{"title": "Fragment Completion in Humans and Machines", "book": "Advances in Neural Information Processing Systems", "page_first": 27, "page_last": 34, "abstract": null, "full_text": "Fragment completion in humans and machines\n\nDavid Jacobs\n\nNEC Research Institute\n\nBas Rokers\n\nPsychology Department at UCLA\n\n4 Independence Way, Princeton, NJ 08540\n\nPO Box 951563, Los Angeles, CA 90095\n\ndwj@research.nj.nec.com\n\nrokers@psych.ucla.edu\n\nArchisman Rudra\n\nCS Department at NYU\n\n251 Mercer St., New York, NY 10012\n\nZili Liu\n\nPsychology Department at UCLA\n\nPO Box 951563, Los Angeles CA 90095\n\narchi@cs.nyu.edu\n\nzili@psych.ucla.edu\n\nAbstract\n\nPartial information can trigger a complete memory. At the same time,\nhuman memory is not perfect. A cue can contain enough information to\nspecify an item in memory, but fail to trigger that item. In the context\nof word memory, we present experiments that demonstrate some basic\npatterns in human memory errors. We use cues that consist of word frag-\nments. We show that short and long cues are completed more accurately\nthan medium length ones and study some of the factors that lead to this\nbehavior. We then present a novel computational model that shows some\nof the \ufb02exibility and patterns of errors that occur in human memory. This\nmodel iterates between bottom-up and top-down computations. These\nare tied together using a Markov model of words that allows memory to\nbe accessed with a simple feature set, and enables a bottom-up process to\ncompute a probability distribution of possible completions of word frag-\nments, in a manner similar to models of visual perceptual completion.\n\n1 Introduction\n\nThis paper addresses the problem of retrieving items in memory from partial information.\nHuman memory is remarkable for its \ufb02exibility in handling a wide range of possible re-\ntrieval cues. It is also very accurate, but not perfect; some cues are more easily used than\nothers. We hypothesize that memory errors occur in part because a trade-off exists between\nmemory accuracy and the complexity of neural hardware needed to perform complicated\nmemory tasks.\nIf this is true, we can gain insight into mechanisms of human memory\nby studying the patterns of errors humans make, and we can model human memory with\nsystems that produce similar patterns as a result of constraints on computational resources.\n\nWe experiment with word memory questions of the sort that arise in a game called su-\nperghost. Subjects are presented with questions of a form: \u2018*p*l*c*\u2019. They must \ufb01nd a\nvalid English word that matches this query, by replacing each \u2018*\u2019 with zero or more letters.\nSo for this example, \u2018place\u2019, \u2019application\u2019, and \u2018palace\u2019 would all be valid answers. In ef-\n\n\ffect, the subject is given a set of letters and must think of a word that contains all of those\nletters, in that order, with other letters added as needed.\n\nMost of the psychological literature on word completion involves the effects of priming\ncertain responses with recent experience (Shacter and Tulving[18]). However, priming is\nonly able to account for about \ufb01ve percent of the variance in a typical fragment completion\ntask (Olofsson and Nyberg[13], Hintzman and Hartry[6]). We describe experiments that\nshow that the dif\ufb01culty of a query depends on what we call its redundancy. This measures\nthe extent to which all the letters in the query are needed to \ufb01nd a valid answer. We show\nthat when we control for the redundancy of queries, we \ufb01nd that the dif\ufb01culty of answering\nquestions increases with their length; queries with many letters tend to be easy only because\nthey tend to be highly redundant. We then describe a model that mimics these and other\nproperties of human memory.