A Chaotic Neural Network Model of Insightful Problem Solving and the Generation Process of Constraints

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1 A Chaotc Neural Networ Model of Insghtful Problem Solvng and the Generaton Process of Constrants Yuchro Wajma Kega Abe Masanor Naagawa Graduate School of Decson Scence Technology, Toyo Insttute of Technology , Ooayama, Meguro-Ku, Toyo, , Japan Abstract The soluton of an nsghtful problem needs a drastc change from the "mpasse" to the "nsght" stage. It s assumed that n ths type of a problem, solvers encounter the mpasse stage because of specal "constrants" le common sense. Abe, Wajma, and Naagawa (23) proposed a model of nsght problem solvng usng a chaotc neural networ. The model successfully smulates an nsght problem. Based on ths system, we developed a new model to explan the generaton process of new constrants. We hypotheszed that once people have solved a problem usng nsght, the experence of the nsght generates new constrants. In order to verfy the above hypothess, we conducted a psychologcal experment and executed a computatonal smulaton of the model. In the experment, partcpants were nstructed to solve the two pctoral puzzles, one was an nsght problem and the other was a non-nsght problem. The expermental results showed that the soluton of the nsght problem generated a new constrant, whle nhbtng the soluton of non-nsght problem. We constructed a new model that represents a renforcement state after solvng the nsghtful problem and several smulatons were executed. The result of model's smulaton showed a close smlarty wth the expermental result. The model successfully smulated the process of generaton of new constrants. Keywords: nsght; constrant; generaton process; neural networ; modelng. Introducton Insght s the process by whch a problem s suddenly solved after an "mpasse," whch s the perod when the solver s unable to solve the problem (Wallas, 1926). The process of nsghtful problem solvng s the change from the state of "mpasse" to the state of "nsght," when problem s solved. In prevous studes, t s generally assumed that the mpasse stage s due to constrants such as common sense (Knoblch, Ohlsson, Hader & Rhenus, 1999). Some constrants can be effectve n gudng how people tacle everyday problems, but for problems that requre some new nsght, such constrants can obstruct the solver from seeng the soluton to the problem. It has been suggested that nsght s the product of breang away from such constrants by dscoverng new effectve drectons durng prevous faled attempts at solvng the problem. Abe, Wajma, and Naagawa (23) proposed a model for nsght problem solvng usng chaotc neural networ. Ths model conssts of two components, namely, a "constrant component," and an "avodance component." In order to represent these components, a system of smultaneous dfferental equatons s proposed, wth each varable denotng a sngle node of a neural networ. In the model, the "constrant component" s represented by controllng the ease wth whch the nodes can actvate, whle the "avodance component" s represented by the term causng the system to move n the drecton n whch the value of an evaluaton functon becomes the largest. Ths movement corresponds to the avodance reacton of humans that s a result of faled trals. The model successfully smulates an nsght problem. However, n the prevous research, suffcent consderaton was not gven to the generaton process of constrants. Furthermore, the prevous research dd not refer to the possblty of the nsght experence generatng a new constrant due to renforcement, once people had experenced solvng a problem usng nsght. We report here the development of a new model based on the nsghtful problem solvng model to represent the generaton of new constrants by hypotheszng that once people solved a problem usng nsght, the experence of the nsght generates new constrants. In order to verfy the above hypothess, we conducted a psychologcal experment and a computatonal smulaton of the present model. Frst, we set up an experment n order to show that new constrants are generated by nsght. In ths experment, partcpants were nstructed to solve a non-nsght problem after solvng an nsght problem to examne how the experence of nsght subsequently changed the performance. Second, we constructed a new model, by addng a mechansm to the prevous model; a new constrant was generated by renforcement of nsght experence, whle computatonal smulatons were executed accordng to the varaton of renforcement values. Hypothess We explan our hypothess of a new constrant generatng process by usng the "T-puzzle" and the "Arrow-puzzle" solvng process. The T-puzzle s an example of an nsght problem (Fgure 1, 2). In the T-puzzle, the solver constructs a "T" usng peces of the followng four shapes: a trangle, a small trapezod, a large-trapezod, and a pentagon wth a notch. 1623

