Elements of a theory of human problem solving

Psychological ReviewVol. 65, No. 3, 1958

ELEMENTS OF A THEORY OF HUMAN PROBLEM SOLVING

ALLEN NEWELL, J. C. SHAW

The RAND Corporation

AND HERBERT A. SIMON

Carnegie Institute of Technology

In this paper we shall set forth theelements of a theory of human problemsolving, together with some evidence forits validity drawn from the currentlyaccepted facts about the nature of prob-lem solving. What questions should atheory of problem solving answer? First,it should predict the performance of aproblem solver handling specified tasks.It should explain how human problemsolving takes place: what processes areused, and what mechanisms performthese processes. It should predict theincidental phenomena that accompanyproblem solving, and the relation ofthese to the problem-solving process.For example, it should account for "set"and for the apparent discontinuities thatare sometimes called "insight." It shouldshow how changes in the attendant con-ditions—both changes "inside" the prob-lem solver and changes in the task con-fronting him—alter problem-solving be-havior. It should explain how specificand general problem-solving skills arelearned, and what it is that the problemsolver "has" when he has learned them.

Information Processing Systems

Questions about problem-solving be-havior can be answered at various levelsand in varying degrees of detail. Thetheory to be described here explainsproblem-solving behavior in terms ofwhat we shall call information processes.If one considers the organism to consistof effectors, receptors, and a control sys-tem for joining these, then this theoryis mostly a theory of the control sys-tem. It avoids most questions of sen-

sory and motor activities. The theorypostulates:

1. A control system consisting of anumber of memories, which containsymbolized information and are inter-connected by various ordering relations.The theory is not at all concerned withthe physical structures that allow thissymbolization, nor with any propertiesof the memories and symbols other thanthose it explicitly states.

2. A number of primitive informationprocesses, which operate on the informa-tion in the memories. Each primitiveprocess is a perfectly definite operationfor which known physical mechanismsexist. (The mechanisms are not neces-sarily known to exist in the humanbrain, however—we are only concernedthat the processes be described withoutambiguity.)

3. A perfectly definite set of rules forcombining these processes into wholeprograms of processing. From a pro-gram it is possible to deduce unequivo-cally what externally observable behav-iors will be generated.

At this level of theorizing, an ex-planation of an observed behavior ofthe organism is provided by a programof primitive information processes thatgenerates this behavior.

A program viewed as a theory ofbehavior is highly specific: it describesone organism in a particular class ofsituations. When either the situationor the organism is changed, the pro-gram must be modified. The programcan be used as a theory—that is, as apredictor of behavior—in two distinct

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152 ALLEN NEWELL, J. C. SHAW, AND HERBERT A. SIMON

ways. First, it makes many precisepredictions that can be tested in detailregarding the area of behavior it isdesigned to handle. For example, thetheory considered in this paper predictsexactly how much difficulty an organ-ism with the specified program will en-counter in solving each of a series ofmathematical problems: which of theproblems it will solve, how much time(up to a proportionality constant) willbe spent on each, and so on.

Second, there will be important quali-tative similarities among the programsthat an organism uses in various situa-tions, and among the programs used bydifferent organisms in a given situation.The program that a human subject usesto solve mathematical problems will besimilar in many respects to the programhe uses to choose a move in chess; theprogram one subject uses for any suchtask will resemble the programs used byother subjects possessing similar train-ing and abilities. If there were no suchsimilarities, if each subject and eachtask were completely idiosyncratic, therecould be no theory of human problemsolving. Moreover, there is some posi-tive evidence, as we shall see, that suchsimilarities and general characteristicsof problem-solving processes do exist.

In this paper we shall limit ourselvesto this second kind of validation of ourtheory of problem solving. We shallpredict qualitative characteristics of hu-man problem-solving behavior and com-pare them with those that have alreadybeen observed and described. Since allof the available data on the psychologyof human problem solving are of thisqualitative kind, no more detailed testof a program is possible at present. Themore precise validation must wait uponnew experimental work.1

1 Several studies of individual and groupproblem-solving behavior with logic problemshave been carried out by 0. K. Moore andScarvia Anderson (5). The problems Moore

In succeeding sections we shall de-scribe an information-processing pro-gram for discovering proofs for theo-rems in logic. We shall compare its be-havior qualitatively with that of humanproblem solvers. In general, the proc-esses that compose the program arefamiliar from everyday experience andfrom research on human problem solv-ing: searching for possible solutions,generating these possibilities out ofother elements, and evaluating partialsolutions and cues. From this stand-point there is nothing particularly novelabout the theory. It rests its claims onother considerations:

1. It shows specifically and in detailhow the processes that occur in humanproblem solving can be compounded outof elementary information processes, andhence how they can be carried out bymechanisms.

2. It shows that a program incorpo-rating such processes, with appropriateorganization, can in fact solve problems.This aspect of problem solving has beenthought to be "mysterious" and unex-plained because it was not understoodhow sequences of simple processes couldaccount for the successful solution ofcomplex problems. The theory dissolvesthe mystery by showing that nothingmore need be added to the constitutionof a successful problem solver.

Relation to Digital Computers

The ability to specify programs pre-cisely, and to infer accurately the be-havior they will produce, derives fromthe use of high-speed digital computers.Each specific theory—each program of

and Anderson gave their subjects are some-what different from those handled by ourprogram, and hence a detailed comparison ofbehavior is not yet possible. We are now en-gaged, with Peter Houts, in replicating andextending the experiments of Moore and An-derson with human subjects and at the sametime modifying our program to predict thehuman laboratory behavior in detail.

THEORY OF HUMAN PROBLEM SOLVING 153

information processes that purports todescribe some human behavior—is codedfor a computer. That is, each primitiveinformation process is coded to be aseparate computer routine, and a "mas-ter" routine is written that allows theseprimitive processes to be assembled intoany system we wish to specify. Oncethis has been done, we can find outexactly what behavior the purportedtheory predicts by having the computer"simulate" the system.

