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UNCORRECTED PROOF
1
2On the cognitive process of human problem solving
3Action editor: Ning Zhong
4Yingxu Wang
*
, Vincent Chiew
5International Center for Cognitive Informatics (ICfCI) and Theoretical and Empirical Software Engineering Research Centre,
6Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary,
72500 University Drive, NW, Calgary, Alberta, Canada T2N 1N4
8Received 25 February 2008; accepted 4 August 2008
9
10 Abstract
11 One of the fundamental human cognitive processes is problem solving. As a higher-layer cognitive process, problem solving interacts
12 with many other cognitive processes such as abstraction, searching, learning, decision making, inference, analysis, and synthesis on the
13 basis of internal knowledge representation by the object–attribute-relation (OAR) model. Problem solving is a cognitive process of the
14 brain that searches a solution for a given problem or finds a path to reach a given goal. When a problem object is identified, problem
15 solving can be perceived as a search process in the memory space for finding a relationship between a set of solution goals and a set of
16 alternative paths. This paper presents both a cognitive model and a mathematical model of the problem solving process. The cognitive
17 structures of the brain and the mechanisms of internal knowledge representation behind the cognitive process of problem solving are
18 explained. The cognitive process is formally described using real-time process algebra (RTPA) and concept algebra. This work is a part
19 of the cognitive computing project that designed to reveal and simulate the fundamental mechanisms and processes of the brain accord-
20 ing to Wang’s layered reference model of the brain (LRMB), which is expected to lead to the development of future generation meth-
21 odologies for cognitive computing and novel cognitive computers that are capable of think, learn, and perceive.
22 Ó2008 Published by Elsevier B.V.
23 Keywords: Cognitive informatics; Cognitive computing; Brain informatics; Computational intelligence; Reference model of the brain; Cognitive proce-
24 sses; Problem solving; Mathematical model; Concept algebra; RTPA
25
26 1. Introduction
27 The attempt to understanding the advance intelligent
28 ability of human beings for problem solving has intrigued
29 researchers from multiple disciplines, which can be traced
30 back to the Aristotle’s era (384–322BC). Problem solving
31 is identified as one of the basic life functions of the natural
32 intelligence of the brain (Polya, 1954; Wallas, 1926; Wang,
33 Wang, Patel, & Patel, 2006; Wilson & Clark, 1988). Most
34 decisions that an individual makes everyday are related
35 to certain problems needed to be solved no matter how
36trivial or critical they are. Problem solving as a process
37may be embodied in many forms, where research itself is
38essentially a typical problem solving paradigm (Beveridge,
391957).
40The history of studies on problem solving has a humble
41beginning (Mayer, 1992; Newell & Simon, 1972; Payne &
42Wenger, 1998; Polya, 1954; Robertson, 2001; Tuma & Reif,
431980; Wallas, 1926; Wang, 2008d; Wilson & Clark, 1988;
44Zadeh, 2008a). Traditional Gestalt psychology related
45human problem solving ability to learning and perception
46(Matlin, 1998; Ormrod, 1999; Wang, 2007e). Since early
471970s, psychologists acknowledged the benefits of using
48computer technologies to simulate the mechanisms of
49problem solving (Goldstein, 1978; Smith, 1991). Matlin
50(1998) expressed that psychology may not prove that a
1389-0417/$ - see front matter Ó2008 Published by Elsevier B.V.
doi:10.1016/j.cogsys.2008.08.003
*
Corresponding author. Present address: Department of Computer
Science, Stanford University, Stanford, CA 94305-9010, USA
Q2 .
E-mail addresses: yingxu@ucalgary.ca,yingxuw@stanford.edu
(Y. Wang), vincent.chiew@ieee.org (V. Chiew).
www.elsevier.com/locate/cogsys
Available online at www.sciencedirect.com
Cognitive Systems Research xxx (2008) xxx–xxx
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51 theory is correct, but may demonstrate so by using com-
52 puter simulations. In 1972, Newell and Simon developed
53 a computer program called the general problem solver
54 (GPS), which was quite successful in solving limited types
55 of carefully defined problems (Newell & Simon, 1972). In
56 his paper entitled ‘‘Developing a Computational Represen-
57 tation for Problem-Solving Skills,”Ira Goldstein men-
58 tioned that computer technology could be of use as
59 personal assistants in problem solving, and to provide a
60 cognitive programming and simulating environments, to
61 achieve a deeper understanding of the subject matter and
62 to explore learning strategies (Goldstein, 1978).