\n\nOur model is based on the idea that a large memory system can gain ef\ufb01ciency by keeping\nthe comparison between input and items in memory as simple as possible. All comparisons\nuse a small, \ufb01xed set of features. To \ufb02exibly handle a range of queries, we add a bottom-\nup process that computes the probability that each feature is present in the answer, given\nthe input and a generic, Markov model of words. So the complexity of the bottom-up\ncomputation does not grow with the number of items in memory. Finally, the system is\nallowed to iterate between this bottom up and a top down process, so that a new generic\nmodel of words is constructed based on a current probability distribution over all words in\nmemory, and this new model is combined with the input to update the probability that each\nfeature is present in the answer.\n\nPrevious psychological research has compared performance of word-stem and word-\nfragment completion. In the former a number of letters (i.e. a fragment) is given beginning\nwith the \ufb01rst letter(s) of the word. In the latter, the string of letters given may begin at any\npoint in the word, and adjacent letters in the fragment do not need, but may, be adjacent\nin the completed word. For example, for stem completion the fragment \u201cstr\u201d may be com-\npleted into \u201cstring\u201d, but for fragment completion also into \u201csatire\u201d. Performance for word-\nfragment completion is lower than word-stem completion (Olofsson and Nyberg[12]). In\naddition words, for which the ending fragment is given, show performance closer to word-\nstem completion than to word-fragment completion (Olofsson and Nyberg[13]).\n\nSeidenberg[17] proposed a model based on tri-grams. Srinivas et al.[21] indicate that as-\nsuming orthographic encoding is in most cases suf\ufb01cient to describe word completion per-\nformance in humans. Orthographic Markov models of words have often been used compu-\ntationally, as, for example, in Shannon\u2019s[19] famous work. Following this work, our model\nis also orthographic. We \ufb01nd that a bigram rather than a trigram representation is suf\ufb01cient,\nand leads to a simpler model.\n\nContradicting evidence exists for the in\ufb02uence of fragment length on word completion.\nOloffsson and Nyberg [12] failed to \ufb01nd a difference between two and three letter fragments\non words of length of \ufb01ve to eight letters. However this might have been due to the fact\nthat in their task, each fragment has a unique completion.\n\nMany recurrent neural networks have been proposed as models of associative memory\n(Anderson[1] contains a review). Perhaps most relevant to our work are models that use\nan input query to activate items from a complete dictionary in memory, and then use these\nitems to alter the activations of the input. For example, in the Interactive Activation model\nof Rumelhart and McClelland[16], the presence of letters activates words, which boost\nthe activity of the letters they contain.\nIn Adaptive Resonance models (Carpenter and\nGrossberg[3]) activated memory items are compared to the input query and de-activated\nif they do not match. Also similar in spirit to our approach is the bidirectional model of\nKosko[10] (for more recent work see, eg., Sommer and Palm[20]). Other models iteratively\ncombine top-down and bottom-up information (eg., Hinton et al.[5], Rao and Ballard[14]),\n\n\falthough these are not used as part of a memory system with complete items stored in\nmemory.\n\nOur model differs from all of these in using a Markov model as an intermediate layer be-\ntween the input and the dictionary. This allows the model to answer superghost queries, and\nleads to different computational mechanisms that we will detail. We \ufb01nd that superghost\nqueries seem more natural to people than associative memory word problems (compare the\nsuperghost query \u201cthink of a word with an a\u201d to the associative memory query \u201cthink of a\nword whose seventh letter is an a\u201d). However, it is not clear how to extend most models of\nassociative memory to handle superghost problems.\n\nOur use of features is more related to feedforward neural nets, and especially the \u201cinforma-\ntion bottleneck\u201d approach of Tishby, Pereira and Bialek[22] (see also Baum, et al.[2]). Our\nwork differs from feedforward methods in that our method is iterative, and uses features\nsymmetrically to relate the memory to input in both directions.