‘I’ Before ‘E’: Marr’s Levels

In his book Vision, David Marr described three “levels of explanation” that we can aspire to when describing biological systems—especially perceptual modules—in computational terms. His levels corresponded to three kinds of questions, with ‘function’ used in its extensional sense.

                        (1)  What is the system computing? Which function is being computed?

                        (2)  How is that function being computed? What algorithm is the system using?

                        (3)  How is the algorithm implemented? How is the procedure physically realized?

The third question presupposes an answer to the second, which presupposes an answer to the first. So Marr suggested that theorists start with some description of the system’s input/output profile, and then revise that description with the aim of describing a computable function F such that we can see (a) why computing F would be useful, given the system’s role in organisms, and (b) how F can be computed. One might start by saying that the visual system maps “arrays of image intensity values” (provided by retinal photoreceptors) onto representations of how objects (at certain scales) are spatially arranged within a field of view. But any such characterization must be made suitably precise, in order to talk about computing functions. And computations are often staged, with outputs of some subcomputations serving as inputs to others. A good Level One description will foster the process of decomposing a computationally recalcitrant input-output profile into a series of tractable functions. Though when composing functions, order often matters: G°F(x) = G[F(x)]; and G[F(x)] may differ from F[G(x)] for many values of ‘x’. So one quickly gets beyond mere extensional description of “the” function being computed.

In an earlier post, I used (4-6) as an example of extensionally equivalent procedures.

(4)  F(x) = |x – 1|                    

(5)  F(x) = ⁺√(x² - 2x + 1)

(6) { ... , <-2, 3>, <-1, 2>, <0, 1>, <1, 0>, <2, 1>, ... }

Likewise, (7) and (8) can be read as descriptions of distinct procedures, each of which determines the set of ordered pairs gestured at with (9).

(7)  G(x) = x + 1

(8)  G(x) = ³√(x³ + 3x² + 3x + 1)

(9)  { ... , <-2, -1>, <-1, 0>, <0, 1>, <1, 2>, <2, 3>, ... }

We can define two composite functions in extension, using lambda expressions extensionally.

(10)  λx.(x + 1)°|x – 1| = { ... , <-2, 4>, <-1, 3>, <0, 2>, <1, 1>, <2, 2>, ... }

(11)  λx.|x – 1|°(x + 1) = { ... , <-2, 2>, <-1, 1>, <0, 0>, <1, 1>, <2, 2>, ... }

If you only consider positive inputs, you won’t see the difference. And you might think the system is computing the identity function. But if the system can compute values for other inputs—even if it normally wouldn’t—that can help tease apart hypotheses about what the black box is doing. Still, even given the input/output profile of (10), there are at least four possibilities concerning how the outputs are being computed, now reading λ-expressions intensionally.

λx.(x + 1)°|x – 1|                                    λx.(x + 1)°⁺√(x² - 2x + 1)

λx.³√(x³ + 3x² + 3x + 1)°|x – 1|          λx.³√(x³ + 3x² + 3x + 1)°⁺√(x² - 2x + 1)        

And given the input/output profile of (11), the procedure λx.|x| might be one’s first hypothesis. If one finds independent evidence that the system cannot compute cube roots, that would tell against some possibilities. If one finds evidence that the system does implement the successor and predecessor functions, but not the square root function, that would be genuine cause for celebration. With these points in mind, let’s return to I-languages, which presumably connect articulations with meanings (in accord with certain constraints) in some staged way.

A caveat: Marr’s talk of levels may apply best to systems that are perceptual, or perhaps modular in Fodor’s sense of encapsulated “input” systems that deliver outputs to an independent “central processor.” And while the human faculty of language (HFL) is used in episodes of perceiving meaningful speech as such, it is also used in other ways, including soliloquy. For some purposes, it may do no harm to describe each mature state of HFL as a machine that (i) yields meanings given articulations as inputs, and (ii) yields articulations given meanings as inputs. Though (ii) seems to presuppose a mental system that can generate meanings; and therein lies at least one mystery. In part for these reasons, Chomsky often speaks of linguistic competence, which gets used in many ways we don’t understand. The “creative” uses of HFL include not just poetry, but actions of deciding to say (or think) something relevant in but not determined by the context; see his review of Skinner, and “Faculty Disputes” by John Collins.

Relatedly, Marr’s questions differ from Chomsky’s big three in Knowledge of Language: what do speakers of a language know; how did they come to know it; and how do they put that knowledge to use? Chomsky’s answer to the first two questions is, roughly, that each speaker of a language knows—where this “knowledge” may just amount to having—a certain I-language, which was acquired by some process of growth and using experience to settle on at least one of the humanly possible I-languages, which conform to generalizations of Universal Grammar. Still, it is useful to think about how Marr’s computational and algorithmic levels, (1) and (2), relate to Chomsky’s knowledge and acquisition levels. As Frege taught us, acquiring knowledge is often a matter of re-presenting. And methodologically, Level One characterization is important. We need some way of thinking about what I-languages generate (viz. articulation-meaning pairs) in order to offer specific proposals about the generative engine.

Moreover, defending specific proposals is hard. So it’s nice to have a fall back position. In my view, a linguist’s grammar is a procedural description of an I-language, not a description of a set. But one can think of a grammar as encoding an immodest claim—about a procedure that speakers have come to know and use in many ways—that theorists can weaken. To write a grammar is to specify a generative procedure P. But one might doubt that speakers implement and use P itself, even if one can’t specify a more likely procedure P' that is no less descriptively adequate than P. (Note: insisting on extensional equivalence to P would be wrong, unless P generates all and only the right articulation-meaning pairs; and inadequacy is the norm.) It can be reasonable to back off the hypothesis that speakers implement P, in favor of the weaker claim that speakers implement some refinement of P: a procedure P' that reflects insights of the initial grammar, as opposed to being a completely different grammar, and is at least as descriptively adequate as P. These conditions are vague, like the general question of when a new theory refines or just replaces an old one. And more needs to be said about what it is to implement P, as opposed a use of P (e.g., in comprehension). But we can talk about strong and weak hypotheses about speakers without talking about strong vs. weak learning, much less learning E-languages.

Let me end by going back to Church. When he said that we can construe λ-expressions extensionally, his point was that we can do so and still say interesting things. (He referred to a particular theorem about functions in extension.) But as he stressed, intensional construal is often needed for claims about computability—e.g., the final theorem in On the Calculi of Lambda Conversion. Maybe there are some interesting generalizations concerning E-languages. Maybe. But the methodological priority of Level One description need not reflect an algorithm-neutral domain of inquiry. It can be tempting to think that one job (for linguists) is to characterize what speakers of a language know extensionally, and that a second job (for psychologists) is to specify a procedure that determines the relevant extension. Resist the temptation. The job is just hard.