We wish to emphasize that we are notusing the computer as a crude analogyto human behavior—we are not com-paring computer structures with brains,nor electrical relays with synapses. Ourposition is that the appropriate way todescribe a piece of problem-solving be-havior is in terms of a program: aspecification of what the organism willdo under varying environmental circum-stances in terms of certain elementaryinformation processes it is capable ofperforming. This assertion has nothingto do—directly—with computers. Suchprograms could be written (now thatwe have discovered how to do it) ifcomputers had never existed.2 A pro-gram is no more, and no less, an analogyto the behavior of an organism than isa differential equation to the behaviorof the electrical circuit it describes.Digital computers come into the picture

2 We can, in fact, find a number of attempts.in the psychological literature to explain be-havior in terms of programs—or the proto-types thereof. One of the most interesting,because it comes relatively close to the mod-ern conception of a computer program, isAdrian de Groot's analysis of problem solvingby chess players (2). The theory of de Grootis based on the thought-psychology of Selz, asomewhat neglected successor to the Wurzburgschool. Quite recently, and apparently inde-pendently, we find the same idea applied byJerome S. Bruner and his associates to thetheory of concept formation (1). Bruner usesthe term "strategy," derived from economicsand game theory, for what we have called aprogram.

only because they can, by appropriateprogramming, be induced to execute thesame sequences of information processesthat humans execute when they aresolving problems. Hence, as we shallsee, these programs describe both hu-man and machine problem solving atthe level of information processes.8

With this discussion of the relation ofprograms to machines and humans be-hind us, we can afford to relax into con-venient, and even metaphoric, uses oflanguage without much danger of mis-understanding. It is often convenientto talk about the behavior implied by aprogram as that of an existing physicalmechanism doing things. This mode ofexpression is legitimate, for if we takethe trouble to put any particular pro-gram in a computer, we have in fact amachine that behaves in the way pre-scribed by the program. Similarly, forconcreteness, we will often talk as ifour theory of problem solving consistedof statements about the ability of acomputer to do certain things.

THE LOGIC THEORIST

We can now turn to an example ofthe theory. This is a program capableof solving problems in a particular do-main—capable, specifically, of discover-ing proofs for theorems in elementarysymbolic logic. We shall call this pro-gram the Logic Theorist (LT).4 Weassert that the behavior of this pro-gram, when the stimulus consists of theinstruction that it prove a particulartheorem, can be used to predict the be-

3 For a fuller discussion of this point see(9).

4 In fact, matters are a little more compli-cated, for in the body of this paper we willconsider both the basic program of LT and anumber of variants on this program. We willrefer to all of these variants, interchangeably,as "LT." This will not be confusing, sincethe exact content of the program we are con-sidering at any particular point will alwaysbe clear from the context.


havior of (certain) humans when theyare faced with the same problem insymbolic logic.

The program of LT was not fash-ioned directly as a theory of human be-havior; it was constructed in order toget a program that would prove theo-rems in logic. To be sure, in construct-ing it the authors were guided by a firmbelief that a practicable program couldbe constructed only if it used many ofthe processes that humans use. Thefact remains that the program was notdevised by fitting it directly to humandata. As a result, there are many de-tails of LT that we would not expect tocorrespond to human behavior. For ex-ample, no particular care was exercisedin choosing the primitive informationprocesses to correspond, point by point,with elementary human processes. Allthat was required in writing the pro-gram was that the primitive processesconstitute a sufficient set and a con-venient set for the type of programunder study.

Since LT has been described in de-tail elsewhere (6, 8), the descriptionwill not be repeated here. It will alsobe unnecessary to describe in detail thesystem of symbolic logic that is used byLT. For those readers who are not fa-miliar with symbolic logic, we may re-mark that problems in the sententialcalculus are at about the same level ofdifficulty and have somewhat the same"flavor" as problems in high school ge-ometry."

Design of the Experiments

First we will describe the overt be-havior of LT when it is presented withproblems in elementary symbolic logic.

5 LT employs the sentential calculus as setforth in Chapters 1 and 2 of A. N. Whiteheadand Bertrand Russell, Principia, Mathematica(10)—the "classic" of modern symbolic logic.A simple introduction to the system of Prin-cipia will be found in (3).

In order to be concrete, we will refer toan experiment conducted on a digitalcomputer. We take an ordinary gen-eral-purpose digital computer,8 and storein its memory a program for interpret-ing the specifications of LT. Then weload the program that specifies LT.The reader may think of this programas a collection of techniques that LThas acquired for discovering proofs.These techniques range from the abil-ity to read and write expressions insymbolic logic to general schemes forhow a proof might be found.

Once we have loaded this programand pushed the start button, the com-puter, to all intents and purposes, isLT. It already knows how to do sym-bolic logic, in the sense that the basicrules of operation of the mathematicsare already in the program (analogouslyto a human's knowing that "equalsadded to equals give equals" in ele-mentary algebra).

We are now ready to give LT a task.We give it a list of the expressions(axioms and previously proved theo-rems) that it may take as "given" forthe task at hand. These are stored inLT's memory. Finally, we present LTwith another expression and instruct itto discover a proof for this expression.

From this point, the computer is onits own. The program plus the taskuniquely determines its behavior. It at-tempts to find a proof—that is, it triesvarious techniques, and if they don't

8 The experiments described here were car-ried out with the RAND JOHNNIAC com-puter. The JOHNNIAC is an automatic dig-ital computer of the Princeton type. It has aword length of 40 bits, with two instructionsin each word. Its fast storage consists of4,096 words of magnetic cores, and its second-ary storage consists of 9,216 words on mag-netic drums. Its speed is about 15,000 op-erations per second. The programming tech-niques used are described more fully in (6).The experiments are reported in more detailin (7).


work, it tries other techniques. If LTfinds a legitimate proof, it prints thisout on a long strip of paper. There is,of course, no guarantee that it will finda proof; after working for some time,the machine will give up—that is, it willstop looking for a proof.