\n\nOur approach is also related to work on visual object recognition that combines per-\nceptual organization and top-down knowledge (see Ullman[23]). Our model is inspired\nby Mumford\u2019s[11] and Williams and Jacobs\u2019[24] use of Markov models of contours for\nbottom-up perceptual completion.\n\nEspecially relevant to our work is that of Grimes and Mozer[4]. Simultaneous with our\nwork ([8]) they use a bigram model to solve anagram problems, in which letters are un-\nscrambled to match words in a dictionary. They also use a Markov model to \ufb01nd letter\norderings that conform with the statistics of English spelling. Their model is quite different\nin how this is done, due to the different nature of the anagram problem. They view anagram\nsolving as a mix of low-level processing and higher level cognitive processes, while it is\nour goal to focus just on lower level memory.\n\n2 Experiments with Human Subjects\n\nIn our experiments, fragments and matching words were drawn from a large standard cor-\npus of English text. The frequency of a word is the number of times it appears in this\ncorpus. The frequency of a fragment is the sum of the frequency of all words that the frag-\nment matches. We used fragments of length two to eight, discarding any fragments with\nfrequency lower than one thousand.\n\nFragments selected for an experiment were presented in random order. In our \ufb01rst exper-\niment we systematically varied the length of the fragments, but otherwise selected them\nfrom a uniform, random distribution. Consequently, shorter fragments tended to match\nmore words, with greater total frequency. In the second experiment, fragments were se-\nlected so that a uniform distribution of frequencies was ensured over all fragment lengths.\nFor example, we used length two fragments that matched unusually few words. As a result\nthe average frequency in experiment two is also much lower than in experiment one.\n\nA fragment was presented on a computer screen with spaces interspersed, indicating the\npossibility of letter insertion. The subject was required to enter a word that would \ufb01t the\nfragment. A subject was given 10 seconds to produce a completion, with the possibility to\ngive up. For each session 50 fragments were presented, with a similar number of fragments\nof each length.\n\nReaction times were recorded by measuring the time elapsed between the fragment \ufb01rst\nappearing on screen and the subject typing the \ufb01rst character of a matching word. Words\nthat did not match the fragment or did not exist in the corpus were marked as not completed.\n\nEach experiment was completed by thirty-one subjects. The subjects were undergraduate\nstudents at Rutgers University, participating in the experiment for partial credit. Total time\n\n\fd\ne\n\nt\n\nl\n\ne\np\nm\no\nC\nn\no\n\n \n\ni\nt\nc\na\nr\nF\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n1\n\n2\n\n3\n\n1\n\n0.8\n\nl\n\nd\ne\nt\ne\np\nm\no\nC\n \nn\no\ni\nt\nc\na\nr\nF\n\n0.6\n\n0.4\n\n0.2\n\n7\n\n8\n\n9\n\n0\n1\n\n2\n\n3\n\n4\n\n5\n\n6\n\nFragment Length\n\n1\n\n4\n\nFragment Length\n\n5\n\n6\n\n7\n\n8\n\n9\n\nR1 \n\nR0 \n\nR2 \n\nl\n\nd\ne\nt\ne\np\nm\no\nC\n \nn\no\n\ni\nt\nc\na\nr\nF\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n1\n\n2\n\n3\n\n4\n\n5\n\n6\n\nFragment Length\n\nR4 \n\nR3 \n\n7\n\n8\n\n9\n\nFigure 1: Fragment completion as a function of fragment length for randomly chosen cues\n(top-left) and cues of equal frequency (top-right). On the bottom, the equal frequency cues\nare divided into \ufb01ve groups, from least redundancy (R0) to most (R5) .\n\nspent on the task varied from 15 minutes to close to one hour.\n\nResults\n\nFor each graph we plot the number of fragments completed divided by the number of\nis the\nis the number of trials. This assumes that all decisions\n\nfragments presented (Figure 1). Error bars are calculated as \npercent correct in the sample, and \r\nare independent and correct with probability \u0002 ; more precise results can be obtained by\n\naccounting for between-subject variance, but roughly the same results hold.