Shared memory multithreading is a popular approach to parallel programming, but also fiendishly hard to get right. We present Liquid Effects, a type-and-effect system based on refinement types which allows for fine-grained, low-level, shared memory multi-threading while statically guaranteeing that a program is deterministic. Liquid Effects records the effect of an expression as a for- mula in first-order logic, making our type-and-effect system highly expressive. Further, effects like Read and Write are recorded in Liquid Effects as ordinary uninterpreted predicates, leaving the effect system open to extension by the user. By building our system as an extension to an existing dependent refinement type system, our system gains precise value- and branch-sensitive reasoning about effects. Finally, our system exploits the Liquid Types refinement type inference technique to automatically infer refinement types and effects. We have implemented our type-and-effect checking techniques in CSOLVE, a refinement type inference system for C programs. We demonstrate how CSOLVE uses Liquid Effects to prove the determinism of a variety of benchmarks.

Programs written in dynamic languages make heavy use of features --- run-time type tests, value-indexed dictionaries, polymorphism, and higher-order functions --- that are beyond the reach of type systems that employ either purely syntactic or purely semantic reasoning. We present a core calculus, System D, that merges these two modes of reasoning into a single powerful mechanism of nested refinement types wherein the typing relation is itself a predicate in the refinement logic. System D coordinates SMT-based logical implication and syntactic subtyping to automatically typecheck sophisticated dynamic language programs. By coupling nested refinements with McCarthy's theory of finite maps, System D can precisely reason about the interaction of higher-order functions, polymorphism, and dictionaries. The addition of type predicates to the refinement logic creates a circularity that leads to unique technical challenges in the metatheory, which we solve with a novel stratification approach that we use to prove the soundness of System D.

We present Low-Level Liquid Types , a refinement type system for C based on Liquid Types . Low-Level Liquid Types combine refinement types with three key elements to automate verification of critical safety properties of low-level programs: First, by associating refinement types with individual heap locations and precisely tracking the locations referenced by pointers, our system is able to reason about complex invariants of in-memory data structures and sophisticated uses of pointer arithmetic. Second, by adding constructs which allow strong updates to the types of heap locations, even in the presence of aliasing, our system is able to verify properties of in-memory data structures in spite of temporary invariant violations. By using this strong update mechanism, our system is able to verify the correct initialization of newly-allocated regions of memory. Third, by using the abstract interpretation framework of Liquid Types, we are able to use refinement type inference to automatically verify important safety properties without imposing an onerous annotation burden. We have implemented our approach in CSOLVE, a tool for Low-Level Liquid Type inference for C programs. We demonstrate through several examples that CSOLVE is able to precisely infer complex invariants required to verify important safety properties, like the absence of array bounds violations and null-dereferences, with a minimal annotation overhead.

We present a refinement type-based approach for the static verification of complex data structure invariants. Our approach is based on the observation that complex data structures are typically fashioned from two elements: recursion (e.g., lists and trees), and maps (e.g., arrays and hash tables). We introduce two novel type-based mechanisms targeted towards these elements: recursive refinements and polymorphic refinements. These mechanisms automate the challenging work of generalizing and instantiating rich universal invariants by piggybacking simple refinement predicates on top of types, and carefully dividing the labor of analysis between the type system and an SMT solver. Further, the mechanisms permit the use of the abstract interpretation framework of liquid type inference to automatically synthesize complex invariants from simple logical qualifiers, thereby almost completely automating the verification. We have implemented our approach in dsolve, which uses liquid types to verify ocaml programs. We present experiments that show that our type-based approach reduces the manual annotation required to verify complex properties like sortedness, balancedness, binary-search-ordering, and acyclicity by more than an order of magnitude.

We present Logically Qualified Data Types, abbreviated to Liquid Types, a system that combines Hindley-Milner type inference with Predicate Abstraction to automatically infer dependent types precise enough to prove a variety of safety properties. Liquid types allow programmers to reap many of the benefits of dependent types, namely static verification of critical properties and the elimination of expensive run-time checks, without the heavy price of manual annotation. We have implemented liquid type inference in DSOLVE, which takes as input an OCAML program and a set of logical qualifiers and infers dependent types for the expressions in the OCAML program. To demonstrate the utility of our approach, we describe experiments using DSOLVE to statically verify the safety of array accesses on a set of OCAML benchmarks that were previously annotated with dependent types as part of the DML project. We show that when used in conjunction with a fixed set of array bounds checking qualifiers, DSOLVE reduces the amount of manual annotation required for proving safety from 31% of program text to under 1%.