2 Fgure 1: Peces comprsng the puzzle Fgure 2: T-puzzle Fgure 4: Hypothess Fgure 3: Arrow-puzzle Ths puzzle has one constrant: "to fll the notch of pentagon." When partcpants solve ths puzzle, they often fll the notch of pentagon (Suzu & Hra, 1997). In order to solve the T-puzzle, partcpants need the nsght to escape from ths constrant. By usng same four peces used n the T-puzzle, an "Arrow" s constructed (Fgure 3). We refer to ths puzzle as the Arrow-puzzle. The Arrow-puzzle has the same constrant as that of the T-puzzle. In order to solve the Arrow-puzzle, partcpants need to fll the notch. The constrant ads solvng the Arrow-puzzle. Thus, the partcpants do not need the nsght acqured whle solvng the T-puzzle to solve the Arrow-puzzle. In order to examne the effect of the prevous nsght on the soluton of the next tas, we nstructed the partcpants to solve the Arrow-puzzle (non-nsght problem) after they had solved the T-puzzle usng nsght. In ths case, we assumed the solvng process as follows (Fgure 4): Frst, due to the constrant,.e., "fll the notch of the pentagon," the T-puzzle was not solved. Insght was acqured, and partcpants dscovered that the notch need not be flled. Renforcement was acqured by solvng the T-puzzle usng nsght, and a new constrant,.e., "do not fll," was generated. Due to the new constrant,.e., "do not fll," t became easy to select "not fll the notch of pentagon" when partcpants solved the Arrow-puzzle after solved the T- puzzle. It s consdered that ths change from fllng the notch to not fllng the notch s one process that generates constrants. Therefore, we focused on the change of frequency of the fllng the notch. Experment In ths experment, we examne how the experence of nsght subsequently changes the performance. Our hypotheses n ths experment are as follows: When partcpants experenced the nsght by escapng from the ntal constrant, a new constrant was generated. Under ths condton, the partcpant's performances whle solvng the next tas changed due to the new constrant. If partcpants dd not experence nsght, the partcpant's performances dd not change. Method Partcpants. The partcpants comprsed 3 graduate and undergraduate students who had never solved the T-puzzle or the Arrow-puzzle. Tass. Tranng tas: Usng 3 peces to mae any shapes other than the "T" or the "Arrow" wthn 3 mnutes. Completon tas: Completng the goal shapes ("T" or "Arrow") by referrng to the goal mage slhouette. Procedure. Partcpants were assgned to two groups ("T- Arrow" or "Arrow-T"). Partcpants n the T-Arrow group were nstructed to frst solve the T-puzzle and then the Arrow-puzzle. On the other hand, partcpants n Arrow-T group were nstructed to frst solve the Arrow-puzzle and then the T-puzzle. Result In ths study, "one tral" s defned as one segment that spans the perod from connectng peces to separatng those peces (Suzu & Hra, 1997). The average tme to solve the Arrow-puzzle was 433.5s for Arrow-T group and 36.33s for T-Arrow group; there s no sgnfcant dfference between the two groups. The average trals count to reach the soluton was 25.4 for the Arrow-T group and for the T-Arrow group; there s no sgnfcant dfference between the two groups. The average of frequency of fllng the notch was.55 for the Arrow-T group and.28 for the T-Arrow group. The t-test revealed a 1624

