The next objection we consider marks an appeal to connectionism; it runs as follows. ``Your argument, I concede, destroys Computationalism. But that suits me just fine: I don't affirm this doctrine, not in the least. For you see, cognition isn't computation; cognition, rather, is suitably organized processing in an architecture which is at least to some degree genuinely neural. And I'm sure you will agree that a Turing Machine is far from neural! Neural nets, however, as their name suggests, are brain-like; and it's on the shoulders of neural nets that the spirit of `Strong' AI and Cognitive Science continues to live, and indeed to thrive."
This objection entails a rejection of Proposition 1''. As such, it succeeds in dodging the Argument From Irreversibility -- since this argument has this proposition as a premise. However, the argument can be rather easily revived, by putting in place of Proposition 1'' a counterpart based on neural nets, viz. (where iff x is an corporeal neural net),
Proposition 2. x is conscious from to passes through some process -- partly determined by causal interaction with the environment -- which is identical to the consciousness x enjoys through
This proposition is affirmed by connectionists (of the ``Strong" variety, anyway). But Proposition 2, combined with a chain of equivalence holding between neural nets, cellular automata, k-tape Turing Machines, and standard TMs, is enough resurrect the Argument from Irreversibility in full force. The chain of equivalence has been discussed in detail by one of us (Bringsjord) elsewhere [2f], in a paper which purports to show that connectionism, at bottom, is orthodox Computationalism in disguise. Here, it will suffice to give an intuitive recapitulation of the chain in question -- a chain which establishes the interchangeability of neural nets and Turing Machines.
Before we sketch this chain, let's pause to make clear that it underlies a certain proposition which can be conjoined with Proposition 2 in order to adapt the Argument From Irreversibility. This proposition is
Proposition 3. passes through some process through , where this computation is identical to the process x enjoys through
It's easy to prove in elementary logic (using such rules as universal elimination and modus ponens) that Proposition 2, conjoined with Proposition 3, reenergizes the Argument From Irreversibility. But why is Proposition 3 true? In order to answer this question, we need to look first at neural nets. After that, even a casual look at ``two-dimensional" TMs should make it plain that Proposition 3 is true.
Neural nets are composed of units or nodes, which are connected by links, each of which has a numeric weight. It is usually assumed that some of the units work in symbiosis with the external environment; these units form the sets of input and output units. Each unit has a current activation level, which is its output, and can compute, based on its inputs and weights on those inputs, its activation level at the next moment in time. This computation is entirely local: a unit takes account of but its neighbors in the net. This local computation is calculated in two stages. First, the input function, , gives the weighted sum of the unit's input values, that is, the sum of the input activations multiplied by their weights:
In the second stage, the activation function, g, takes the input from the first stage as argument and generates the output, or activation level, :
One common (and confessedly elementary) choice for the activation function (which usually governs all units in a given net) is the step function, which usually has a threshold t that sees to it that a 1 is output when the input is greater than t, and that 0 is output otherwise. (This is supposed to look ``brain-like" to some degree, given the metaphor that 1 represents the firing of a pulse from a neuron through an axon, and 0 represents no firing.)
As you might imagine, there are many different kinds of neural nets. The main distinction is between feed-forward and recurrent nets. In feed-forward nets, as their name suggests, links move information in one direction, and there are no cycles; recurrent nets allow for cycling back, and can become rather complicated. But no matter what neural net you care to talk about, Proposition 3's deployment of its universal quantifier remains justified. Proving this would require a paper rather more substantial than the present one, but there is a way to make the point in short order. The first step is to note that neural nets can be viewed as a series of snapshots capturing the state of its nodes. For example, if we assume for simplicity that we have a 3-layer net (one input layer, one ``hidden" layer, and one output layer) whose nodes, at any given time, or either ``on" (filled circle) or ``off" (blank circle), then here is such a snapshot:
As the units in this net compute and the net moves through time, snapshots will capture different patterns. But Turing Machines can accomplish the very same thing. In order to show this, we ask that you briefly consider two-dimensional TMs. We saw k-tape TMs above; and we noted the equivalence between these machines and standard TMs. One-head two-dimensional TMs are simpler than k-tape machines, but (in the present multidisciplinary context) more appropriate than their standard k-tape cousins. Two-dimensional TMs have an infinite two-dimensional grid instead of a one-dimensional tape. As an example, consider a two-dimensional TM which produces an infinite ``swirling" pattern. We present a series of snapshots (starting with the initial configuration, in which all squares are blank, and moving through the next eight configurations produced by the ``swirl" program) of this machine in action. Here's the series:
The point of this series of snapshots is to convey that snapshots of a neural net in action can be captured by a one-head two-dimensional TM (and, more easily, by a k-tape, k-head machine). Hopefully you can see why this is so, even in the absence of the proof itself. The trick, of course, is to first view the neural net in question as an array (we ignore the part of the infinite grid beyond this finite array), as we did above. Of course, it's necessary to provide a suitably enlarged alphabet for our neural-net-copying TM: it will need to have an alphabet which contains a character corresponding to all the states a node can be in. For this reason, our swirl TM is a bit limited, since it has but a binary alphabet. But it's easy enough to boost the alphabet (and thereby produce some entrancing pyrotechnics).
The proponent of Objection 7 might say that a sequence of configurations of a Turing Machine M proposed to capture a neural net N as it passes through time isn't complete, because such a sequence includes, at best, only discrete ``snapshots" of the continuous entity N. But suppose that the sequence includes snapshots of N at t (= ) and t+1 (= ), and that our opponent is concerned with what is left out here, i.e., with the states of N between t and t+1. The concern is easily handled: one can make the interval of time during which the state of N is ignored arbitrarily small. Doing so makes it the case that the states of N which had formerly been left out are now captured.