The Cognitive Processes of Formal Inferences

Transcription

1 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December The Cognitive Processes of Formal Inferences Yingxu Wang, University of Calgary, Canada Abstract Theoretical research is predominately an inductive process; while applied research is mainly a deductive process. Both inference processes are based on the cognitive process and means of abstraction. This article describes the cognitive processes of formal inferences such as deduction, induction, abduction, and analogy. Conventional propositional arguments adopt static causal inference. This article introduces more rigorous and dynamic inference methodologies, which are modeled and described as a set of cognitive processes encompassing a series of basic inference steps. A set of mathematical models of formal inference methodologies is developed. Formal descriptions of the four forms of cognitive processes of inferences are presented using Real-Time Process Algebra (RTPA. The cognitive processes and mental mechanisms of inferences are systematically explored and rigorously modeled. Applications of abstraction and formal inferences in both the revilement of the fundamental mechanisms of the brain and the investigation of next generation cognitive computers are explored. Keywords: abduction; abstraction; analogy; cognitive informatics; cognitive processes; deduction; induction; inference; LRMB; mathematical models; OAR; RTPA; the brain INTRODUCTION Inferences are a formalized cognitive process that reasons a possible causal conclusion from given premises based on known causal relations between a pair of cause and effect proven true by empirical observations, theoretical inferences, and/or statistical regulations (Bender, 1996; Wang, 2007a; Wilson & Keil, Formal logic inferences may be classified as causal argument, deductive inference, inductive inference, abductive inference, and analogical inference (Hurley, 1997; Schoning, 1989; Smith, 2001; Sperschneider & Antoniou, 1991; Tomassi, 1999; Wang, Wang, Patel, & Patel, 2006; Wilson & Keil, Theoretical research is predominately an inductive process; while applied research is mainly a deductive process. Abstraction is a powerful means of philosophy and mathematics. It is also a preeminent trait of the human brain identified in cognitive informatics studies (Wang, 2005, 2007c; Wang et al., All formal logical inferences and reasonings can only be carried out on the basis of abstract

2 76 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December properties shared by a given set of objects under study. Definition 1: Abstraction is a process to elicit a subset of objects that shares a common property from a given set of objects and to use the property to identify and distinguish the subset from the whole in order to facilitate reasoning. Abstraction is a gifted capability of human beings. Abstraction is a basic cognitive process of the brain at the meta cognitive layer according to the Layered Reference Model of the Brain (LRMB (Wang, 2003a, 2007c; Wang, Liu, & Wang, 2003, Only by abstraction can important theorems and laws about the objects under study be elicited and discovered from a great variety of phenomena and empirical observations in an area of inquiry. Definition 2: Inferences are a formal cognitive process that reasons a possible causality from given premises based on known causal relations between a pair of cause and effect proven true by empirical arguments, theoretical inferences, or statistical regulations. Mathematical logic, such as propositional and predicate logic, provide a powerful means for logical reasoning and inference on truth and falsity (Hurley, 1997; Schoning, 1989; Sperschneider & Antoniou, 1991; Van Heijenoort, Definition 3. An argument is an assertion that yields ( a proposition Q called the conclusion from a given finite set of propositions known as the premises P 1, P 2,, P n, that is: BL (P 1 BL P 2 BL P n BL QBLBL (1 where the argument and all propositions are in type Boolean (BL. Hence, BL = T called a valid argument, otherwise it is a fallacy, that is, BL = F. Equation 1 can also be denoted in the following inference structure: Premises BL P1 BL P2 BL... Pn BL BL ConclusionBL QBL (2 Example 1 The following expressions are concrete arguments: a. A concrete deductive argument Information processing is an intelligent behavior ( P 1. 1 BL Computer is able to process information ( P 2. Computer is an itelligent machine ( Q. b. A concrete inductive argument (3 Human is able to process information ( P 1. Computer is able to process information ( P2. 2 BL Information processing is a common property of itelligence ( Q. (4 Example 2 The following expressions are abstract arguments: a. Abstract deductive arguments x S, P( x a S 3 BL x a, P( a n N, n n 1 4 BL n 1, 1 2 (5 N (6 where N represents the type of natural numbers. b. Abstract inductive arguments 3 x a S, P( a, x b S P( b BL, x c S P( c (7, x S P( x