Now the experimenters know exactlywhat is in the memory of LT when itstarts—indeed, they created the pro-gram. This, however, is quite differ-ent from saying that the experimenterscan predict everything LT will do. Inprinciple this is possible; but in factthe program is so complex that the onlyway to make detailed predictions is toemploy a human to simulate the pro-gram by hand. (A human can do any-thing a digital computer can do, al-though it may take him considerablylonger.)

1. As the initial experiment, we storedthe axioms of Principia Mathematica,together with the program, in the mem-ory of LT, and then presented to LTthe first 52 theorems in Chapter 2 ofPrincipia in the sequence in which theyappear there. LT's program specifiedthat as a theorem was proved it wasstored in memory and was available,along with the axioms, as material forthe construction of proofs of subsequenttheorems. With this program and thisorder of presentation of problems, LTsucceeded in proving 38 (73%) of the52 theorems. About half of the proofswere accomplished in less than a minuteeach; most of the remainder took fromone to five minutes. A few theoremswere proved in times ranging from 15minutes to 45 minutes. There was astrong relation between the times andthe lengths of the proofs—the time in-creasing sharply (perhaps exponentially)with each additional proof step.

2. The initial conditions were now re-stored by removing from LT's memorythe theorems it had proved. (Trans-late: "A new subject was obtained who

knew how to solve problems in logic butwas unfamiliar with the particular prob-lems to be used in the experiment.")When one of the later theorems ofChapter 2 (Theorem 2.12) was pre-sented to LT, it was not able to finda proof, although when it had held theprior theorems in memory, it had foundone in about ten seconds.

3. Next, an experiment was performedintermediate between the first two. Theaxioms and Theorem 2.03 were stored inmemory, but not the other theoremsprior to Theorem 2.12, and LT wasagain given the task of proving the lat-ter. Now, using Theorem 2.03 as oneof its resources, LT succeeded—in fif-teen minutes—where it had failed in thesecond experiment. The proof requiredthree steps. In the first experiment,with all prior theorems available, theproof required only one step.

Outcome of the Experiments

From these three series of experimentswe obtain several important pieces ofevidence that the program of LT isqualitatively like that of a human facedwith the same task. The first, andmost important, evidence is that LTdoes in fact succeed in finding proofsfor a large number of theorems.

Let us make this point quite clear.Since LT can actually discover proofsfor theorems, its program incorporates asufficient set of elementary processes ar-ranged in a sufficiently effective strategyto produce this result. Since no otherprogram has ever been specified forhandling successfully these kinds ofproblem-solving tasks, no definite al-ternative hypothesis is available. Weare well aware of the standard argu-ment that "similarity of function doesnot imply similarity of process."ever useful a caution this may bshould not blind us to the fact thatspecification of a set of nVeBVfelJsms

m . . . , , :noDm,BDuanu9nsufficient to produce observed benay^r


is strong confirmatory evidence for thetheory embodying these mechanisms, es-pecially when it is contrasted with theo-ries that cannot establish their suffi-ciency.

The only alternative problem-solvingmechanisms that have been completelyspecified for these kinds of tasks aresimple algorithms that carry out ex-haustive searches of all possibilities,substituting "brute force" for the selec-tive search of LT. Even with the speedsavailable to digital computers, the prin-cipal algorithm we have devised as analternative to LT would require timesof the order of hundreds or even thou-sands of years to prove theorems thatLT proves in a few minutes. LT's suc-cess does not depend on the "bruteforce" use of a computer's speed, buton the use of heuristic processes likethose employed by humans.7 This canbe seen directly from examination of theprogram, but it also shows up repeat-edly in all the other behavior exhibitedby LT.

The second important fact thatemerges from the experiments is thatLT's success depends in a very sensitiveway upon the order in which problemsare presented to it. When the sequenceis arranged so that before any particu-lar problem is reached some potentiallyhelpful intermediate results have al-ready been obtained, then the task iseasy. It can be made progressivelyharder by skipping more and moreof these intermediate stepping-stones.Moreover, by providing a single "hint,"as in the third experiment (that is,"Here is a theorem that might help"),we can induce LT to solve a problemit had previously found insoluble. Allof these results are easily reproduced intTte laboratory with humans. To com-{Jafe't'LT's behavior with that of a hu-

e analysis of the power of thecorporated in LT will be found in

man subject, we would first have totrain the latter in symbolic logic (thisis equivalent to reading the programinto LT), but without using the specifictheorems of Chapter 2 of PrincipiaMathematica that are to serve as prob-lem material. We would then presentproblems to the human subject in thesame sequence as to LT. For each newsequence we would need naive subjects,since it is difficult to induce a humansubject to forget completely theoremshe has once learned.

PERFORMANCE PROCESSES IN THELOGIC THEORIST

We can learn more about LT's ap-proximation to human problem solvingby instructing it to print out some ofits intermediate results—to work itsproblems on paper, so to speak. Thedata thus obtained can be comparedwith data obtained from a human sub-ject who is asked to use scratch paperas he works a problem, or to thinkaloud.8 Specifically, the computer canbe instructed to print out a record ofthe subproblems it works on and themethods it applies, successfully and un-successfully, while seeking a solution.We can obtain this information at anylevel of detail we wish, and make acorrespondingly detailed study of LT'sprocesses.

To understand the additional informa-tion provided by this "thinking aloud"procedure, we need to describe a little

8 Evidence obtained from a subject whothinks aloud is sometimes compared with evi-dence obtained by asking the subject to theo-rize introspectively about his own thoughtprocesses. This is mieleading. Thinking aloudis just as truly behavior as is circling the cor-rect answer on a paper-and-pencil test. Whatwe infer from it about other processes goingon inside the subject (or the machine) is, ofcourse, another question. In the case of themachine, the problem is simpler than in thecase of the human, for we can determine ex-actly the correspondence between the internalprocesses and what the machine prints out.


more fully how LT goes about solvingproblems. This description has twoparts: (a) specifying what constitutesa proof in symbolic logic; (b) describ-ing the methods that LT uses in find-ing proofs.