3 sgnfcant dfference between the Arrow-T group and T- Arrow group, (t(22) = 3.97, p =.7 >.1). In order to examne the changes n the frequency of fllng the notch, all partcpants' trals for each group was dvded nto 4 sectons n the tme seres, and the average of the frequency of fllng the notch n each secton was plotted (Fgure 5). In the case of the Arrow-T group, the frequency of fllng the notch n the Arrow-puzzle was approxmately.55 n all sectons. In the case of the T-Arrow group, the frequency was approxmately.2 n t1 ~ t3, and t rapdly ncreased to.61 n t4. For the Arrow-puzzle, the frequency change of fllng the notch was examned by the analyss of varance, by consderng the "Wthn factor" secton and the "Between factor" group. The results ndcated sgnfcant effects of the secton, group, and ther nteracton. For the effect of group, F(1,27) = , p <.1; for the effect of secton, F(3,81) = 7.894, p <.1; and for the nteracton effect, F(3,81) = 3.224, p <.5. Ths result shows that the change of the frequency of fllng the notch was affected by the experence of solvng the T-puzzle. For solvng the T-puzzle, the average tme to obtan the soluton was s for T-Arrow group and s for Arrow-T group, and thus no sgnfcant dfference was observed between the groups. The average trals count to soluton was for the T-Arrow group and 81. for the Arrow-T group, and no sgnfcant dfference was observed between the groups. The average of frequency of fllng the notch was.5 for the T-Arrow group and.45 for the Arrow-T group; there was no sgnfcant dfference between both the groups. For the frequency of fllng the notch n the T-puzzle, we obtaned three common tendences between both groups: the frequency was approxmately.5 n t1, approxmately.4 n t4, and decreased n t1 ~ t4 (Fgure 6). For the T-puzzle, the frequency change of fllng the notch was examned by the analyss of varance by consderng the "Wthn factor" secton and the "Between factor" group. The results ndcated sgnfcant effects of secton only. For the effect of secton, F = (3,81) =.28, p <.5. Ths result shows that the change of the frequency of fllng the notch was not affected by experence of solvng the Arrow-puzzle. Dscusson of experment There s a sgnfcant dfference between the averages of the frequency of fllng the notch for the T-Arrow group and the Arrow-T group. The experment shows that the experence of nsght affects the performances whle solvng the next tas. For the Arrow-puzzle, n frst secton (t1), there s a consderable dfference of the frequences between the T- Arrow group and Arrow-T group. Ths result ndcates that the constrant, "to fll the notch" s changed to the constrant, "not fllng the notch," by the experence of nsght n solvng the T-puzzle. Addtonally, the frequency n the last secton (t4) ncreased greatly. Ths ncreased result meant Fgure 5: Result of experment for Arrow-puzzle Fgure 6: Result of experment for T-puzzle the escapng from the new generated constrant, "not fllng the notch" n solvng the T-puzzle. Conversely, wth regard to the T-puzzle, there s no sgnfcant dfference between the T-Arrow group and the Arrow-T group for the soluton tme, tral count, and the frequency of fllng the notch. Ths result shows that the experence of solvng the Arrow-puzzle does not affect the performances whle solvng the next tas. The result of the experment clarfes that a new constrant s generated by the experence of nsght. However, ths result does not show whether there s renforcement by the nsght. In the next secton, we construct a new model, whch has a mechansm for the generaton of a new constrant by renforcement, and we smulate the generaton process of constrants. Model In ths study, we construct a new model that ncludes a component for the generaton of new constrants (constrant generatng component), based on the prevous nsghtful problem solvng model (Abe, Wajma, & Naagawa, 23). Prevous Model The model of nsghtful problem solvng has two man components, the "Constrant component" and the "Avodance component." Along wth both the components, the model s represented by a system of smultaneous 1625

4 dfferental equatons, wth each varable denotng each node of the neural networ as follows: Ev && x + g( x&, t, x ) = β wju j + θ + γ j u 1 u = 1x 1 e 2 g ( x&, t, x ) = ( d sn( ω t) + d ) x& + d x& sgn ( x& ) Ev = n η log U d U : x sgn ( x) = 1 : x > where x, u and θ denote the nner state of node, the output of node, and the threshold of node, respectvely, w j means the weght from j to, U d represents the problem state of n th tral, and U represents the current state. "Constrant component" represents constrants faced by humans whle solvng problems. "Constrant component" was constructed to control the ease wth whch nodes are actvated. In ths model, the larger θ s, the easer t s for node to be actvated. β represents the effect of the "Constrant component." "Avodance component" represents the avodance for faled