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\begin{document}

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\section*{bunched logic}

\hypertarget{bunched_logic}{}\section*{{Bunched logic}}\label{bunched_logic}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{bunched_implications}{Bunched implications}\dotfill \pageref*{bunched_implications} \linebreak
\noindent\hyperlink{relevance_logic}{Relevance logic}\dotfill \pageref*{relevance_logic} \linebreak
\noindent\hyperlink{classical_bunched_implication}{Classical bunched implication}\dotfill \pageref*{classical_bunched_implication} \linebreak
\noindent\hyperlink{categorical_semantics}{Categorical semantics}\dotfill \pageref*{categorical_semantics} \linebreak
\noindent\hyperlink{related_pages}{Related pages}\dotfill \pageref*{related_pages} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[logic]], and more specifically in [[sequent calculus]]/[[natural deduction]], a \textbf{bunched logic} is a logic in which the formulas in the [[context]] are not just a [[list]] or [[set]] but have some additional, usually [[tree]]-like, structure.

This can be indicated syntactically by the use of two or more punctuation symbols, such as comma and semicolon, along with parentheses for grouping. Thus for instance a sequent with bunches might be written like

\begin{displaymath}
A,B,(C;(D,E);F),(G;H) \vdash K
\end{displaymath}
The contexts put together with both commas and semicolons are called \emph{bunches}. The general phrase \emph{bunched logic} is not entirely standard, although the word ``bunches'' has been used with more than one logic of this form (the original is ``bunched implication (BI)'', below).

Each type of punctuation also comes with a nullary version. The punctuation symbols like comma and semicolon are sometimes called \emph{structural connectives}, since they are part of the judgmental structure (like [[structural rules]]) but are closely related to the [[logical connectives]] such as conjunction and disjunction.

Usually the reason for using a bunched logic is that the different structural connectives obey different [[structural rules]]. For instance, the semicolon might allow the [[contraction rule]] and/or the [[weakening rule]], while the comma does not.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{bunched_implications}{}\subsubsection*{{Bunched implications}}\label{bunched_implications}

In \textbf{bunched implications} logic (\hyperlink{BI}{BI}), the semicolon allows contraction and weakening while the comma does not. This allows defining both an [[additive conjunction]] and a [[multiplicative conjunction]], internalizing the semicolon and the comma respectively, such that both [[distributive law|distribute]] over the [[additive disjunction]] and moreover both come with a corresponding implication: the [[additive implication]] and the [[multiplicative implication]].

\hypertarget{relevance_logic}{}\subsubsection*{{Relevance logic}}\label{relevance_logic}

Some forms of [[relevance logic]] can be presented with a bunched sequent calculus that is similar to BI, but in which the comma also has contraction (though not weakening) and there is no additive implication. See for instance (\hyperlink{Mints}{Mints}).

\hypertarget{classical_bunched_implication}{}\subsubsection*{{Classical bunched implication}}\label{classical_bunched_implication}

A logic of \textbf{classical bunched implication} is like BI but with arbitrary bunches on the right as well as on the left. On the right, the semicolon represents the [[additive disjunction]] and the comma represents a [[multiplicative disjunction]], and there are both an additive and a multiplicative [[negation]] that move formulas back and forth. Both negations are ``classical'' with respect to their corresponding connectives, e.g. we have $\sim\sim A \multimap A$ (where $\sim$ is the multiplicative negation and $\multimap$ the multiplicative implication) and also $\neg\neg A \to A$ (where $\neg$ is the additive negation and $\to$ the additive implication). See \hyperlink{Pym2002}{Pym 2002} and \href{https://nforum.ncatlab.org/discussion/4004/paraconsistent-logic/?Focus=57557#Comment_57557}{this discussion}.

\hypertarget{categorical_semantics}{}\subsection*{{Categorical semantics}}\label{categorical_semantics}

Bunched logics naturally have semantics in categories with more than one [[monoidal structure]], so that a bunch such as $(A,(B;C),((D,E);F))$ can be interpreted as $A \otimes (B\boxtimes C) \otimes ((D\otimes E)\boxtimes F)$. Frequently (e.g. if one kind of bunch admits contraction and weakening) one of the two monoidal structures is a [[cartesian monoidal category|cartesian]] one. A typical and motivating example of a model for BI is the [[category of presheaves]] $[C^{op},Set]$ over a monoidal category $C$, which comes equipped both with the ordinary [[ccc structure on presheaves]] as well as the closed monoidal structure given by [[Day convolution]].

\hypertarget{related_pages}{}\subsection*{{Related pages}}\label{related_pages}

\begin{itemize}%
\item [[linear logic]]
\item [[relevance logic]]
\item [[separation logic]]
\item [[display logic]]
\item [[substructural logic]]
\item [[monoidal topos]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Peter W. O'Hearn and David J. Pym, \emph{The Logic of Bunched Implications}. \href{http://www.lsv.ens-cachan.fr/~demri/OHearnPym99.pdf}{PDF}

\end{itemize}
\begin{itemize}%
\item G. E. Mints. \emph{Cut-elimination theorem for relevant logics}, Zap. Nauchn. Sem. LOMI, 1972, Volume 32, Pages 90--97. (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=2569&option_lang=eng}{math-net.ru}). An English translation appears in the Journal of Soviet Mathematics 6 (1976) pp.422-8. (\href{http://doi.org/10.1007/BF01084083}{doi})

\end{itemize}
\begin{itemize}%
\item Bodil Biering, \emph{On the logic of bunched implications - and its relation to separation logic}, \href{http://www.itu.dk/people/biering/papers/speciale.ps}{masters thesis}


\item David Pym, \emph{The Semantics and Proof Theory of the Logic of Bunched Implications}, \href{https://books.google.com/books/about/The_Semantics_and_Proof_Theory_of_the_Lo.html?id=0bAfqhzDuOcC&redir_esc=y}{Google books}



\end{itemize}
\begin{itemize}%
\item Brotherston and Calcagno, \emph{Classical BI: Its Semantics and Proof Theory}, \href{http://arxiv.org/abs/1005.2340}{arxiv}

\end{itemize}
[[!redirects bunched logics]] [[!redirects bunch]] [[!redirects bunches]] [[!redirects bunched implication]] [[!redirects bunched implications]] [[!redirects BI]] [[!redirects structural connective]] [[!redirects structural connectives]]



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