- ...IRREVERSIBILITY
- We are indebted to
Bill Rapaport, Pat Hayes, Ken Ford, Marvin Minsky, Jim Fahey,
two anonymous
referees (who provided particularly insightful comments), and
many Rensselaer students.
These people provided trenchant objections which saw to the
evolution of the present version from a rather inauspicious primogenitor.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...computation.
- But due to space constraints, nowhere
in this paper do we provide a
detailed, comprehensive account of Computationalism. Such an account
would probably need to include careful versions of at least the following
five propositions.
- A function
*f*is effectively computable if and only if*f*is Turing-computable (Church's Thesis). - Proposition 1.
- The Turing Test is valid.
- Computationalists will succeed in building persons.
- Computationalists will succeed in building Turing Test-passing artifacts. (This proposition is presumably entailed by its predecessor.)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- A function
- ...precise,
- Formally, keeping in mind that after a while,
in both directions, the tape is populated by infinitely many blanks,
a configuration is
a member of the set 11#11{
*h*}12#12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...1's.
- To the uninitiated, this computation
will doubtless sound remarkably unimpressive, but initiated ought to note that
this machine is the result of Genetic Algorithm-powered search
(engineered by Gordon Greene) in the
(enormous) space of 6-state TMs for ``Busy Beaver" candidates in the
``quaduple" formalism. (Currently, the most productive known 6-state
machine can produce 21 1's. The machine - built by
Chris Nielsen - can be seen (in flow graph form) and
obtained by linking
through Bringsjord's web site
(URL above), under the course
*Symbolic Logic*. Alternatively, go directly to http://csli-www.stanford.edu/hp/Beaver.html.) In the ``Busy Beaver" function, 23#23 (**N**here denotes the natural numbers), 24#24 yields the greatest number of 1's an*n*-state TM, starting on a black tape, can leave after halting. This function, and the corresponding proof that it's uncomputable, is due to Rado [32], who used 25#25 for 26#26. Busy Beaver candidates are those*n*-state TMs which appear to produce 24#24 1's.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...TM.
- See
[2f] for a discussion of the consequences of this fact for AI.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...one.
- The
interested reader can consult an octet of books we find useful: For
broad coverage of the basic material, see [24],
[15], [9], and
[21]. For a nice comprehensive discussion of
computability theory that includes succinct coverage of uncomputability,
including the Arithmetic Hierarchy, see [10] and the
difficult but rewarding [40]. [29] contains a very
nice discussion of the Chomsky Hierarchy. And, of course, there's always the
classic [33].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...World"41#41,
- The
``Turing's World"41#41 software comes with [3].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...theorem:
- Note that by Church's Thesis Theorem 1 is interchangeable
with this theorem (which is easy to prove
*without*CT):**Theorem 1'**. For every computation 42#42, there exists a computation 43#43 which can be obtained from the original computation via some TM 44#44.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...consciousness.
- This
is borne out by, among other things,
looking at proposed comprehensive models of cognition in
cognitive science: these models often include some component claimed to
account for consciousness. For example,
Schacter [34] gives us the the picture of cognition
shown in Figure 2.
For a survey of models like this one, see
[1]; for a penetrating
analysis of such
models (including Schacter's) see [8]. For those interested in
charting the first-order formalization of our argument,
the relevant sentence in first-order logic would be a full symbolization of
50#50, where `cognizing' is understood to indicate
the full scope of human cognition as purportedly captured in models like
Schacter's.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...life.
- It's important to realize that we're
talking about your
*mental*life: we're not talking about reversing some sequence of physical actions which can be described in such as way as to imply that it is a discrete chain. For example, suppose that you move block*a*from on top of block*b*to a position next to block*c*, and then move block*c*to on top of block*b*. You might say that this sequential action is easily reversed by first moving block*c*to its original position, and then moving block*a*on top of block*b*. Though as a matter of fact - as we shall see below - you haven't truly reversed the sequence*qua*physical process, the present point is that we are concerned with reversing a stretch of conscious experience, not a sequence of ostensibly discrete bodily actions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...possible.
- Compare this sort of indivisibility with the type
Descartes famously ascribed (perhaps incorrectly) to the mind when he said:
In order to begin this examination, then, I here say, in the first place, that there is a great difference between mind and body, inasmuch as body is by nature always divisible, and the mind is entirely indivisible. For, as a matter of fact, when I consider the mind, that is to say, myself inasmuch as I am only a thinking thing, I cannot distinguish in myself any parts, but apprehend myself to be clearly one and entire; and although the whole mind seems to be united to the whole body, yet if a foot, or an arm, or some other part, is separated from my body, I am aware that nothing has been taken away from my mind. ([13], p. 196)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...irreversible.
- Additional
objections are possible, but not powerful. For example, someone
might claim that appeals to
thermodynamics have no place here since neither Max nor the system being
manipulated need be closed. The short answer to this is that only
equilibrium thermodynamics requires closed systems. Indeed, our
arguments depend on recognizing important differences between equilibrium
and non-equilibrium situations. Consciousness necessarily involves
departure from equilibrium; computation doesn't.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...brains.
- One of us
(Bringsjord) conducts a lot of ``Weak" AI, in the form of an attempt
to engineer systems capable of autonomously generating stories.
See, for example, the programs featured in [2a].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...computation.
