Memory for Goals in Means-ends Behavior. Erik M. Altmann George Mason University Fairfax, VA

Transcription

1 Memory for Goals 1 Memory for Goals in Means-ends Behavior Erik M. Altmann George Mason University Fairfax, VA J. Gregory Trafton Naval Research Laboratory Washington, DC 7/30/99 Please do not quote or distribute without permission Running head: Memory for Goals Correspondence: Erik M. Altmann George Mason University Psychology, MS 2E5 Fairfax, VA (voice) (fax) altmann@gmu.edu

2 Memory for Goals 2 Abstract Means-ends behavior arises when goals (the ends) need to be suspended to achieve subgoals (the means) and then resumed again. One traditional explanation of such behavior is that cognition stores goals on a stack (the task-goal stack, or TGS). However, the TGS lacks face validity, and the authors propose the memory-as-goal-store (or MAGS) approach instead. A MAGS model explains means-ends performance better than a published TGS model implemented in the same cognitive architecture (ACT-R). Experimental data are presented that contradict even weak predictions of the TGS and support the MAGS view. For example, pending goals intrude more often than chance as a function of active maintenance. The computational processes of the MAGS model are specific enough to make fine-grained predictions and general enough to explain memory for goals in diverse contexts.

3 Memory for Goals 3 Memory for Goals in Means-ends Behavior Means-ends behavior involves suspending a goal that is currently unachievable (the ends), working on its subgoals for a while (the means), and when the subgoals are achieved returning to the pending goal. Such behavior is fundamental in how we approach problems that are too large or complex to solve directly (Miller, Galanter, & Pribram, 1960). From a cognitive perspective, means-ends behavior requires keeping track of pending goals in order to select or reactivate them at the right time (e.g., Brandimonte, Einstein, & McDaniel, 1996). This in turn requires storing those goals where they can be retrieved when necessary. Often this storage is internal in memory, but it can also be external, say in the form of a written note. In both cases the situation is more complex, however, in that each storage medium can depend on the other to be functional. For example, a goal stored in memory (say to tell someone something) might be associated with an environmental cue (seeing that person). Conversely, retrieving a goal written down on a piece of paper depends on remembering that the goal is in fact written down, and then remembering where (Altmann & John, 1999). Thus means-ends behavior depends critically on goals stored in some combination of internal and external representations. From the internal perspective, the standard explanation of how people store goals during means-ends behavior is that they use a stack (Anderson & Lebiére, 1998; John & Kieras, 1996; Lehman, Laird, & Rosenbloom, 1998; Newell, 1990; Newell & Simon, 1972). The appeal of this explanation is linked to the stack s accuracy in describing the means-ends structure of many tasks that humans face. For example, traveling to a conference may entail taking an airplane, and whereas getting to the conference is the higher-level goal, getting on the airplane has to happen first. Thus the order in which steps are planned (conference, then airplane) is opposite the order in which steps are executed (airplane, then conference). As shown in Figure 1, this is precisely

4 Memory for Goals 4 the kind of reversal supplied by a stack, if goals are pushed in the order they are planned and popped in the order they are achieved. Cognitive theorists have often assumed that because people can accomplish means-ends tasks, cognition must have a task-goal stack available. This assumption was embedded in the problem behavior graph and the General Problem Solver (Newell & Simon, 1972) as representations of human behavior, was later incorporated in cognitive architectures (Anderson, 1983; Laird, 1984; Laird, Rosenbloom, & Newell, 1986), and lives on today in many forms. The goal stack is a feature of task analysis tools like GOMS (John & Kieras, 1996) and remains a core mechanism in cognitive architectures like ACT-R (Anderson & Lebiére, 1998) and Soar (Lehman et al., 1998; Newell, 1990; Newell, Yost, Laird, Rosenbloom, & Altmann, 1991). The purpose of the stack is typically to store goals associated with the task task goals as long as progress on them is blocked. However, descriptive adequacy for task structure does not imply descriptive adequacy for cognitive performance. Indeed, interpreted literally as cognitive theory the task-goal stack has little face validity. Stacked goals are immediately available for error-free retrieval, with their order perfectly preserved, regardless of when they were stacked, and regardless of how much or how little they were referenced in the meantime. Moreover, specific implementations have their own unique problems; the architectural stack in ACT-R is difficult to reconcile with human reactivity, and the architectural stack in Soar makes conflicting predictions concerning the interaction of task decomposition and learning (John, 1996). This article examines the task-goal stack critically and offers an alternative approach to modeling goals in means-ends behavior. The alternative, which we refer to as memory-as-goalstore or MAGS approach, assumes that goals are ordinary declarative memory elements and then