63 Cognitive informatics is the transdisciplinary study of
64 cognitive and information sciences that investigate into
65 the internal information processing mechanisms and pro-
66 cesses of the natural intelligence – human brains and minds
67 – and their engineering applications ( Wang, 2002b, 2003a,
68 2007b; Zadeh, 2008a; Zadeh, 2008b; Zhong, 2006). Cogni-
69 tive informatics provides a coherent framework of contem-
70 porary theories for explaining human cognitive processes,
71 such as problem solving, learning, decision making, and
72 consciousness. A layered reference model of the brain
73 (LRMB) is developed in Wang et al. (2006), which reveals
74 that the brain and human intelligent behaviors may be
75 explained by 39 cognitive processes at six layers known
76 as the sensation, memory, perception, action, meta-cogni-
77 tion, and higher cognition layers. It is recognized that, in
78 order to enable rigorous modeling and description of the
79 brain and its cognitive mechanisms, the formal structures
80 of internal knowledge representation and new forms of
81 denotational mathematical means are essential and neces-
82 sary. The former is implemented by the object–attribute-
83 relation (OAR) model (Wang, 2007c) on the basis of neural
84 informatics (Wang, 2007b). The latter is expressed by a col-
85 lection of denotational mathematics (Wang, 2007a, 2007d,
86 2008a), such as concept algebra (Wang, 2008b) and real-
87 time process algebra (RTPA) (Wang, 2002a, 2003b,
88 2008c), which is a category of expressive mathematical
89 structures for dealing complex mathematical entities shared
90 by many modern scientific and engineering disciplines.
91 In cognitive informatics, problem solving is identified as
92 a cognitive process of the brain at the higher cognitive layer
93 that searches a solution for a given problem or finds a path
94 to reach a given goal (Wang, 2007b). Problem solving is
95 one of the 39 fundamental cognitive processes modeled in
96 the LRMB model (Wang et al., 2006). It is recognized that
97 there is a need to seek an axiomatic and rigorous model of
98 the cognitive process of human problem solving in order to
99 develop a solid and coherent theoretical foundation for
100 integrating various theories, models, and practices of prob-
101 lem solving (Wang, 2007b).
102 This paper presents a formal model of human problem
103 solving and its cognitive process. It will proceed in Section
104 2with literature surveys on problem solving and related
105 work developed in psychology, cognitive science, and com-
106 putational intelligence. In Section 3, a set of cognitive,
107 mathematical, and process models of problem solving will
108be developed and elaborated. Section 4introduces useful
109paradigms of denotationnal mathematics in the forms of
110concept algebra and RTPA. Based on them, the process
111model of problem solving is formalized for machine simu-
112lations in computational intelligence. Applications of this
113work in brain simulation and cognitive informatics are dis-
114cussed toward the development of future generation cogni-
115tive computers.
1162. Cognitive psychology of problem solving
117This section surveys related studies on problem solving
118and its cognitive foundations in psychology and computa-
119tional intelligence. The approaches to and factors affecting
120problem solving are explored. The cognitive characteristics
121of problem solvers and the impact of their knowledge in
122problem solving are discussed. A set of empirical problem
123solving procedures proposed in the field of psychology is
124comparatively analyzed, after the presentation of the con-
125text framework of problem solving in form of the LRMB
126model.
1272.1. Approaches and factors in problem solving
128It is proposed that a problem consists of three compo-
129nents known as the givens,goals,andoperations (Ormrod,
1301999; Polya, 1954). The givens are information available as
131part of the problem. The goals are defined as the desired
132termination state of a solution to the problem. The opera-
133tions are potential actions that can be executed to achieve
134the goals of a solution. For any given problem there is
135an associated problem space (Bender, 1996; Wilson &
136Clark, 1988), which is all the possible goals and paths
137potentially related to the problem known by a problem sol-
138ver. A solution to a certain problem might not exist within
139the solver’s current solution space. This could be caused by
140many factors, such as that the problem could be ill-defined,
141expected goals are ambiguous, and/or no method (path) is
142available that links the give problem object to the goal(s).
143A similar perception on the problem space is presented
144by Tuma and Reif (1980), as well as Payne and Wenger
145(1998), which identified two elements in problem solving:
146(a) adescription of all possible states of the task and prob-
147lem solver (representation); and (b) alist of the ways of
148moving among those states (search). The first element sup-
149ports a problem solver to understand the problem by
150abstraction and identification. The second element enables
151the problem solver to search for a possible solution in
152memory.
153A various of approaches to problem solving have been
154studied and proposed in psychology, cognitive informatics,
155and computational intelligence (Matlin, 1998; Ormrod,
1561999; Rubinstein & Firstenberg, 1995; Wang et al., 2006;
157Wang & Ruhe, 2007) as follows, inter alia:
158Direct facts –finding a direct solution path based on
159known solutions.
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160 Heuristic –adopting rule of thumb or the most possible
161 solutions.
162 Analogy –reducing a new problem to an existing or
163 similar one for which solutions have already been
164 known.
165 Hill climbing –making any move that approaches closer
166 to the problem goal step by step.
167 Algorithmic deduction –applying a known and well
168 defined solution for a problem.
169 Exhaustive search –using a systematic search for all pos-
170 sible solutions.
171 Divide-and-conquer –solving a whole problem via
172 decomposing it into a set of subproblems.
173 Analysis and synthesis –reducing a given problem to a
174 known category and then finding particular solutions.
175
176 Adoptions of the above approaches to problem solving
177 may not guarantee a problem goal to be reached, espe-
178 cially when a solution is not within the problem solver’s
179 solution space. There are a number of factors that may
180 hinder the process of problem solving (Matlin, 1998;
181 Smith, 1991) such as: (a) mental set in which a fixed or
182 improper method is adopted for a new problem while eas-
183 ier solutions could have been utilized; (b) meta-cognition
184 in which a problem solving process may require the sup-
185 port of other metacognitive processes to achieve the solu-
186 tion goal; and (c) lack of knowledge in which either the
187 problem or the goal could not be well represented or
188 modeled, and no method or solution could be applied
189 to the problem.