3 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December BL 2 1n 1n 4 1n 5 n N N N N i 1 i 1 4 i 1 5 i 1 i i n( n ( ( ( (8 where N represents the type of natural numbers. In the previous examples the premier propositions should be arranged in a list that the most general ones are put in the front. This condition preserves the deductive chain in reasoning. It is noteworthy that propositional arguments can be classified as a kind of causal and static inference. More rigorous and dynamic inferences may be modeled and described as a set of mental processes encompassing a series of basic inference steps (Wang, 2007a; Wang & Wang, Further, the cognitive processes and mental mechanisms of inferences need to be systematically explored and rigorously modeled (Wang, 2002b, 2007b; Wang & Kinsner, 2006; Wang et al., This article describes the cognitive processes of formal inferences, which are the foundations of human reasoning, thinking, learning, and problem solving. A formal treatment of the mechanisms of inferences is presented. Mathematical models of four forms of inferences known as deduction, induction, abduction, and analogy are rigorously developed. Formal descriptions of all inference processes are developed using RTPA (Wang, 2002a, 2003b, 2006a, 2007a. Applications of the formal inference methodologies and processes in dealing with complicated problems in both the revilement of the fundamental mechanisms of the brain and the investigation of next generation cognitive computers are explored. MATHEMATICAL MODELS OF FORMAL INFERENCES Inferences are a formal cognitive process that reasons a possible causal conclusion from given premises based on known causal relations between a pair of cause and effect proven true by empirical arguments, theoretical inferences, or statistical regulations. Inferences may be classified into the deductive, inductive, abductive, and analogical categories (Hurley, 1997; Tomassi, 1999; Wang et al., 2006; Wilson & Keil, For seeking generality and universal truth, either the objects or the relations can only be abstractly described and rigorously inferred by abstract models rather than real world details. Deduction Definition 4: Deduction is a cognitive process by which a specific conclusion necessarily follows from a set of general premises. Deduction is a reasoning process that discovers or generates new knowledge based on generic beliefs one already holds such as abstract rules or principles. The validity of a deductive inference depends on its conformity to the validity of generic principle; at the same time, the generic principle that the deduction is based on is evaluated during the deductive practice. Theorem 1: A generic inference formula of logical deduction states that, given an arbitrary non-empty finite set X, let p(x be a proposition for x X, a specific conclusion on a X, p(a can be drawn as follows: x X, p(x a X, p(a (9 where denotes yield or a causal relation. Acomposite form of propositions of Equation 2 can be given as follows:

4 78 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December ( x X, p(x q(x ( a X, p(a q(a (10 Any valid logical statement, established mathematical formula, or proven theorem can be used as the generic promise for facilitating the above deductive inferring process. Corollary 1: A sound deductive inference is yielded if all premises are true and the argument is valid. Corollary 1 may be used to avoid any deductive dilemma and falsity in logical reasoning. Induction Definition 5: Induction is a cognitive process by which a general conclusion is drawn from a set of specific premises based mainly on experience or experimental evidences. Induction is a reasoning process that derives a general rule, pattern, or theory from summarizing a series of stimuli or events. Contrary to the deductive inference approach, induction may introduce uncertainty during the extension of limited observations into general rules. Inductive inferences encompass rule learning, category formation, generalization, and analogy. Theorem 2: A generic inference formula of logical induction states that, if a, k, succ(k X, p(a and p(k p(succ(k are three valid predicates, then a generic conclusion on x X, p(x can be drawn as follows: (( a X, p(a ( k, succ(k X, (p(k p(succ(k x X, p(x (11 where succ(k denotes the next element of k in X. A composite form of Equation 11 can be given as follows: (( a X, p(a q(a ( k, succ(k X, ((p(k q(k (p(succ(k q(succ(k x X, p(x q(x (12 Theorem 2 indicates that for a finite list or an infinite sequence of recurring patterns, three samplings (two determinate and one random are sufficient to determine the behavior of the given list or sequence of patterns. Therefore, logical induction is a tremendous powerful and efficient cognitive and inferring tool in science and engineering, as well as in everyday life. It is noteworthy that because of the limitation of samples, logical induction may result in faulty proofs or conclusions. Therefore, as a rule of thumb, the inference results of logic inductions need to be evaluated or validated by more random samples. Corollary 2: A cogent inductive inference is yielded if all premises are true and the argument is valid. Corollary 2 may be used to avoid any inductive dilemma in logical reasoning. Abduction Definition 6: Abduction is a cognitive process by which an inference to the best explanation or most likely reason of an observation or event is resulted. Abduction is widely used in causal reasoning, particularly when a change of events need to be traced back where not all of the events have been observed. Theorem 3: A generic inference formula of logical abduction states that based on a general implication x X, p(x q(x, a specific conclusion on a X, p(a can be drawn as follows:

