We develop the first theory of control-flow graphs from first principles, and use it to create an algorithm for automatically synthesizing many variants of control-flow graph generators from a language’s operational semantics. Our approach first introduces a new algorithm for converting a large class of small-step operational semantics to an abstract machine. It next uses a technique called “abstract rewriting” to automatically abstract the semantics of a language, which is used both to directly generate a CFG from a program (“interpreted mode”) and to generate standalone code, similar to a human-written CFG generator, for any program in a language. We show how the choice of two abstraction and projection parameters allow our approach to synthesize several families of CFG-generators useful for different kinds of tools. We prove the correspondence between the generated graphs and the original semantics. We provide and prove an algorithm for automatically proving the termination of interpreted-mode generators. In addition to our theoretical results, we have implemented this algorithm in a tool called Mandate, and show that it produces human-readable code on two medium-size languages with 60-80 rules, featuring nearly all intraprocedural control constructs common in modern languages. We then show these CFG-generators were sufficient to build two static analyses atop them. Our work is a promising step towards the grand vision of being able to synthesize all desired tools from the semantics of a programming language.