For more than thirty years, the parallel programming community has used the dependence graph as the main abstraction for reasoning about and exploiting parallelism in "regular" algorithms that use dense arrays, such as finite-differences and FFTs. In this paper, we argue that the dependence graph is not a suitable abstraction for algorithms in new application areas like machine learning and network analysis in which the key data structures are "irregular" data structures like graphs, trees, and sets.To address the need for better abstractions, we introduce a datacentric formulation of algorithms called the operator formulation in which an algorithm is expressed in terms of its action on data structures. This formulation is the basis for a structural analysis of algorithms that we call tao-analysis. Tao-analysis can be viewed as an abstraction of algorithms that distills out algorithmic properties important for parallelization. It reveals that a generalized form of data-parallelism called amorphous data-parallelism is ubiquitous in algorithms, and that, depending on the tao-structure of the algorithm, this parallelism may be exploited by compile-time, inspector-executor or optimistic parallelization, thereby unifying these seemingly unrelated parallelization techniques. Regular algorithms emerge as a special case of irregular algorithms, and many application-specific optimization techniques can be generalized to a broader context. These results suggest that the operator formulation and taoanalysis of algorithms can be the foundation of a systematic approach to parallel programming.

In this paper we address the problem of shape analysis for concurrent programs. We present new algorithms, based on abstract interpretation, for automatically verifying properties of programs with an unbounded number of threads manipulating an unbounded shared heap. Our algorithms are based on a new abstract domain whose elements represent thread-quantified invariants: i.e., invariants satisfied by all threads. We exploit existing abstractions to represent the invariants. Thus, our technique lifts existing abstractions by wrapping universal quantification around elements of the base abstract domain. Such abstractions are effective because they are thread modular: e.g., they can capture correlations between the local variables of the same thread as well as correlations between the local variables of a thread and global variables, but forget correlations between the states of distinct threads. (The exact nature of the abstraction, of course, depends on the base abstraction lifted in this style.) We present techniques for computing sound transformers for the new abstraction by using transformers of the base abstract domain. We illustrate our technique in this paper by instantiating it to the Boolean Heap abstraction, producing a Quantified Boolean Heap abstraction. We have implemented an instantiation of our technique with Canonical Abstraction as the base abstraction and used it to successfully verify linearizability of data-structures in the presence of an unbounded number of threads.

Abstract. Predicate abstraction and canonical abstraction are two finitary abstractions used to prove properties of programs. We study the relationship between these two abstractions by considering a very limited case: abstraction of (potentially cyclic) singly-linked lists. We provide a new and rather precise family of abstractions for potentially cyclic singlylinked lists. The main observation behind this family of abstractions is that the number of shared nodes in linked lists can be statically bounded. Therefore, the number of possible "heap shapes" is also bounded. We present the new abstraction in both predicate abstraction form as well as in canonical abstraction form. As we illustrate in the paper, given any canonical abstraction, it is possible to define a predicate abstraction that is equivalent to the canonical abstraction. However, with this straightforward simulation, the number of predicates used for the predicate abstraction is exponential in the number of predicates used by the canonical abstraction. An important feature of the family of abstractions we present in this paper is that the predicate abstraction representation we define is far more practical as it uses a number of predicates that is quadratic in the number of predicates used by the corresponding canonical abstraction representation. In particular, for the most abstract abstraction in this family, the number of predicates used by the canonical abstraction is linear in the number of program variables, while the number of predicates used by the predicate abstraction is quadratic in the number of program variables. We have encoded this particular predicate abstraction and corresponding transformers in TVLA, and used this implementation to successfully verify safety properties of several list manipulating programs, including programs that were not previously verified using predicate abstraction or canonical abstraction.

For more than thirty years, the parallel programming community has used the dependence graph as the main abstraction for reasoning about and exploiting parallelism in "regular" algorithms that use dense arrays, such as finite-differences and FFTs. In this paper, we argue that the dependence graph is not a suitable abstraction for algorithms in new application areas like machine learning and network analysis in which the key data structures are "irregular" data structures like graphs, trees, and sets.To address the need for better abstractions, we introduce a datacentric formulation of algorithms called the operator formulation in which an algorithm is expressed in terms of its action on data structures. This formulation is the basis for a structural analysis of algorithms that we call tao-analysis. Tao-analysis can be viewed as an abstraction of algorithms that distills out algorithmic properties important for parallelization. It reveals that a generalized form of data-parallelism called amorphous data-parallelism is ubiquitous in algorithms, and that, depending on the tao-structure of the algorithm, this parallelism may be exploited by compile-time, inspector-executor or optimistic parallelization, thereby unifying these seemingly unrelated parallelization techniques. Regular algorithms emerge as a special case of irregular algorithms, and many application-specific optimization techniques can be generalized to a broader context. These results suggest that the operator formulation and taoanalysis of algorithms can be the foundation of a systematic approach to parallel programming.

Abstract.One of the continuing challenges in abstract interpretation is the creation of abstractions that yield analyses that are both tractable and precise enough to prove interesting properties about real-world programs. One source of difficulty is the need to handle programs with different behaviors along different execution paths. Disjunctive (powerset) abstractions capture such distinctions in a natural way. However, in general, powerset abstractions increase space and time costs by an exponential factor. Thus, powerset abstractions are generally perceived as very costly. In this paper, we partially address this challenge by presenting and empirically evaluating a new heap abstraction. The new heap abstraction works by merging shape descriptors according to a partial isomorphism similarity criteria, resulting in a partially disjunctive abstraction. We implemented this abstraction in TVLA-a generic system for implementing program analyses.We conducted an empirical evaluation of the new abstraction and compared it with the powerset heap abstraction. The experiments show that analyses based on the partially disjunctive heap abstraction are as precise as the ones based on the powerset heap abstraction. In terms of performance, analyses based on the partially disjunctive heap abstraction are often superior to analyses based on the powerset heap abstraction. The empirical results show considerable speedups, up to 2 orders of magnitude, enabling previously non-terminating analyses, such as verification of the Deutsch-Schorr-Waite scanning algorithm, to terminate with no negative effect on the overall precision. Indeed, experience indicates that the partially disjunctive shape abstraction improves performance across all TVLA analyses uniformly, and in many cases is essential for making precise shape analysis feasible.