5 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December ( x X, p(x q(x ( a X, q(a p(a (13 A composite form of Equation 13 can be given as follows: ( x X, p(x q(x r(x q(x ( a X, q(a (p(a r(a (14 Abduction is a powerful inference technique for seeking the most likely cause(s and reason(s of an observed phenomenon in causal analyses. Analogy Definition 7. Analogy is a cognitive process by which an inference about the similarity of the same relations holds between different domains or systems, and/or examines that if two things agree in certain respects then they probably agree in others. Analogy is a mapping process that identifies relation(s in order to understand one situation in terms of another. Analogy can be used as a mental model for understanding new domains, explaining new phenomena, capturing significant parallels across different situations, describing new concepts, and discovering new relations. Theorem 4. A generic inference formula of logical analogy states that based on a specific predicates a X, p(a, a similar specific conclusion can be drawn iff x X, p(x as follows: x X, p(x a X, p(a b X b a, p(b (15 A composite form of Equation 15 can be given as follows: x X, p(x ( a X, p(a q(a ( b X b a, p(b q(b (16 Analogy is widely used to predict a similar phenomenon or consequence based on a known observation. Table 1. Summary of the mathematical models of formal inferences No. Inference technique Primitive form Formal description Composite form Usage 1 Deduction (Equations 9/10 x X, p(x a X, p(a ( x X, p(x q(x a X, p(a ( a X, p(a q(a To derive a conclusion based on a known and generic premise 2 Induction (Equations 11/12 (( a X, P(a ( k, k+1 X, (P(k P(k+1 x X, P(x (( a X, p(a q(a ( k, k+1 X, ((p(k q(k (p(k+1 q(k+1 x X, p(x q(x To determine the generic behavior of the given list or sequence of recurring patterns by three samples 3 Abduction (Equations 13/14 ( x X, p(x q(x ( a X, q(a p(a ( x X, p(x q(x r(x q(x ( a X, q(a (p(a r(a To seek the most likely cause(s and reason(s of an observed phenomenon 4 Analogy (Equations 15/16 a X, p(a b X, p(b ( a X, p(a q(a ( b X, p(b q(b To predict a similar phenomenon or consequence based on a known observation

6 80 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December The four inference methodologies, deduction, induction, abduction, and analogy, form a set of fundamental cognitive processes of the natural intelligence, which are modeled in LRMB (Wang et al., A summary of the formal definitions and mathematical models of the four forms of inference techniques is provided in Table 1. THE COGNITIVE PROCESS OF FORMAL INFERENCES This section formally describes the four cognitive processes of deduction, induction, abduction, and analogy using RTPA (Wang, 2002a, 2003b, 2006a, 2007a. RTPA is designed for describing the architectures and static and dynamic behaviors of software systems as well as human cognitive behaviors and sequences of actions. In the discussions, a generic model, the Object-Attribute-Relation (OAR model (Wang et al., 2003; Wang, 2007c, is adopted to describe internal knowledge representation. The RTPA description of the cognitive processes of inferences provides rigorous models of the mental processes, which enable accurate and precise reasoning of the natural intelligence. The formal models also enable computer simulations of human thinking mechanisms as a set of cognitive processes (Wang, 2006b, 2007b, 2007d. The Cognitive Process of Deduction Based on the mathematical model of deduction as described in Equations 9 and 10, the cognitive process of deduction is presented in Figure 1. The deduction process is divided into three sub-processes known as: (1 to form the deductive goal; (2a to search and validate primitive predicate in memory; (2b to search and validate composite predicate in memory; and (3 to represent and memorize the deduction result. The input of the deduction process is the deductive goal as and the abstract properties p(xbland q(xbl.. The output of the deduction process is the validation of the deduction result p(a a XBL and the memorization of the updated OAR ST in memory. In the deduction process, step (i forms one or multiple deductive goal(s os by identifying the object and abstracting its property or category; steps (ii.a and (ii.b search and validate primitive and/or composite deductive predicates in memory in parallel. The former repetitively searches all related objects x that is equivalent ( to b until the search is successful or is given up. If the premise is true based on the search, the conclusion p(o bbl is validated. The latter does the same with the composite deduction. Finally, step (iii represents the deduction result by a sub-oar model (o, A, R ST and memorizes it by a composition operation OARST soarst Figure 1. The cognitive process of deduction in RTPA The Deduction Process Deduction (I:: as; p(xbl, q(xbl; O:: p(a a XBL, OAR ST { I. Form deductive goal(s os := as ObjectIdentification (os // Set deductive goal Abstraction (X o X // Abstract the property or category of o II.a Search and valid primitive prodecate in memory Sat(p BL= T Giveup BL= T ( R ( Search (p(x bbl = T Sat(p F Evaluate (p ( p(x bbl = T // If premise is true p(o bbl := T <o, p> ~ // Otherwise p(o bbl := F II.b Search and valid composite predicate in memory Sat(p q BL= T Giveup BL= T ( R ( Search (p(x bbl = T Sat(p q F } p(x cbl = T Evaluate (p Evaluate (q ( (p(x bbl = T p(x cbl= TBL = T ~ (p(o bbl := T p(o cbl= TBL := T <o, p(o q(o> (p(o bbl := F p(o cbl= TBL := F III. Represent and memorize deduction result soarst := (o, A, R ST // Form new OAR model Memorization (OARST soarst