\n\n\u0001\u0003\u0002\u0005\u0004\u0006\u0002\b\u0007\n\t\f\u000b\u000e\n\n, where \u0002\n\nFor random, uniformly chosen fragments, there is a U-shaped dependence of performance\non length. Controlling for frequency reduces performance because on average lower fre-\nquency fragments are selected. The U-shaped curve is \ufb02attened, but persists; hence U-\nshaped performance is not just due to frequency\n\nFinally, we divide the fragments from the two experiments into \ufb01ve groups, according to\ntheir redundancy. This is a rough measure of how important each letter is in \ufb01nding a cor-\nrect answer to the overall question. It is the probability that if we randomly delete a letter\nfrom the fragment and \ufb01nd a matching word, that this word will match the full fragment.\n(total frequency\n\nSpeci\ufb01cally, let \u000f denote the frequency of a query fragment of length \u0010\nof words that match it). Let \u000f\u0012\u0011 denote the frequency of the fragment that results when we\n\t .\ndelete the\u0013 \u2019th letter from the query (note, \u000f\n\u000f ). Then redundancy is: \u0010\u0017\u0016\u0018\u000f\n\u000f\u001b\u001a\u000e\u001c\u001e\u001d\u001f\u001d \u001d!\u001c\"\u000f\u0012#\n\nIn all cases where there is a signi\ufb01cant difference, greater redundancy leads to better per-\nformance. In almost all cases, when we control for redundancy performance decreases with\nlength. We will discuss the implications of these experiments after describing correspond-\ning experiments with our model.\n\n\u0011\u0015\u0014\n\n\u000b\u0019\u0001\n\n\f3 Using Markov Models for Word Retrieval\n\nWe now describe a model of word memory in which matching between the query and\nmemory is mediated by a simple set of features. Speci\ufb01cally, we use bigrams (adjacent\npairs of letters) as our feature set. We denote the beginning and end of a word using\nthe symbols \u20180\u2019 and \u20181\u2019, respectively, so that bigram probabilities also indicate how often\nindividual letters begin or end a word. Bottom up processing of a cue is done using this as\na Markov model of words. Then bigram probabilities are used to trigger words in memory\nthat might match the query.\n\nOur algorithm consists of three steps. First, we compute a prior distribution on how likely\neach word in memory is to match our query. In our simulations, we just use a uniform\ndistribution. However, this distribution could re\ufb02ect the frequency with which each word\noccurs in English. It could also be used to capture priming phenomena; for example, if a\nword has been recently seen, its prior probability could increase, making it more likely that\nthe model would retrieve this word. Then, using these we compute a probability that each\nbigram will appear if we randomly select a bigram from a word selected according to our\nprior distribution.\n\nSecond, we use these bigram probabilities as a Markov model, and compute the expected\nnumber of times each bigram will occur in the answer, conditioned on the query. That is,\nas a generic model of words we assume that each letter in the word depends on the adjacent\nletters, but is conditionally independent of all others. This conditional independence allows\nus to decompose our problem into a set of small, independent problems. For example,\nconsider the query \u2018*p*l*c*\u2019. Implicitly, each query begins with \u20180\u2019 and ends with \u20181\u2019, so\nthe expected number of times any bigram will appear in the completed word is the sum of\nthe number of times it appears in the completions of the fragments: \u20180*p\u2019, \u2018p*l\u2019, \u2018l*c\u2019, and\n\u2018c*1\u2019.\n\nTo compute this, we assume a prior distribution on the number of letters that will replace a\nletters\n\n\u001a (in practice we truncate \r at 5 and normalize the probabilities). A similar model\n\n\u2018*\u2019 in the completed word. We use an exponential model, setting the probability of \r\nto be \u001a\n\u0007\u0001\ncompleted cue. For a \ufb01xed \n\nis used in the perceptual completion of contours ([11, 24]). Using these priors, it becomes\nstraightforward to compute a probability distribution on the bigrams that will appear in the\n\u0003 bigrams,\nand each bigram depending on only its neighbors. The conditional probability of each\nbigram given its neighbor comes from the Markov model, and we can solve the problem\nwith belief propagation.