Uniformly reasonable source encoding is often practically impossible

The decoding of efficiently encoded messages, from either probabilistic, nonprobabilistic, or unknown message sources, is shown to be often practically impossible. If
\tau(S)
is a running-time bound on the computational effort of a decoder
\Psi
accepting a codeword
P
for message
S
, and
\gamma[K_{\Psi}(S)]
is an upper bound to acceptable codeword length
\mid P \mid
when the shortest codeword for
S
has length
K_{\Psi}(S)
, then for many message sources
\mathcal{M}
there exist messages
S \in \mathcal{M}
such that: 1) if the encoder satisfies
\gamma
, then the decoder violates
\tau
; 2) if the decoder satisfies
\tau
, then the encoder violates
\gamma
. These conclusions remain valid even when we allow the decoder to reconstruct only an approximation
S \prime
in a neighborhood
\delta(S)
of
S
. 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.