Semantics

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(→‎Additive structure: diagonal)
m (notation for isomorphismes \cong instead of \sim)
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\begin{array}{rclcrcl}
  
\begin{array}{rclcrcl}
 
A\biorth &=& A\\
  
A\biorth &=& A\\
−
(A\tens B)\orth &\sim& A\orth\parr B\orth &\quad& \one\orth &\sim& \bot\\
 +
(A\tens B)\orth &\cong& A\orth\parr B\orth &\quad& \one\orth &\cong& \bot\\
−
(A\parr B)\orth &\sim& A\orth\tens B\orth &\quad& \bot\orth &\sim& \one\\
 +
(A\parr B)\orth &\cong& A\orth\tens B\orth &\quad& \bot\orth &\cong& \one\\
−
(A\with B)\orth &\sim& A\orth\plus B\orth &\quad& \top\orth &\sim& \zero\\
 +
(A\with B)\orth &\cong& A\orth\plus B\orth &\quad& \top\orth &\cong& \zero\\
−
(A\plus B)\orth &\sim& A\orth\with B\orth &\quad& \zero\orth &\sim& \top\\
 +
(A\plus B)\orth &\cong& A\orth\with B\orth &\quad& \zero\orth &\cong& \top\\
−
(\oc A)\orth &\sim& \wn A\orth\\
 +
(\oc A)\orth &\cong& \wn A\orth\\
−
(\wn A)\orth &\sim& \oc A\orth\\[1ex]
 +
(\wn A)\orth &\cong& \oc A\orth\\[1ex]
−
A\limp B &\sim& A\orth\parr B\\
 +
A\limp B &\cong& A\orth\parr B\\
−
A\limp B &\sim& B\orth\limp A\orth\\
 +
A\limp B &\cong& B\orth\limp A\orth\\
 
\end{array}
  
\end{array}
 
</math>
  
</math>
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<math>
  
<math>
 
\begin{array}{rcl}
  
\begin{array}{rcl}
−
A\tens\one &\sim& \one\tens A\sim A\\
 +
A\tens\one &\cong& \one\tens A\cong A\\
−
A\parr\bot &\sim& \bot\parr A\sim A\\
 +
A\parr\bot &\cong& \bot\parr A\cong A\\
−
A\with\top &\sim& \top\with A\sim A\\
 +
A\with\top &\cong& \top\with A\cong A\\
−
A\plus\zero &\sim&\zero\plus A\sim A\\
 +
A\plus\zero &\cong&\zero\plus A\cong A\\
 
\end{array}
  
\end{array}
 
</math>
  
</math>
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<math>
  
<math>
 
\begin{array}{rcl}
  
\begin{array}{rcl}
−
A\tens B &\sim& B\tens A\\
 +
A\tens B &\cong& B\tens A\\
−
A\parr B &\sim& B\parr A\\
 +
A\parr B &\cong& B\parr A\\
−
A\with B &\sim& B\with A\\
 +
A\with B &\cong& B\with A\\
−
A\plus B &\sim& B\plus A\\
 +
A\plus B &\cong& B\plus A\\
 
\end{array}
  
\end{array}
 
</math>
  
</math>
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<math>
  
<math>
 
\begin{array}{rcl}
  
\begin{array}{rcl}
−
(A\tens B)\tens C &\sim& A\tens(B\tens C)\\
 +
(A\tens B)\tens C &\cong& A\tens(B\tens C)\\
−
(A\parr B)\parr C &\sim& A\parr(B\parr C)\\
 +
(A\parr B)\parr C &\cong& A\parr(B\parr C)\\
−
(A\with B)\with C &\sim& A\with(B\with C)\\
 +
(A\with B)\with C &\cong& A\with(B\with C)\\
−
(A\plus B)\plus C &\sim& A\plus(B\plus C)\\
 +
(A\plus B)\plus C &\cong& A\plus(B\plus C)\\
 
\end{array}
  
\end{array}
 
</math>
  
</math>
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<math>
  
<math>
 
\begin{array}{rclcrcl}
  
\begin{array}{rclcrcl}
−
A\tens(B\plus C) &\sim& (A\tens B)\plus(A\tens C) &\quad&
 +
A\tens(B\plus C) &\cong& (A\tens B)\plus(A\tens C) &\quad&
−
A\tens\zero &\sim& \zero\\
 +
A\tens\zero &\cong& \zero\\
−
A\parr(B\with C) &\sim& (A\parr B)\with(A\parr C) &\quad&
 +
A\parr(B\with C) &\cong& (A\parr B)\with(A\parr C) &\quad&
−
A\parr\top &\sim& \top\\
 +
A\parr\top &\cong& \top\\
 
