|Abstract:||K (Roşu and Şerbănuţă, 2010) is a rewrite-based executable semantic framework in which programming languages, type systems and formal analysis tools can be de ned using con gurations, computations and rules. In this paper we de ne IsaK, a reference semantics for K, which was developed through discussion with the K team to meet their expectations of a semantics of K. IsaK is de ned in the interactive theorem prover Isabelle/HOL (Paulson, 1990), and, to the best of our knowledge, is the most complete of any existing K speci cation. IsaK de nes the full behavior of K, a useful sort system for K and suggests several undesirable behaviors in the current K implementations (K 3.6 and K 4.0). We also provide an OCaml based executable K interpreter generated automatically from the K speci cation in Isabelle. By using a prede ned K parser, the K interpreter is suitable to interpret major K de nitions for large languages such as the LLVM semantics in K, the Java semantics in K and the C semantics in K. We ran a test suite including 13 speci cations and 356 programs to test our K interpreter and we are able to compile all 13 speci cations and run the 338 programs not requiring keyboard input. As a utility of IsaK, we also formally prove a CTL application in IsaK correct.