We present a constraint-based approach to computing winning strategies in two-player graph games over the state space of infinite-state programs. Such games have numerous applications in program verification and synthesis, including the synthesis of infinite-state reactive programs and branching-time verification of infinite-state programs. Our method handles games with winning conditions given by safety, reachability, and general Linear Temporal Logic (LTL) properties. For each property class, we give a deductive proof rule that --- provided a symbolic representation of the game players --- describes a winning strategy for a particular player. Our rules are sound and relatively complete. We show that these rules can be automated by using an off-the-shelf Horn constraint solver that supports existential quantification in clause heads. The practical promise of the rules is demonstrated through several case studies, including a challenging "Cinderella-Stepmother game" that allows infinite alternation of discrete and continuous choices by two players, as well as examples derived from prior work on program repair and synthesis.

We present an analysis to automatically determine if a program represents a continuous function, or equivalently, if infinitesimal changes to its inputs can only cause infinitesimal changes to its outputs. The analysis can be used to verify the robustness of programs whose inputs can have small amounts of error and uncertainty---e.g., embedded controllers processing slightly unreliable sensor data, or handheld devices using slightly stale satellite data. Continuity is a fundamental notion in mathematics. However, it is difficult to apply continuity proofs from real analysis to functions that are coded as imperative programs, especially when they use diverse data types and features such as assignments, branches, and loops. We associate data types with metric spaces as opposed to just sets of values, and continuity of typed programs is phrased in terms of these spaces. Our analysis reduces questions about continuity to verification conditions that do not refer to infinitesimal changes and can be discharged using off-the-shelf SMT solvers. Challenges arise in proving continuity of programs with branches and loops, as a small perturbation in the value of a variable often leads to divergent control-flow that can lead to large changes in values of variables. Our proof rules identify appropriate ``synchronization points'' between executions and their perturbed counterparts, and establish that values of certain variables converge back to the original results in spite of temporary divergence. We prove our analysis sound with respect to the traditional epsilon-delta definition of continuity. We demonstrate the precision of our analysis by applying it to a range of classic algorithms, including algorithms for array sorting, shortest paths in graphs, minimum spanning trees, and combinatorial optimization. A prototype implementation based on the Z3 SMT-solver is also presented.

We present a new technique for parameter synthesis under boolean and quantitative objectives. The input to the technique is a "sketch" --- a program with missing numerical parameters --- and a probabilistic assumption about the program's inputs. The goal is to automatically synthesize values for the parameters such that the resulting program satisfies: (1) a {boolean specification}, which states that the program must meet certain assertions, and (2) a {quantitative specification}, which assigns a real valued rating to every program and which the synthesizer is expected to optimize. Our method --- called smoothed proof search --- reduces this task to a sequence of unconstrained smooth optimization problems that are then solved numerically. By iteratively solving these problems, we obtain parameter values that get closer and closer to meeting the boolean specification; at the limit, we obtain values that provably meet the specification. The approximations are computed using a new notion of smoothing for program abstractions, where an abstract transformer is approximated by a function that is continuous according to a metric over abstract states. We present a prototype implementation of our synthesis procedure, and experimental results on two benchmarks from the embedded control domain. The experiments demonstrate the benefits of smoothed proof search over an approach that does not meet the boolean and quantitative synthesis goals simultaneously.

We show that the reachability problem for recursive state machines (or equivalently, pushdown systems), believed for long to have cubic worst-case complexity, can be solved in slightly subcubic time. All that is necessary for the new bound is a simple adaptation of a known technique. We also show that a better algorithm exists if the input machine does not have infinite recursive loops.

A common workflow for developing parallel software is as follows: 1) start with a sequential program, 2) identify subcomputations that should be converted to parallel tasks, 3) insert synchronization to achieve the same semantics as the sequential program, and repeat steps 2) and 3) as needed to improve performance. Though this is not the only approach to developing parallel software, it is sufficiently common to warrant special attention as parallel programming becomes ubiquitous. This paper focuses on automating step 3), which is usually the hardest step for developers who lack expertise in parallel programming. Past solutions to the problem of repairing parallel programs have used static-only or dynamic-only approaches, both of which incur significant limitations in practice. Static approaches can guarantee soundness in many cases but are limited in precision when analyzing medium or large-scale software with accesses to pointer-based data structures in multiple procedures. Dynamic approaches are more precise, but their proposed repairs are limited to a single input and are not reflected back in the original source program. In this paper, we introduce a hybrid static+dynamic test-driven approach to repairing data races in structured parallel programs. Our approach includes a novel coupling between static and dynamic analyses. First, we execute the program on a concrete test input and determine the set of data races for this input dynamically. Next, we compute a set of "finish" placements that prevent these races and also respects the static scoping rules of the program while maximizing parallelism. Empirical results on standard benchmarks and student homework submissions from a parallel computing course establish the effectiveness of our approach with respect to compile-time overhead, precision, and performance of the repaired code.