Concurrency

Concurrency is the study of programmes in which multiple computations proceed simultaneously and interact. The central difficulty is that the set of possible executions grows combinatorially with the number of interacting components: n independent actions admit n! interleavings, and a bug may appear in only one of them. Mathematical tools for taming this complexity include labelled transition systems, bisimulation equivalence, Mazurkiewicz trace theory, event structures, and the happens-before partial order. Correctness conditions (linearizability, sequential consistency, causal consistency) are predicates over the partial orders of execution events. Logical clocks (Lamport scalar, vector, hybrid) make causal structure observable at runtime.