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The decoding of efficiently encoded messages, from either probabilistic, nonprobabilistic, or unknown message sources, is shown to be often practically impossible. If is a running-time bound on the computational effort of a decoder accepting a codeword for message , and is an upper bound to acceptable codeword length when the shortest codeword for has length , then for many message sources there exist messages such that: 1) if the encoder satisfies , then the decoder violates ; 2) if the decoder satisfies , then the encoder violates . These conclusions remain valid even when we allow the decoder to reconstruct only an approximation in a neighborhood of . The compatibility of these results with those of information theory rests upon the fact that we are inquiring into the detailed properties of coding systems for individual messages and not into the ensemble average properties.