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...nd Yang 2007; Nanevski et al. 2006; Benton 2006; Peterson et al. 2008] for further applications of that semantics. An approach to reasoning about objects using separation logic has been developed in [=-=Parkinson and Bierman 2005-=-; Parkinson 2005]. Instead of hiding resource invariants, as here, there the approach is to let the client use abstract predicates (i.e., predicate variables) when reasoning about the use of an object...

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...neralize our more significant results on concurrency. Abstract predicates and higher-order separation logic have been used to approach modules, while the treatment here avoids higher-order predicates =-=[26, 6]-=-. Finally, it would be desirable to go beyond algebra and formulate the essence of local action at a categorical level, perhaps on the level of the general theory of effects [19, 29]. Rather than shoo...

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...llow one to express that a predicate depends only on certain objects of a heap. It has been used to reason about classical invariants of modules with one instance [49]. Although Parkinson and Bierman =-=[50]-=- extend separation logic to an object-oriented language, they do not address how to prove invariants. Applying separation logic to reason about invariants for layered object structures is considered o...

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...work [8] is a first attempt to deal with information hiding in separation logic. The solution does not scale to dynamic modularity, i.e. it deals only with single instances of a hidden data structure =-=[22]-=-. Thus, it is not suitable for the dynamic modularity of object orientation in which the solution must usually be applied to arbitrarily many objects. Parkinson and Bierman [22] provide a much more co...

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...ula R1.f ⊆ R2 says that for every object in R1, if it has field f with non-null value then that value is in R2. The formula x.f = E is like the points-to predicate in object-oriented separation logic =-=[8]-=-; it says that x is non-null and the value of its f field is E. Here is the grammar. 3 θ :: = E = E | x.f = E | R ⊆ R | R # R | type(N , R) | R.f ⊆ R | R.f # R | ∀x : T · θ | θ ∧ θ | ¬θ The semantics ...

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..., moreover, defined the coherent semantics of idealized algol typed with this system. One shortcoming of our type system is that the higherorder frame rules in the system allow only static modularity =-=[12]-=-. With the higher-order frame rules alone, we cannot capture all the the information hiding aspect of dynamically allocated data structures as needed for modeling abstract data types. However, it is w...

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...number of annotations/LOC ratio) for our examples is around 7%. VI. RELATED WORK Separation Logic. The general framework of separation logic [16], [10] is highly expressive but undecidable. Likewise, =-=[13]-=- formalised the proof rules for handling abstract predicates (with scopes on visibility of predicates) but provided no automated procedure for checking the user supplied specifications. In the search ...

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...ular data structure, not accessed by other modules. If the structure is manipulated only by module functions, it can be represented just in terms of its abstract properties, using abstract predicates =-=[21]-=-. For example, we can give a specification for a set using an abstract predicate to assert that “the set is {5, 6}”. A client can then reason about the set without reasoning about its internal impleme...