Linda (coordination language)

We abstract the concrete representation of tuples. We just assume that we have a set of tuples ${\displaystyle t\in {\mathcal {T}}}$ and we are allowed to form and apply a substitution function ${\displaystyle \sigma }$ on tuples substituting variables for terms that yields a tuple. For example, given we have a tuple ${\displaystyle t=({\mathit {hello}},x)}$, then applying a substitution ${\displaystyle \sigma =\{{\mathit {world}}/x\}}$ on ${\displaystyle t}$ yields ${\displaystyle ({\mathit {hello}},{\mathit {world}})}$

The Linda-calculus processes are defined by the following grammar.

${\displaystyle P,Q::=t\;|\;\mathrm {outeval} (P).Q\;|\;\mathrm {rd} (t).P\;|\;\mathrm {in} (t).P\;|\;P+Q\;|\;\mathbf {rec} X.P\;|\;X}$

The syntax includes the aftermentioned Linda operations, non-deterministic choice, and recursion. The substitution function is extended to processes recursively.

A tuple space is represented as a multiset of the processes. We write ${\displaystyle M\,|\,P}$ for ${\displaystyle M\cup \{P\}}$ where ${\displaystyle M}$ is a multiset, ${\displaystyle \{P\}}$ a singleton multiset, and ${\displaystyle \cup }$ is the multiset union operation. The semantics is then defined as a reduction relation on a multiset ${\displaystyle M\rightarrow M'}$ as follows.

${\displaystyle {\begin{array}{lrcll}{\text{(outeval)}}&M\,|\,\mathrm {outeval} (P).Q&\rightarrow &M\,|\,P\,|\,Q&\\{\text{(read)}}&M\,|\,\mathrm {rd} (t).P\,|\,t'&\rightarrow &M\,|\,P\sigma \,|\,t'&{\text{if exists }}\sigma {\text{ such that }}t\sigma =t'\\{\text{(input)}}&M\,|\,\mathrm {in} (t).P\,|\,t'&\rightarrow &M\,|\,P\sigma &{\text{if exists }}\sigma {\text{ such that }}t\sigma =t'\\{\text{(left choice)}}&M\,|\,P+Q&\rightarrow &M\,|\,P&\\{\text{(right choice)}}&M\,|\,P+Q&\rightarrow &M\,|\,Q&\\{\text{(recursion)}}&M\,|\,\mathbf {rec} X.P&\rightarrow &M\,|\,P\{\mathbf {rec} X.P/X\}&\\\end{array}}}$

Note that (input) consumes the tuple ${\displaystyle t'}$ from the tuple space whereas (read) only reads it. The resulting operational semantics is synchronous.