Memory & Cognition1975, Vol, 3 (4;,434-444

Simplicity, symmetry, and syntely:Stimulus measures of binary pattern structure

JOSEPH PSOTKAYale University, Ntw Haven, Connecticut 06520

A simple parameter free algorithm based solely on the sequential run and alternation structure ofbinary, sequences is elaborated: T~e algorithm is designed to measure the sequence's syntely: the degreeto whlc? past conseque,nces ~lthlD a sequence converge on the continuation of that sequence's terminalru~. This sy~telY algorithm IS shown to predict subjects' expectancies in a sequential prediction taskusmg short bmary sequences. Other algorithms measuring the symmetry and simplicity (If short binarypatterns are demonstrated, and their measures shown to be correlated with each other but notcorrelated with syntely. Goodness and strength of expectancy are unrelated. The syntely algorithm is~own to be successful in predicting the error profiles of subjects learning short recurrent patterns ofbmary se~uences. The s~ntely algorithm is based on the straightforward principle of induction byenumeratIOn: the composite of past events controls the expectancy of future events. The success of thisalgorithm, even though it may embody the principle of induction only imperfectly, provides goodevidence that this principle is a useful normative guide for understanding the human processing ofcontingencies in binary sequences, making complicated schemes of rules and hypotheses unnecessary.

An intelligent observer is quite able to use manydifferent aspects of the information available in astimulus. To begin the analysis of such information at asynoptic level, a simple dichotomy may usefully becreated to distinguish two kinds of information byidentifying information with structure and illustratingtwo sorts of structure, intrinsic and extrinsic (Garner,1974). Extrinsic structure occurs when the stimulusdenotes or signifies something other than itself, like themeaning contained in the word "dog," or the arithmeticprogression inherent in the sequence of numbers(I 3 5 79 ...). This sort of structure is beyond thescope of this paper. Instead, the subjective use ofintrinsic structure as given by perceptible spatiotemporalregularities of the stimulus, like the optical and spatialfrequency properties of "dog" or the single alternationstructure of (0 X 0 X 0 X ...), will be analyzed at avery general level.

Information theory offers at its core the strikinglysimplifying notion that information or structure can bediscussed completely generally as it is embodied in theform of a binary time series like (X 0 X X 0 X ...). Theactual elements used in the series are completelyirrelevant unless they designate some external structure.Since this is a real consideration with most, if not all,available elements, special precautions must be used inany practical case. Usually the simplest procedure toeliminate the effects of extrinsic structure on the resultsobtained with one sequence like the example above is tomake sure that the same results are also obtained with its

I am grateful to W. R. Gamer and Michael Kubovy for theircomments on an earlier draft. This research was supported byGrant MH 14229 from the National Institute of Mental Healthto Yale University. The author is now at the University ofWaterloo. Waterloo. Canada N2L 3G1.

complementary sequence, in this case(0 X 0 0 X 0 . ' .). The results are then dependent onthe intrinsic structure of the sequence, and the elementsthemselves can be considered to be intrinsically neutral.

MEMORY AND PROBABILITY LEARNING

There already exists a large and established body ofwork on one aspect of the intrinsic structure of binarytime series. This is the literature on partial reinforcementconditioning and probability learning (Jenkins &Stanley, 1950). More recent results in this tradition aresummarized by Jones (1971). One of the clearest resultsof probability learning experiments with humans onbinary patterns was the finding that the mean resultsacross subjects and blocks of trials indicated that eventpredictions occurred in the same proportion as events inthe sequence: probability matching occurred. Althoughthis result presented some difficulty for an explanationbased on a simple minded decision theory that predictedsubjects should predict the more frequent alternativeconsistently on every trial, it quickly became apparentthat subjects were responding to patterns of events, andthe stimulus for responding on each trial was acomposite of whatever experiences they had on previoustrials (Hake & Hyman, 1953; Nies, 1962).

An explanation of probability matching in terms ofthe ability of subjects to use subsequences of a series ascues to the future behavior of the series presupposedthat they were able to remember at least short repeatingsequences of binary events. That humans have thisability has been shown in many experiments forsequences of single alternation (Anderson, 1960), doublealternation (Schoonard & Restle, 1961), and shortsequences like (X X 0) (Goodnow & Pettigrew, 1956) or

434

(00 X 0 X) (Galanter & Smith, 1958), or many othersequences (Garner & Gottwald, 1967; Restle, 1967;Royer & Gamer, 1966) at varying rates, starting points,and modalities of presentation (Gamer & Gottwald,1968; Handel & Buffardi, 1968; Preusser, Garner, &Gottwald, 1970), and with a variety of tasks to testmemory (Alexander & Carey, 1968; Glanzer & Clark,1962). In an experiment on probability learning using acomplicated Markov sequence, Feldman and Hanna(1966) were able to show that event predictionsmatched the conditional probabilities following eventstates up to a length of five events at least.

In spite of this very clear evidence that subjects arequite capable of remembering event sequences of allsorts of patterns, attempts at making use of thisinformation in order to understand the patterns ofexpectancy in probability learning experiments haveused only a tiny proportion of it: memory for runs ofidentical events. The clearest expression of a model todescribe subjects' expectancy based on memory for runswas given by Restle (1961). This model had only limitedsuccess (Restle, 1966). Modified versions of Restle's runsmodel have fared little better (Gambino & Myers, 1967).It seems necessary to take into account the basic factthat subjects are able to remember and respond topatterns other than runs (Butler, Myers, & Myers, 1969;Jones, 1971) if an adequate description of choicebehavior in a binary prediction situation is to beachieved.

