A lightweight logical approach to race-free sharing of heap storage between concurrent threads is described, based on the notion of permission to access. Transfer of permission between threads, subdivision and combination of permission is discussed. The roots of the approach are in Boyland's [3] demonstration of the utility of fractional permissions in specifying non-interference between concurrent threads. We add the notion of counting permission, which mirrors the programming technique called permission counting. Both fractional and counting permissions permit passivity, the specification that a program can be permitted to access a heap cell yet prevented from altering it. Models of both mechanisms are described. The use of two different mechanisms is defended. Some interesting problems are acknowledged and some intriguing possibilities for future development, including the notion of resourcing as a step beyond typing, are paraded.

We propose a novel approach to proving the termination of heapmanipulating programs, which combines separation logic with cyclic proof within a Hoare-style proof system. Judgements in this system express (guaranteed) termination of the program when started from a given line in the program and in a state satisfying a given precondition, which is expressed as a formula of separation logic. The proof rules of our system are of two types: logical rules that operate on preconditions; and symbolic execution rules that capture the effect of executing program commands.Our logical preconditions employ inductively defined predicates to describe heap properties, and proofs in our system are cyclic proofs: cyclic derivations in which some inductive predicate is unfolded infinitely often along every infinite path, thus allowing us to discard all infinite paths in the proof by an infinite descent argument. Moreover, the use of this soundness condition enables us to avoid the explicit construction and use of ranking functions for termination. We also give a completeness result for our system, which is relative in that it relies upon completeness of a proof system for logical implications in separation logic. We give examples illustrating our approach, including one example for which the corresponding ranking function is non-obvious: termination of the classical algorithm for in-place reversal of a (possibly cyclic) linked list.