Nature of a Proof

A proof in symbolic logic (and inother branches of logic and mathemat-ics) is a sequence of statements suchthat each statement: (a) follows fromone or more of the others that precedeit in the sequence, or (b) is an axiom orpreviously proved theorem.9 Here "fol-lows" means "follows by the rules oflogic."

LT is given four rules of inference:Substitution. In a true expression

(for example, "[p or p] implies p")there may be substituted for any vari-able a new variable or expression, pro-vided that the substitution is madethroughout the original expression.Thus, by substituting p or q for p inthe expression "(p or p) implies p," weget: "([p or q] or [p or q]) implies(p or q)" but not: "([p or q] or p)implies p."

9 The axioms of symbolic logic and the theo-ries that follow from them are all tautologies,true by virtue of the definitions of their terms.It is their tautological character that giveslaws of logic their validity, independent ofempirical evidence, as rules of inductive in-ference. Hence the very simple axioms thatwe shall use as examples here will have anappearance of redundancy, if not triviality.For example, the first axiom of Principiastates, in effect, that "if any particular sen-tence (call it p) is true, or if that same sen-tence (p) is true, then that sentence (p) is,indeed, true"—for example, "if frogs are fish,or if frogs are fish, then frogs are fish." The"if—then" is trivially and tautologically trueirrespective of whether p is true, for in truthfrogs are not fish. Since our interest here isin problem solving, not in logic, the readercan regard LT's task as one of manipulatingsymbols to produce desired expressions, andhe can ignore the material interpretations ofthese symbols.

Replacement. In a true expression aconnective ("implies," etc.) may be re-placed by its definition in terms of otherconnectives. Thus "A implies B" is de-nned to be "not-A or B"; hence the twoforms can be used interchangeably.

Detachment. If "A" is a true expres-sion and "A implies B" is a true expres-sion, then B may be written down as atrue expression.

Syllogism (Chaining). It is possibleto show by two successive applicationsof detachment that the following is alsolegitimate: If "a implies b" is a trueexpression and "b implies c" is a trueexpression, then "a implies c" is also atrue expression.

Proof Methods

The task of LT is to construct aproof sequence deriving a problem ex-pression from the axioms and the previ-ously proved theorems by the rules ofinference listed above. But the rulesof inference, like the rules of any mathe-matical system or any game, are per-missive, not mandatory. That is, theystate what sequences may legitimatelybe constructed, not what particular se-quence should be constructed in orderto achieve a particular result (i.e., toprove a particular problem expression).The set of "legal" sequences is exceed-ing large, and to try to find a suitablesequence by trial and error alone wouldalmost always use up the available timeor memory before it would exhaust theset of legal sequences.10

To discover proofs, LT uses methodswhich are particular combinations of in-formation processes that result in co-ordinated activity aimed at progress ina particular direction. LT has fourmethods (it could have more): substi-

10 See (7). The situation here is like thatin chess or checkers where the player knowswhat moves are legal but has to find in a rea-sonable time a move that is also "suitable"—that is, conducive to winning the game.


tution, detachment, forward chaining,and backward chaining. Each methodfocuses on a single possibility for achiev-ing a link in a proof.

The substitution method attempts toprove an expression by generating itfrom a known theorem employing sub-stitutions of variables and replacementsof connectives.

The detachment method tries to workbackward, utilizing the rule of detach-ment to obtain a new expression whoseproof implies the proof of the desiredexpression. This possibility arises fromthe fact that if B is to be proved, andwe already know a theorem of the form"A implies B," then proof of A is tan-tamount to proof of B.

Both chaining methods try to workbackward to new problems, using therule of syllogism, analogously to thedetachment method. Forward chaininguses the fact that if "a implies c" isdesired and "a implies b" is alreadyknown, then it is sufficient to prove "bimplies c." Backward chaining runs theargument the other way: desiring "aimplies c" and knowing "b implies c"yields "a implies b" as a new problem.

The methods are the major organiza-tions of processes in LT, but they arenot all of it. There is an executiveprocess that coordinates the use of themethods, and selects the subproblemsand theorems upon which the methodsoperate. The executive process also ap-plies any learning processes that are tobe applied. Also, all the methods uti-lize common subprocesses in carryingout their activity. The two most im-portant subprocesses are the matchingprocess, which endeavors to make twogiven subexpressions identical, and thesimilarity test, which determines (onthe basis of certain computed descrip-tions) whether two expressions are"similar" in a certain sense (for de-tails, cf. 8).

LT can be instructed to list its at-

attempts, successful and unsuccessful, touse these methods, and can list the newsubproblems generated at each stage bythese attempts. We can make this con-crete by an example:

Suppose that the problem is to prove"p implies p." The statement "(p orp) implies p" is an axiom; and "p im-plies (p or p)" is a theorem that hasalready been proved and stored in thetheorem memory. Following its pro-gram, LT first tries to prove "p im-plies p" by the substitution method, butfails because it can find no similar theo-rem in which to make substitutions.

Next, it tries the detachment method.Letting B stand for "p implies p," sev-eral theorems are found of the form "Aimplies B." For example, by substitu-tion of not-p for q, "p implies (q or p)"becomes "p implies (not-p or p)"; thisbecomes, in turn, by replacement of"or" by "implies": "p implies (p im-plies p)." Discovery of this theoremcreates a new subproblem: "Prove A"—that is, "prove p." This subproblem,of course, leads nowhere, since p is nota universally true theorem, hence can-not be proved.