INDUCTION AND EXPECTANCY

Expectancy in a binary prediction paradigm mayprofitably be viewed as the result of inductive inference.The principle of induction may be stated simply in aform that is very similar to Newton's first law of motion.An induction is made by assuming that the present stateof nature will continue into the future, and anyconsequences of events in the past will follow the sameevents in the future. Induction is to be distinguishedfrom cross induction (Reichenbach, 1968) or analogywhere the consequences of one set of events arepredicted from another set of similar or different events.An example of a cross induction would be to predict thecontinuation of (0 X X 0 0 0 X X X X ... ?) on thebasis of the learned arithmetic sequence (1 2 3 4 ... ?).The distinction between induction and cross induction isrelated to the distinction between intrinsic and extrinsicstructure. Inductions are based on the intrinsic structureof events, whereas cross inductions are based on theirextrinsic structure. Since this paper is only concernedwith intrinsic structure, it makes use only of theprinciple of induction. This basic principle that underliesall scientific achievement finds a ready application in theanalysis of subjects' expectancy when predicting binaryseries.

Consider a short sequence of eight equally likely

SIMPLICITY, SYMMETRY, AND SYNTELY 435

binary events like (0 X X X 0 0 X 0) as it is read fromleft to right. What inference may be drawn from thissequence about what will happen next(0 X X X 0 0 X 0 ... ?)? Using the basic principle ofinduction, events the same as the right-hand or terminalportion of the sequence may be sought in the left-handor preceding portion of the sequence, their followingevents or consequences noted, and the inference drawnthat these same consequences will next occur. This isinduction by enumeration (Reichenbach, 1968). Thefrequencies of events following the terminal state in thepast are evaluated and expected to occur with the samefrequencies in the future.

This principle of induction is so simple and clear, itwould seem to be nicely suited to determining thestrength of expectancies by numerical calculation.However, it should be clear from the outset that thiscalculation cannot be performed on the basis of idealequations. The necessary matching of the terminal stateto all previous states within a sequence of finite and, infact, relatively short length guarantees that no generalmathematical solution is possible and an algorithm, astep by step calculation, is necessary. Also, an algorithmthat tries to put into practice the theoretical principle ofinduction has to take into account practical details inorder to deal with the performance of real people thattheory may easily discount, like the accuracy of memoryfor past events. These are practical problems that musteventually be subjected to experimental investigationthemselves, but in the beginning they can only behandled through careful experimental control and bydrawing certain assumptions. The assumptions necessaryfor applying the principle of induction to short binarysequences will be considered next, after a definition ofthe terms used to describe the structure of binarypatterns.

DEfINITIONS

In order to describe the intrinsic structure of binarypatterns and the operation of algorithms designed tomeasure this structure, a number of terms will be usedrepeatedly in the ensuing discussion. Some of thesewords have been used in several different ways before inthe sequential processing literature. It would seemadvisable to provide definitions of these particular termsas they are used in this paper.

AlternationAn alternation occurs when an element is followed by

the complementary element to its right in the binarysequence (for example, either XO or OX).

RunA run occurs when an element is followed by the

identical element to its right in the binary sequence (forexample, either XX or 00).

436 PSOTKA

SYNTELY: THE STRENGTH OFSTIMULUS CONTINUATION

\\hidl past cllnscqueth:t's L'onvcrge on tht: continuationllf the tt:rminal run, t:xpressed as a number ranging fromo tll I. This neologism, syntely, has been coined toprovidt: a word whose meaning relates to the stimulus ina similar way lhat the strength of expectancy relates tothe organism. Thus, syntely. like symmetry, describessomething about the stimulus. The similarity of thisanalytic syslt:m to a symmetry algorithm is thenexposed. and their relation to the simplicity of thepattern explored. Experimental evidence corroboratingIhis analysis is given.

An algorithm is like a game: It is easy to showsomeone how to play, but a formal description of therules is often unwieldy and cumbersome. Here are therules for calculating the syntely of any short binarypattern.

I. Record the length of the terminal run. Forexample. in (a) (X 0 X X X 0 0 0), the terminal run,underlined, is three elements long. In(b) (0 X 0 X 0 X 0 20 the terminal run, underlined, isonly one element long.

2. Compare all of the rightmost or terminal segmentof the original sequence with the subsequencesimmediately to the left by shifting a duplicate sequenceunderneath and to the left by the number of elements inthe terminal run:

3. The subsequences may now be matched verticallybeginning with the rightmost element in the lower row.Since expectancy is assumed to depend on the pattern ofruns and alternations in the sequence (not on theelements of the sequence themselves), match therightmost, first element in the duplicate automaticallyand then look only to see if the pattern to the left in theduplicate is the same as in the to-be-matchedsubsequences in the original. This is true of bothexamples, since in a the second element from the right inthe lower row duplicate forms part of a run with therightmost element, as do the two elements directlyabove them, and in b the same two pairs formalternations. If matches are found, as in these examples,the vertical comparison continues to the left; otherwise,it stops. In the examples Sequence b matches itselfperfectly after the first shift to the left (all seven verticalcomparisons match). Sequence a matches only the firstfour elements from the right.