190 2.2. Cognitive characteristics of problem solvers
191 The typical psychological traits that may be of use and
192 of benefit to a successful problem solver are as follows:
193 To correctly identify problem goals, to be persistent, to
194 adopt efficient strategies in search, and to be able to trace
195 back to a certain previous point in the solution process.
196 In cognitive psychology, what differentiates an expert
197 from a novice problem solver is studied. It is observed
198 that not everyone possesses the same ability for problem
199 solving. These differences may provide insight to explain
200 the nature of problem solving. The most significant traits
201 between experts and novices in problem solving are iden-
202 tified as follows (Payne & Wenger, 1998; Polya, 1954;
203 Smith, 1991): scope of knowledge on accumulated infor-
204 mation, problem solving schemas, skills, expertise, mem-
205 ory capacity, problem representation ability, abstraction,
206 and categorization abilities, analysis, and synthesis skills,
207 long-term concentration ability, motivation, efficiency,
208 and accuracy.
209 It is interesting to contrast and analyze the differences
210 between professionals and amateurs in software engineer-
211 ing problem solving. Professional software engineers are
212 persons with professional cognitive models and knowledge
213 on software engineering. They are trained with: (a) funda-
214 mental knowledge that governs software and software engi-
215neering practices; (b) basic principles and laws of software
216engineering; (c) proven algorithms; (d) problem domain
217knowledge; (e) problem solving experience; (f) knowledge
218about program developing tools/ environments; (g) solid
219programming skills in multiple programming languages;
220and (h) aglobal and insightful view on system develop-
221ment, including its required functionalities as well as excep-
222tion handling and fault-tolerance strategies. However,
223amateurish programmers are persons who know only one
224or a couple of programming languages but lack formal
225training. They may be characterized as follows: (a) ad
226hoc structure of programming knowledge; (b) limited pro-
227gramming experience and skills; (c) eager to try what is
228directly required before a system architecture is designed;
229and (d) tend to focus on details without a global and sys-
230tematic view.
231In programming, Richard Mayer identified four aspects
232of knowledge required for solving programming problems
233known as the syntactic, semantic, schematic, and strategic
234knowledge (Mayer, 1992). He reported that there exist sig-
235nificant differences between the knowledge structures of
236experts and novices in all the four categories.
237It is noteworthy that unlike a machine, human abilities
238for problem solving do change depending on ages and
239external influences (Payne & Wenger, 1998). A senior per-
240son may have a broader knowledge base leading to better
241performance in problem solving. However, elderliness
242may reduce the efficiency in problem solving physiologi-
243cally and psychologically. In addition, personal motiva-
244tion, attitude, and external influences such as social
245pressure and environment inconvenience may hamper
246human problem solving efficiency and effectiveness (Wilson
247& Clark, 1988).
2482.3. LRMB: the context framework for problem solving
249It is noteworthy that the cognitive process of problem
250solving is not a trivial and isolated mental process rather
251than a complex and dynamic one interacting with almost
252all other cognitive processes at the meta-cognition, higher
253cognition, and lower layers according to the LRMB model
254as shown in Fig. 1 (Wang et al., 2006).
255A variety of life functions and their cognitive processes
256have been identified in cognitive informatics, cognitive neu-
257ropsychology, cognitive science, and neurophilosophy. In
258order to formally and rigorously describe a comprehensive
259and coherent set of mental processes and their relation-
260ships, the hierarchical LRMB model is established, which
261encompasses 39 cognitive processes at seven layers known
262as the sensation, memory, perception, action, meta-cogni-
263tive, meta-inference, and higher cognitive layers from the
264bottom up.
265LRMB reveals that the hierarchical life functions of the
266brain can be divided into two categories: the subconscious
267and conscious life functions. The former encompasses the
268layers of sensation, memory, perception, and action (Lay-
269ers 1–4). The latter includes the layers of meta-cognitive,
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270 meta-inference, and higher cognitive functions (Layers 5–
271 7).
272 The subconscious layers of the brain are inherited, fixed,
273 and relatively mature when a person was born. Therefore,
274 the subconscious function layers are not directly controlled
275 by the conscious life function layers. This is why it is called
276 the unconscious life functions in some literature (Matlin,
277 1998; Payne & Wenger, 1998). However, the conscious lay-
278 ers of the brain are acquired, highly plastic, and can be con-
279 trolled intentionally based on willingness, goals, and
280 motivations.
281 LRMB provides a coherent framework to explain the
282 functional mechanisms and cognitive processes of the nat-
283 ural intelligence and human mind. According to LRMB,
284 the functioning of any higher-layer process is supported
285 by the lower layer processes from bottom up. For
286 instance, the problem solving process at the top layer
287 invokes lower layer processes’ support such as object
288 identification (Layer 5), abstraction (Layer 5), search
289 (Layer 5), memorization (Layer 5/2), and a number of
290 inference processes (Layer 6). It also interacts with pro-
291 cesses at the same layer such as comprehension, learning,
292 and decision making. Detailed discussions on the relation-
293 ships and interactions of the problem solving process with
294 other processes in the context of LRMB will be extended
295 in Section 4.3.