\end{array}
  
\end{array}
 
</math>
  
</math>
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<math>
  
<math>
 
\begin{array}{rclcrcl}
  
\begin{array}{rclcrcl}
−
\oc(A\with B) &\sim& \oc A\tens\oc B &\quad& \oc\top &\sim& \one\\
 +
\oc(A\with B) &\cong& \oc A\tens\oc B &\quad& \oc\top &\cong& \one\\
−
\wn(A\plus B) &\sim& \wn A\parr\wn B &\quad& \wn\zero &\sim& \bot\\
 +
\wn(A\plus B) &\cong& \wn A\parr\wn B &\quad& \wn\zero &\cong& \bot\\
 
\end{array}
  
\end{array}
 
</math>
  
</math>

Revision as of 10:45, 19 October 2009

Linear Logic has numerous semantics some of which are described in details in the next sections. We give here an overview of the common properties that one may find in most of these models. We will denote by A\longrightarrow B the fact that there is a canonical morphism from A to B and by A˜B the fact that there is a canonical isomorphism between A and B. By "canonical" we mean that these (iso)morphisms are natural transformations.

Contents

Linear negation

\begin{array}{rclcrcl} A\biorth &=& A\\ (A\tens B)\orth &\cong& A\orth\parr B\orth &\quad& \one\orth &\cong& \bot\\ (A\parr B)\orth &\cong& A\orth\tens B\orth &\quad& \bot\orth &\cong& \one\\ (A\with B)\orth &\cong& A\orth\plus B\orth &\quad& \top\orth &\cong& \zero\\ (A\plus B)\orth &\cong& A\orth\with B\orth &\quad& \zero\orth &\cong& \top\\ (\oc A)\orth &\cong& \wn A\orth\\ (\wn A)\orth &\cong& \oc A\orth\\[1ex] A\limp B &\cong& A\orth\parr B\\ A\limp B &\cong& B\orth\limp A\orth\\ \end{array}

Neutrals

\begin{array}{rcl} A\tens\one &\cong& \one\tens A\cong A\\ A\parr\bot &\cong& \bot\parr A\cong A\\ A\with\top &\cong& \top\with A\cong A\\ A\plus\zero &\cong&\zero\plus A\cong A\\ \end{array}

Commutativity

\begin{array}{rcl} A\tens B &\cong& B\tens A\\ A\parr B &\cong& B\parr A\\ A\with B &\cong& B\with A\\ A\plus B &\cong& B\plus A\\ \end{array}

Associativity

\begin{array}{rcl} (A\tens B)\tens C &\cong& A\tens(B\tens C)\\ (A\parr B)\parr C &\cong& A\parr(B\parr C)\\ (A\with B)\with C &\cong& A\with(B\with C)\\ (A\plus B)\plus C &\cong& A\plus(B\plus C)\\ \end{array}

Multiplicative semi-distributivity

\begin{array}{rcl} A\tens(B\parr C) &\longrightarrow& (A\tens B)\parr C\\ \end{array}

Multiplicative-additive distributivity

\begin{array}{rclcrcl} A\tens(B\plus C) &\cong& (A\tens B)\plus(A\tens C) &\quad& A\tens\zero &\cong& \zero\\ A\parr(B\with C) &\cong& (A\parr B)\with(A\parr C) &\quad& A\parr\top &\cong& \top\\ \end{array}

Additive structure

\begin{array}{rcl} A\with B \longrightarrow A &\quad& A\with B \longrightarrow B\\ (C\limp A)\with(C\limp B) &\longrightarrow& C\limp(A\with B)\\ A &\longrightarrow& A\with A\\ A \longrightarrow A\plus B &\quad& B \longrightarrow A\plus B\\ (A\limp C)\with(B\limp C) &\longrightarrow& (A\plus B)\limp C\\ A\plus A &\longrightarrow& A\\ \end{array}

Exponential structure

\begin{array}{rclcrcl} \oc A &\longrightarrow& A &\quad& A&\longrightarrow&\wn A\\ \oc A &\longrightarrow& 1 &\quad& \bot &\longrightarrow& \wn A\\ \oc A &\longrightarrow& \oc A\tens\oc A &\quad& \wn A\parr\wn A &\longrightarrow& \wn A\\ \oc A &\longrightarrow& \oc\oc A &\quad& \wn\wn A &\longrightarrow& \wn A\\ \end{array}

Monoidality of exponential

\begin{array}{rclcrcl} \oc A\tens\oc B &\longrightarrow& \oc(A\tens B) &\quad& \one &\longrightarrow& \oc\one\\ \end{array}

The exponential isomorphism

\begin{array}{rclcrcl} \oc(A\with B) &\cong& \oc A\tens\oc B &\quad& \oc\top &\cong& \one\\ \wn(A\plus B) &\cong& \wn A\parr\wn B &\quad& \wn\zero &\cong& \bot\\ \end{array}

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