4. The number of different elements matched in thisfashion are, then, the number of matched subsequencesin the sequence that have the same consequent andmatch the terminal segment. The consequent, either a

(b) OXOXOXOXOXOXOXOX

(a) XOXXXOOO

XOXXXOOO

ASSUMPTIONS OF THESYNTELY ALGORITHM

The cognitively salient features of a binary sequenceare the runs and alternations (Butler, Myers, & Myers,1969; Jones, 1971; Royer & Garner, 1966). It isassumed that the induction is based on the pattern ofruns and alternations, so that the prediction. is one ofcontinuing a run or an alternation, and the states in thepast that are evaluated are subsequences of runs andalternations.

It is assumed that terminal runs are unitized in agestalt sense on the basis of identity and proximity. It isassumed that as a unit no inductions about the futurecan be drawn from itself, unless nothing else is available.Thus, the continuation of (XOXOXXXX) is based on(XOXO ....) and not on (. ... XXXX).

Using this simple principle of induction, the patternof a sequence provides the intrinsic structure from whichthe alternatives governing expectancy are chosen. Theexpected continuation of a pattern may thus be derivedfrom a straightforward analysis of the pattern itself. It isimportant to observe, however, that this analysis doesnot take into account any asymmetry in the frequencyof the elements, since it deals only with the pattern ofruns and alternations. Since it is clear that relativefrequencies of events have a strong effect on expectancy,it remains for ftlture analysis to show how relativefrequencies and conditional relative frequencies interact.

The remainder of this paper provides a detailedsystem· for the analysis of the intrinsic structure ofbinary patterns as this structure affects expectancy. Thisanalytic system is given in the form of a syntelyalgorithm, where syntely (from the Greek sun, same ortogether, plus telos, completion or end) is the degree to

Matching SubsequencesSubsequences within a pattern can be said to match if

they form the identical sequence of runs andalternations. Thus, within (XXOXOOXO) there arc avery large number of subsequences of various lengthsthat are matched at least once within the pattern. As anillustration, the longest subsequences that match withinthis pattern are XXOX and OOXO, each composed offour elements in length. But each of these subsequencesmay be further broken down into two subsequences ofLength 3 that match, three subsequences of Length 2that match, and the degenerate case containing no runsor alternations of four subsequences of Length I thatmatch.

Terminal RunA string of adjacent identical elements at the

rightmost extreme of the binary sequence preceded bythe complementary element constitutes the terminal runof the sequence (for example, the string of four Xs inXOXOXXXX).

SIMPLICITY, SYMMETRY, AND SYNTELY 437

EXPERIMENT I

XXOXXOOOmm - alt

mm - altSyntely =0/4 =.0

MethodSince the syntely algorithm as proposed so far does not yet

take asymmetric event frequencies into account, it was decidedto use sequences of equally frequent alternatives to test theadequacy of the present conception. Since the syntely algorithm

Using this straightforward algorithm, the syntely ofany binary pattern, no matter what its length, may befound. A simple experiment was conducted to determinewhether subjects' expectancies are described by thisalgorithm.

OXXXXOOOm mm - alt

mmmm - runSyntely = 4/7 = .57

OXXXOOXOmm-altm-alt

mm-runm-alt

m-runmm-runm-alt

Syntely = 5/10 = .5

OXOXOXOXmmmmmmm-altmmmmmm-altmmmmm-altmmmm--altmmm-altm m -altm-alt

Syntely =0/28 =.0

XXXXOOOOmmmm - alt

Syntely =0/4 =.0

Finally, consider a pattern in which the unity of theterminal run must be broken because matches withsubsequences of equal run length are not available(X X 0 X X 0 0 0). On the first shift, a match withRun Length 2 is accepted. In general, since the algorithmproceeds from right to left in its matching operations,matches are accepted shorter than the terminal run onlyif the match is equal or greater than any match found tothe right. With this example, the stepwise matchingoperation of the syntely algorithm produces thefollowing result:

The previous examples illustrated the operation of thebasic algorithm. Next consider the following sequences.These patterns illustrate the gestalt unitization of theterminal run. Since the terminal run is to be consideredas a unit, it has no matching subsequences within itself.Similarly, in these examples, since the match of th.eterminal run is immediately available, the gestalt umtremains intact in its matching operations and is notsubdivided. The stepwise matching shifts of thealgorithm produce the follOWing results:

(a) XOXXX-OOOXOXXXOOO

m m m m - alternation

(a) X 0 X X X 000XOXXXOOO

(mm - run)

(b) OXOXOXOXOXOXOXOX

m m m m m m - alternation

In Example a there are only two matching elements,which is less than the length of the terminal run of thesequence (Length 3) and less than the number ofmatching elements in the previous line (four). Therefo.re,the matches on this line are deleted (indicated by placmgthem in brackets). The duplicate is then shifted left byone element, the procedure continued, and so on untilthe left end of the sequence is reached by the right endof the duplicate.

6. After the matching process is completed, thenumber of recorded subsequences preceding a run arecumulated and divided by the total number of recordedmatching subsequences to provide the numeric value ofthe syntely of the pattern. The entire algorithm mayeasily be computerized.