2962.4. Problem solving procedures
297Wallas (1926) and Polya (1954) proposed two sets of
298classical problem solving procedures. The former studied
299problem solving in a creativity context; while the latter pro-
300posed a generic empirical process of problem solving.
301Wallas’ creative problem solving procedure is known as
302follows (Wallas, 1926):
303(a) Preparation:defining the problem and gathering
304information relevant to its solution.
305(b) Incubation:thinking about the problem at a subcon-
306scious level while engaging in other activities.
307(c) Inspiration:having a sudden insight into the solution
308of the problem.
309(d) Verification:checking to be certain that the solution
310is correct.
311
312
Following Wallas’ work, an influential problem solving
313procedure was proposed by Polya as described below
314(Polya, 1954):
315(a) Understanding the problem:identifying the problem’s
316knowns (givens) and unknowns and, if appropriate,
317using suitable notation, such as mathematical symbols,
318to represent the problem.
319(b) Devising a plan:determining appropriate actions to
320take to solve the problem.
321(c) Carrying out the plan:executing the actions that have
322been determined to solve the problem and checking their
323effectiveness.
324(d) Looking backward:evaluating the overall effective-
325ness of the approach to the problem, with the intention
326of learning something about how similar problems may
327be solved on future occasions.
328
329
It is apparent that both Wallas and Polya derive their
330conceptual procedures for problem solving based on intro-
331spections and informal observations. However, mathemat-
332ical abstraction, symbolic reasoning, and formalism have
333not been introduced to rigorously model the cognitive pro-
334cess of problem solving in order to enable machine simula-
335tion of the human cognitive process. These important
336topics will be addressed in the following sections through-
337out this article.
3383. Cognitive informatics models of problem solving
339As reviewed in previous sections, the lack of suitable
340mathematical models and formal inference treatments has
341kept studies on problem solving at the empirical level based
342on observations and subjective interpretations. In this sec-
343tion, a rigorous study on problem solving as one of the fun-
344damental cognitive mechanisms of the brain is described.
345As a result, the mathematical model and process model
346of problem solving will be formally created on the basis
Conscious
life
functions
Subconscious
life
functions
Layer 7
Higher cognitive functi ons
Layer 5
Meta-cognitive functions
Level 3
Perception
Layer 4
Action
Level 2
Memory
Layer 1
Sensation
Layer 6
Meta-inference functions
Fig. 1. The layered reference model of the brain (LRMB).
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347 of empirical studies in cognitive psychology and computa-
348 tional intelligence.
349 3.1. The mathematical model of problem solving
350
351
352
353 Definition 1. A problem space, or solution space, His a
354 Cartesian product of a nonempty set of problem objects X,
355 a nonempty set of paths P,and a nonempty set of goals G,
356 i.e.:
H¼
^XPG ð1Þ
358358
359 where represents a Cartesian product.
360 Definition 2. Assuming the layout of a problem solving
361 process is a function f:X??Yon H,the problem q
362 is the domain of f,X, in general, and a specific instance
363 x,x2X, in particular, i.e.:
364
365
q¼
^ðXjf:X! !YÞ;q2X ð2Þ
367367
368 Eq. (2) denotes that, in problem solving, a problem qis
369 the fix point of a denotational function in general, and the
370 input of the function in particular. The former represents
371 the broad sense of the problem, and the latter is the narrow
372 sense of the problem.
373 Problem solving is a process that seeks the generic func-
374 tion for a layout of problem, and determines its domain
375 and codomain. Then, a solution in problem solving can
376 be perceived as a concrete instance of the given function
377 for the layout of the problem.
378 Definition 3. Problem solving is a cognitive process of the
379 brain that searches or infers a solution for a given problem
380 in the form of a set of paths to reach a set of expected
381 goals.
382 Definition 4. A goal G in problem solving is the terminal
383 result Yof satisfactory in the solution space of the problem
384 q, which deduces Xto Yby a sequence of inference in finite
385 steps, i.e.:
386
387
388
G¼
^ðYjX!!YÞ;G2G ð3Þ
390390
391 Definition 5. Apath P in problem solving on His a 3-tuple
392 with a nonempty finite set of problem inputs X, a nonempty
393 finite set of traces T, and a nonempty finite set of goals G,
394 i.e.:
395
396
397
P¼
^ðX;T;GÞ¼XTGð4Þ
399399
400 where the a trace t 2Tis an internal node or subpath,
401 t:X
t
?Y
t
, that maps an intermediate subproblem X
t
to a
402 subgoal Y
t
.
403 According to Definitions 1–5, there are two categories of
404 problems in problem solving: (a) the convergent problem
405 where the goal of problem solving is given, but the path
406 of problem solving is unknown; and (b) the divergent prob-
407 lem where the goal of problem solving is unknown and the
408 path of problem solving are either known or unknown.
409The combination of the above cases in problem solving
410can be summarized in Table 1, which identifies four types
411of problem solving, i.e., proof, instance, case study, and
412explorative/creative problem solving. A special case in
413Table 1 is that when both the goal and path are known,
414the case is only a solved instance for a given problem. In
415a related work (Wang, 2008d), the cognitive process of cre-
416ation is recognized as a novel or unexpected but useful
417solution to a given problem. Therefore, a creation may
418be perceived as a special novel solution where the problem,
419goal, and/or path are usually unknown. With this view, the
420study of the generic theory of creativity can be reduced to
421the theory of problem solving.
