A Restricted Definition of the Magic Wand to Soundly Combine Fractions of a Wand

Many separation logics support fractional permissions to distinguish between read and write access to a heap location, for instance, to allow concurrent reads while enforcing exclusive writes. The concept has been generalized to fractional assertions. $A^p$ (where $A$ is a separation logic assertion and $p$ a fraction between $0$ and $1$) represents a fraction $p$ of $A$. $A^p$ holds in a state $\sigma$ iff there exists a state $\sigma_A$ in which $A$ holds and $\sigma$ is obtained from $\sigma_A$ by multiplying all permission amounts held by $p$. While $A^{p + q}$ can always be split into $A^p * A^q$, recombining $A^p * A^q$ into $A^{p+q}$ is not always sound. We say that $A$ is combinable iff the entailment $A^p * A^q \models A^{p+q}$ holds for any two positive fractions $p$ and $q$ such that $p + q \le 1$. Combinable assertions are particularly useful to reason about concurrent programs, for instance, to combine the postconditions of parallel branches when they terminate. Unfortunately, the magic wand assertion $A \mathbin{-\!\!*} B$, commonly used to specify properties of partial data structures, is typically not combinable. In this entry, we formalize a novel, restricted definition of the magic wand, described in a paper at CAV 22, which we call the combinable wand. We prove some key properties of the combinable wand; in particular, a combinable wand is combinable if its right-hand side is combinable.