7 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December between the newly established soarst and the entire OARST (Wang, 2006c, 2007e. The Cognitive Process of Induction Based on the mathematical model of induction as described in Equations 11 and 12, the cognitive process of induction is presented in Figure 2. The induction process is divided into two sub-processes known as: (i.a to check primitive predicate; (i.b to check composite predicate; and (ii to represent and memorize the induction result. The input of the induction process is a set of inductive samples XSET and the abstract properties p(abl and q(abl.. The output of the induction process is the validation of the induction result p(x x XBL and the memorization of the updated OAR ST in memory. In the induction process, step (i.a checks the Figure 2. The cognitive process of induction in RTPA The Induction Process Induction (I:: XSET; p(xbl, q(xbl; O:: p(x x XBL, OAR ST { I.a Check primitive predicate ( ( p(a a XBL = T p(k k XBL = T p(succ(k succ(k XBL = T p(x x XBL = T o := X <X, p(x x X> ~ } p(x x XBL = F o := II.b Check composite predicate ( ( p(a a XBL = T q(a a XBL = T p(k k XBL = T q(k k XBL = T p(succ(k succ(k XBL = T q(succ(k succ(k XBL = T p(x x XBL = T q(x x XBL = T o := X <X, p(x x X> ~ p(x x XBL = F o := II. Represent and memorize induction result soarst := (o, A, R ST // Form new OAR model Memorization (OARST soarst primitive induction by three samples in X, that is, xs = as (a specific, usually the first, element, xs = ks (a random element, and xs = succ(ks (the next element following k. If all three samples confirm p(x x XBL is true, an induction result p(x x XBL = T is achieved. Step (i.b does the same reasoning with the composite induction in parallel. Finally, step (ii represents the induction result by a sub- OAR model (o, A, R ST and memorizes it by a composition operation OARST soarst between the newly established soarst and the entire OARST (Wang, 2006c, 2007e. The Cognitive Process of Abduction Based on the mathematical model of abduction as described in Equations 13 and 14, the cognitive process of abduction is presented in Figure 3. The abduction process is divided into four sub-processes known as: (i to form the abductive goal; (ii.a to search abductive predicate; (ii.a to search composite abductive predicate; (iii to valid the abductive predicate; and (iv to represent and memorize the abduction result. The input of the abduction process is the abductive goal as and the abstract properties p(xbl, q(xbl,, and r(xbl.. The output of the abduction process is the validation of the abduction result p(q(a p(a a XBL and the memorization of the updated OAR ST in memory. In the abduction process, step (i forms one or multiple abductive goal(s os by identifying the object and abstracting its property or category; steps (ii.a and (ii.b search primitive and/or composite abductive propositions in memory in parallel. The former repetitively searches all related objects x that validates proposition p(x bbl = T Þ q(x cbl = T, until the search is successful or gave up. The latter does the same with the composite deduction. Step (iii validates the abduction result. If the premise is true based on the search in steps (ii.a and/or (ii.b, the conclusion p(q(a p(a a XBL is validated. Finally, step (iv represents the abduction result by a sub- OAR model (o, A, R ST and memorizes it by