5 Memory for Goals 5 asks what other information and processes are necessary and available to support means-ends behavior. To test the MAGS approach we revisit the Tower of Hanoi, a means-ends task often taken to reveal the cognitive reality of the task-goal stack. We discuss a published stack-based model of the Tower of Hanoi (the TGS model) and contrast it with a MAGS-based model that makes stronger predictions and explains means-ends behavior in greater detail. Both models are constructed in ACT-R (Anderson & Lebiére, 1998); however, whereas the TGS model stores task-goals on the architectural stack, MAGS stores them in unstructured declarative memory and uses heuristics and perceptual cues for retrieval. We then report an experiment for which the MAGS view and the task-goal stack make different predictions. Even a weak interpretation of the task-goal stack fails to explain the results, but the MAGS explanations are straightforward and consistent with multiple measures. In the general discussion we argue that care should be taken before the architectural stack is discarded outright from computational theories like ACT-R and Soar, because it has evolved to support important processes other than storing task goals. We conclude that the encoding and retrieval processes described in this article build on ACT-R s memory theory and will transfer to explain memory behavior in other contexts. Memory for Goals in the Tower of Hanoi To examine means-ends behavior in detail we revisit the Tower of Hanoi, a foundational task in the study of problem solving and problem representation (Anzai & Simon, 1979; Egan & Greeno, 1973; Karat, 1982; Kotovsky, Hayes, & Simon, 1985; Ruiz & Newell, 1989; Simon, 1975; Svendsen, 1991; VanLehn, 1991; Zhang & Norman, 1994). Human performance on the Tower of Hanoi has often been analyzed on the assumption that cognition has a stack available for storing task goals (Anderson, Kushmerick, & Lebiére, 1993; Anderson & Lebiére, 1998; Egan & Greeno, 1973; Simon, 1975). Moreover, although working memory limitations have

6 Memory for Goals 6 been proposed as factors in Tower of Hanoi performance (Karat, 1982; Kotovsky et al., 1985; Simon, 1975), models actually grounded in memory theory are rare. A model implemented in the 3CAPS architecture attributes performance errors to activation loss (Just, Carpenter, & Hemphill, 1996), but errors are not represented functionally in the model and there is no constraint from empirical response times. Thus the stage is set for a detailed model of how people really manage goals as they perform a means-ends task. The initial state of a typical Tower of Hanoi problem is shown in Figure 2, together with the final destination for one of the disks (4). The arrows in the figure illustrate the goal-recursive algorithm (Simon, 1975) applied to deriving the first move from the initial state. The high-level goal is to move 4 from its source peg (A) to its final destination (C). However, this move is blocked because all smaller disks are in the way one disk is said to block another if it is smaller than the other and rests on the other s source or destination peg. The goal-recursive algorithm focuses on each smaller disk in turn, asking where that disk needs to be to unblock the previous disk (the unfilled arrows in Figure 2). Eventually the algorithm arrives at 1, which can be moved immediately (the filled arrow). After 1 is moved, 2 can be moved, but then 3 is blocked, so the algorithm must be invoked again. In this example we have purposely omitted final destinations for disks other than 4 to make a distinction between final and intermediate destinations. Smaller disks have more intermediate destinations than larger disks, because they have to move more often to unblock larger disks. Thus 1 has a new intermediate destination on every second move, whereas the largest disk (4 in this example) moves only once; after it has been moved, it can be completely ignored. A task goal in this context is an association between a disk D and a destination peg P. We designate a task goal (and a move, where appropriate) as D:P. The first task goal produced by the

7 Memory for Goals 7 goal-recursive algorithm is 4:C, the second 3:B, and the third 2:C. The first of these (4:C) is readily inferred from the display by comparing the current state of the puzzle to the final state. However, the recursive task goals formulated in service of 4:C are not perceptually available. These must be re-planned after every move using the goal-recursive algorithm or they must be stored in memory and retrieved at the right time. For example, once 1:B has been made, the next move could be planned by focussing on 4 again, then on 3, and then on 2. Alternatively, if a task goal for 2:C had been stored in memory during the first pass of the algorithm, and if that task goal could be retrieved now, then this second planning episode could be spared; memory would indicate 2:C. A Model Using the Task-Goal Stack A stack-based model of the Tower of Hanoi is illustrated in Figure 3, which shows the first eight moves from the initial state shown in Figure 2. This eight-move sequence is the optimal sequence of steps for unblocking 4 and moving it to its destination. Changes in the task-goal stack over time are shown at the top of the figure, the move number is shown in the middle, and the state of the puzzle after a move is shown at the bottom. From the initial state, cognition pushes task goals for 4:C, 3:B, and 2:C in turn as it plans 1:B. After making 1:B (move 1), cognition examines the stack to see what move is on top. The top move is 2:C, which is now clear to make. If instead the top move had been blocked, cognition would have pushed a new goal to move the blocking disk. For example, once 2:C is made, 3:B (underneath 2:C on the stack) is blocked because disk 1 is at peg B, so cognition pushes a goal for 1:C. The behavior trace in Figure 3 shows that cognition gains an implausible advantage from having task goals perfectly preserved on a stack. For example, after making 2:C (move 2), the task goal for 3:B is immediately available. However, the task goal for 3:B was pushed before