422Definition 6. Asolution s to a given problem qon His an
423instance of a set of selected relation or function, S, which is
424a subset of the solution paths in P, i.e.:
425
S¼^ðX;T;GÞP;X;T;G6¼;
s2S
ð5Þ427427
428
According to Definition 6, in case #X=0, #G=0, or
429#T= 0, there is no solution for the given problem. For a
430convergent problem, i.e. #G1, the number of possible
431solutions is #X #T.This leads to the following theorem.
432Theorem 1. The polymorphic solution principle for problem
433solving states that the size of the solution space (SS, SS v
434H), N
SS
, of a given problem qis a product of the numbers of
435problem inputs N
x
, traces N
t
, and goals N
g
, i.e.:
436
NSS ¼
^NxNtNg¼#X#T#Gð6Þ438438
439
Theorem 1 provides a fundamental mathematical model
440for problem solving, which reveals that the factors deter-
441mining a solution to a given problem are the Cartesian
442space of all possible goals G, problem inputs X, and solu-
443tion paths Pof the problem.
444The polymorphic characteristic of the solution space
445contributes greatly to the complexity of problem solving.
446It is noteworthy that the path p(x,t,g)2Pin Definition
4475can be a simple or a complex function. A complex func-
448tion that mapping a given problem into a solution goal
449may be very complicated depending on the nature of the
450problem.
451Corollary 1. The divide-and-conquer principle for problem
452solving states that the efficiency gain or complexity reduction
Table 1
Classification of problems and problem solving methods
Type of
problem
Goal Path Type of solution
Convergent Known Unknown Proof (specific)
Known Known Instance (specific)
Divergent Unknown Known Case study (open-ended)
Unknown Unknown Explorative/creative (open-
ended)
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453 for a given problem qis proportional to how many
454 subproblems, n, that qis partitioned, and/or how many path
455 segments, m, that the solution path P is partitioned, i.e.:
gq¼1n
nðn1Þ¼11
n1
gP¼1Pm
i¼1NPi
Qm
i¼1NPi
1mNP
ðNPÞm¼1m
ðNPÞm1
8
>
<
>
:
ð7Þ
457457
458 where NPis the average size or alternatives of all subpaths in
459 SS.
460 Corollary 1 provides a mathematical model that for-
461 mally explains the long-existing empirical heuristic princi-
462 ple of divide-and-conquer in problem solving for both
463 human and computing systems. According to Corollary
464 1, it can be expected that the greater the partitions, the
465 higher the efficiency gains in both ways. For instance, given
466 n¼3;m¼3 and NP¼5;the efficiency gains of qand P
467 partitions are g
q
=50% and g
P
= 88%, respectively. How-
468 ever, when n¼5;m¼5 and NP¼5;the efficiency gains
469 of qand Ppartitions are g
q
= 75% and g
P
= 99.2%,
470 respectively.
471 3.2. The cognitive process of problem solving
472 On the basis of the mathematical models of problem
473 solving as established in Section 3.1, the cognitive process
474 of problem solving can be derived logically in this subsec-
475 tion. In the problem solving process, the representation
476 of the problem is crucial. Problem representation includes
477 a description of the given situation, predefined operators
478 for changing the situation, and assessment criteria to deter-
479 mine whether the goal has been achieved. In most problems
480 of interest, the solution space can be too large to be
481 searched exhaustively, because it was noted in (Wang,
482 Liu, & Wang, 2003) that the maximum human memory
483 capacity is up to 10
8.432
bits in the brain. Typical
484 approaches to reduce search complexity in problem solv-
485 ing, according to Corollary 1, are to adopt divide-and-con-
486 quer and heuristic principles in order to simplify the
487 problem into subproblems and/or to reduce the paths into
488 a set of short ones.
489 On the basis of Definitions 1–6, a generic problem solv-
490 ing process is modeled as shown in Fig. 2, where the cogni-
491 tive process of problem solving can be divided into the
492 following five steps:
493 (a) To define the problem: This step describes the prob-
494 lem and its input layout Xby identifying its object O
X
495 and attributes A
X
, in a sub-OAR model (Wang, 2007c).
496 (b) To search the solution goals and paths: In this step,
497 the brain performs a parallel search for possible goals
498 Gand paths Tfor the solution. External memory and
499 resources may be searched if there is no available or suf-
500 ficient G or Tin the internal knowledge of the problem
501 solver.
502(c) To generate solutions according to Eq. (6): This step
503forms a set of possible solutions by a Cartesian product
504S=XTG.
505(d) To select suitable solutions: This step evaluates each
506possible solutions s2S=XTGas obtained in
507Step (c). Recursive searching actions may be executed
508if the obtained solution(s) could not satisfy the problem
509solver’s expectation.
510(e) To represent the problem solving result: This step
511incorporates and memorizes the solution(s), s2S,as
512the result of problem solving into the entire OAR model
513in the long-term memory of the problem solver. Before
514memorization, the solution Sis represented as a part
515of the relations, R, in the sub-OAR model.
516
517
It is noteworthy in Fig. 2 that a number of lower layer
518cognitive processes, as represented by double-ended boxes,
519are adopted to carry out the problem solving process.