- We
make veiled reference here to a distinction between
*simulation*and*replication*. Details of the distinction aren't necessary for this paper; a rough-and-ready characterization suffices. Accordingly, we say that to simulate some human chess player via AI techniques would be to produce a computer program whose overt behavior (i.e., actual chess moves) resembles that of some human, whereas to replicate Kasparov would be to construct an artifact that literally has the same inner life he has, the same emotions, plans, experiences, mental images, memories, etc.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...consciousness,
- In
[2g]
we devise and exploit a variation on the
classic brain-in-a-vat gedankenexperiment in order to make the case for
this view. In [2c] one of us (Bringsjord) argues,
*contra*Harnad [17], that Turing Testing, in order to test for consciousness, needn't include a test for the ability of a would-be AI to interact with its environment.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...
*M*. - One method
for such encoding comes via gödel
numbers. For example, see [15].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...2.
- We place scare quotes around
`Theorem' because we describe the result very informally. For a more
precise statement of the theorem, as well as a more precise account of the
proof than what we provide below,
see [24].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...true.
- We leave aside a complexity-based
construal of `power.' Obviously, a physical TM with
multiple tapes could sometimes solve problems
*faster*than an ordinary one-tape TM. However, there are no problems which are unsolvable by a standard TM yet solvable by a multi-tape machine.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...TM.
- It's easy enough to imagine the details of
this conversion
in some cases. For example, suppose that we present you with a 4-tape TM
roughly in the the form of a model railroad set. That is, suppose that:
the tapes are divided into squares; the
read/write head, for each tape, is a lone boxcar; there is some way to give
simple instructions to this railroad TM; etc. Now imagine converting
this physical TM to a one-tape TM via the trick at the heart of Theorem 2.
You would have to find some way to link the tracks together into one unit.
If you have any experience with model railroading, you probably can
visualize yourself tackling the process.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...(iii)}.
- This is as good
a place as any to
point out that our argument is of necessity a good deal trickier
than this one: ``TMs are abstract entities that do not exist
in space/time and between which only logical relations obtain, while
minds are causal systems in space/time. Logical processes are
reversible wile causal processes are not). Therefore minds are not
TMs." Our argument is trickier because Computationalism doesn't
hold that minds (or persons) are abstract entities. On the contrary,
computationalists as a rule hold that minds are physical things.
They must also hold, however, that minds, as physical things, abide
by the principles of computation - and this is what gets them
into trouble, as we are in the process of showing. The simplest
distillation of our argument is that it is a proof of inconsistency
holding between four propositions, as we indicated at the outset,
and as we explain in more detail in the present section.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...1''.
- They are
*not*free to reject Theorem 1, however. Two different and determinate levels may be addressed by Theorem 1 (and the like): perhaps there is the purely mathematical level of the theorem itself, and and then perhaps also what might be called the ``logic gate level" of computer engineering - a level indispensable for Weak AI. Computer engineers have on hand physical instantiations of Turing Machines which they can combine in order to generate increasingly sophisticated computational artifacts, but there is no guarantee, say, that such engineers will be familiar with the purely formal set theoretic definition of a TM and a TM configuration. Nonetheless, both theoretical computer science*and*computer engineering is constrained and informed by Theorem 1 and its relatives. Work in AI and Cog Sci, whether of the ``Weak" or ``Strong" variety, is carried out at both of these levels. To abandon these levels and carry out a research program which in its entirety is ``below" them, say research exclusively at the neuromolecular level, is to abandon AI for a variant on bio-engineering.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...processes.
- Again,
for a detailed discussion of Max, ballistic computers,
and other, similar devices, see [6] and
[4].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...impotent.
- This
is as good a place as any to register
our prediction that at this juncture those desperate to dodge our
argument might say about the Sandra thought-experiment what those
who defend the coherence of
time travel have said about the so-called ``Grandfather
Paradox" (GP, in a word, being that if time travel if possible, then
you could go back in time and kill your grandfather, but then how
could you be around to do the killing?), namely that the Sandra case
just won't ever happen. This move is laughably
*ad hoc*. Besides, Sandra's P-consciousness isn't really any different than garden variety stretches of P-consciousness evoked by watching a movie played from a VCR whose digital clock keeps normal time.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...anyway).
- One of
us (Bringsjord) distinguishes between various sorts of computationalists
and connectionists in [2f].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...disguise.
- This paper is based on
the classic statement of connectionism given by Paul
Smolensky [38], [39].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...otherwise.
- McCulloch
and Pitts [27] showed long ago that such a simple
activation function allows for the representation of the basic
Boolean functions of AND, OR and NOT.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...action.
- The
machine itself, as a series of quadruples (with 0 for blank,
1 for filled, R and L for ``right"
and ``left," resp., A for ``any," etc.)
is (read left to write, top to bottom):
91#91

The implementation of this machine - in Turing's World41#41 - sent upon request.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...pyrotechnics).
- Readers wanting to see some of them
are encouraged to consult Poundstone's [30]
classic discussion of Conway's [7] Game of Life.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...it.
- Don't
worry about how the chess board and pieces got there in the
first place. Perhaps our props were launched into space centuries ago,
when the human race still existed. Perhaps the board and the pieces
were formed from mud by the random winds invoked below104#104
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...evaporates.
- It is at any rate undeniable,
given this gedankenexperiment, that the ontological status of chess
games
*is*a bit mysterious.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...taking.
- It is the route
one of us (Bringsjord) follows in
the attempt to engineer systems which
*appear*to be creative; see [2a].. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fri Sep 6 11:58:56 EDT 1996