8 Memory for Goals 8 move 1, perhaps five to 10 seconds previously (assuming a typical interactive Tower of Hanoi display). Despite this lag and the intervening cognitive activity, there is no deterioration in memory for 3:B. Moreover, the position of 3:B with respect to its neighbors on the stack is also perfectly preserved; there is no possibility of confusing one task goal with another. Finally, though an item may remain essentially immediately available for five seconds or so, the stack predicts perfect memory for task goals that are arbitrarily old. However, despite this lack of face validity the task-goal stack has been used in recent models in precisely the manner shown in Figure 3 (Anderson et al., 1993; Anderson & Lebiére, 1998). A Model Using Memory as Goal Store Representing means-ends behavior without a task-goal stack requires alternative storage for task goals, which we take to be ordinary declarative memory used in combination with retrieval heuristics and perceptual cues. Storing goals in ordinary unprivileged memory poses two functional challenges. First, items decay over time if not actively maintained. For example, an old goal such as 3:B in the example above (Figure 3) could be difficult to retrieve given that intervening cognitive activity inhibits the use of memory strategies like rehearsal. Second, after one goal is achieved another must be selected. The last-in, first-out (LIFO) ordering provided by the task-goal stack is optimal for goal selection in means-ends tasks like the Tower of Hanoi, and without a stack a useful selection order must be indicated by some other information source. Below we discuss an encoding process that encodes goals to resist decay and a retrieval process that uses heuristics and cues instead of a stack to retrieve task goals in an efficient order. Both processes are implemented in a MAGS-based model that uses ACT-R s architectural stack only to shift mental attention briefly to a retrieval cue. The model explains empirical data accurately and in detail, but first we discuss the underlying predictive processes.

9 Memory for Goals 9 The encoding process. Remembering something in the future requires paying attention to it in the present. One way to interpret paying attention operationally is as the encoding of a durable memory representation of the attended items in the Tower of Hanoi, a disk and its destination. We propose that as people plan a sequence of steps for moving a blocked disk, they pay attention to each step as they plan it, encoding the intermediate blocking disk and its destination peg as a task goal in memory. When the intermediate disk needs to be moved in the future, the task goal (if available) makes re-planning unnecessary. In previous computational cognitive models the encoding process responsible for paying attention has not been represented in detail. For example, Altmann and John (1999) represented encoding as one all-or-nothing step implemented by Soar s symbolic learning mechanism. Similarly, in the Tower of Hanoi model of Anderson and Lebiére (1998) encoding is represented as a single, non-functional production acting as a placeholder for the actual process. The cornerstone of the MAGS approach is an analysis of the encoding process at the 50 msec level of cognitive operations. At this atomic level, memory constrains the encoding process by requiring it to cope with noise and decay. These constraints imply two logical components to the encoding process: strengthening the item being encoded to resist noise and decay, interleaved with testing to check when the strength level is high enough. We discuss these two stages in detail in this section. The purpose of the MAGS encoding process is to increase the activation of the item being encoded, or target. Activation in general is a construct that represents the availability of memory elements. In ACT-R, activation has two components, base-level activation and source activation (Anderson & Lebiére, 1998). Source activation, which captures the effect of context, will be

10 Memory for Goals 10 discussed later. Base-level activation represents history of use, where one use is a retrieval of the target from memory or a creation of the target in memory. As a predictor of future need, base-level activation increases with number of uses and decreases as lag increases from the previous use. In the example shown in Figure 4, the encoding process causes a steep increase in the target s activation by using it rapidly, once per 100 msec (Altmann & Gray, 1999a, 1999b). Encoding stops after about 10 uses, based on a functional test we describe below, and decay follows as cognition carries on with other processing and stops using the target for the time being. The target remains retrievable for about three seconds, before it falls below the base-level threshold. How does cognition know when to stop encoding? From an everyday perspective, the answer is important for explaining the contingent or strategic way in which people encode targets. Examples of contingent encoding involve mundane actions like turning off the coffee before leaving the house, or locking the front door on the way out. Such actions may be habitual or reflexive, but this does not necessarily make them memorable; indeed, habitual actions may be harder to remember precisely because they do not require one s full attention. Episodic memory for such an action (e.g., I am sure I locked the door this morning ) is a function of whether sufficient attention was paid to the action as it occurred. With deliberate attention (e.g., subvocalizing I am locking the door ), memory for the action is likely to succeed. Conversely, a distraction at the time of the action will make memory more likely to fail. Such phenomena beg the question of how cognition determines how much encoding is enough. From a theoretical perspective, it is important to answer address the question of when to stop encoding without proposing a homunculous. However, ACT-R theory specifies no separate mechanism for monitoring the encoding process and stopping it when appropriate (whatever