\n\n, we structure this problem as a belief net with \n\nBeginning the third step of the algorithm, we know the expected number of times that each\nbigram appears in the completed cue. Each bigram then votes for all words containing that\nbigram. The weight of this vote is the expected number of times each bigram appears in\nthe completed cue, divided by the prior probability of each bigram, computed in step 1. We\ncombine these votes multiplicatively. We update the prior for each word as the product of\nthese votes with the previous probability. We can view this an approximate computation of\nthe probability of each word being the correct answer, based on the likelihood that a bigram\nappears in the completed cue, and our prior on each word being correct.\n\nAfter the third step, we once again have a probability that each word is correct, and can\niterate, using this probability to initialize step one. After a small number of iterations, we\nterminate the algorithm and select the most probable word as our answer. Empirically, we\n\ufb01nd that the answer the algorithm produces often changes in the \ufb01rst one or two iterations,\nand then generally remains the same. The answer may or may not actually match the input\ncue, and by this we judge whether it is correct or incorrect.\n\nWe can view this algorithm as an approximate computation of the probability that each\n\n\u0002\n\u001c\n\fd\ne\n\nt\n\nl\n\ne\np\nm\no\nC\nn\no\n\n \n\ni\nt\nc\na\nr\nF\n\n1\n\n0.9\n\n0.8\n\n0.7\n\n0.6\n\n0.5\n\n0.4\n1\n\n2\n\n3\n\n4\n\nFragment Length\n\n5\n\n6\n\n1\n\n0.9\n\n0.8\n\n0.7\n\n0.6\n\nl\n\nd\ne\nt\ne\np\nm\no\nC\n \nn\no\ni\nt\nc\na\nr\nF\n\n0.5\n\n0.4\n1\n\n2\n\n3\n\n1\n\n0.9\n\n0.8\n\n0.7\n\n0.6\n\nl\n\nd\ne\nt\ne\np\nm\no\nC\n \nn\no\ni\nt\nc\na\nr\nF\n\n7\n\n8\n\n9\n\n0.5\n\n0.4\n1\n\nR2 R3 \n\nR1 \n\n2\n\n3\n\nR4 \n\n4\n\n5\n\n6\n\n7\n\n8\n\n9\n\nFragment Length\n\nR0 \n\n4\n\n5\n\n6\n\n7\n\n8\n\n9\n\nFragment Length\n\nFigure 2: Performance as a function of cue length, for cues of frequency between 4 and 22\n(top-left) and between 1 and 3 (top-right). On the bottom, we divide the \ufb01rst set of cues\ninto \ufb01ve groups ranging from the least redundant (R0) to the most (R4).\n\n(with a bigram representation,\n\nblanks replacing a \u2018*\u2019 by \n\nword matches the cue, where the main approximation comes from using a small set of fea-\ntures to bring the cue into contact with items in memory. Denote the number of features\nby\n(ie., the word length plus one), the number of words by\n\n, and the maximum number of\n. Then steps one and three require O(mw) computation, and step\ntwo requires O(Fn) computation. In a neural network, the primary requirement would be\nbidirectional connections between each feature (bigram) and each item in memory. There-\nfore, computational simplicity is gained by using a small feature set, at the cost of some\napproximation in the computation.\n\n), the number of features in each word by\n\n\u0002\u0001\u0004\u0003\u0006\u0005\b\u0007\n\nExperiments\n\nWe have run experiments to compare the performance of this model to that of human sub-\njects. For simplicity, we used a memory of 6,040 words, each with eight characters. First,\nwe simulated the conditions described in Olofsson and Nyberg[12] comparing word stem\nand word fragment completion. To match their experiments, we used a modi\ufb01ed algorithm\nthat handled cues in which the number of missing letters can be speci\ufb01ed. We used cues\nthat speci\ufb01ed the \ufb01rst three letters of a word, the last three letters, or three letters scattered\nthroughout the word. The algorithm achieved accuracy of 95% in the \ufb01rst case, 87% in the\nsecond, and 80% in the third. This qualitatively matches the results for human subjects.\nNote that our algorithm treats the beginning and end of words symmetrically. Therefore,\nthe fact that it performs better when the \ufb01rst letters of the word are given than when the last\nare given is due to regularities in English spelling, and is not built into the algorithm.