Machine Models for Cognitive Science

computational devices are required to comprehend the behaviors of actual computers or the analogy will not go through; and as a consequence the computationalist hypothesis in so far forth will have no support. Likewise, the other analogies may be understood as specifying features models must satisfy. In all cases it is fairly easy to see that Turing machines are adequate to the analogies. They seem to be inadequate, however, for other reasons I will mention later in section 6. Given this background, the gist of Pylyshyn's argument is that machine operations count as computations, of which cognitive processes are a subclass, only if they are transformations over symbolic expressions that are semantically interpretable. Inasmuch as this condition (5) does not hold of FSA (in a special sense introduced by Pylyshyn and detailed below), they are not adequate for describing any kind of computational process. The question of semantical interpretability is closely related to one of automaton structure ((3) and (4) above). FSA are structurally wrong for modeling cognitive processes. Pylyshyn's point here is that input strings of basic symbols are bound-up with state-to-state transitions. Input symbols are not literally processed by FSA but merely serve to drive its operation from state to state. Moreover a model must, in effect, separate programs from data. What is required for semantical interpretations is that the processing of symbols be factorable out of the transition process-as is true in the case with Turing machines. FSA fail to satisfy processing and separability requirements. Third, even if we count FSA as computers, the class of FSA computable functions is a very small class of the functions programmable on a full digital computer (item (1) above). Theoretically, FSA can add, compute the successor function, and compute compositions of functions. Multiplication, and a fortiori mathematical algorithms that arise in science as well as string operations in data processing and artificial intelligence are beyond their range. They fail to meet a universality requirement. So although in a certain formal sense FSA do compute finitely specifiable functions (2), they fail to support analogies (1), (3), (4) and (5); therefore they are not suitable for cognitive science. In other words, they do not capture the relevant properties of computers. It is somewhat tempting to ignore all of this, even after having gone through the tedious exercise of displaying the analogical underpinnings. For as we shall see, Pylyshyn's argument depends on a gratuitous redefinition of "finite state automaton" and fails to do justice to the scope of the FSA concept. Moreover, the redefinition is not used consistently MACHINE MODELS FOR COGNITIVE SCIENCE 395 in Pylyshyn's argument itself. So in my opinion the issues raised result from misinterpretations of the model and unless they are brought forward and thoroughly reviewed the unwary reader might take them to be relevant to the very kind of automaton concept used by computationalists from Turing's classical paper (Turing 1950) to the present.5 I doubt this is what Pylyshyn intends. At any rate the issues are serious ones. FSA models should be put in the proper perspective. In the following I propose to clarify the complex issues raised by Pylyshyn's treatment of FSA, given the background just sketched, and to show where I think he goes wrong. I begin with (5) and then work through the conditions backwards to the universality requirement (1). 3. Interpretability. The standard definition of "finite state automaton" (Hopcroft and Ulman 1979, p. 18) is the following: (I) A finite state automaton is a system consisting of two disjoint finite, nonempty sets S and X; a distinguished element s, of S; a nonempty subset S' of S; a function M defined on the Cartesian product S X X with domain S. S is the set of states; X is the set of inputs or vocabulary; s, is the initial state; S' is the set of final or accepting states; and M is the transition function. An FSA is essentially a device for implementing a decision procedure on sets of sequences of elements, i.e. strings of X. Rather than load up the reader with more definitions, let us proceed by examples and introduce the necessary formalities as we go along. Pylyshyn's own example (1984, p. 68) of an FSA appears in Figure 1 in the form of a directed graph, where directed edges represent state transitions. In the example the FSA in initial state s, transits to state s, under input x, or to 52 under y; and then to S3 from S2 under x, and so forth. This graph is simply a way of displaying the transition function M considered as a finite machine table in the usual sense. We will indicate the set of 5Contrary to what seems to be the popular view, in his 1950 paper Turing does not explicitly appeal to Turing machines, but to "discrete state machines", which are finite state automata with output (sequential machines) in the present day technical sense (defined below). He makes no reference to unbounded two-way tape machines. He does allude to digital computers as "universal" machines; but they are universal only if one thinks of them as having been supplied with potentially infinite memories. The facts are, a stored program digital computer is a finite state automaton with output (See section 3). Moreover, although a standard minicomputer or mainframe usually has two-way tape (or virtually two-way tape on discs or drums) as auxiliary memory, the tape is bounded. And every two-way bounded tape machine is equivalent to a finite state automaton (Nelson 1978).