Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Programming language for Computable Functions, or PCF, is a typed functional language introduced by Gordon Plotkin in 1977. It is based on the Logic of Computable Functions (LCF) by Dana Scott. It can be considered as a simplified version of modern typed functional languages such as ML. A fully abstract model for PCF was first given by Milner (1977). However, since Milner's model was essentially based on the syntax of PCF it was considered less than satisfactory (Ong, 1995). The first two fully abstract models not employing syntax were formulated during the 1990s. These models are based on game semantics (Hyland and Ong, 2000; Abramsky, Jagadeesan, and Malacaria, 2000) and Kripke logical relations (O'Hearn and Riecke, 1995). For a time it was felt that neither of these models was completely satisfactory, since they were not effectively presentable. However, Ralph Loader demonstrated that no effectively presentable fully abstract model could exist, since the question of program equivalence in the finitary fragment of PCF is not decidable.