Many of the proposals embodied in his numerous equations in fact turned out to be empirically refutable, but is of some historical interest that his formula for the growth of stimulus-response habits can be seen as a primitive form of the generalized delta rule for back- propagation in neural nets. Sutton and Barto (1981, p. 155) examined in some detail the relationship between an equation introduced by Rescorla and Wagner (1972) and the rule for an adaptive element proposed by Widrow and Hoff (1960) and concluded that “the Widrow-Hoff rule is essentially identical to the Rescorla-Wagner equation”; and Rescorla and Wagner themselves introduced this as “a modification of Hull's account of the growth of sHr” (Rescorla and Wagner, 1972; p. 75). Versions of the relevant formulae are given below.  

Hull (1943; pp. 115-120)
H = f(M — H) (1)


Rescorla and Wagner (1972, p. 75)
Va = xa yr (Mr — Vax ) (2)


Rumelhart et al, 1986a; p. 322)
p Wij = nIpi (Tpj — Opj ) (3)

Hull wrote out equation (1) in a more elaborate form, but his procedure for calculating the strength of a habit was very straightforward. A maximum or target value (M) was selected - the theoretical significance of this is that its level was determined by the effectiveness of the teaching agent. The habit starts at zero, but on every occasion when it is strengthened (when the conjunction of stimulus input and response output is followed by motivational effects) the increment to the connection is a constant fraction (f) of the difference between the previous value (H) and the maximum (M).

This was partly a matter of convenience for Hull: he took observed response habits to correspond to connection strengths “hidden within the complex structure of the nervous system” and the equation was based on behavioural data.

Rescorla and Wagner (1972) were also concerned to give an account of behavioural results in Pavlovian conditioning paradigms. Hull’s equation is for one stimulus and one response. A wide variety of phenomena are observed when combinations of stimuli are studied in animal learning experiments. Of relevance here is a reliable empirical result, known as “blocking”, seen if one stimulus, say a buzzer, B, is paired with food many times before a combination of B plus A (say a light) are thereafter joint signals for the food — the light if subsequently tested separately has little or no behavioural effect (Mackintosh 1983). Equation (2) was designed to accommodate this result. The constants x and y are fractions which determine learning rate for a given stimulus “A” the event “R” it predicts, and can be ignored. There is a terminological change in that measured behaviours are attributed to the “associative value” of stimulus A (Va) rather than to a habit. Its maximum is determined by the predicted or “teaching” event “R”. But the major alteration to Hull’s formula is that associative increments are proportional to the difference between the maximum (Mr) and the strength of all stimulus elements present at the time (Vax). Thus if one stimulus element, X, already at the maximum, always accompanies A, nothing will happen to A and this takes care of the blocking phenomenon. The details of the relationship between the modified Hullian equation (2) and the rule for an adaptive element proposed by Widrow and Hoff are explained fully by Sutton and Barto (1981). Clearly the superficial similarity is simply that incremental learning is defined as proportional to the difference between a current value and some target. It is for that reason that the rule is often referred to as a “delta” rule (Rumelhart et al. 1986b, p.53)

Equation (3) is the generalized delta rule used by Rumelhart et al to show that multi- layered networks with hidden units can learn to accomplish, albeit extremely laboriously, the task of producing the correct output when presented with the set of stimuli for the ‘exclusive or’ relationship, and the others which Minsky and Pappert (1969) proved to be impossible for single- layered perceptron networks. It is used to modify all the connection weights in a given network by “back propagation” of the difference between an actual output vector and a desired target output. The similarity between this and the simpler equations is that incremental learning is again defined as proportional to the difference between a current value and some target.

The implications of defining a learning rule in terms of the difference between current and target values are clearly more far-reaching in the case of the generalized delta rule than in Hull’s relatively straightforward growth function, but the formal similarity adds poignancy to the fact that Hull would certainly have been in sympathy with the attempt to model complex behaviour by mathematical treatments of the consequences of strengthening multiple connections.

Hull also provides a link between Pavlov’s connectionism and hardware simulations, since he was explicitly in favour of constructing machines which could mimic psychological phenomena (Hull, 1930, 1937). His own efforts in this direction were technologically limited, but he collaborated in two - reasonably successful - attempts to simulate the rudimentary phenomena of Pavlovian conditioning with electrical circuitry. The first used a serial and parallel arrangement of mechanical switches which allowed current from a battery to charge up a set of capacitors which subsequently showed topographical generalization (Kreuger and Hull, 1931) while in the second (Baernstein and Hull,1931) ‘stimuli’ fed the heaters of mercury-toluene regulators, the contacts of which were arranged so that only forward conditioning was possible (empirically accurate under some but not all circumstances).

Difficulties in Hullian theory and mediating responses.

Many of the principles adopted by Hull turned out to be capable of empirical disconfirmation. He was wedded to a single teaching principle - the reduction of drives related to biological needs - and this is neither necessary nor sufficient. Both excitatory and inhibitory learning were tied to response performance, but the detection of events, or their absence, is sufficient for learning in many instances (Mackintosh, 1974; Walker, 1987).

A more general and serious problem for the whole approach is that it is impossible to account for even fairly ordinary forms of animal behaviour in terms of directly testable stimulus-response connections. Generations of experimental psychologists have rigorously tested the capacities of laboratory rats for negotiating mazes: at first sight the data suggest they possess structured knowledge of the location of objects in space (“cognitive maps”), which are consulted in pursuit of goals, rather than a collection of fixed habits, since they take short-cuts, detour around barriers, and avoid revisiting recently depleted food locations (Tolman, 1932, 1948; Olton, 1979). A simple example which exercised Hull (1934) was the behaviour of rats trained always to run some distance passed a closed door, and then come back to it, before it was opened to allow them to proceed towards a goal. It is not particularly surprising that under some conditions rats waste time trying to scratch their way through the door as they go past it, and perform the novel act of going through on the first pas if it is left open. But neither of these behaviours is directly predictable from Hull’s first principles, in which the underlying mechanisms are entirely frequency- sensitive and response based.

Hull therefore developed an account of the “hidden processes” (1930, p. 511) which intervene in human cognition, and when “the principle of frequency must be overridden” (1934, p.134) in humbler creatures. In order to retain the stimulus- response format, knowledge, purpose and expectancy were attributed to internal “pure stimulus acts”, these being the loci of associations other than between receptor and effector, and the naturalistic basis of symbolism and thought (1930, p. 517). For human linguistic processes fleeting references were made to “sub-vocal speech pure stimulus acts” (Hull, 1952; p. 151). For animal behaviour pure stimulus acts were typically internal anticipations of goals (“fractional antedating goal reactions”: Hull, 1952; p. 151). Also in the context of novel or geographically informed performances of rats Hull elaborated something called the “habit-family hierarchy” to accommodate systematicity and apparent inference: “It is believed, for example, that the habit-family hierarchy constitutes the dominant physical mechanism which mediates such tests of truth and error as organisms employ -- that it provides the basis for a purely physical theory of knowledge” (Hull, 1934, p. 40).