520Based on the discussions in Section 2.3, the relationship
521among the problem solving process and other cognitive
522processes as modeled in LRMB will be discussed in Section
5234.3.
Search (Alternative
goals - G)
Representation
(OAR)
Begin
Identify
(
Object - O
X
)
Identify
(
Attributes - A
X
)
Search (Alternative
paths - T)
Evaluate
(Adequacy of T)
Sele ct ( Solu tio n - S )
S = X T G
Evaluate
(Satisfaction of S)
Memorize
(OAR)
End
Yes
No
Yes
Yes
No
Quantify (G) Quantify (T)
No
Evaluate
(Adequacy of G)
×
×
Fig. 2. The cognitive process of problem solving.
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524 The impact of individual’s knowledge on the outcomes
525 and efficiencies of problem solving has been recognized in
526 cognitive informatics and psychology. It is often true that
527 a problem to somebody is not one to others; and a hard
528 problem to somebody is not so tough at all to others. These
529 phenomena may be explained empirically by the difference
530 of expertise, knowledge, and problem solving skills. They
531 can be more rigorously explained by the differences of indi-
532 vidual’s solution space SS as given in Theorem 1 and Cor-
533 ollary 1. That is, although a problem as given is the same to
534 everybody, the solution spaces to the problem may be
535 greatly different due to the differences of each problem sol-
536 ver’s knowledge, experience, skills, and strategies in prob-
537 lem solving.
538 4. Formal description of the cognitive process of problem
539 solving in denotational mathematics
540 Before the formal description of problem solving is pre-
541 sented, the structures of denotational mathematics (Wang,
542 2007a, 2007d, 2008a), particularly RTPA (Wang, 2002a,
543 2003b, 2008c) and concept algebra (Wang, 2008b), are
544 introduced in this section.
545 Definition 7. Denotational mathematics is a category of
546 expressive mathematical structures that deals with high-
547 level complex mathematical entities beyond numbers and
548 sets, such as abstract objects, complex relations, behavioral
549 information, concepts, knowledge, processes, and systems.
550 4.1. Formal modeling of human behaviors using RTPA
551
552 Definition 8. RTPA is a denotational mathematical struc-
553 ture for algebraically denoting and manipulating system
554 and human behavioral processes and their attributes by a
555 triple, i.e.:
556
RTPA ¼
^ðT;P;NÞð8Þ
558558
559 where Tis a set of 17 primitive types for modeling system
560 architectures and data objects, Pa set of 17 meta-processes
561 for modeling fundamental system behaviors, and Ra set of
562 17 relational process operations for constructing complex
563 system behaviors.
564 Definition 9. The RTPA type system Tencompasses 17
565 primitive types elicited from fundamental computing
566 needs, i.e.:
567
ð9Þ
569569
570 where all types in Thave been defined in Wang (2007a).
571 Ameta-process in RTPA is a primitive computational
572 operation that cannot be broken down to further individ-
573 ual actions or behaviors. A meta-process serves as a basic
574 building block for modeling software behaviors. Complex
575processes can be composed from meta- processes using pro-
576cess relational operations.
577Definition 10. The RTPA meta-process system Pencom-
578passes 17 fundamental computational operations elicited
579from the most basic computing needs, i.e.:
580
581
582
ð10Þ
584584
585
where the 17 meta-processes stand for assignment, evalua-
586tion, addressing, memory allocation, memory release, read.
587write, input, output, timing, duration, increase, decrease,
588exception detection, skip, stop, and system, respectively.
589Aprocess relation in RTPA is an algebraic operation
590and a compositional rule between two or more meta-pro-
591cesses in order to construct a complex process, which is
592elicited from fundamental algebraic and relational opera-
593tions in computing in order to build and compose complex
594processes in the context of real-time software behaviors.
595Definition 11. The RTPA process relation system Rencom-
596passes 17 fundamental algebraic and relational operations
597elicited from basic computing needs, i.e.:
598
ð11Þ
600600
601
where the 17 relational operators of RTPA stand for se-
602quence, jump, branch, while-loop, repeat-loop, for-loop,
603recursion, function call, parallel, concurrence, interleave,
604
pipeline, interrupt, time-driven dispatch, event-driven dis-
605patch,and interrupt-driven dispatch, respectively.
606RTPA provides a coherent notation system and a for-
607mal engineering methodology for modeling both software
608and intelligent systems. RTPA can be used to describe both
609logical and physical models of systems, where logic views of
610the architecture of a software system and its operational
611platform can be described using the same set of notations.
612
When the system architecture is formally modeled, the sta-
613tic and dynamic behaviors that perform on the system
614architectural model, can be specified by a three-level refine-
615ment scheme at the system, class, and object levels in a top-
616down approach. Detailed syntaxes and formal semantics of
617RTPA meta-processes and process relations may be
618referred to Wang (2002a, 2007a, 2008c).
619
4.2. Formal treatment of internal knowledge in problem
620solving by concept algebra
621Aconcept is a cognitive unit to identify and/or model a
622concrete entity in the real-world and an abstract object in
623the perceived-world. Before an abstract concept is defined,
624the semantic environment or context of concepts is intro-
625duced below.
