Symbolic Semantics
\[\newcommand\ve[1]{\tilde{#1}}
\newcommand\sort[1]{\mathsf{Sort}(#1)}
\newcommand\entails\vdash
\newcommand\chEq{\stackrel{\cdot}{\leftrightarrow}}
\newcommand\frameEq\simeq
\newcommand\compose\otimes
\newcommand\Unit{\mathbf{1}}\]
Requirements on psi-calculus parameters
Channel properties
\[\begin{split}\text{Channel Symmetry}\; & \Psi\entails M \chEq N \implies \Psi\entails N\chEq M \\
\text{Channel Transitivity}\; & \Psi\entails M \chEq N \land \Psi\entails N \chEq L \implies \Psi\entails M\chEq L\end{split}\]
Assertion properties
\[\begin{split}\text{Composition} & \Psi\frameEq\Psi' \implies \Psi\compose\Psi'' \frameEq \Psi'\compose\Psi'' \\
\text{Identity} & \Psi\compose\Unit \frameEq \Psi \\
\text{Associativity} & (\Psi\compose\Psi')\compose\Psi'' \frameEq \Psi\compose(\Psi'\compose\Psi'') \\
\text{Commutativity} & \Psi\compose\Psi' \frameEq \Psi'\compose\Psi\end{split}\]
Entailement properties
\[\text{Wakening}\; \Psi\entails\varphi \implies \Psi\compose\Psi'\entails\varphi\]
Name are terms
\[\mathcal{N} \subseteq \mathbf{T}\]
Broadcast requirements
\[\Psi\entails M \stackrel{\cdot}{\prec} K \implies n(K) \subseteq n(M)\]\[\Psi\entails K \stackrel{\cdot}{\succ} M \implies n(K) \subseteq n(M)\]
Sorted Substitution Requirements
Substitution \([\ve{a} := \ve{N}]\) is well formed if
\[\sort{a_i}\prec\sort{N_i}\]
substitution must land in the same datatype it started with.
requirements, \(T \in \mathbf{T, A, C}\)
\[T[\ve{a} := \ve{N}] = ((\ve{a}\,\ve{b})\cdot T)[\ve{b} := \ve{N}]\]
alpha-renaming of substituted variables, and
\[\sort{M\sigma} \leq \sort{M}\]
References
- Johansson, B. Victor, and J. Parrow. Computing strong and weak bisimulations for psi-calculi - with proofs. Technical Report 2011-018, Department of Information Technology, Uppsala University, Aug. 2011.
- Bengtson, M. Johansson, J. Parrow, and B. Victor. Psi-calculi: a framework for mobile processes with nominal data and logic. Logical Methods in Computer Science, 7(1:11), 01 2011.