Sunday, December 05, 2010

impossible flawless imitation

The inherent limitations to the analysis of a Turing Machine (TM) include a consequence that deserves more attention. A TM cannot be imitated perfectly through a generalized algorithm. Stated differently, a TM's full "description" cannot be inferred from an arbitrary amount of its output ("tape"). No empirical method can be all-sufficient for determining the exact source TM. You can't deduce a TM by its "footprints".

The reasoning is similar to other TM arguments, e.g. the inability to compute when/if a TM will halt. Suppose we know that some TM "S" produces a specific sequence of finite bits, and thereby we contrive our own TM "I" that produces the same sequence. Well, what about the next bit after that sequence (or the next erasure, etc.)? "I", according to its present configuration, produces either a 1 or 0. But given that the output is only assured of matching up to the last bit, there's no theoretical reason why "S" couldn't be identical to "I" in every way except for the configuration for the next bit, in which case our otherwise flawless imitation is ruined. For example, even if "S" has steadily produced alternating bits, it still might produce "00" at position 30, or 35, or 1,462.

Moreover, the situation could be quite possibly worse in a multitude of creative ways. Perhaps "S" is really a complex specimen of Universal Turing Machine with two "source" TM that are much different (analogous to a "piece-wise" mathematical function). "S" executes source TM "A" and then increments a count. When the count exceeds a predetermined value, "S" branches to a configuration that executes the other source TM "B". One may elaborate on "S" with further counts, branching, and source TMs. The point is to reiterate that although we can invent an "I" to imitate "S", we can never conclude that "I" and "S" are exact matches, and in fact we can't figure the ultimate similarity at all as execution stretches to infinity. Failure of "I" is due to missing information about the "S" TM, but worse is the more subtle problem: there's no way to know how much information is missing at any time!

So...
  • Generally speaking, clean-room or black-box re-implementations of software can't be guaranteed to succeed in every detail. For instance, exceptional conditions or highly specific combinations of input could trigger diverging outcomes. The imitation of the previous software's bugs can be particularly troublesome.
  • Whenever we compute/emulate anything that requires TM-level processing, we can't be absolutely sure whether we ever achieve total fidelity. This doesn't imply that a conceptual TM is the most convenient theoretical model of a phenomenon (it usually isn't!), but merely that the chosen model can somehow run on a "computer". Or in more formal terms, the model outlines an explicit procedure that yields predictions of one or more modeled quantities. In some open-and-shut cases, the model's error is persistently trivial under all experiments and we therefore have no justification for doubting its effectiveness. Yet history is filled with times in which a serviceable first model (circular planetary orbits with epicycles) is later replaced by one with still greater nuance and realism (elliptical).
  • Third-person human brains (and behavior) undoubtedly fall under these conclusions. That is, any verbalized or intuitive model of a specific brain cannot be completely understood. Sympathy has its limits. While careful observation combined with genuine insight yields an impressive array of knowledge about "what makes his or her brain tick", especially among species evolved to be highly social, the application of the knowledge often proves how superficial it truly is. Biographers can detect what they call formative experiences, but tasking the biographer to reliably predict what a living subject shall do next will illustrate that the analysis works best retroactively. Of course, if someone theorizes that the human brain performs computations beyond the power of a TM, then the argument is correspondingly strengthened. Two such brains surely cannot be synchronized when two lowly TMs cannot. The proposed mechanism of brain-based super-computation, e.g. quantum effects of exotic pieces/states of matter, would increase not decrease the difficulty. 
  • More surprisingly, first-person human brains (and behavior) are also prone. There's no "imitation" of oneself, but there's a self-concept. The very process of answering the self-evident question, "Why did I think or do that?", necessitates the mental model of self. The brain attempts to develop a "TM" whose workings imitate the actual and mysterious workings of itself. Unfortunately, it's sabotaged by bias toward a pleasing answer! Transparency and self-knowledge are notoriously plagued by mistakes. Countless fictional works have relied on the humor if not the tragedy of self-delusion.

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