11 Memory for Goals 11 appropriate might mean). Such a mechanism is an idiom (Lallement & John, 1998) or model increment (Howes & Young, 1997) that must be represented in terms of operations the architecture does provide. Our hypothesis is that cognition determines when to stop encoding in a functional manner by trying to retrieve the target at encoding time. In the MAGS encoding process, the baseline criterion is simply for the target to be strong enough to retrieve at encoding time if the target cannot be retrieved then, it is unlikely to be retrievable later. A functional test like this avoids implicating or positing any new mechanisms beyond those architectural mechanisms that govern retrieval generally. In Figure 4, this first criterion is met when the target s activation crosses the base-level threshold, the point at which the target is retrievable from memory (subject to noise). To allow for strategic encoding beyond this baseline criterion, the encoding process must be able to represent relevant differences between the encoding and retrieval contexts, if there is knowledge or experience available to anticipate such differences. For example, a student might study differently for an examination to be taken later that day (short retrieval interval) than for one to be taken later that week (long retrieval interval). Similarly, if the student can anticipate being allowed to use a reference card or cheat sheet during the examination, he or she might encode material by relating it to cues on that sheet. Thus the encoding process must be able to represent knowledge about the retrieval context for cognition to be able to encode strategically. In Figure 4, knowing that the target will have to stay available for a few seconds without active maintenance causes the encoding process to continue strengthening the target beyond the baselevel threshold. In the Tower of Hanoi, strategic encoding is possible because retrieval cues and retention intervals can be anticipated (to some extent) from the task environment. Cognition can anticipate

12 Memory for Goals 12 retrieval cues because it knows what information will be available visually at retrieval time, and retention intervals are the same from trial to trial. With such information cognition can adjust the encoding process to continue long enough to accommodate decay without consuming unnecessary cognitive cycles or over-learning the target. The ACT-R productions that implement the encoding process are described in Figure 5 in pseudo-code and in Figure 6 in terms of control flow. Encoding begins (see Figure 6) with strengthen-goal firing repeatedly. Each firing involves one use of the task goal that increases its base-level activation. Another production, simulate-retrieval, is always selected from the conflict set before strengthen-goal. However, simulate-retrieval depends on being able to retrieve the task goal from memory, and this retrieval will fail if the task goal is too weak. When simulate-retrieval fails, ACT-R selects strengthen-goal to fire again. When simulate-retrieval finally does fire, it means that the task goal s activation has reached the base-level threshold (Figure 4). At this point the task goal is strong enough to be retrieved now, before any decay occurs. However, decay will occur between now and when the task goal is needed, assuming that cognition is otherwise occupied in between. Therefore, the encoding process must continue for some amount of time long enough to keep the target above the base-level threshold until it is needed, but no longer. To prolong encoding by a controlled amount, the model uses a mechanism that we refer to as focussed retrieval. This mechanism is grounded in source activation (or priming), the other component of activation in ACT-R. Source activation, which represents the effect of context, spreads from cues in the current focus of attention to memory elements related to the cues. In ACT-R the current focus of attention can contain several cues, but there is a fixed amount of source activation divided equally among the cues. For example, if there is only one

13 Memory for Goals 13 cue in focus, say doctor, then maximal source activation will reach the memory element representing the doctor s name (as well as other facts about the doctor). However, if the focus contains an unrelated cue as well, like smoke or perhaps fire, then that cue will reduce the source activation reaching doctor and the chances of retrieving the doctor s name will decrease. Focussed retrieval works by placing particular cues in the focus of attention and then trying to retrieve the task goal. Focussed retrieval also exploits ACT-R s architectural stack in a limited and principled way. In ACT-R the current focus of (mental) attention is always the top element on the architectural stack. In a TGS model the stack would be used to store task goals and to automatically maintain focus on the most recent one. However, in MAGS the stack contains at most a main focus and a temporary focus, where the temporary focus concentrates attention on a particular retrieval cue. For example, in Figure 7 the main focus contains assorted information necessary to support problem solving, including the fact that disk 4 must move to peg C but is blocked. At this point in the encoding process, memory contains the task goal 4:C and the task goal is strong enough to cause simulate-retrieval to fire. Simulate-retrieval constructs a temporary focus containing disk 4 as a cue and pushes the new focus on top of the main focus. The temporary focus concentrates source activation on the retrieval cue to test whether the task goal can be retrieved using that cue. The cue makes the retrieval more likely to succeed than if it were absent in that it primes the task goal. However, there is also a dummy cue or sink in the temporary focus. This sink is unrelated to the task goal, and its purpose is to reduce the amount of source activation reaching the task goal to represent a rough estimate of how much decay in base-level activation will occur before the task goal is needed. The number and nature of sinks in the temporary focus, for example the degree to which they are related to the task goal,

14 Memory for Goals 14 represent domain-specific knowledge about the retention interval. Cognition uses these sinks, which are simply declarative memory elements, to control the stringency of the test on when to stop encoding. Focussed retrieval succeeds when the test-retrieval production successfully retrieves the task goal (see Figure 7). At this point the test-passes production pops the temporary focus off the architectural stack and halts the encoding process. Focussed retrieval fails if test-retrieval fails, which it will if the target is not strong enough to withstand a noisy retrieval. In the event of failure, the test-fails production pops the temporary focus, but encoding continues strengthengoal resumes firing until simulate-retrieval resets the test. The main focus and the temporary focus complement each other to support strategic encoding, in that each provides functionality that the other does not. For cognition to encode strategically, it must be able to produce high activation levels reliably and be able to verify those levels against anticipated demands. However, these constraints cannot be met within a single focus. The main focus contains all the information necessary for the encoding process to strengthen a memory representation for the target. In terms of the productions involved, strengthen-goal can fire reliably because it requires no failure-prone memory retrievals; all the information that makes up the target is immediately available for matching to the production s conditions. However, precisely this availability makes it impossible for cognition to assess whether a narrower focus (a smaller set of cues) will serve to prime the target at retrieval time. The focus will always be narrower at retrieval time, because otherwise there would be no need to retrieve; all the information making up the target would already be in focus. Thus retrieving the target in the main focus cannot verify that it is strong enough to be retrieved using the retrieval cue alone. In contrast, focussed retrieval in the temporary focus does have predictive value, but