Some Complete ExamplesThe stepwise matching shifts of the algorithm produce

the following results in some real examples, where eachline indicates a stepwise shift of the terminal segmentone element to the left, the fiS indicate subsequencesmatching the terminal segment pattern, and run and altto the right of a dash indicate that a run or analternation, respectively, was a consequent of thematched subsequences:

(b) 0 X 0 X 0 X 0 - XOXOXOXOX

m m m m m m m - alternation

5. The duplicate sequence is then shifted left by oneelement and the same procedure of matching is followedagain. However, if the number of matched elements isless than the length of the terminal run and less than thenumber of matched elements on any previous line, it isnot recorded. In the two examples this shift results in:

run or an alternation, is determined by whether theelement to the right of the matched subsequences in theupper row continues a run or an alternation with thosematched subsequences. In the two examples theseconsequents are both alternations, as indicated belowwith matched subsequences underlined and matchesindicated by ms:

438 PSOTKA

Table 1Subjects' Judgments and Stimulus Measures for

Complexity, Symmetry, and Syntely

35 PatternsComplexity Symllletry S~ Hldy

----

of 8 Binary Hcode \1 J \1Elements (:':.5 ) (:':.5 )

XOXOXOXO* 121 6.00 695 .38 .50 .50OXXOXOXO 525 18.27 426 .25 .19 .37XXOOXOXO* 535 18.27 329 .22 .19 .29OXOXXOXO* 350 19.69 738 .38 .12 .25XOOXXOXO 535 18.27 408 .25 .23 .27OOXXXOXO 595 16.00 310 .25 .00 .14XXOXOOXO 520 19.69 364 .22 .25 .21XOXXOOXO* 315 18.27 425 .19 .06 .08OXXXOOXO* 550 17.00 351 11 .06 .00XXXOOOXO* 365 16.00 281 .28 .31 .05OXOXOXXO* 400 18.27 429 .25 .12 .30XOOXOXXO* 575 19.69 389 .19 .25 .17OOXXOXXO 505 17.00 390 .25 .3\ .04XOXOOXXO 520 18.27 365 .25 .06 .04OXXOOXXO* 405 7.00 720 .38 .25 .25XXOOOXXO* 430 17.00 409 .28 .31 .23OOXOXXXO* 545 16.00 225 .22 .19 .14OXOOXXXO 575 17.00 320 .22 .31 .05XOOOXXXO 350 16.00 385 .25 .13 .25OOOXXXXO* 300 6.81 376 .34 .31 .30XXOXOXOO 540 18.27 379 .25 .50 .50XOXXOXOO 555 19.69 336 .22 .44 .50OXXXOXOO 575 16.00 231 .22 .06 .07XXXOOXOO* 430 17.00 274 .31 .19 .25XOXOXXOO* 480 18.27 350 .22 .50 .50OXXOXXOO* 510 17.00 420 .25 .44 .50XXOOXXOO 250 7.00 595 .25 .50 .50OXOXXXOO* 535 16.00 285 .25 .19 .10XOOXXXOO 545 17.00 343 .28 .i9 .06OOXXXXOO 255 6.81 730 .44 .06 .10XXXOXOOO 420 16.00 380 .25 .44 .50XXOXXOOO* 540 17.00 279 .31 .37 .50XOXXXOOO* 395 16.00 300 .28 .50 .50OXXXXOOO* 320 6.81 380 .34 .13 .07XXXXOOOO 147 6.81 740 .38 .44 .50

ResultsThe proportion of responses that were runs

(continuations of the terminal run) were tabulated foreach sequence. and the two complementary sets werecompared. The correlation coefficient for theproportions of run responses between the twocomplementary sets of 35 sequences was .89, p < .001 .indicating a reliable effect relatively uncontaminated bythe nature of the elemcnts of the scquences, butdependent instead on thc patterns of the sequences.Tl~ese proportions were averaged for the complementarypairs and wrrelated with the measured syntely of thesequences, yielding a coefficient of .86, p < .00 I. Thescattergram for this comparison is shown in Figure I .Each point is based on an N of 70.

It should be clear from the scattergram and thecorrelation coeftlcient that the syntely algorithmprovides an excellent fit to the expectancy predictionsof the subjects in this experiment. The fit is this good inspite of the fact that the algorithm is parameter free, andno minor adjustments can be made to accommodate thedata. No other models based on in trinsic structure existthat make explicit predictions for comparison with theseresults other than a probability matching model, whichwould predict a run proportion response of .5 for eachsequence, and Restle's (1961) model, which isinapplicable to J 1 of the 35 patterns which have no runlengths equal or longer than the terminal run andpredicts run proportions of .0, .67, or 1.0 for the rest ofthe sequences. Although it is inappropriate for almostone-third of the sequences, Restle's model does take intoaccount a portion of the intrinsic structure of thepatterns, namely, run lengths, and so it does have some

Figure 1. The proportion of subjects predicting a "run" foreach of the 35 sequences of eight binary elements as a functionof the syntely of those sequences, where syntely is a stimulusmeasure of pattern continuation.

0.2 0.4 0.6SYNTELY

o

0.8

o

o

r a .86

oo

o

o

o

0.0

:0.8 ~

z i~ f-0::

0.6 ~0

~

z if- I 0 00

0

u ......a I 0w

000

a:: 0.4 r- 0c.. i

z ~ 00 I 0

00

i 00..... 0

a:: 0.2 r-- @0 Ill. I 00a::ll.

0.0 i

Note-J = judged, M = measured*The first sel'en elements from the left were lIsed as stimuliin Experiment II.

is designed to measure all of the intrinsic structure available in apattern that influences expectancy, it was decided to present thepatterns in such a way as to reduce as much as possible anydisturbing influences of faulty memory. Therefore, each patternwas presented all at once rather than sequentially. All 70possible sequences of four Xs and four Os were used. These weresegregated into two sets of 35 sequences, so that in either set allsequences were different and each had a complementarysequence in the other set. For each of these sets of 35 sequences,35 different orders of the sequences were printed in a verticalcolumn on computer sheets so that each sequence appeared onceat each position of the vertical column. An ordered set of thesesequences is shown in Table I.