8 82 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December a composition operation OARST soarst between the newly established soarst and the entire OARST (Wang, 2006c, 2007e. The Cognitive Process of Analogy Based on the mathematical model of analogy as described in Equations 15 and 16, the cognitive process of analogy is presented in Figure 4. The analogy process is divided into three sub-processes known as: (i to form the analogical goal; (ii.a to search primitive analogy predicate; (ii.b to search composite analogy predicate; and (iii to represent and memorize the analogue result. The input of the analogy process is the analogy goal as and the abstract properties p(xbl and q(xbl.. The output of the analogy Figure 4. The cognitive process of analogy in RTPA Figure 3. The cognitive process of abduction in RTPA The Abduction Process Abduction (I:: as; p(xbl, q(xbl, r(xbl; O:: p(q(a p(a a XBL, OAR ST { I. Form abductive goal(s os := as ObjectIdentification (os // Set abductive goal Abstraction (X o X // Abstract the property or category of o II.a Search primitive abductive predicate Sat(p q BL= GiveUp = ( R T BL T ( Search (p(x bbl = T Sat(p q BL F q(x cbl = T Evaluate (p q II.b Search composite abductive predicate Sat(p q r q BL= T GiveUp BL= T ( R ( Search ( Sat(p q r q BL F } p(x bbl = T q(x cbl = T Search (r(x dbl = T q(x cbl = T Evaluate (p q Evaluate (r q III. Valid abductive predicate ( (p(x bbl = T q(x cbl = TBL = T (q(o cbl = T p(o bbl = T BL = T <o, q(o p(o> // Primitive abduction (p(x bbl = T q(x cbl = TBL = T (r(x dbl = T q(x cbl = TBL = T (q(o ~ cbl = T (p(o bbl = T r(o dbl = T BL = T <o, q(o p(o r(o> // Composite abduction IV. Represent and memory abduction result soarst := (o, A, R ST // Form new OAR model Memorization (OARST soarst The Analogy Process Analogy (I:: as; p(xbl, q(xbl; O:: p(b b XBL, OAR ST { I. Form analogical goal(s os := as ObjectIdentification (os // Set deductive goal Abstraction (X os X // Abstract the property or category of o II.a Search primitive predicate Sat(p BL= T GiveUp BL= T ( R ( Sat(p F Search (p(b kbl = T Evaluate (p ( p(b kbl = T // If premise is true p(o kbl = T p(b kbl = T <(a, b, (p(a p(b> ~ // Otherwise p(o kbl = T p(b kbl = T II.b Search composite predicate Sat(p q BL= GiveUp = ( T BL T ( Sat(p q F Search (p(b kbl = T } q(b kbl = T Evaluate (p q ( (p(b kbl = T q(b kbl = TBL = T ~ (p(o kbl = T q(o kbl = TBL = T (p(b kbl = T q(b kbl = TBL = T <(a, b, (p(a q(a (p(a q(a> (p(o kbl = T q(o kbl = TBL = T (p(b kbl = T q(b kbl = TBL = T III. Represent and memory analogue result soarst := (o, A, R ST // Form new OAR model Memorization (OARST soarst

