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\begin{document}

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\section*{phase semantics}

\hypertarget{phase_semantics}{}\section*{{Phase semantics}}\label{phase_semantics}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Girard's \emph{phase semantics} is a way of building [[star-autonomous category|star-autonomous]] [[posets]] with [[exponential modalities]], hence models of classical [[linear logic]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $M$ be a commutative [[monoid]], and $\bot\subseteq M$ a specified subset. ($M$ equipped with $\bot$ is sometimes called a \emph{phase space}.) Then $P M$, the [[powerset]] of $M$, is a commutative [[quantale]], with tensor product

\begin{displaymath}
X\otimes Y = \{ m n \mid m\in X \wedge n \in Y \}
\end{displaymath}
and internal-hom

\begin{displaymath}
X\multimap Y = \{ m \mid \forall n \in X, m n \in Y \}.
\end{displaymath}
Indeed, if we regard $M$ as a discrete [[poset]], hence as a category enriched over $\mathbf{2} = \{0\le 1\}$, then $P M$ is its $\mathbf{2}$-enriched [[presheaf category]], and this is its [[Day convolution]] monoidal structure.

Now $\bot$ is an object of $P M$, hence induces a contravariant self-[[adjunction]] $(-\multimap\bot) \dashv (-\multimap \bot)$ of $P M$, which is [[idempotent adjunction|idempotent]] since $P M$ is a poset. We define a \textbf{fact} to be a [[fixed point of an adjunction|fixed point]] of this adjunction, i.e. a subset $X\in P M$ of the form $Y\multimap \bot$, or equivalently such that $X = (X\multimap\bot)\multimap\bot$.

\begin{utheorem}
The poset of facts is [[star-autonomous category|star-autonomous]].

\end{utheorem}
\begin{proof}
It is closed under $\multimap$, since $(X\multimap (Y\multimap\bot)) = (X\otimes Y\multimap \bot)$. And it is reflective, with reflector $(-\multimap\bot)\multimap\bot$. Thus, it is closed symmetric monoidal with tensor product $((X\otimes Y)\multimap\bot)\multimap\bot$. Since it also contains $\bot$, as $\bot = (I\multimap \bot)$, it is therefore star-autonomous by construction.

\end{proof}
The poset of facts is also, of course, a [[complete lattice]], since it a reflective sub-poset of the complete lattice $P M$. In addition, it admits [[exponential modalities]] $!$ and $?$. There are different ways to obtain these, but perhaps the simplest (see \hyperlink{GirardSS}{here}) is to take

\begin{displaymath}
!X = ((X \cap Idem \cap 1)\multimap\bot)\multimap\bot
\end{displaymath}
where $Idem= \{ m \mid m m = m \}$ is the set of idempotents in $M$, and $1 = (\{1\}\multimap\bot)\multimap\bot$ is the unit object of the monoidal category of facts.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Phase semantics is complete for provability in linear logic, i.e. if a sequent of linear logic is valid in the phase semantics for all choices of $M$ and $\bot$, then it is provable. This follows from a construction of a particular $M$ and $\bot$ out of the syntax: let $M$ be the free commutative monoid on the formulas of linear logic subject to the relation that formulas of the form $?A$ are idempotent, and let $\bot$ be the set of $\Gamma\in M$ that are provable when regarded as one-sided sequents $\vdash\Gamma$, and interpret a formula $A$ by the set of $\Gamma\in M$ such that $\vdash \Gamma,A$ is provable. See \hyperlink{Girard}{Girard} for details.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

When phrased categorically as above, there is an obvious generalization to the case when $M$ is a commutative monoidal \emph{poset}, with $P M$ replaced by its $\mathbf{2}$-enriched presheaf category, i.e. the poset $D M$ of downsets in $M$. We can also generalize to other enrichments, although in that case the idempotence of the adjunction $(-\multimap\bot) \dashv (-\multimap \bot)$ is no longer automatic but has to be assumed.

If $M$ is assumed only to be a [[promonoidal category|promonoidal]] $\mathbf{2}$-enriched category, i.e. equipped with a suitable [[relation]] $M\times M   &#8696; M$, then it is a [[ternary frame]]. This can be used to construct models of more general [[substructural logics]].

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

\begin{itemize}%
\item [[Jean-Yves Girard]], \emph{Linear logic}, Theoretical Computer Science 50:1, 1987. (\href{http://iml.univ-mrs.fr/~girard/linear.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item [[Jean-Yves Girard]], \emph{Linear logic, its syntax and semantics} (\href{http://www.cs.brandeis.edu/~cs112/2006-cs112/docs/girard-intro.pdf}{pdf})

\end{itemize}
[[!redirects phase space semantics for linear logic]] [[!redirects linear logic phase space]] [[!redirects linear logic phase spaces]] [[!redirects phase space for linear logic]] [[!redirects phase spaces for linear logic]] [[!redirects Girard phase space]] [[!redirects Girard phase spaces]]



\end{document}