15 Memory for Goals 15 by design this retrieval can fail. When it does fail, the temporary focus lacks the information (again by design) to strengthen the target. Control reverts to the main focus, which continues to increase activation monotonically until simulate-retrieval fires to reset the test. With complementary interplay between main and temporary focus, encoding will terminate no matter how stringent the retrieval test. We interpret this guarantee to mean that people can pay as much attention to an item as they deem necessary, with increasing (if diminishing) returns. In sum, the MAGS encoding process involves strengthening an item to increase its activation, interleaved with a test that factors in anticipated decay and anticipated retrieval cues. The process is implemented with precision at the 50 msec level, is functionally sound in that it affords strategic encoding based on knowledge of the task environment, and is constrained by theory to make empirical predictions about encoding time. The retrieval process. In using ordinary memory to store a task goal, the second challenge is to retrieve the task goal when the information it contains could be useful. Retrieving a task goal in MAGS involves first selecting the disk to work on next (after a successful move) and then attempting to retrieve a task goal using that disk as a cue. Selecting the best disk to work on next (which we refer to as the cue disk) is crucial to efficient performance in the absence of a task-goal stack, and is discussed in detail below. Once a cue is selected, the actual retrieval is by design very similar to the simulated retrieval that occurs during encoding. A production establishes a new focus containing only the retrieval cue, allowing maximum source activation to reach the task goal. If retrieval succeeds and the cue disk can be moved to the indicated destination, the model makes the move. If retrieval succeeds but the cue disk is blocked, the model invokes the goal-recursive algorithm to unblock it. If retrieval fails, the model invokes the goal-recursive algorithm starting

16 Memory for Goals 16 with the Largest Out-Of-Place disk, which we refer to as the LOOP disk. Thus the task goal, if available, reduces problem-solving effort by indicating either the best move to make or the best move to plan to make. Selecting the cue disk is done by the heuristics shown in Figure 8 and Figure 9. The first heuristic, select-next, represents the model asking itself (after a move) where it wanted to move the disk it just uncovered. Focussing on the uncovered disk makes sense because the purpose of the just-made move was to unblock a disk, and the goal-recursive algorithm, inversely applied, suggests that the disk to be unblocked was the uncovered one. A special case of select-next occurs when the just-made move empties a peg. In this case, the heuristic selects the disk one larger than the just-moved disk. This also makes sense, because the purpose of clearing a peg is to move a larger disk onto it. In this special case, select-next assumes that people attend to the size of the disk they are moving (Karat, 1982) in order to infer which is the next larger. The second heuristic, retrieved-stale, allows the model to recover if it retrieves a stale task goal. Some task goals become stale because their disks move more than once as a given LOOP disk is being unblocked. For example, disk 2 moves twice in the course of unblocking disk 4 (moves 2 and 6 in Figure 3). The first time the task goal is retrieved it indicates the correct intermediate destination (C at move 2). The second time the task goal is retrieved it indicates the same destination, but the disk is already there. Retrieved-stale recognizes this situation by comparing the retrieved destination with the cue disk s current location. If the two are the same, retrieved-stale changes the cue disk to be the next larger disk. Thus retrieved-stale assumes a cognitive-perceptual comparison, but this would be easily accommodated in the several seconds it takes to make a move (empirical response times are presented later).

17 Memory for Goals 17 The remaining heuristics save the step of retrieving a task goal, by indicating a disk and a destination directly. The don t-undo heuristic says to avoid moving the disk 1 twice in a row, because the two moves could have been combined into one. This applies on every other move; because odd-numbered moves always involve disk 1, don t-undo rules out disk 1 on evennumbered moves. With disk 1 fixed, the rules of the task dictate that of the two remaining disks, the smaller must move atop the larger. There is evidence that don t-undo is learned so easily that the actual learning event cannot be traced (VanLehn, 1991). Because of this, and because the underlying efficiency considerations are so basic, we assume that don t-undo always applies. Thus even-numbered moves in our model are fully determined. The fourth heuristic, one-follows-two, says that after disk 2 is moved disk 1 should be moved on top of it. Like don t-undo, one-follows-two is admissible in that it always recommends a correct move. Moreover, it also applies often (every fourth move), making it a common pattern that participants are likely to notice with experience; like don t-undo, one-follows-two is quickly learned (VanLehn, 1991). However, unlike don t-undo, one-follows-two is a task-specific heuristic, and we assume that it competes with the other task-specific strategy of retrieving a task goal for the next move. Thus when one-follows-two could apply, the model incurs the cost of memory retrieval half the time anyway. The four heuristics described above are efficient in guiding behavior and plausible in terms of cognitive and perceptual processing. However, the question remains whether participants actually use them. We can only argue that if participants had them then they used them in the trials from which the empirical data reported below were taken. The data were taken from perfect trials, in which participants made no unnecessary moves, which are the trials most likely to reflect the deployment of all available skills.