trals. Naagawa's psychologcal avodance behavor model has the term that moves the system n the drecton n whch the value of the Ev becomes the largest (Naagawa, 1978). Ths move for the term corresponds to the avodance of faled trals. To construct the "Avodance component," ths model was appled. γ represents the effect of the "Avodance component." After repeatng some faled trals, because of competton between "Constrant component" and "Avodance component", the model falls nto a nd of local mnmum where the model can not chose any adequate operator. In order to escape from ths local mnmum, we appled the chaotc steepest descent method (CSD) (Tan, 1991). CSD has the chaotc term that unstablzes the component when t reaches a local mnmum. Therefore, the component escapes from a local mnmum. The competton of the "Constrant component" and "Avodance component" s solved by usng CSD. T-Puzzle and Arrow-Puzzle Solvng Model We constructed the T-puzzle and Arrow-puzzle solvng model as an applcaton of the above mentoned "nsghtful problem solvng model." Ths model selects 2 of the 4 peces, and decdes 6 attrbutes' states: the drecton of the fgure to connect, the drecton of the fgure to be connected, fll or clear of the pentagon notch, types of connectons, lengths of connected lnes are equal or unequal, and f the same angle as the goal object exsts or not. And the nodes u1k u 15 represent attrbutes' states (Fgure 7). In order to choose only one length and only one drecton, w j s set as follows, O w = O θ s set, as shown n Fgure 7, so that 'fll of the pentagon notch' s easly chosen. Then, the model connects the selected two peces based on the decded 6 attrbutes' states. After connectng the peces three tmes, the shape of connected peces s checed to determne f t matches the "T" shape or the "Arrow" shape. The above operatons are repeated for the model untl the shape of connected peces matches the "T" or the "Arrow." Proposed Constrant Generatng Component In the new model, the constrant generatng component ncreases some θ values of attrbutes that were actvated by nsght. Consequently t becomes easy to choose those attrbutes n the followng problem solvng. For example, n the model, as the ntal state of the notch of pentagon, the "fll" node s easy to actvate, whle the "clear" node s dffcult to actvate. However, after solvng the T puzzle usng nsght, the state changes to mae the "clear" node easy to actvate. Ths process s represented by a mechansm that lowers new the threshold of the "clear" node ( θ 8 = θ 8 α ). Ths process represents the generaton of new constrants that maes the Arrow-puzzle dffcult to solve, because the soluton of the Arrow-puzzle needs the state n whch the "clear" node s dffcult to actvate (Fgure 8). Smulaton Smulaton 1 Frst, a smulaton that solves the Arrow-puzzle was executed for β =.1, γ = 1 and ω =. 7. In ths case, the soluton tme was t = , the tral count to soluton 1626

5 Fgure 7: T-puzzle and Arrow-puzzle solvng model Fgure 9: Result of smulaton 1 Fgure 8: Constrant generaton component was 33, and the average of frequency of fllng the notch was.51. The change n frequency for all sectons was neglgble (Fgure 9). Smulaton 2 Several smulatons that solved the Arrow-puzzle after solvng T-puzzle were executed, changng the α, where β, γ,and ω were mantaned. In the case of α =.1, soluton tme was t = 1813., tral count for the soluton was 52, and the average of frequency of fllng the notch was.33. The frequency was approxmately.2 n t1 ~ t3, and ncreased rapdly to.61 n t4. Smulaton 3 In the case of α =.2, the soluton tme was t = , the tral count for the soluton was 41, and the average of frequency of fllng the notch was.51. The frequency change was neglgble n all sectons (Fgure 1). Comparson between the experment result and the smulaton result When the Arrow-puzzle was solved wthout the experence of the solvng the T-puzzle, the experment results showed that the average trals count was 25.4 and the frequency of fllng the notch was.55. The result of the smulaton of solvng the Arrow-puzzle wthout the experence to solve the T-puzzle was almost dentcal to the experment, the average trals count was 33 and the frequency of fllng the notch was.51. Addtonally, the frequency of fllng the notch n both the experment and the smulaton showed neglgble change n all sectons. Snce the result of 1627