These 70 lists were presented to 70 subjects of anintroductory psychology class at Southern Connecticut StateCollege. The subjects were not informed about the purpose ofthe experiment. They were told simply to read each sequence ofeight elements from left to right and, for each of the 35sequences on their page, to write the element (either an X or an0) that they thought should come next in the sequence if thesequence was continued. They were then told not to worry orspend a lot of time on the sequences, but just to read them andput down whatever they felt should happen next.

predictive power. For the 24 sequences where it isappropriate, the correlation coefficient of Restle's modelwith subjects' predictions is .54, whereas the syntelycorrelation with subjects' results is .81. Somehowsubjects are responding to the intrinsic structure of thesequences by matching their predictions with thefrequency of the alternative consequences as provided ina syntely analysis based on the inductive principle.

A plausible process by which a subject chooses aprediction from the set of alternatives in the sequencepattern is given by stimulus sampling theory: Thematched subsequence consequents are available in a poolfor each subject from which they are sampled. Thepredictions of a large group of subjects thus reflects theavailability of the alternatives.

However, it is just as plausible to suppose that at somelevel, somehow, subjects are individually calculating thesyntely of each pattern and making their response on thebasis of this calculation and some idiosyncratic decisionrule. If this were the case, extensive observations on asingle subject might eventually shed light on the natureof this process.

But within the present frame of our ignorance, it isunnecessary to take either of these explanatory steps. Itis sufficient to show that a functional relationship existsbetween the intrinsic structure of the patterns as derivedby the syntely analysis and the predictions of a group ofuntrained subjects, and that is what this experiment hasdone.

Other approaches not based on a direct measure ofthe stimulus structure, using instead the responsepatterns of subjects to the stimulus, are possible. Onesuch method has been proposed by Lordahl (1968,1970) to account for sequential predictions by theweighted sum of several hypotheses. A very good fit tothe data may be achieved through the selection ofhypotheses and weighting parameters. The parsimony ofthis approach depends inversely on the number ofparameters needed. Lordahl found it necessary to use 40different parameters, corresponding to particularhypotheses related to perceived pattern structures in thebinary stimulus sequences.

Simplicity and ExpectancyA still different strategy for understanding expectancy

with binary patterns might argue that subjects predictthe next element that makes the resulting sequence insome sense simpler or less complex. Several models ofthe simplicity of binary patterns have been proposed.These models can be partitioned into two classes, thosebased on response measures and those based on stimulusmeasures (Vitz, 1968). Among response measures, themean verbal length of the description of the sequencesby human subjects has been used (Glanzer & Clark,1962), as has the response uncertainty of the preferred

SIMPLICITY, SYMMETRY, AND SYNTELY 439

start point of the pattern (Royer & Garner, 1966). Theseanalyses will not be treated in detail since the primaryconcern of this paper lies with stimulus measures ofpattern structure. A stimulus measure of patterncomplexity has been proposed by Vitz (1968) based onan information measure of the run structure of thepatterns, and this measure will b~ used to test theargument first.

None of the models has been applied to sequentialprediction data, and no specific numeric assessment ofsubjects' predictions has been drawn from them.However, all three models correlate highly with eachother and with the number of runs in a sequence. Infact, Vitz's measure Hrun-span is simply log2 of the totalnumber of runs in a pattern. It follows that they shouldpredict that, in order to make the resulting pattern assimple as possible, subjects should mainly continue theterminal run of a sequence. In fact, only for 14 of the 3Spatterns tested did the majority of subjects continue theterminal run, a number not different from chance. Thisalso eliminates the possibility that subjects werecontinuing the sequences in such a way as to make theresulting sequences as complex as possible in terms ofthe number of runs in the sequence.

A related model of the complexity of binary andtrinary sequences based on stimulus measures has beenoffered by Vitz and Todd (1969). They have providednumeric values for the complexity of binary patterns ofLength 8. These are reproduced in Table I under thecolumn headed Hcode ' A detailed description of how toevaluate Hcode can be tound in the original paper. Unlya description of the principles involved will be givenhere. Essentially, Vitz and Todd constructed aninformation theoretic measure of pattern complexitybased on the number and length of coded elements.These elements are found in a hierarchic series ofencoding levels, and they are based on the detection inthe sequences of runs of individual symbols and runs ofcoded subpatterns at each coding level. As a result, thereis a very high correlation between their informationmeasure and a simple count of runs (Simon, 1972). Infact, the correlation between Hcode and the number ofruns in a sequence varies from .79 to .92 for differentsets of sequences (Simon, 1972, p. 377). Accordingly, asimilar result may be expected for the Hcode measure aswas found for the run measure already tested. However,Hcode correlates higher with direct measures ofcomplexity than does the run length of the pattern forall sets of sequences, so Hcode may be a better measureof complexity and produce different results. In order totest the possibility that the previous measure ofcomplexity, the number of runs in the sequence, wasinadequate and that subjects really were continuing thesequences with the element that made the resultingsequence in some sense simpler, another simpleexperiment was carried out.