18 Memory for Goals 18 Model Results and Discussion The data to which we fit our model, and to which Anderson and Lebiére (1998) fit their TGS model, came from a study in which participants were essentially instructed to use the goalrecursive algorithm (Anderson et al., 1993). 1 Empirical response times from that study and simulated response times from our model are shown in Figure 10 for four-disk problems and Figure 11 for five-disk problems. Both empirical and simulated response times are for trials solved in the optimal number of steps (15 steps for four-disk problems and 31 steps for five-disk problems). The MAGS model. Our model provides an excellent fit, accounting for 99% of the variance in four-disk response times across moves and 95% of the variance in five-disk response times. (The TGS model fits equally well, as we discuss below.) In examining how the model explains response times we address only the four-disk data (Figure 10) because the same patterns occur in both data sets. The first pattern consists of the large peaks at moves 1, 9, and 13. In the simulation, these peaks are due to the encoding process, which the model triggers when planning to unblock the LOOP (largest out-of-place) disk. That is, the model generates a plan and saves each step, retrieving individual steps later as it needs them. When one plan has been completely executed, the model generates and saves a new plan for the next LOOP disk. Encoding a task goal to criterion takes one to two seconds depending on the size of the disk, adding up to the height of the large peaks above the baseline of about two seconds per move. The second pattern is that the large peaks are smaller later in the trial. In the simulation, the peaks shrink because there is one fewer step to each successive plan once a LOOP disk

19 Memory for Goals 19 reaches its final destination it can be ignored. Thus at move 1 the model encodes three task goals (for disks 4, 3, and 2), but at move 9 it encodes only two (for disks 3 and 2). The third pattern is that even-numbered moves are all equally fast. In the simulation, the correct move at even-numbered moves is indicated immediately by the don t-undo heuristic. The fourth pattern is the small peaks at moves 3, 7, and 11. In the simulation, these peaks reflect retrieval and planning time when the one-follows-two heuristic is not selected. Onefollows-two always applies on those moves, and if it were always selected those moves would be as fast as don t-undo moves. However, don t-undo is the more generic heuristic, and viewed as a skill is probably more practiced than one-follows-two. In the simulation, don t-undo is selected whenever it applies, but one-follows-two is selected only half the time it applies, with the cue disk being used the rest of the time (Figure 9). Using the cue disk takes longer on average because memory for its task goal can fail due to noise and decay. Also, the cue disk is blocked on small-peak moves, so the model has to use the goal-recursive algorithm to unblock it. The fifth and final pattern is the medium peak at move 5. In the simulation, medium peaks reflect a combination of factors. First, there is no heuristic giving the next best move; the move is odd-numbered so don t-undo fails to apply, and the just-moved disk is 3 so one-follows-two fails to apply. Thus the model always falls back on the cue disk. Second, the cue disk is blocked, so the goal-recursive algorithm is necessary. Third, the cue disk is larger on this move than on small-peak moves, so the goal-recursive algorithm has to run longer. The MAGS model also predicts errors (moves that stray from the optimal path) due to noise in memory. If the model happens to retrieve the wrong task goal (due to noise in activation levels), then it can make stray moves. For example, in the problem in Figure 3, the correct task goal after move 2 is 3:B (in the figure, 3:B is on top of the task-goal stack). If the MAGS model

20 Memory for Goals 20 incorrectly recalls 3:C instead, say from a previous trial, then instead of making 1:C next it would make 1:B and veer off the optimal path. Figure 12 shows predicted, observed, and optimal path lengths for four-disk and five-disk problems. Predictions are from Monte Carlo simulations of 2000 trials. Predicted average path lengths are 17.7 and 50.6, both within 10% of the observed path lengths of 19.2 and 53.5 (Anderson et al., 1993). Optimal path lengths are 15 and 31; the much higher error rate on five-disk trials suggests a combinatorial effect of interference among task goals. In our model we used default values for ACT-R parameters when they were available and values from other models when not. Default values were used for goal activation (W = 1.0), latency factor (F = 1.0), and base-level learning (d = 0.5). Transient activation noise (s = 0.3) and the value for ACT-R s retrieval threshold (τ = 4.0) were taken from a model of a different but related goal-management task (Altmann & Gray, 1999b). Perceptual encoding time (185 msec) was taken from ACT-R models of menu scanning and the Sperling task (Anderson, Matessa, & Lebiere, 1997). The one free parameter is move time, which we set to the same value as the Anderson and Lebiére (1998) TGS model (2.15 sec). The code for the MAGS model is available at The task-goal stack model. The TGS-based Anderson and Lebiére (1998) model is equally accurate in describing the response time data. It also accounts for 99% of the variance in four-disk response times and 95% in five-disk response times. However, its explanations for the first two patterns described above are less detailed, and its explanations for patterns three, four, and five reflect traditional use of the task-goal stack. Also, unlike the MAGS model, the TGS model does not predict errors, because it performs every trial perfectly due to the perfect memory provided by the stack.