At a later stage in its search LT triesthe chaining method. Chaining for-ward, it finds the theorem "p implies (por p)" and is then faced with the newproblem of proving that "(p or p) im-plies p." This it is able to do by thesubstitution method, when it discoversthe corresponding axiom.

All of these steps, successful and un-successful, in its proof—and the oneswe have omitted from our exposition, aswell—can be printed out to provide uswith a complete record of how LT exe-cuted its program in solving this par-ticular problem.

SOME CHARACTERISTICS OF THEPROBLEM-SOLVING PROCESS

Using as our data the informationprovided by LT as to the methods it


tries, the sequence of these methods,and the theorems employed, we can askwhether its procedure shows any re-semblance to the human problem-solv-ing process as it has been described inpsychological literature. We find thatthere are, indeed, many such resem-blances, which we summarize under thefollowing headings: set, insight, con-cept formation, and structure of theproblem-subproblem hierachy.

Set

The term "set," sometimes defined as"a readiness to make a specified re-sponse to a specified stimulus" (4, p.65), covers a variety of psychologicalphenomena. We should not be sur-prised to find that more than one as-pect of LT's behavior exhibits "set,"nor that these several evidences of setcorrespond to quite different underlyingprocesses.

1. Suppose that after the programhas been loaded in LT, the axioms anda sequence of problem expressions areplaced in its memory. Before LT un-dertakes to prove the first problem ex-pression, it goes through the list ofaxioms and computes a description ofeach for subsequent use in the "simi-larity" tests. For this reason, the proofof the first theorem takes an extra in-terval of time amounting, in fact, toabout twenty seconds. Functionallyand phenomenologically, this computa-tion process and interval represent apreparatory set in the sense in whichthat term is used in reaction-time ex-periments. It turns out in LT that thispreparatory set saves about one thirdof the computing time that would other-wise be required in later stages of theprogram.

2. Directional set is also evident inLT's behavior. When it is attemptinga particular subproblem, LT tries firstto solve it by the substitution method.If this proves fruitless, and only then,

it tries the detachment method, thenchaining forward, then chaining back-ward. Now when it searches fortheorems suitable for the substitutionmethod, it will not notice theorems thatmight later be suitable for detachment(different similarity tests being appliedin the two cases). It attends single-mindedly to possible candidates for sub-stitution until the theorem list has beenexhausted; then it turns to the detach-ment method.

3. Hints and the change in behaviorthey induce have been mentioned ear-lier. Variants of LT exist in which theorder of methods attempted by LT, andthe choice of units in describing expres-sions, depend upon appropriate hintsfrom the experimenter.

4. Effects from directional set occurin certain learning situations—as illus-trated, for example, by the classical ex-periments of Luchins. Although LT atthe present time has only a few learn-ing mechanisms, these will producestrong effects of directional set if prob-lems are presented to LT in appropriatesequences. For example, it requiredabout 45 minutes to prove Theorem2.48 in the first experiment becauseLT, provided with all the prior theo-rems, explored so many blind alleys.Given only the axioms and Theorem2.16, LT proved Theorem 2.48 in about15 minutes because it now considered aquite different set of possibilities.

The instances of set observable in thepresent program of LT are natural andunintended by-products of a programconstructed to solve problems in an effi-cient way. In fact, it is difficult to seehow we could have avoided such effects.In its simplest aspect, the problem-solv-ing process is a search for a solution ina very large space of possible solutions.The possible solutions must be examinedin some particular sequence, and if theyare, then certain possible solutions willbe examined before others. The par-


ticular rule that induces the order ofsearch induces thereby a definite set inthe ordinary psychological meaning ofthat term.

Preparatory set also arises from theneed for processing efficiency. If cer-tain information is needed each time apossible solution or group of solutionsis to be examined, it may be useful tocompute this information, once and forall, at the beginning of the problem-solving process, and to store it insteadof recomputing it each time.

The examples cited show that set canarise in almost every aspect of theproblem-solving process. It can governthe sequence in which alternatives areexamined (the "method" set), it can se-lect the concepts that are used in classi-fying perceptions (the "viewing" set),and it can consist in preparatory proc-esses (the description of axioms).

None of the examples of set in LT re-late to the way in which information isstored in memory. However, one wouldcertainly expect such set to exist, andcertain psychological phenomena bearthis out—the set in association experi-ments, and so-called "incubation" proc-esses. LT as it now stands is inade-quate in this respect.

Insight

In the psychological literature, "in-sight" has two principal connotations:(a) "suddenness" of discovery, and (b)grasp of the "structure" of the problem,as evidenced by absence of trial anderror. It has often been pointed outthat there is no necessary connectionbetween the absence of overt trial-and-error behavior and grasp of the prob-lem structure, for trial and en or maybe perceptual or ideational, and no ob-vious cues may be present in behaviorto show that it is going on.

In LT an observer's assessment ofhow much trial and error there is willdepend on how much of the record of

its problem-solving processes the com-puter prints out. Moreover, the amountof trial and error going on "inside"varies within very wide limits, depend-ing on small changes in the program.

The performance of LT throws somelight on the classical debate betweenproponents of trial-and-error learningand proponents of "insight," and showsthat this controversy, as it is usuallyphrased, rests on ambiguity and con-fusion. LT searches for solutions tothe problems that are presented it.This search must be carried out in somesequence, and LT's success in actuallyfinding solutions for rather difficultproblems rests on the fact that the se-quences it uses are not chosen casuallybut do, in fact, depend on problem"structure."

To keep matters simple, let us con-sider just one of the methods LT uses—proof by substitution. The numberof valid proofs (of some theorem) thatthe machine can construct by substitu-tion of new expressions for the variablesin the axioms is limited only by its pa-tience in generating expressions. Sup-pose now that LT is presented with aproblem expression to be proved by sub-stitution. The crudest trial-and-errorprocedure we can imagine is for themachine to generate substitutions in apredetermined sequence that is inde-pendent of the expression to be proved,and to compare each of the resultingexpressions with the problem expres-sion, stopping when a pair are identical(cf. 7).

