The main claim of this paper is that notions of implementation based on an isomorphic correspondence between physical and computational states are not tenable. Rather, "implementation" has to be based on the notion of "bisimulation" in order to be able to block unwanted implementation results and incorporate intuitions from computational practice. A formal definition of implementation is suggested, which satisfies theoretical and practical requirements and may also be used to make the functionalist notion of "physical realization" precise. The upshot of this new definition of implementation is that implementation cannot distinguish isomorphic bisimilar from non-isomporphic bisimilar systems anymore, thus driving a wedge between the notions of causal and computational complexity. While computationalism does not seem to be affected by this result, the consequences for functionalism are not clear and need further investigations.
@article{scheutz01mm,
title={Causal vs. Computational Complexity?},
author={Scheutz, Matthias},
year={2001},
journal={Minds and Machines},
volume={11},
pages={543--566}
url={https://hrilab.tufts.edu/publications/scheutz01mm.pdf}
doi={10.1023/A:1011855915651}
}