21 Memory for Goals 21 In the TGS model, patterns one and two (the large response-time peaks and their slope, respectively) are explained by an encoding process, as they are in the MAGS model. However, as represented in the TGS model, the encoding process is not functional. The model fires a single production for each disk that is not at its final destination. However, this production does not actually create, modify, or strengthen any kind of memory representation. Its firing time of 560 msec, estimated to fit the empirical data, is an order of magnitude longer than the ACT-R default of 50 msec, reflecting the production s role as a placeholder for a more complex process. Thus the TGS model does not ground encoding in the constraints of the architecture. Moreover, because encoding is treated as a free parameter and is not constrained by theory, a degree of freedom must be used to estimate its duration. The TGS model explains patterns three, four, and five with the task-goal stack. On evennumbered moves, the top disk on the stack is unblocked and can be moved immediately. On moves with small peaks, the top disk is blocked by one disk, so the goal-recursive algorithm has to run one cycle. On moves with medium peaks, the top disk is blocked by more than one disk, so the goal-recursive algorithm has to run multiple cycles. On each cycle, the model pushes the blocked disk onto the task-goal stack, where it is safely preserved for as long as the unblocking process might take. These pushes are illustrated in Figure 3 as stack operations taking place between successive moves. The TGS explanation of patterns three, four, and five is elegant in its simplicity, but in theoretical and methodological terms this is a liability more than an asset. The TGS model's encoding process does nothing because nothing is required of it: the stack does all the memory management. Thus the explanation of patterns one and two, in addition to being a free parameter, has no functional relationship to the explanation of patterns three, four, and five. In contrast, the

22 Memory for Goals 22 MAGS encoding and retrieval processes are tightly constrained and functionally related by the underlying assumptions about memory: both processes are effortful because memory is subject to noise and decay. On one hand this comparison shows only that the task-goal stack masks important detail. On the other hand, the task-goal stack may have lived beyond its usefulness as a simplifying assumption precisely because of its representational elegance. Summary. We have shown that a task-goal stack is not necessary to explain means-ends behavior. The Tower of Hanoi has often been taken to induce cognitive goal stacking and the TGS model of Anderson and Lebiére (1998) incorporates this premise. However, the MAGS view, implemented computationally in the same cognitive architecture as the TGS model, is an improvement on several dimensions. First, it fits response-time data equally well while dispensing with the task-goal stack. Second, it represents the encoding and retrieval processes in explicit detail using existing architectural sub-theories. Third, it goes beyond the TGS model to predict errors as well as response times. Fourth, it uses fewer free parameters, with no need to estimate encoding time empirically. Thus the MAGS approach essentially subsumes the taskgoal stack as an explanation of means-ends behavior on this task. Moreover, the processes that accommodate strategic encoding and focussed retrieval are generic. Indeed, they were adapted from a model of a serial attention task with a dramatically different structure. (A closed-form algebraic version of this model is described in Altmann & Gray, 1999b.) Thus we expect these processes to transfer to essentially any cognitive activity involving fine-grained episodic memory. Nonetheless, our case for the MAGS approach has rested on competitive argumentation over two models that both explain a data set. The MAGS approach wins on face validity, theoretical

23 Memory for Goals 23 constraint, and breadth of coverage, but to improve our case we need to show that the task-goal stack is inaccurate as well as implausible. Memory for Goals in the Red Tape Task Interpreted literally the stack predicts the absence of cost and error in memory for goals. Interpreted less literally, the stack makes more interesting predictions. It seems quite reasonable to suppose that the power of human problem solving derives in part from special processing of items linked in means-ends relationships. Such processing would subserve canonically human behavior like planning and task decomposition, much like the opposable thumb subserves uniquely human manipulations of the environment. One way to interpret the stack realistically is as a retrieval structure, or mental scaffolding constructed at encoding time to support transient, task-related information (Altmann & John, 1999; Ericsson & Kintsch, 1995). Cognitive effort is required to construct the scaffolding and to retrieve an item from it later. However, little active maintenance is required in between because the scaffolding (once constructed) is a relatively permanent memory structure. The operations defined on the task-goal stack, namely pushing and popping, map directly to those defined on the retrieval structure, namely encoding and retrieval. Viewed as a kind of retrieval structure, the task-goal stack predicts that memory for goals and goal order should be reliable as long as the processes responsible for encoding (pushing a task goal) and retrieval (popping to the next older task goal) are allowed to proceed without restriction or interference. These predictions are plausible but different than those of the MAGS approach, as contrasted schematically in Figure 13. The strong interpretation of the stack predicts two error-free processes, pushing and popping. The weak interpretation of stack as retrieval structure accommodates the possibility of cost and error but predicts that once a stack is fully constructed