Suppose, now, that the generator ofsubstitutions is constructed so that it isnot independent of the problem expres-sion—so that it tries substitutions indifferent sequences depending on thenature of the latter. Then, if the de-pendence is an appropriate one, theamount of search required on the aver-age can be reduced. A simple strategyof this sort would be to try in the


axioms only substitutions involving vari-ables that actually appear in the prob-lem expression.

The actual generator employed by LTis more efficient (and hence more "in-sightful" by the usual criteria) thanthis. In fact, it works backward fromthe problem expression, and takes intoaccount necessary conditions that a sub-stitution must satisfy if it is to work.For example, suppose we are substitut-ing in the axiom "p implies (q or p),"and are seeking to prove "r implies (ror r)." Working backward, it is clearthat if the latter expression can be ob-tained from the former by substitutionat all, then the variable that must besubstituted for p is r. This can be seenby examining the first variable in eachexpression, without considering the restof the expression at all (cf. 7).

Trial and error is reduced to stillsmaller proportions by the method forsearching the list of theorems. Onlythose theorems are extracted from thelist for attempted substitution whichare "similar" in a defined sense to theproblem expression. This means, inpractice, that substitution is attemptedin only about ten per cent of the theo-rems. Thus a trial-and-error search ofthe theorem list to find theorems simi-lar to the problem expression is substi-tuted for a trial-and-error series of at-tempted substitutions in each of thetheorems.

In these examples, the concept ofproceeding in a "meaningful" fashionis entirely clear and explicit. Trial-and-error attempts take place in some"space" of possible solutions. To ap-proach a problem "meaningfully" is tohave a strategy that either permits thesearch to be limited to a smaller sub-space, or generates elements of the spacein an order that makes probable thediscovery of one of the solutions earlyin the process.

We have already listed some of themost important elements in the pro-gram of LT for reducing search totolerable proportions. These are: (a)the description programs to select theo-rems that are "likely" candidates forsubstitution attempts; (b) the processof working backwards, which uses in-formation about the goal to rule outlarge numbers of attempts withoutactually trying them. In addition tothese, the executive routine may selectthe sequence of subproblems to beworked on in an order that takes up"simple" subproblems first.

Concepts

Most of the psychological research onconcepts has focused on the processesof their formation. The current ver-sion of LT is mainly a performance pro-gram, and hence shows no concept for-mation. There is in the program, how-ever, a clearcut example of the use ofconcepts in problem solving. This isthe routine for describing theorems andsearching for theorems "similar" to theproblem expression or some part of itin order to attempt substitutions, de-tachments, or chainings. All theoremshaving the same description exemplifya common concept. We have, for ex-ample, the concept of an expressionthat has a single variable, one argumentplace on its left side, and two argumentplaces on its right side: "p implies (por p)" is an expression exemplifyingthis concept; so is "q implies (q im-plies q)."

The basis for these concepts is purelypragmatic. Two expressions having thesame description "look alike" in someundefined sense; hence, if we are seek-ing to prove one of them as a theorem,while the other is an axiom or theoremalready proved, the latter is likely con-struction material for the proof of theformer.


Hierarchies of Processes

Another characteristic of the behav-ior o'f LT that resembles human prob-lem-solving behavior is the hierarchicalstructure of its processes. Two kindsof hierarchies exist, and these will bedescribed in the next two paragraphs.

In solving a problem, LT breaks itdown into component problems. Firstof all, it makes three successive at-tempts: a proof by substitution, a proofby detachment, or a proof by chaining.In attempting to prove a theorem byany of these methods, it divides its taskinto two parts: first, finding likely rawmaterials in the form of axioms or theo-rems previously proved; second, usingthese materials in matching. To findtheorems similar to the problem expres-sion, the first step is to compute a de-scription of the problem expression; thesecond step is to search the list of theo-rems for expressions with the same de-scription. The description-computingprogram divides, in turn, into a pro-gram for computing the number of lev-els in the expression, a program forcomputing the number of distinct vari-ables, and a program for computing thenumber of argument places.

LT has a second kind of hierarchyin the generation of new expressions tobe proved. Both the detachment andchaining methods do not give proofs di-rectly but, instead, provide new alter-native expressions to prove. LT keepsa list of these subproblems, and, sincethey are of the same type as the origi-nal problem, it can apply all its prob-lem-solving methods to them. Thesemethods, of course, yield yet other sub-problems, and in this way a large net-work of problems is developed duringthe course of proving a given logic ex-pression. The importance of this typeof hierarchy is that it is not fixedin advance, but grows in response tothe problem-solving process itself, and

shows some of the flexibility and trans-ferability that seem to characterize hu-man higher mental processes.

The problem-subproblem hierarchy inLT's program is quite comparable withthe hierarchies that have been discov-ered by students of human problem-solving processes, and particularly byde Groot in his detailed studies of thethought methods of chess players (2,pp. 78-83, 105-111). Our earlier dis-cussion of insight shows how the pro-gram structure permits an efficient com-bination of trial-and-error search withsystematic use of experience and cuesin the total problem-solving process.

SUMMARY OF THE EVIDENCE

We have now reviewed the principalevidence that LT solves problems in amanner closely resembling that exhibitedby humans in dealing with the sameproblems. First, and perhaps most im-portant, it is in fact capable of findingproofs for theorems—hence incorporatesa system of processes that is sufficientfor a problem-solving mechanism. Sec-ond, its ability to solve a particularproblem depends on the sequence inwhich problems are presented to it inmuch the same way that a human sub-ject's behavior depends on this sequence.Third, its behavior exhibits both pre-paratory and directional set. Fourth,it exhibits insight both in the sense ofvicarious trial and error leading to "sud-den" problem solution, and in the senseof employing heuristics to keep the to-tal amount of trial and error within rea-sonable bounds. Fifth, it employs sim-ple concepts to classify the expressionswith which it deals. Sixth, its programexhibits a complex organized hierarchyof problems and subproblems.