Y. Wang, V. Chiew / Cognitive Systems Research xxx (2008) xxx–xxx 7
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626 Definition 12. Let Obe a finite nonempty set of objects,
627 and Abe a finite nonempty set of attributes, then a semantic
628 environment or context Hof all concepts in the discourse is
629 denoted as a triple, i.e.:
630
H¼^ðO;A;RÞ
¼R:O ! OjO ! AjA ! OjA ! A ð12Þ
632632
633 where Ris a set of relations between Oand A,as well as
634 their reflective relations, and jdenotes alternative relations.
635 Concepts in denotational mathematics are an abstract
636 structure that carries certain meaning in almost all cogni-
637 tive processes such as problem solving, learning, and
638 reasoning.
639 Definition 13. An abstract concept c on His a 5-tuple, i.e.:
640
c¼
^ðO;A;Rc;Ri;RoÞð13Þ
642642
643 where
644 Ois a nonempty set of objects of the concept,
645 O¼fo1;o2;...;omg·O,where ·Odenotes a power
646 set of O.
647 Ais a nonempty set of attributes, A¼fa1;a2;...;
648 ang·A.
649 R
c
=OAis a set of internal relations.
650 R
i
A0A,A0vC0^Avc, is a set of input relations,
651 where C0is a set of external concepts, C0H.For con-
652 venience, R
i
=A0Amay be simply denoted as
653 R
i
=C0c.
654 R
o
cC0is a set of output relations.
655 Concept algebra is an abstract mathematical structure
656 for the formal treatment of concepts and their algebraic
657 relations, operations, and associative rules for composing
658 complex concepts.
659 Definition 14. Aconcept algebra CA on a given semantic
660 environment His a triple, i.e.:
661
CA ¼
^ðC;OP ;HÞ¼ðfO;A;Rc;Ri;Rog;fr;cg;HÞð14Þ
663663
664 where OP ={
r
,
c
} are the sets of relational and composi-
665 tional operations on abstract concepts.
666 Definition 15. The relational operations
r
in concept
667 algebra encompass eight comparative operators for manip-
668 ulating the algebraic relations between concepts, i.e.:
669
670
671
r¼
^f$;&;;;¼;ffi;;,gð15Þ
673673
674 where the relational operators stand for related, indepen-
675 dent, subconcept, superconcept, equivalent, consistent, com-
676 parison,and definition, respectively.
677 Definition 16. The compositional operations
c
in concept
678 algebra encompass nine associative operators for manipu-
679 lating the algebraic compositions among concepts, i.e.:
680
c¼
^f);)
;)
þ;)
;];w;t;‘;7!g ð16Þ
682682
683
where the compositional operators stand for inheritance,
684tailoring, extension, substitute, composition, decomposition,
685aggregation, specification,and instantiation, respectively.
686Detailed descriptions of the relational and composi-
687tional operations of concept algebra may be referred to
688(Wang, 2008b). Concept algebra provides a powerful deno-
689tational mathematical means for algebraic manipulations
690of abstract concepts. Concept algebra can be used to
691model, specify, and manipulate generic ‘‘to be”type prob-
692lems, particularly system architectures, knowledge bases,
693and detail-level system designs, in cognitive informatics,
694computing, software engineering, computational intelli-
695gence, and system engineering.
6964.3. The formal model of the cognitive process of problem
697solving
698Based on RTPA and concept algebra, the cognitive pro-
699cess of problem solving as elaborated in Fig. 2 can be for-
700mally described as presented in Fig. 3. According to the
701OAR model (Wang, 2008c) of internal knowledge repre-
702sentation in the brain, the result of a solution produced
703in problem solving in the mind of a problem solver is a
704new sub-OAR model, which will be used to update the
705entire OAR model of knowledge by concept composition
706(Wang, 2008b) in the long-term memory of the problem
707solver.
708The center in the formal model of the problem solving
709process is that knowledge about the problem and its solu-
710tion(s) are represented internally in the brain as an OAR
711model as shown in Fig. 4, where the external world is rep-
712resented by real entities (RE), and the internal world by vir-
713tual entities (VE) and objects (O). The internal world can be
714divided into two layers: the image layer and the abstract
715layer.
716The virtual entities are direct images of the external
717real-entities located at the image layer. The objects are
718abstract artifacts located at the abstract layer. The
719abstract layer is an advanced property of human brains.
720It is noteworthy that animal species have no such abstract
721layer in their brains. Therefore, they have no indirect or
722abstract thinking capability (Wang, 2007b). In other
723words, abstract thinking is a unique power of the human
724brain known as the qualitative advantage of human
725brains. The other advantage of the human brain is the tre-
726mendous capacity of long-term memory in the cerebral
727cortex known as the quantitative advantages. On the basis
728of these two principal advantages, mankind gains the
729power as human beings.
730There are meta-objects (O) and derived objects (O0) at the
731abstract layer in the OAR model. The former are concrete
732objects directly corresponding to the virtual entities and
733then to the external world. The latter are abstracted objects
734that are derived internally and have no direct connection
735with the virtual entities or images of the real-entities such
736as abstract concepts, notions, ideas, and states of feelings.