9 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December process is the validation of the analogy result p(b b XBL and the memorization of the updated OAR ST in memory. In the analogy process, step (i forms the analogical goal as and the abstract properties p(xbl and q(xbl. Steps (ii.a searches the primitive analogical propositions in memory for p(a kbl = T Þ p(b kbl = T, until the search is successful or gave up. The existence of p(b kbl = T validates the analogy based on p(a kbl = T. Step (ii.b does the same analogical reasoning with the composite analogy in parallel with the primitive analogy. Finally, step (iii represents the analogy result by a sub-oar model (o, A, R ST and memorizes it by a composition operation OARST soarst between the newly established soarst and the entire OARST (Wang, 2006c, 2007e. APPLICATIONS OF THE FORMAL INFERENCE PROCESSES The formal modeling of the cognitive processes of formal inference is not only important for revealing the fundamental mechanisms of the brain, but also inspiring for the investigation of the next generation intelligent computers known as the cognitive computers (CC. This section describes the applications of the formal inference processes in the design of cognitive computers (Wang, 2006b. Definition 8: The architecture of a CC is a parallel structured inference engine (IE and a perception engine (PE, that is as shown in Box 1. As shown in Definition 8, CC is not centered by a CPU for data manipulation as that of the conventional computers with the Von Neumann architecture. However, CC is centered by the concurrent IE and PE for cognitive knowledge processing and autonomic perception based on abstract concept inferences and empirical stimulus perception. In the architecture of CCs, IE is designed for formal inferences and thinking based on the four cognitive inference processes and for concept/knowledge manipulation based on concept algebra (Wang, 2006c, 2006d, particularly the nine concept operations for knowledge acquisition, creation, and manipulation. PE is designed for feeling and perception processing based on RTPA (Wang, 2002a, 2003b, 2006a, 2007a and the formally described cognitive process models of the perception layers as defined in the LRMB model (Wang et al., Definition 9. Concept algebra is an abstract mathematical structure for the formal treatment of concepts and their algebraic relations, operations, and associative rules for composing complex concepts. Associations of concepts, R, defined in concept algebra form a foundation to denote Box 1. CC (IE PE = ( KPU // The Knowledge Processing Unit BPU // The Behavior Processing Unit ( BPU // The Behavior Perception Unit EPU // The Event Perception Unit (17

10 84 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December complicated relations between concepts in knowledge representation. The associations between concepts can be classified into nine categories, such as (1 inheritance, (2 extension, (3 tailoring, (4 substitute, (5 composition, (6 decomposition, (7 aggregation, (8 specification, and (9 instantiation, that is: + R= {,,,,,,,, } (18 According to concept algebra, a concept is the basic unit of thinking and formal inference (Wang, 2006c, 2006d. Human knowledge can be formally represented by concept networks in the form of the OAR model (Wang, 2007c. The formal modeling of concepts and their manipulation in formal inference form a foundation for machine intelligence beyond conventional data processing. The formal inference processes are frequently applied by other higher-layer cognitive processes such as those of knowledge presentation, comprehension, learning, decision making, and problem solving (Wang, 2007a; Wang et al., CONCLUSION This article has explained how rigorous thinking may be carried out by formal inferences. It has demonstrated that formal inferences in the brain may be embodied by the cognitive processes of deduction, induction, abduction, and analogy. Formal descriptions of the four forms of cognitive processes of inferences have been presented using RTPA. A set of rigorous and dynamic inference methodologies has been introduced, which are modeled and described as a set of cognitive processes encompassing a series of basic inference steps. It has been recognized that theoretical research is predominately an inductive process; while applied research is mainly a deductive process. All forms of formal inference processes are based on the cognitive process and means of abstraction and symbolic representation, because the basic unit of human language is abstract concepts. In order to seek generality and universal truth, either the objects or the relations can only be abstractly described and rigorously inferred by abstract models rather than real-world details. Applications of the abstraction and formal inferences in revealing the fundamental mechanisms of the brain and in investigating the next generation cognitive computers have been explored. ACKNOWLEDGMENT The author would like to acknowledge the Natural Science and Engineering Council of Canada (NSERC for its support to this work. We would like to thank the anonymous reviewers for their valuable comments and suggestions. REFERENCES Bender, E. A. (1996. Mathematical methods in artificial intelligence. Los Alamitos, CA: IEEE CS Press. Hurley, P. J. (1997. A concise introduction to logic (6th ed.. London: Wadsworth. Lipschutz, S. (1964. Schaum s outline of theories and problems of set theory and related topics. New York: McGraw-Hill. Schoning, U. (1989. Logic for computer scientists. Boston: Birkhauser. Smith, K. J. (2001. The nature of mathematics (9th ed.. CA: Brooks/Cole, Thomson Learning Inc. Sperschneider, V., & Antoniou, G. (1991. Logic: A foundation for computer science. Reading, MA: Addison-Wesley. Tomassi, P. (1999. Logic. London and New York: Routledge. Van Heijenoort, J. (1997. From Frege to Godel, A source book in mathematical logic Cambridge, MA: Harvard University Press. Wang, Y. (2002a. The real-time process algebra (RTPA. Annals of Software Engineering: An International Journal, 14, Wang, Y. (2002b. On cognitive informatics (Keynote speech. Proceedings of the 5th IEEE Interna-