COMPARISON WITH OTHER THEORIESWe have proposed a theory of the

higher mental processes, and have shownhow LT, which is a particular exemplar


of the theory, provides an explanationfor the processes used by humans tosolve problems in symbolic logic. Whatis the relation of this explanation toothers that have been advanced?

Associationism

The broad class of theories usuallylabelled "associationist" share a gener-ally behaviorist viewpoint and a com-mitment to reducing mental functionsto elementary, mechanistic neural events.We agree with the associationists thatthe higher mental processes can be per-formed by mechanisms—indeed, we haveexhibited a specific set of mechanismscapable of performing some of them.

We have avoided, however, specify-ing these mechanisms in neurologicalor pseudo-neurological terms. Problemsolving—at the information-processinglevel at which we have described it—has nothing specifically "neural" aboutit, but can be performed by a wide classof mechanisms, including both humanbrains and digital computers. We donot believe that this functional equiva-lence between brains and computers im-plies any structural equivalence at amore minute anatomical level (e.g.,equivalence of neurons with circuits).Discovering what neural mechanisms re-alize these information-processing func-tions in the human brain is a task foranother level of theory construction.Our theory is a theory of the informa-tion processes involved in problem solv-ing, and not a theory of neural or elec-tronic mechanisms for information proc-essing.

The picture of the central nervoussystem to which our theory leads is apicture of a more complex and activesystem than that contemplated by mostassociationists. The notions of "trace,""fixation," "excitation," and "inhibi-tion" suggest a relatively passive elec-trochemical system (or, alternatively, apassive "switchboard"), acted upon by

stimuli, altered by that action, and sub-sequently behaving in a modified man-ner when later stimuli impinge on it.

In contrast, we postulate an informa-tion-processing system with large stor-age capacity that holds, among otherthings, complex strategies (programs)that may be evoked by stimuli. Thestimulus determines what strategy orstrategies will be evoked; the contentof these strategies is already largely de-termined by the previous experience ofthe system. The ability of the systemto respond in complex and highly se-lective ways to relatively simple stimuliis a consequence of this storage of pro-grams and this "active" response tostimuli. The phenomena of set and in-sight that we have already describedand the hierarchical structure of the re-sponse system are all consequences ofthis "active" organization of the cen-tral processes.

The historical preference of behavior-ists for a theory of the brain that pic-tured it as a passive photographic plateor switchboard, rather than as an ac-tive computer, is no doubt connectedwith the struggle against vitalism. Theinvention of the digital computer hasacquainted the world with a device—obviously a mechanism—whose responseto stimuli is clearly more complex and"active" than the response of more tra-ditional switching networks. It has pro-vided us with operational and unobjec-tionable interpretations of terms like"purpose," "set," and "insight." Thereal importance of the digital computerfor the theory of higher mental proc-esses lies not merely in allowing us torealize such processes "in the metal"and outside the brain, but in providingus with a much profounder idea thanwe have hitherto had of the character-istics a mechanism must possess if it isto carry out complex information-proc-essing tasks.


Gestalt Theories

The theory we have presented resem-bles the associationist theories largely inits acceptance of the premise of mecha-nism, and in few other respects. It re-sembles much more closely some of theGestalt theories of problem solving, andperhaps most closely the theories of "di-rected thinking" of Selz and de Groot.A brief overview of Selz's conceptionsof problem solving, as expounded byde Groot, will make its relation to ourtheory clear.

1. Selz and his followers describeproblem solving in terms of processesor "operations" (2, p. 42). These areclearly the counterparts of the basicprocesses in terms of which LT is speci-fied.

2. These operations are organized ina strategy, in which the outcome of eachstep determines the next (2, p. 44).The strategy is the counterpart of theprogram of LT.

3. A problem takes the form of a"schematic anticipation." That is, it isposed in some such form as: Find an Xthat stands in the specified relation Rto the given element E (2, pp. 44-46).The counterpart of this in LT is theproblem: Find a sequence of sentences(X) that stands in the relation of proof(R) to the given problem expression(E). Similarly, the subproblems posedby LT can be described in terms ofschematic anticipations: for example,"Find an expression that is 'similar' tothe expression to be proved." Manyother examples can be supplied of"schematic anticipations" in LT.

4. The method that is applied to-ward solving the problem is fully speci-fied by the schematic anticipation. Thecounterpart in LT is that, upon receiptof the problem, the executive programfor solving logic problems specifies thenext processing step. Similarly, whena subproblem is posed—like "prove the

theorem by substitution"—the responseto this subproblem is the initiation ofa corresponding program (here, themethod of substitution).

5. Problem solving is said to involve(a) finding means of solution, and (b)applying them (2, pp. 47-53). A coun-terpart in LT is the division between thesimilarity routines, which find "likely"materials for a proof, and the matchingroutines, which try to use these mate-rials. In applying means, there areneeded both ordering processes (to as-sign priorities when more than onemethod is available) and control proc-esses (to evaluate the application) (2,p. SO).

6. Long sequences of solution meth-ods are coupled together. This couplingmay be cumulative (the following stepbuilds on the result of the preceding)or subsidiary (the previous step wasunsuccessful, and a new attempt is nowmade) (2, p. 51). In LT the former isillustrated by a successful similaritycomparison followed by an attempt atmatching; the latter by the failure ofthe method of substitution, which isthen followed by an attempt at detach-ment.

7. In cumulative coupling, we can dis-tinguish complementary methods fromsubordinated methods (2, p. 52). Theformer are illustrated by successive sub-stitutions and replacements in succes-sive elements of a pair of logic expres-sions. The latter are illustrated by therole of matching as a subordinate proc-ess in the detachment method.