737The objects on the brain’s abstract layer can be further
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738 extended into a network of objects, attributes, and rela-
739 tions according to the OAR model as shown in Fig. 4.In
740 Fig. 4, the connections between objects/attributes (O/A)
741 via relations are partially connected. In other words, it is
742 not necessary to find a relation among all pairs of objects
743 or attributes.
744 The OAR model can be used to describe the dynamic
745 state changes of the problem solver during the problem
746 solving process. The problem solving process is essentially
747a set of representation and search operations. According to
748the OAR model, a problem can be represented as an object
749OS.The problem object is identified once it has been com-
750prehended as the problem of interest XðOS;AST;RSTÞST.
751The problem object XST is a structure as a part of the
752entire OARST model of the problem solver’s knowledge
753base or the solution space.
754All meta-objects, attributes, and relations within the
755OARST solution space will be exhaustively searched.
Fig. 3. Formal description of the cognitive process of problem solving in RTPA.
Y. Wang, V. Chiew / Cognitive Systems Research xxx (2008) xxx–xxx 9
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756 Upon satisfactory of a parallel search on GSand TST,a
757 set of solutions (SXS;TST;GSTÞST will be generated
758 and represented, which is memorized by using the concept
759 composition operations ]in order to update the solutions
760 to the given problem (Wang, 2008b).
761The relationships and interactions between the problem
762solving process and other cognitive processes can be
763explained according to the LRMB model as illustrated in
764Fig. 5. As a top layer process of the conscious life functions
765
of the brain in LRMB, the problem solving process inter-
The internal world
Real Entities
RE
1
Virtual Entities
Real Entities
Os/As
Relations
Os/As
Virtual Entities
The external world
The external world
The Image L ayer
The Image Layer
The Abstract Layer
Derived
objects
Meta
objects
…
…
…
…
…
…
…
…
…
…
O
1
VE
1
VE
1
O
1
RE
1
O
p
O
’1
O’
2
O’
m
O
2
VE
n
VE
2
VE
n
VE
2
RE
n
RE
2
RE
2
RE
n
O’
1
O
2
O’
2
O’
m
O
n
Other
internal
relations
The concrete
space
The abstract
space
Fig. 4. The extended OAR model of the brain.
L7 –
Higher
cognition
L5 –
Meta-cognition
L2 –
Memory
L1 –
Sensation
L6 –
Meta-inference
L4 – L3 –
Perception Action
L
R
M
B
Problem
Solving
Learning Decission mak ing …
SBM STM LTM ABM CPM
Deduction Anal
y
sis S
y
nthesis …
…
… …
Identif
y
Ob
j
Abstraction Search Memoriz
e
Vision Audition Smell Tactilit
y
Tast
e
Attention Emotions
Fig. 5. Interaction between problem solving and other cognitive processes in LRMB.
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766 acts with other higher cognitive processes at layer 7, such as
767 the learning and comprehension processes, It also involves
768 lower layer processes such as those of abstraction, search,
769 and memorization.
770 According to the cognitive model of the brain (Wang,
771 2007b), the thinking engine of the brain may be referred
772 to as a real-time natural intelligence system (NI-Sys). In
773 NI-Sys, the pre-determined operating system is defined as
774 the natural intelligence operating system (NI-OS) and a
775 set of acquired life applications is known as the natural
776 intelligence applications (NI-App). Subsequently, the
777 human memory has been classified into both short-term
778 memory (STM) and long-term memory (LTM). Since
779 problem solving is an acquired life function, it is classified
780 as a part of NI-App. However, the problem solving process
781 dynamically invokes the support of lower layer processes in
782 NI-OS, as well as STM and LTM, where the acquired
783 knowledge about the solution of a given problem is repre-
784 sented and retained in the logical form of an OAR model.
785 5. Conclusions
786 This paper has presented a generic and formal model of
787 the fundamental cognitive process of problem solving on
788 the basis the layered reference model of the brain (LRMB)
789 and the object–attribute-relation (OAR) model. With the
790 exploration of empirical studies on problem solving in cog-
791 nitive psychology and computational intelligence, a set of
792 formal and rigorous cognitive, mathematical, and process
793 models of problem solving as a cognitive process has been
794 developed. The cognitive structures of the brain and the
795 mechanisms of internal knowledge representation behind
796 the cognitive process of problem solving have been
797 explained. In order to facilitate computer simulations in
798 cognitive informatics and computational intelligence, the
799 denotational mathematical structures of real-time process
800 algebra (RTPA) and concept algebra have been intro-
801 duced, which provide a powerful tool to formally model
802 and manipulate human and system architectures and
803 behaviors in rigorous approaches.
804 Future research direction related to this work will be on
805 the modeling and explanation of the rest of the remaining
806 cognitive processes of LRMB using denotational mathe-
807 matics and formal means in hope to better understand
808 how the brain works as a whole. The entire project will lead
809 to the development of future generation cognitive comput-
810 ers and methodologies for cognitive computing that are
811 capable of think, learn, and perceive.
812 6. Uncited reference
813 Wang (in press).Q1
814 Acknowledgements
815 This work is partially sponsored by Natural Sciences
816 and Engineering Research Council of Canada (NSERC).
817The authors would like to thank Prof. Ning Zhong, Jiming
818Liu, Yiyu Yao, and the anonymous reviewers for their
819valuable suggestions and comments that have greatly im-
820proved the quality of the final version of this article.
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