Kronecker Algebra-based Deadlock Analysis in the Linux Kernel

Multi-core technology has moved concurrent programming to the forefront of computer science. The C programming language explicitly supports concurrent programming through POSIX threads. Threads execute in parallel and communicate via shared objects that can be locked using synchronized access to achieve mutual exclusion. However, with concurrent programming comes a new set of problems that can hamper the quality of the software. Deadlocks are category from such a problem. The Linux kernel currently deploys on many different architectures. The Linux kernel source now has 12 million lines of code. The code of Linux kernel is mostly written in C. As of now, The Linux kernel source-code defines 45 kernel threads. For several kernel threads, multiple threads in sharing are executing in parallel at runtime. The Linux kernel-level synchronization primitives are software mechanisms provided by the operating system for the purpose of supporting kernel thread synchronization. Examples are memory barriers, atomic operations, mutexes, spinlocks. Erroneous use of synchronization primitives is difficult to detect by testing. It leads to deadlocks in the Linux kernel on multi-core processor architectures. This report describes a static analysis for C-code to detect deadlocks due to erroneous use of spinlocks. The intended application is the detection of deadlocks in the Linux kernel. A major obstacle is the sheer size of the Linux kernel source-code. We analyze the entire source code of all 45 concurrent threads defined in the Linux kernel. The sizes of the control flow graphs (CFGs) of the underlying kernel threads are intractable for static analysis. We thus devise a reduction mechanism to eliminate CFG nodes which are not relevant for deadlock detection. We define graph rewrite rules to make problem sizes tractable. Deadlocks constitute a locking hierarchy violation. In our analysis, we show that spinlocks which can be shown to adhere to a given locking hierarchy are irrelevant for deadlock detection. This allows us to omit such locks and thus further reduce the problem size. Kronecker algebra is a matrix calculus that has been proven useful for analysing multi-threaded programs. Our analysis for deadlock detection among Linux kernel threads is based on Kronecker algebra. We employ Kronecker algebra on the entire Linux kernel source-tree to detect deadlocks.