11 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December tional Conference on Cognitive Informatics (ICCI 06 (pp IEEE CS Press. Wang, Y. (2003a. On cognitive informatics. Brain and Mind: A Transdisciplinary Journal of Neuroscience and Neurophilosophy, 4(2, Wang, Y. (2003b. Using process algebra to describe human and software behaviors. Brain and Mind: A Transdisciplinary Journal of Neuroscience and Neurophilosophy, 4(2, Wang, Y. (2005. The cognitive processes of abstraction and formal inferences. In Proceedings of the 4th IEEE International Conference on Cognitive Informatics (ICCI 05 (pp IEEE CS Press. Wang, Y. (2006a. On the informatics laws and deductive semantics of software. IEEE Transactions on Systems, Man, and Cybernetics, 36(2, Wang, Y. (2006b. Cognitive informatics Towards the future generation computers that think and feel (Keynote speech. In Proceedings of the 5th IEEE International Conference on Cognitive Informatics (ICCI 06 (pp IEEE CS Press. Wang, Y. (2006c. On concept algebra and knowledge representation. In Proceedings of the 5th IEEE International Conference on Cognitive Informatics (ICCI 06 (pp IEEE CS Press. Wang, Y. (2006d. Cognitive informatics and contemporary mathematics for knowledge representation and manipulation. In Proceedings of the 1st International Conference on Rough Set and Knowledge Technology (RSKT 06, Lecture Notes on Artificial Intelligence, Wang, Y. (2007a. Software engineering foundations: A software science perspective. CRC Series of Software Engineering: Vol. 2. New York: CRC Press. Wang, Y. (2007b. The theoretical framework of cognitive informatics. The International Journal of Cognitive Informatics and Natural Intelligence, 1(1, pp Wang, Y. (2007c. The OAR model of neural informatics for internal knowledge representation in the brain. The International Journal of Cognitive Informatics and Natural Intelligence, 1(3, Wang, Y. (2007d. Toward theoretical foundations of autonomic computing. The International Journal of Cognitive Informatics and Natural Intelligence, 1(3, Wang, Y. (2007e. Formal description of the cognitive process of memorization. In Proceedings of the 6th IEEE International Conference on Cognitive Informatics (ICCI 07. IEEE CS Press. Wang, Y., & Kinsner, W. (2006. Recent advances in cognitive informatics. IEEE Transactions on Systems, Man, and Cybernetics, 36(2, Wang, Y., Liu, D., & Wang, Y. (2003. Discovering the capacity of human memory. Brain and Mind: A Transdisciplinary Journal of Neuroscience and Neurophilosophy, 4(2, Wang, Y., & Wang, Y. (2006. Cognitive informatics models of the brain. IEEE Transactions on Systems, Man, and Cybernetics, 36(2, Wang, Y., Wang, Y., Patel, S., & Patel, D. (2006. A layered reference model of the brain (LRMB. IEEE Transactions on Systems, Man, and Cybernetics, 36(2, Wilson, R. A., & Keil, F. C. (2001. The MIT encyclopedia of the cognitive sciences. Cambridge, MA: MIT Press. Yingxu Wang is professor of cognitive informatics and software engineering, director of the International Center for Cognitive Informatics (ICfCI, and director of the Theoretical and Empirical Software Engineering Research Center (TESERC at the University of Calgary. He received a PhD in software engineering from The Nottingham Trent University, UK, in 1997, and a BSc in electrical engineering from Shanghai

12 86 Int l Journal of Cognitive Informatics and Natural Intelligence, 1(4, 75-86, October-December Tiedao University in He was a visiting professor in the computing laboratory at Oxford University during 1995, and has been a full professor since He is editor-in-chief of International Journal of Cognitive Informatics and Natural Intelligence (IJCINI, editor-in-chief of the IGI book series of Advances in Cognitive Informatics and Natural Intelligence, and editor of CRC book series in Software Engineering. He has published over 300 papers and 10 books in software engineering and cognitive informatics, and won dozens of research achievement, best paper, and teaching awards in the last 28 years, particularly the IBC 21st Century Award for Achievement in recognition of outstanding contribution in the field of Cognitive Informatics and Software Science.