We could continue this list a gooddeal further. Our purpose is not tosuggest that the theory of LT can orshould be translated into the languageof "directed thinking." On the con-trary, the specification of the programfor LT clarifies to a considerable ex-tent notions whose meanings are onlyvague in the earlier literature. Whatthe list illustrates is that the processes


that we observe in LT are basically thesame as the processes that have beenobserved in human problem solving inother contexts.

PERFORMANCE AND LEARNING

LT is primarily a performance ma-chine. That is to say, it solves prob-lems rather than learning how to solveproblems. However, although LT doesnot learn in all the ways that a humanproblem solver learns, there are a num-ber of important learning processes inthe program of LT. These serve toillustrate some, but not all, of the formsof human learning.

Learning in LT

By learning, we mean any more orless lasting change in the response ofthe system to successive presentationsof the same stimulus. By this defini-tion—which is the customary one—LTdoes learn.

1. When LT has proved a theorem,it stores this theorem in its memory.Henceforth, the theorem is available asmaterial for the proof of subsequenttheorems. Therefore, whether LT isable to prove a particular theorem de-pends, in general, on what theorems ithas previously been asked to prove.

2. LT remembers, during the courseof its attempt to prove a theorem, whatsubproblems it has already tried tosolve. If the same subproblem is ob-tained twice in the course of the at-tempt at a proof, LT will rememberand will not try to solve it a secondtime if it has failed a first.

3. In one variant, LT rememberswhat theorems have proved useful inthe past in conjunction with particularmethods and tries these theorems firstwhen applying the method in question.Hence, although its total repertory ofmethods remains constant, it learns toapply particular methods in particularways.

These are types of learning thatwould certainly be found also in humanproblem solvers. There are other kindsof human learning that are not yet rep-resented in LT. We have already men-tioned one—acquiring new methods forattacking problems. Another is modi-fying the descriptions used in searchesfor similar theorems, to increase the effi-ciency of those searches. The latterlearning process may also be regardedas a process for concept formation. Wehave under way a number of activitiesdirected toward incorporating new formsof learning into LT, but we will post-pone a more detailed discussion of theseuntil we can report concrete results.

What is Learned

The several kinds of learning nowfound in LT begin to cast light onthe pedagogical problems of "what islearned?" including the problems oftransfer of training. For example, ifLT simply stored proofs of theorems asit found these, it would be able to provea theorem a second time very rapidly,but its learning would not transfer atall to new theorems. The storage oftheorems has much broader transfervalue than the storage of proojs, since,as already noted, the proved theoremsmay be used as stepping stones to theproofs of new theorems. There is nomystery here in the fact that the trans-ferability of what is learned is depend-ent in a very sensitive way upon theform in which it is learned and remem-bered. We hope to draw out the impli-cations, psychological and pedagogical,of this finding in our subsequent re-search on learning.

CONCLUSION

We should like, in conclusion, only todraw attention to the broader impli-cations of this approach to the studyof information-processing systems. Theheart of the approach is describing the


behavior of a system by a well speci-fied program, defined in terms of ele-mentary information processes. In thisapproach, a specific program plays therole that is played in classical systemsof applied mathematics by a specificsystem of differential equations.

Once the program has been specified,we proceed exactly as we do with tra-ditional mathematical systems. We at-tempt to deduce general properties ofthe system from the program (the equa-tions) ; we compare the behavior pre-dicted from the program (from theequations) with actual behavior ob-served in experimental or field settings;we modify the program (the equations)when modification is required to fit thefacts.

The promise of this approach is sev-eral-fold. First, the digital computerprovides us with a device capable ofrealizing programs, and hence, of actu-ally determining what behavior is im-plied by a program under various en-vironmental conditions. Second, a pro-gram is a very concrete specification ofthe processes, and permits us to seewhether the processes we postulate arerealizable, and whether they are suffi-cient to produce the phenomena. Thevaguenesses that have plagued the the-ory of higher mental processes and otherparts of psychology disappear when thephenomena are described as programs.

In the present paper we have illus-trated this approach by beginning theconstruction of a thoroughly operationaltheory of human problem solving. Thereis every reason to believe that it will

prove equally fruitful in application tothe theories of learning, of perception,and of concept formation.

REFERENCES

1. BRUNER, J. S., GOODNOW, J., & AUSTIN, G.A study of thinking. New York: Wiley,19S6.

2. DE GROOT, A. Het Denken van den Schaker.Amsterdam: Noord-Hollandsche Uit-gevers Maatschappij, 1946.

3. HILBERT, D., & ACKERMANN, W. Princi-ples of mathematical logic. New York:Chelsea, 1950.

4. JOHNSON, D. M. The psychology ofthought and judgment. New York:Harper, 19SS.

5. MOORE, O. K., & ANDERSON, S. B. Searchbehavior in individual and group prob-lem solving. Amer. social. Rev., 195S,19, 702-714.

6. NEWELL, A., & SHAW, J. C. Program-ming the logic theory machine. Pro-ceedings Western Joint Computer Con-ference (Institute of Radio Engineers),1957, 230-240.

7. NEWELL, A., SHAW, J. C., & SIMON, H. A.Empirical explorations with the logictheory machine. Proceedings WesternJoint Computer Conference (Instituteof Radio Engineers), 1957, 218-230.

8. NEWELL, A., & SIMON, H. A. The logictheory machine: A complex informationprocessing system. Transactions on in-formation theory (Institute of RadioEngineers), 1956, Vol. IT-2, No. 3,61-79.

9. SIMON, H. A., & NEWELL, A. Models,their uses and limitations. In L. D.White (Ed.), The state of the socialsciences. Chicago: Univer. ChicagoPress, 1956. Pp. 66-83.

10. WHITEHEAD, A. N., & RTJSSELL, B. Prin-cipia matkematica. Vol. I. (2nd ed.)Cambridge: Cambridge Univer. Press,1925.

(Received July 10, 1957)

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Elements of a theory of human problem solving