The authors propose a formal definition of the two-phase locking class derived from the semantic description of the two-phase locking protocol, and prove that this definition is equivalent to that given by C.H. Papadimitriou (1979). They present: (1) a precise definition of the two phase locking; (2) a clarification of the occurrence and the order of all events such as lock points, unlock points, read operations, and write operations of conflicting transactions; and (3) by relaxing some conditions in the given definition, the derivation of a new class called restricted-non-two-phase locking (RN2PL), which is a superset of the class two-phase locking (2PL) but a subset of the class D-serializable (DSR) given by Papadimitriou. >