Learning Nonlinear Loop Invariants with Gated Continuous Logic Networks (Extended Version)

Verifying real-world programs often requires inferring loop invariants with nonlinear constraints. This is especially true in programs that perform many numerical operations, such as control systems for avionics or industrial plants. Recently, data-driven methods for loop invariant inference have shown promise, especially on linear invariants. However, applying data-driven inference to nonlinear loop invariants is challenging due to the large numbers of and magnitudes of high-order terms, the potential for overfitting on a small number of samples, and the large space of possible inequality bounds. In this paper, we introduce a new neural architecture for general SMT learning, the Gated Continuous Logic Network (G-CLN), and apply it to nonlinear loop invariant learning. G-CLNs extend the Continuous Logic Network (CLN) architecture with gating units and dropout, which allow the model to robustly learn general invariants over large numbers of terms. To address overfitting that arises from finite program sampling, we introduce fractional sampling---a sound relaxation of loop semantics to continuous functions that facilitates unbounded sampling on real domain. We additionally design a new CLN activation function, the Piecewise Biased Quadratic Unit (PBQU), for naturally learning tight inequality bounds. We incorporate these methods into a nonlinear loop invariant inference system that can learn general nonlinear loop invariants. We evaluate our system on a benchmark of nonlinear loop invariants and show it solves 26 out of 27 problems, 3 more than prior work, with an average runtime of 53.3 seconds. We further demonstrate the generic learning ability of G-CLNs by solving all 124 problems in the linear Code2Inv benchmark. We also perform a quantitative stability evaluation and show G-CLNs have a convergence rate of $97.5\%$ on quadratic problems, a $39.2\%$ improvement over CLN models

Analyzing Array Manipulating Programs by Program Transformation

We explore a transformational approach to the problem of verifying simple array-manipulating programs. Traditionally, verification of such programs requires intricate analysis machinery to reason with universally quantified statements about symbolic array segments, such as "every data item stored in the segment A[i] to A[j] is equal to the corresponding item stored in the segment B[i] to B[j]." We define a simple abstract machine which allows for set-valued variables and we show how to translate programs with array operations to array-free code for this machine. For the purpose of program analysis, the translated program remains faithful to the semantics of array manipulation. Based on our implementation in LLVM, we evaluate the approach with respect to its ability to extract useful invariants and the cost in terms of code size

Connecting Program Synthesis and Reachability: Automatic Program Repair using Test-Input Generation

We prove that certain formulations of program synthesis and reachability are equivalent. Specifically, our constructive proof shows the reductions between the template-based synthesis problem, which generates a program in a pre-specified form, and the reachability problem, which decides the reachability of a program location. This establishes a link between the two research fields and allows for the transfer of techniques and results between them. To demonstrate the equivalence, we develop a program repair prototype using reachability tools. We transform a buggy program and its required specification into a specific program containing a location reachable only when the original program can be repaired, and then apply an off-the-shelf test-input generation tool on the transformed program to find test values to reach the desired location. Those test values correspond to repairs for the original program. Preliminary results suggest that our approach compares favorably to other repair methods