6 Fgure 1: Result of smulaton 2 and 3 smulaton 1 s smlar to that of the experment of the Arrow-puzzle n the Arrow-T group, our model of solvng the Arrow-puzzle represent human performance of solvng the Arrow-puzzle. When the Arrow-puzzle s solved after solvng the T- puzzle, the experment results showed that average trals count was 81. and the frequency of fllng the notch was.45. The results of smulaton 2 ( α =. 1) of the Arrowpuzzle beng solved after solvng T-puzzle showed that average trals count was 52 and the frequency of fllng the notch was.33. The experment result of the changes of frequency of fllng the notch was almost dentcal the smulaton 2's result. These frequences dd not change n secton t1 ~ t3 at.2, and ncreased rapdly to.61 n t4. Snce the result of smulaton 2 s smlar to that of the Arrow-puzzle solvng experment n the T-Arrow group, our model of solvng the Arrow-puzzle, where α =.1, represented human performance of solvng the Arrowpuzzle after solvng the T-puzzle.Ths result showng that the frequences n t1 ~ t3 were lower and ncreased rapdly n t4 was caused by a new constrant "Not fllng the notch" that was generated by nsght. In the case of smulaton 3 ( α =. 2 ), average trals count was 41 and the frequency of fllng the notch was.51. The change n frequency was neglgble at approxmately.5 n all sectons. Ths result of smulaton 3 showed a dfference from results of the experment n the T-Arrow group. Thus, the smulaton for α =. 2 cannot represent human performance to solve the Arrow-puzzle after solvng the T-puzzle. In the case of α =.2, the nodes of fllng the notch after operatng the constrant generaton component was easer to actvate than that of not fllng the notch. Ths means a new constrant was not generated. In the case of α =.1, t s easer to actvate the nodes of not fllng the notch after operatng the constrant generaton component than those of fllng the notch. In ths state of the nodes, the frequency of not fllng the notch ncreases. Ths process s dentcal that observed n the human renforcement process. These result show that there s a renforcement process generated by nsght. Concluson In ths study, the experment showed that a new constrant s generated by experence of nsght. A new constrant was generated when partcpants experenced nsght, whle t was not generated wthout the experence of nsght. We constructed a new model that has a mechansm where a new constrant s generated by renforcement when nsght s experenced. Smulaton results showed that there s a renforcement process generated by nsght. The results also ndcated that renforcement caused by nsght generates a new constrant. People usually experence a strong emoton (happness, surprse, etc.) when they have solved a problem usng nsght (Davdson, 1995). It s consdered that ths strong emoton s very mportant for renforcement by nsght. We propose the hypothess that the renforcement s a result of the reward of the above emotonal experences, and a new constrant s generated by the renforcement. In order to verfy the above hypothess, we are plannng a new experment that wll examne a new constrant that s not generated f a partcpant does not experence a strong emoton whle acqurng nsght. Acnowledgements The present research s supported by Toyo Insttute of Technology 21COE Program, "Framewor for Systematzaton and Applcaton of Large-scale Knowledge Resources." Reference Abe, K., Wajma, Y. & Naagawa, M. (23). The Chaotc Neural Networs Model of Insght Problem Solvng. Proceedngs of the 22nd Annual Meetng of the Japanese Cogntve Scence Socety, Davdson, J. E. (1995). The suddenness of nsght, n Sternberg, R. J. and Davdson, J. E. eds., The nature of nsght, Cambrdge, MA: MIT Press. Knoblch, G., Ohlsson, S., Hader, H. & Rhenus, D. (1999). Constrant relaxaton and chun decomposton n nsght problem solvng. Journal of Expermental Psychology : Learnng, Memory, and Cognton, 25(6), Naagawa, M. (1978). A mathematcal model of approach and avodance behavor n psychologcal feld. Japanese Psychologcal Reserch, 29(2), Suzu, H. & Hra, K. (1997). Constrants and ther relaxaton n the processes of nsght. ETL Techncal Report, TR Tan, J. (1991). Proposal of Chaotc Steepest Descent Method for Neural Networs and Analyss of Ther Dynamcs. The transactons of the Insttute of Electroncs and Communcaton Engneers of Japan, J74-A, 8, Wallas, G. (1926). The art of thought, Harcout, Brace, and World. 1628