partow@bham.ac.uk}{https://orcid.org/0009-0003-9652-9469}{}


\authorrunning{J. Open Access and J.\,R. Public} %TODO mandatory. First: Use abbreviated first/middle names. Second (only in severe cases): Use first author plus 'et al.'

\Copyright{Jane Open Access and Joan R. Public} %TODO mandatory, please use full first names. LIPIcs license is "CC-BY";  http://creativecommons.org/licenses/by/3.0/

%\ccsdesc[100]{\textcolor{red}{Replace ccsdesc macro with valid one}} %TODO mandatory: Please choose ACM 2012 classifications from https://dl.acm.org/ccs/ccs_flat.cfm 

\keywords{Operational semantics, Higher-order GSOS, Extended combinatory logic}

\category{} %optional, e.g. invited paper

\relatedversion{} %optional, e.g. full version hosted on arXiv, HAL, or other respository/website
%\relatedversiondetails[linktext={opt. text shown instead of the URL}, cite=DBLP:books/mk/GrayR93]{Classification (e.g. Full Version, Extended Version, Previous Version}{URL to related version} %linktext and cite are optional

%\supplement{}%optional, e.g. related research data, source code, ... hosted on a repository like zenodo, figshare, GitHub, ...
%\supplementdetails[linktext={opt. text shown instead of the URL}, cite=DBLP:books/mk/GrayR93, subcategory={Description, Subcategory}, swhid={Software Heritage Identifier}]{General Classification (e.g. Software, Dataset, Model, ...)}{URL to related version} %linktext, cite, and subcategory are optional

%\funding{(Optional) general funding statement \dots}%optional, to capture a funding statement, which applies to all authors. Please enter author specific funding statements as fifth argument of the \author macro.

%\acknowledgements{I want to thank \dots}%optional

\nolinenumbers %uncomment to disable line numbering



%Editor-only macros:: begin (do not touch as author)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\EventEditors{John Q. Open and Joan R. Access}
%\EventNoEds{2}
%\EventLongTitle{42nd Conference on Very Important Topics (CVIT 2016)}
%\EventShortTitle{CVIT 2016}
%\EventAcronym{CVIT}
%\EventYear{2016}
%\EventDate{December 24--27, 2016}
%\EventLocation{Little Whinging, United Kingdom}
%\EventLogo{}
%\SeriesVolume{42}
%\ArticleNo{23}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\hideLIPIcs

\def\subjclassHeading{}
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\begin{document}
\allowdisplaybreaks
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\maketitle

%TODO mandatory: add short abstract of the document
\begin{abstract}
 
%One remarkable aspect of our approach is its potential for implementation in functional programming languages and proof assistants, 
%to atomate SOS specification transformations and reasoning about them.
\end{abstract}



%

%
%\section{Mathematical Preliminaries}%\label{sec:}


\paragraph{Compositionality}
Given a labeled transition systems $(S,L,\to)$, where $S$ is a set of states, $L$ is a set of labels, and $\to\subseteq S\times L\times S$ is a set of labeled transitions, a \emph{simulation} is a relation $r\subseteq S\times S$ such that the following sentence holds:
\begin{gather*}
	(x,y)\in r, x\xto{l} x' \Rightarrow \exists y', y\xto{l} y' \;\text{ and } (x',y')\in r
\end{gather*}  
The greatest simulation relation over $S$ is called \emph{similarity} and shown with $\leq$. A simulation relation $r$ on $S$ is a \emph{bisimulation} iff the following sentence holds: 
\begin{gather*}
	(x,y)\in r, y\xto{l} y' \Rightarrow \exists x', x\xto{l} x' \;\text{ and } (x',y')\in r
\end{gather*} 
The greatest bisimulation relation over $S$ is called \emph{bisimilarity} and shown with $\sim$. This is the traditional definition of bisimilarity. There are many different notions. For this definition, we can say that the symmetric similarity is the bisimilarity. But it is not always true for different notions.

\paragraph{Coalgebraic Bisimulation}
For an endofunctor $F$ over a category $\BC$, a coalgebra, is a pair $(X,\alpha)$, where $X$ is an object of $\BC$ and $\alpha\c X\to FX$ is a morphism in $\BC$. Coalgebras serve as an abstraction of variant transition systems. For example, a labeled transition system $(S,L,\to)$ is a coalgebra $(S,\gamma)$, where $\gamma\c S\to\mathcal{P}(L\times S)$ and $\mathcal{P}$ is the powerset functor.

$S$ can be the set of the terms of a programming language given by a signature $\Sigma$, and $\to$ can be the set of labeled inductions of the language, given by an operational semantics. Following that, a context $C$ is defined as:
\begin{gather*}
	C\Coloneqq\Box\mid f(C,\bar{t})\mid f(\bar{t},C)\mid f(\bar{u},C,\bar{s})
\end{gather*}
$f\in\Sigma$ and by $\bar{t}$, $\bar{s}$ and $\bar{u}$ we mean vectors of terms in $S$. $\Box$ is a placeholder. Assuming $t\in S$, then $C[t]\in S$. We call a relation $r$ congruence iff for terms $t$ and $s$, and a context $C$, $t\mathrel{r}s$ gives $C[t]\mathrel{r}C[s]$. A language with its operational semantics is compositional iff the bisimilarity relation over the terms of the language is a congruence.

\paragraph{Howe's method}
Howe's method has been traditionally used for compositionality results. \emph{Howe closure} of a relation $r$ is shown by $\hat{r}$ and is defined with the following inference rule: 
\begin{gather*}
	\infer{f(\bar{t})\mathrel{\hat{r}} s}{\bar{t}\mathrel{\hat{r}} \bar{s} & f(\bar{s})\mathrel{r} s}
\end{gather*}
Assuming that $r$ is reflexive, then $r\subseteq\hat{r}$, and $\hat{r}$ is a congruence. Additionally, if $r$ is reflexive and symmetric, then $\hat{r}^\star$, the transitive closure of $\hat{r}$ is symmetric (the transitive closure trick). So, to prove that bisimilarity is a congruence it is sufficient to prove that $\hat{\sim}$ is a bisimulation. Given the non-symmetric nature of the closure, it is more common to prove that $\hat{\sim}$ is a simulation. It is usually expected from a symmetric simulation to be a bisimulation. But it has not always been the case.

\paragraph{Relators}



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