440 PSOTKA

EXPERIMENT II

MethodNineteen different sequences of four Xs and three Os were

selected randomly for one list, and their complementary set of19 sequences of three Xs and four Os formed the other list.These 19 patterns are shown with asterisks in Table 1. For eachlist 16 random orders were created and printed, each in a verticalcolumn of a computer output sheet. Each of 32 uninformedsubjects of an introductory psychology class at SouthernConnecticut State College was given one of these 32 lists and,with the same instructions as given in Experiment I, was asked towrite on the sheet the element he thought should come next inthe sequences.

ResultsThe correlation of the proportion of run responses

between the two sets of 19 sequences, for a sequenceand its complement, was .84, p < .001, indicating againa reliable response to the pattern of each sequence. Thecorrelation of the mean of these two sets of proportionswith the syntely of each sequence as determined by theapplication of the algorithm was .83, p < .001. In spiteof the inequality in the frequency of the alternativeelements in a sequence (4 : 3), the syntely algorithm stillprovided a good fit to the data. Moreover, looking onlyat whether the proportion of subjects responding runwas greater or less than .5 when the syntely measure wasgreater or less than .5 for each sequence, the sign was thesame for 18 of the 19 patterns. In the remainingsequence the syntely measure had a value of .5 exactly.

Using these results, the Hcode values provided by Vitzand Todd (1969) for binary patterns of Length 8 can beused to see if subjects are continuing each sequence withthe element that makes the resulting sequence somehowsimpler. If this is the case, then the majority of subjectsshould select the simpler of the two possible sequencesas measured by Hcode ' In fact, this was true for only 9of the 19 sequences. This result makes it implausible aswell that subjects select the element that makes theresulting sequence more complex.

These results raise the general question of whetherthere is any relation between the syntely measure orsubjects' predictions and the simplicity of the binarypatterns. It is clear that subjects do not select the nextelement in order to make the resulting sequence simpleror more complex. But perhaps there exists nevertheless arelation between the simplicity of the sequence and thedegree of agreement among subjects about what shouldcome next. Before trying to answer this question, therelation between syntely and symmetry will beestablished.

Symmetry and SyntelyIn an analysis of the results of the famous Zenith

radio experiments on telepathy, Goodfellow (1938)showed convincingly that the bilateral symmetry of thefive-element-Iong binary patterns had a strong influence

on the frequency of their choice by listeners of the radioprogram. The listeners shunned the symmetric sequencesas possible answers to the telepathic message. Using avery similar analysis which they chose to callsubsymmetry, Alexander and Carey (1968) showed thatthe cognitive simplicity of 35 binary patterns ofLength 7 was almost perfectly accounted for by therelative numbers of subsymmetries in the differentpatterns.

The syntely measure proposed in this paper bearssome striking similarities to this measure of symmetry.Syntely is in fact based technically on a kind ofsymmetry, translational symmetry. The finding ofmatches to the terminal segment by moving thatsegment laterally detlnes lateral subsymmetries.Symmetry in the more common sense of the word isbilateral symmetry, defined about the median axis of thesequence. This is the sense of symmetry used in thispaper.

A simple algorithm provides a numeric measure of thesymmetry of binary patterns that is very similar toGoodfellow's measure and correlates perfectly withAlexander and Carey's subsymmetry. The similarity withthe syntely algorithm lies basically in the use of anautocorrelational technique to find matchingsubsequences; but the matching for symmetry is doneabout the median axis instead of from the rightmostend, the pattern is matched against its reflected selfrather than against a duplicate sequence, and thematching is performed on the elements themselves ratherthan on the pattern of runs and alternations, as in thesyntely algorithm. Autocorrelation mechanisms havebeen proposed by a number of theorists as ideal ways ofdescribing certain processes in perception (Dodwell ,1971) and in memory (Anderson, 1973). The symmetryand syntely algorithms proposed in this paper extend theuse of autocorrelation to a new area of perception andto the grand domain of future expectancy.

Symmetry AlgorithmThe steps for carrying out the symmetry algorithm

begin with taking the binary pattern and aligning itsreflected self below it as in the following example:

XXOOXOXOOXOXOOXX

Compare the reflection of this pattern vertically,indicating a match with an m and a mismatch with adash. Matching here' is defined by a physical match ofthe elements, not a match of the runs and alternations,as in the syntely algorithm. Continue the comparison byshifting the top pattern one-half space to the left and thereflected pattern one·half space to the right. By shiftingeach one-half space the results are aligned vertically:

-mm--mm---mmm--m----mmmmmm

SIMPLICITY, SYMMETRY, AND SYNTELY 441

Table 2Correlation Coefficients Among Subjects' Judgments andStimulus Measures of Complexity, Symmetry, and Syntely:

r(p < .05) = .30

mmm

m

And then begin again and reverse the shifts of eachpattern:

Judged ComplexityHcode ComplexityJudged SymmetryMeasured SymmetryJudged Syntely (±.5)

SymmetryHcodeCom- J :>1

plexity

.74 -.70-.68-.58 -.71

.71

Syntely

J M(±.5) (±.5)

-.30 -.31-.17 -.12

.12 .29

.06 .II.77

;---mm--mm­m--m---m-mmmm-­m-m-m

-m­mmm

Since the comparisons are completely symmetrical, theycan be reflected on themselves and the median axisremoved:

--mm ---mm-­

mm

m

-mm­m-­-mm

m-

m

Finally, the matched, and thus symmetric, elements arecounted on each line in from the right until the fIrstdash is encountered, at which point the algorithm shiftsto the next line. The sum of these matches (theunderlined ms in the above example) is divided by thetotal number of comparisons for the pattern, and thequotient is then the numeric measure of the symmetryof the pattern. In this example, the number ofsymmetric elements is 7 (the underlined ms) and thetotal number of comparisons is 32. The symmetry ofthis pattern is then 7/32 =.22. This algorithm may easilybe extended to any binary pattern of any length todetermine its bilateral symmetry, as a proportion rangingfrom 0 to I.

