The intuitionistic modal logic of necessity is based on the judgmental notion of categorical truth. In this article we investigate the consequences of relativizing these concepts to explicitly specified contexts. We obtain contextual modal logic and its type-theoretic analogue. Contextual modal type theory provides an elegant, uniform foundation for understanding metavariables and explicit substitutions. We sketch some applications in functional programming and logical frameworks.

Abstract. We present a novel model of concurrent computations with shared memory and provide a simple, yet powerful, logical framework for uniform Hoarestyle reasoning about partial correctness of coarse-and fine-grained concurrent programs. The key idea is to specify arbitrary resource protocols as communicating state transition systems (STS) that describe valid states of a resource and the transitions the resource is allowed to make, including transfer of heap ownership. We demonstrate how reasoning in terms of communicating STS makes it easy to crystallize behavioral invariants of a resource. We also provide entanglement operators to build large systems from an arbitrary number of STS components, by interconnecting their lines of communication. Furthermore, we show how the classical rules from the Concurrent Separation Logic (CSL), such as scoped resource allocation, can be generalized to fine-grained resource management. This allows us to give specifications as powerful as Rely-Guarantee, in a concise, scoped way, and yet regain the compositionality of CSL-style resource management. We proved the soundness of our logic with respect to the denotational semantics of action trees (variation on Brookes' action traces). We formalized the logic as a shallow embedding in Coq and implemented a number of examples, including a construction of coarse-grained CSL resources as a modular composition of various logical and semantic components.

In previous work we have proposed a Dependent Hoare Type Theory (HTT) as a framework for development and reasoning about higher-order functional programs with effects of state, aliasing and nontermination. The main feature of HTT is the type of Hoare triples {P }x:A{Q} specifying computations with precondition P and postcondition Q, that return a result of type A. Here we extend HTT with predicative type polymorphism. Type quantification is possible in both types and assertions, and we can also quantify over Hoare triples. We show that as a consequence it becomes possible to reason about disjointness of heaps in the assertion logic of HTT. We use this expressiveness to interpret the Hoare triples in the "small footprint" manner advocated by Separation Logic, whereby a precondition tightly describes the heap fragment required by the computation. We support stateful commands of allocation, lookup, strong update, deallocation, and pointer arithmetic.

Efficient concurrent programs and data structures rarely employ coarse-grained synchronization mechanisms (i.e., locks); instead, they implement custom synchronization patterns via fine-grained primitives, such as compare-and-swap. Due to sophisticated interference scenarios between threads, reasoning about such programs is challenging and error-prone, and can benefit from mechanization.In this paper, we present the first completely formalized framework for mechanized verification of full functional correctness of fine-grained concurrent programs. Our tool is based on the recently proposed program logic FCSL. It is implemented as an embedded domain-specific language in the dependently-typed language of the Coq proof assistant, and is powerful enough to reason about programming features such as higher-order functions and local thread spawning. By incorporating a uniform concurrency model, based on state-transition systems and partial commutative monoids, FCSL makes it possible to build proofs about concurrent libraries in a thread-local, compositional way, thus facilitating scalability and reuse: libraries are verified just once, and their specifications are used ubiquitously in client-side reasoning. We illustrate the proof layout in FCSL by example, and report on our experience of using FCSL to verify a number of concurrent algorithms and data structures.

Abstract. This paper proposes a novel method of harnessing existing SAT solvers to verify reachability properties of programs that manipulate linked-list data structures. Such properties are essential for proving program termination, correctness of data structure invariants, and other safety properties. Our solution is complete, i.e., a SAT solver produces a counterexample whenever a program does not satisfy its specification. This result is surprising since even first-order theorem provers usually cannot deal with reachability in a complete way, because doing so requires reasoning about transitive closure. Our result is based on the following ideas: (1) Programmers must write assertions in a restricted logic without quantifier alternation or function symbols. (2) The correctness of many programs can be expressed in such restricted logics, although we explain the tradeoffs. (3) Recent results in descriptive complexity can be utilized to show that every program that manipulates potentially cyclic, singly-and doubly-linked lists and that is annotated with assertions written in this restricted logic, can be verified with a SAT solver. We implemented a tool atop Z3 and used it to show the correctness of several linked list programs.