We consider the duality between computation and measurement emerging from the thermodynamics of Turing machines. This duality suggests a new measure of complexity (physical complexity) which corresponds to our intuition and appears to be practical enough to estimate the complexity of genomes. From an automata theoretic point of view, this complexity is just the mutual information between a thermodynamic Turing machine and the equilibrated tape that constitutes its ``universe''. By the same token, the complexity is dual to the mutual entropy between the observer and the system in the information theory of measurement. The complexity of a bit-string, therefore, is the number of bits that can be used to lower the entropy of a closed system. Random strings have vanishing physical complexity (even though they have maximal Kolmogorov complexity), as they cannot be the result of a (classical) measurement or computation.