Given this measure of the intrinsic structure of binarypatterns, how does it relate to the judged complexity of

Note-J = judged, M = measured

the sequences? Is there a stronger relation with thejudged symmetry of these patterns? Does it bear anyrelation to the syntely of the patterns? If there doesexist a relation between symmetry and syntely, it maybe that, like the relation between simplicity and syntely,it is not straightforward.

EXPERIMENT III

MethodTwenty-four Yale undergraduates fulfilling an introductory

psychology course requirement were tested individually asmembers of three experimental groups of eight. They were eachpresented with 70 cards 2% x 3% in. in size, each of which hadone of the 70 possible patterns of four Xs and four Os centeredon it, and were asked to go through the cards one at a time, eachgroup doing only one of the three followinl!: thinl!:S: (1) React thepattern aloud from left to right and judge its complexity bygiving a number from 1 to 50 that reflected how complex thepattern seemed, the higher the number the greater thecomplexity. (2) Read the pattern aloud from left to right andjudge its symmetry by giving a number from 1 to 50 thatrenected the symmetry ot the pattern, the greater the symmetrythe greater the number. (3) Read the pattern aloud from left toright and then say the element, either an X or an 0, they feltshould come next if the sequence continued (judge syntely).

ResultsThe responses of all three groups were fIrst sorted

arbitrarily into two complementary sets of 35 sequencesso that in each set all the sequences were different andeach had a complementary sequence in the other set.Then, for each sequence, the eight complexityjudgments were summed, the eight symmetry judgmentswere summed, and the proportion of "run" responseswas subtracted from .5 and given an absolute value. Thecorrelations between the two complementary sets of 35sequences for the judged complexity, symmetry, and thedeviation of the syntely judgments from .5 were .88,.96, and .79, respectively, indicating again a highreliability of the subjects' responses to the structure ofthe patterns.

These response measures were then collapsed acrosscomplementary sequences and are presented in Table I,along with three stimulus measures of complexity,symmetry, and syntely. What is of interest are theinterrelations among this set of response and stimulus

442 PSJTKA

PATTERN LOCATION

o

o.

0.0

0.5 ~I

04 rI

0.3 rVl

~ 02 ~a:: I

~ 0.1 L~ 00 ~o ~I I~ ----1_ILo i

~ 05 r /\~ I I'a:: 0.4 r- / ,~. I ,o 0.3 ~ I' •a:: ~ .I./e , ./Il. 0.2 rf/ '\' ./, ./r----O

0.1 ./.,,/e ""b-'''''o

Figure 2. Error profiles for a simple and a complex pattern.The fiUed circles indicate the proportion of the total errors forthe pattern at each location, from Garner and Gottwald (1967).The open circles indicate the proportion given by syntelyanalysis normalized for the pattern.

syntely at each point of a repetitive string of the basicsequence. The empirical result of such an application ofthe syntely algorithm is that there exists a pointsomcwhcre in the string after which the syntely measureis greater than .5 when there is a run in the string, andless than .5 when there is an alternation in the string.TIle syntely algorithm may be said to be correctlypredicting the string from that point on. The number ofelements needed in the string to reach this point can be .taken as a simple estimate of the number of trials tocriterion. The absolute deviation of the syntely measurefrom .5 may be taken as an estimate of the likelihood ofan error at this point. If these absolute deviations arenormalized for the first full cycle of the sequence that iscorrectly predicted by the syntely algorithm, at eachlocation of the pattern, the resulting profile may becompared directly with the subjects' actual errorprofIles. In this way, the syntely algorithm may be usedto predict the relative number of errors at each locationand the estimated trials to criterion for any patternwithout any conversion parameters.

The first application of the syntely algorithm is to theerror data provided by Garner and Gottwald (1967).They asked their subjects to predict each element of arecurrent binary sequence as it was presented visually atthe rate of one element every 4 sec. They used only twosequences, an easy one (X X X 0 0) and a difficult

measures. In Table 2 are shown the correlations amongthis set.

The most striking result is that the significantco.rrelations split simply into two groups, either dealingwIt~ p~ttern simplicity or else with patternc~ntmuatlOn. The measures in each group correlatehIghly with each other but not with the measures in theother group. It seems that pattern simplicity orsymmetry is not related to the strength of judgedpattern expectancy or syntely.

The syntely meaSUre correlates well only with thejudged syntely, when both are taken as deviations from.5. The symmetry meaSUre correlates well with bothjUdged complexity and judged symmetry as well asHcode , all measures of pattern simplicity. The onlyfurther interest lies in the slight but still significantcorrelations of the judged syntely and the syntelymeasure with judged complexity, but not with the othermeasures. This represents a relatively slight dissociationbetween the relative contributions of sequentialregularity and symmetry to the simplicity of patterns,insofar as syntely measures sequential regularity and notsymmetry.