24 Memory for Goals 24 in memory, with all its goals in place, then no further active processing is required until retrieval time. In contrast, the MAGS view predicts that active maintenance between encoding and retrieval is an important component of memory for goals, particularly the absence of perceptual cues and the chance to use relevant background knowledge to keep track of goals. To test the predictions of the weak stack, we developed the Red Tape task, which strips away the opportunity to use knowledge and cues to organize goals coherently. Nonetheless, the task has a very salient means-ends structure. Participants are told up front the order in which task goals have to be achieved and how many there will be. The number of task goals that have to be retained at once is relatively few, but the incentive to retain them is strong the task requires a correct response at every step, however long this might take to generate. Thus if there were a task-goal stack available to cognition that is distinct from domain-specific knowledge and cues, then we would expect it to be deployed here. Method Participants. Ten George Mason University undergraduates participated for course credit. Two were male and eight were female. Ages ranged from 18 to 27 years. Materials. Stimulus presentation, response recording, and time measurement were controlled by MEL Professional software (Schneider, 1996) running on an IBM-compatible computer with an Intel Pentium processor, a standard keyboard, and a VGA monitor. Procedure. Participants were tested individually. Each participant was read a cover story that provides clear cues to the means-ends structure of the task:

25 Memory for Goals 25 Your task is to complete a number of forms by taking each form to a number of different offices for their approval. For each form, the number of offices you have to visit will be indicated when you start to work on the form. To visit an office, simply type its name on the keyboard. Some offices require approval from other offices before they will approve the form themselves. However, you won t know until you get to an office whether it requires approval from another office. So when you get to an office, you will see one of two messages. If the message says the office has signed off, then the office has approved your form and you don t have to come back. If instead the message says something like Go to X, then the office you re at won t approve the form until office X does. Order is very important, so after you receive approval from one office you must go back to the office that sent you there. The computer ensures that you do this, and will not let you leave an office until you correctly type in the office that sent you there. A list of all the offices in the company will always be visible on the screen. From time to time you will be asked to do a second task, which is to remember a sequence of letters presented one at a time on the screen. After a sequence of letters has appeared, you will be asked to type the letters on the keyboard in the order in which they were presented. To start with there will be four practice forms, two with the letter-remembering task and two without. After that there will be 48 forms total, with two short breaks in between.

26 Memory for Goals 26 The experimenter stayed in the room through the practice forms to answer questions, then left the participant to work on his or her own. The experiment software administered breaks of at least two minutes duration after 16 th and 32 nd forms of the session proper. Each form, or trial, can be described in terms of five phases. The first phase was simply an initial screen indicating the trial size, or how many offices had to be visited to complete the new form. Trial size was between two and five, with each size occurring 12 times per session in random order. The participant dismissed the trial-size screen with the space bar. In the second phase, a series of Goto screens appeared, each of which displayed the message Go to X where X was the name of an office (Figure 14a). The participant responded to each screen by typing the name of the office to go to. If the name was typed incorrectly, the screen remained unchanged and the participant tried again. When the name was typed correctly, the screen went blank for 500 msec and the next screen appeared. The full set of 10 office names (the company listing referred to in the instructions) appeared at the bottom of every Goto screen to give the list of allowable options. The same 10 names were always used, but name order was random from screen to screen to discourage the use of spatial cues for remembering goal order. The third phase was a retention interval. The contents of this interval defined the two conditions of the experiment. During a Rehearsal interval, the screen was blank for seven seconds, and to afford the opportunity for active maintenance no action or response was required of the participant. During a Distractor interval, four letters were presented consecutively for 1 sec each, with the first letter displayed as soon as the interval began and with each letter replacing the previous one on the screen. Letters were selected randomly with no letter appearing twice in the same series. After the fourth letter, the message Please enter the letters appeared and the

27 Memory for Goals 27 participant had three seconds to type the letters in presentation order. If the letters were typed correctly in three seconds or less the message Right! appeared for the balance of the three seconds. If one or more letters were wrong or out of place, or if it took longer than three seconds to type four letters, the computer generated a 400 msec tone. During both kinds of interval, the screen background turned to red (in all other phases the background was black; letters were always white). Condition (Rehearsal or Distractor) was chosen randomly for each trial, constrained only in that the same condition could not occur on more than three consecutive trials. Condition information was withheld prior to the retention interval to prevent people from changing their encoding strategies during the Goto phase. In the fourth phase, a series of Whereto screens appeared, each displaying the message X signs off. Where to now? (Figure 14b) where X was the most recent office visited. The participant responded to each screen by typing the name of the office that had referred the participant to the office that had just signed off. If the correct name was not typed, the screen remained unchanged until the participant tried again. When the correct name was typed, the screen went blank for 500 msec and the next screen appeared. During the Whereto phase participants were required to generate all but the last of the names presented during the Goto phase. (The last office presented in the Goto phase automatically signed off to begin the Whereto phase.) As in the Goto phase, all 10 names were displayed on every screen but in random order. Thus candidate names were available at each step in case guessing was necessary, but order information was not. Fifth and finally, after the last Whereto screen the message Form completed appeared. This message was dismissed with the space bar, at which point the next trial began.