1. A multi-processing system with a main storage which is common to a plurality of processor means, each processor means having a buffer storage, the system comprising: the main storage comprising a plurality of blocks; a plurality of block valid memories, each corresponding to a different one of the buffer storages, and each having at least one valid bit, the valid bit of each block valid memory corresponding to a one block of the main storage and being representative of whether the corresponding one block is stored in a buffer storage which corresponds to a one of the plurality of block valid memories; means for detecting whether the valid bit of the one block valid memory corresponding to the one block is valid; means for precluding access to the buffer storage corresponding to the one block valid memory when said detecting means detects that the valid bit is invalid; means for accessing the buffer storage corresponding to the one block valid memory when said detecting means detects that the valid bit is valid to facilitate control for invalidating the block stored in the corresponding buffer storage; and, a block group valid table (BGVT) having at least one group valid bit, the group valid bit corresponding to a block group comprising a predetermined number of blocks, and the group valid bit indicating whether at least one of a plurality of valid bits of block valid memories corresponding to the predetermined number of blocks contained in the block group is valid wherein all of the plurality of blocks belong to a corresponding block group and each group valid bit is provided for a different one of all block groups, said block group valid table being provided for each of said plurality of processor means so that each of said processor means can reference a corresponding BGVT at first when a store access from said each processor means to said main storage occurs, and that said each processor means can reference said block valid memories only in a case where reference to the corresponding BGVT by said each processor means is hit.