It is clear that both Hcode and the symmetry measureevaluate some psychologically relevant aspects of thepatterns for their simplicity. Whether they evaluate thesame, different, or overlapping aspects of the patterns isnot clear. The high positive correlation between Hcodeand the symmetry measure gives reason to believe thatthey measure the same aspect, but it may be useful inthe future to search for patterns that prove to be simplebut not sy mmetric and vice versa, in order to determinewhether Hcode and the symmetry measure are sensitiveto different aspects of the patterns. There is littleevidence in the present data that these patterns can befound. However, it would seem likely that for muchlonger sequences the correlation between symmetry andsimplicity should break down, while Hcode may still beuseful.

Errors in the Tracking of RecurrentBinary Sequences

The syntely algorithm has another application thatmay be used to test its descriptive power. Consider thesituation in which a subject predicts each next elementof an unknown binary sequence as it is presentedrecurrently one element at a time. The subject will err inhis predictions at those points where his expectancydoes not match the pattern, and those errOrs willcontinue until the subject remembers the pattern fully.The errors at each point may be accumulated andnormalized (divided by the total number of errors forthe whole pattern) to provide an error profile for eachpattern. The number of elements presented up to thelast error constitutes the number of trials to criterion.

A similar proftle may be derived from the syntelyalgorithm in a straightforward way by measuring the

SIMPLICITY, SYMMETRY, AND SYNTELY 443

Figure 3. Error profIles for two binary patterns. The fIlledcircles indicate the proportion of the total errors for eachpattern at each location found by Restle (1967). The opencircles indicate the normalized proportion at each location givenby a syntely analysis.

sequence (X X 0 X 0) and began each pattern at each ofthe five different starting points for different subjects.The actual locations of errors were recorded for eachpattern. The mean proportion of the total errors foreach pattern, pooled over all starting points, are shownat each location in Figure 2. The normalized errorprofile as given by the syntely algorithm is shown inFigure 2 for comparison. The fit of the two profIles isexcellent for the simple pattern, and the error transitionsare followed faithfully in the complex pattern. Garnerand Gottwald recorded a number of other responsemeasures as well, among them the mean location in thesequence of the element or trial following the last error.These mean numbers of trials to criterion were 16.3 forthe simple pattern and 25.8 for the complex pattern.The corresponding lengths of string used by the syntelyalgorithm before perfect prediction were 17 and 25,respectively.

Another set of error profIles is provided by Restle(1967). Subjects again predicted recurrent cycles of twounknown sequences, a hard one (0 0 X X 0 X X 0 X)and an easy one (00 X X 00 X 0 X). The sequenceswere presented visually at the rate of about one elementevery 2 sec. The actual normalized error profIles alongwith the normalized profIles given by the syntelyalgorithm are shown in Figure 3. Again a very good fit tothe profIle transition is apparent. The mean location ofthe trial following the last error is not provided for thereal subjects, but the length of string needed for the

The accumulated success of the syntely measure inpredicting the subjects' responses in a sequentialprediction task and their error profIles when theypredict recurrent sequences, its relation to a provenmeasure of the symmetry and thus the simplicity ofshort binary sequences, and its potential usefulness fordissociating separate sequential and symmetric aspects ofthe intrinsic structure of binary patterns provides goodevidence that the syntely algorithm, and the inductiveprinciple upon which it is based, measurespsychologically relevant aspects of intrinsic stimulusstructure.

It must be emphasized that the algorithm, intended tomeasure the syntely of binary sequences, is, as it ispresented in this paper, only a first step, a firstapproximation to the practical application of theprinciple of induction to the problems of understandinghuman exp~ctancy. The key to understanding humanexpectancy, at least with binary patterns, lies inmeasuring the intrinsic structure of the patterns. Theprinciple of induction provides the essential rule forproducing this measure: match the present events to thepast and weigh the past consequences of those eventsthat match. This matching operation effectively definesthe process of autocorrelation, a process which has beenproposed by many as a basic biologic activity eitherdirectly (Kabrisky, 1964) or indirectly as part of aholographic system (van Heerden, 1968). It is verysatisfying indeed to extend the applicability of thispotentially powerful analogy from the two mainpsychological domains where its usefulness has beendemonstrated, the past and the present, memory andperception, to the third major domain ot psychology,the future, expectancy. The process of autocorrelationis, however, subservient to the general principle ofinduction and may turn out not to be the best means ofimplementing this principle in the long run.

As it has been applIed in its present form, theprinciple of induction has been shown to be analtogether adequate basis for describing the inferencessubjects make after reading short binary sequences, evenwhen the intrinsic structure of those sequences has notbeen constrained in any artificial or extrinsic way. Thisis an important consideration and represents a steptoward greater generality beyond the current inferentialmodels outlined and summarized so beautifully bySimon (1972), which depend on the existence ofregularities within the binary sequence based on rules.The success of the syntely algorithm shows that subjectscan be responsive to the regularities of a pattern withoutresorting to the formulation of rules, but simply as the

CONCLUSION

syntely algorithm was 26 elements for the easy patternand 28 elements for the hard pattern.

oo

PATTERN LOCATION

o 0

R R f~1\ '\ I\~

~ ~00110010

0.00

0.20

0.05

0.15UlII:o 0.10II:II:l&J 0.05...J

~ 0.00ol-

lLoz 0.20o

~ 0.15ol1.

~ 0.10l1.

-J..-l.4 pson;,.\

result of a basi.: inferl?ntial pW.:I?SS that Jepl?JlJs ul1

memory, as long as thl? regularitil?s Jre expresseL! in thl?intrinsic structure of the stimulus.

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(Received for publication September 5,1974;revision accepted October 27,1974.)