\n\nNext we simulated conditions comparable to our own experiments on human subjects, us-\ning superghost cues. First we selected cues of varying length that match between four\nand twenty-two words in the dictionary. Figure 2-top-left shows the percentage of queries\nthe algorithm correctly answered, for cues of lengths two to seven. This \ufb01gure shows a\nU-shaped performance curve qualitatively similar to that displayed by human subjects.\n\n\n\t\n\n\fWe also ran these experiments using cues that matched one to three words (Figure 2-top-\nright). These very low frequency cues did not display this U-shaped behavior. The algo-\nrithm performs differently on fragments with very low frequency because in our corpus\nthe shorter of these cues had especially low redundancy and the longer fragments had es-\npecially high redundancy, in comparison to fragments with frequencies between 4 and 22.\nNext (Figure 2-bottom) we divided the cues into \ufb01ve groups of equal size, according to\ntheir redundancy. We can see that performance increases with redundancy and decreases\nwith cue length.\n\nDiscussion\n\nOur experiments indicate two main effects in human word memory that our model also\nshares. First, performance improves with the redundancy of cues. Second, when we control\nfor this, performance drops with cue length. Since redundancy tends to increase with cue\nlength, this creates two con\ufb02icting tendencies that result in a U-shaped memory curve. We\nconjecture that these factors may be present in many memory tasks, leading to U-shaped\nmemory curves in a number of domains.\n\nIn our model, the fact that performance drops with cue length is a result of our use of\na simple feature set to mediate matching the cue to words in memory. This means that\nnot all the information present in the cue is conveyed to items in memory. When the\nlength of a cue increases, but its redundancy remains low, all the information in the cue\nremains important in getting a correct answer, but the amount of information in the cue\nincreases, making it harder to capture it all with a limited feature set. This can account for\nthe performance of our model; similar mechanisms may account for human performance\nas well. On the other hand, the extent to which redundancy grows with cue length is really\na product of the speci\ufb01c words in memory and the cues chosen. Therefore, the exact shape\nof the performance curve will also depend on these factors. This may partly explain some\nof the quantitative differences between our model and human performance.\n\nFinally, we also point out that our measure of redundancy is rather crude. In particular, it\ntends to saturate at very high or very low levels. So, for example, if we add a letter to a cue\nthat is already highly redundant, the new letter may not be needed to \ufb01nd a correct answer,\nbut that is not re\ufb02ected by much of an increase in the cue\u2019s redundancy.\n\n4 Conclusions\n\nWe have proposed superghost queries as a domain for experimenting with word memory,\nbecause it seems a natural task to people, and requires models that can \ufb02exibly handle\nsomewhat complicated questions. We have shown that in human subjects, performance on\nsuperghost improves with the redundancy of a query, and otherwise tends to decrease with\nword length. Together, these effects results in a U-shaped performance curve.\n\nWe have proposed a computational model that uses a simple, generic model of words to\nmap a superghost query onto a simple feature set of bigrams. This means that somewhat\ncomplicated questions can be answered while keeping comparisons between the fragments\nand words in memory very simple. Our model displays the two main trends we have found\nin human memory. It also does better at word stem completion than word fragment com-\npletion, which agrees with previous work on human memory. Future work will investigate\nthe modi\ufb01cation of our model to account for priming effects in memory.\n\nReferences\n\n[1] J. Anderson. An Introduction to Neural Networks, MIT Press, Cambridge MA. 1995.\n\n\f[2] E. Baum, J. Moody and F. Wilczek. \u201cInternal Representations for Associative Memory,\u201dBio-\n\nlogical Cybernetics, 59:217-228, 1988.\n\n[3] G. Carpenter, and S. 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