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Mathematics and Artificial Intelligence, two branches of the same tree

Abstract

Unfortunately, in the learning of Mathematics and Computer Science, they appear often as disconnected areas, when they are indeed two necessary and complementary branches of the same tree. Either of them alone produces only ethereal structures, or routines and ad-hoc programs. For this reason, it would be preferable to study, progressively, from the lower educational levels, both disciplines as naturally linked. So, it will be overrated the pure mechanistic of only give informatics to usuary level, as mere blind instructions, either too abstract pure mathematical constructs.

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WCES2010

Mathematics and Artificial Intelligence, two branches of the same

tree

Angel Garrido

Faculty of Sciences, UNED, Madrid, Spain

Received October 9, 2009; revised December 18, 2009; accepted January 6, 2010

Abstract

Unfortunately, in the learning of Mathematics and Computer Science, they appear often as disconnected areas, when they are

indeed two necessary and complementary branches of the same tree. Either of them alone produces only ethereal structures, or

routines and ad-hoc programs. For this reason, it would be preferable to study, progressively, from the lower educational levels,

both disciplines as naturally linked. So, it will be overrated the pure mechanistic of only give informatics to usuary level, a s mere

blind instructions, either too abstract pure mathematical constructs.

Keywords: Mathematical Education; Computer Science; Artificial Intelligence; Game Theory; Graph Theory; Heuristics; Mathematics.

1. Introduction to mathematical games

From remote times, the history of the human being is developed by a successive chain of steps and sometimes

jumps, until the relative sophistication of the modern brain and its culture. We will describe here the evolution and

application of certain games, being a useful tool not only as the "drosophila melanogaster fly" of AI, but also in

Mathematical Education, as can be considering Chess and Go. Many great thinkers should study this difficult

question: to reach the more efficient heuristic, i.e. strategies to win. Certainly, it will be a challenge for AI, and

therefore, a mathematical problem, solved in the case of Chess, but it remain as a new challenge in the case of Go,

as a many more complex game. For this reason, this appears as the new frontier of AI. So, the Go is waiting, and

defy as an open mathematical problem to be solved, with new techniques, useful on another open problems.

Has been developed different programs to play it. But they are very far of the level of a great master. Because

until now we haven´t an efficient heuristic evaluation function, to solve it. Not occurs as in the case of Chess, by the

Alpha-Beta Pruning Method. But it does not appear as valid in this new situation. Therefore, we dispose of a new

and very important challenge for the current AI research, a bench-work for future research.

Another interesting example, useful motivating the students, as connected with its diary life, can be the Sudoku

puzzle. It has achieved worldwide popularity in recent times, and attracted great attention of the computational

intelligence community. Sudoku is always considered as Satisfiability Problem or Constraint Satisfaction Problem.

* Angel Garrido. Tel.: 00-34-916103797; fax: 00-34-913987237.

E-mail address: agarrido@mat.uned.es.

Procedia Social and Behavioral Sciences 2 (2010) 1133–1136

doi:10.1016/j.sbspro.2010.03.160

Open access under CC BY-NC-ND license.

1134 Angel Garrido / Procedia Social and Behavioral Sciences 2 (2010) 1133–1136

It is possible [Chen, 2009] to focus on the essential graph structure underlying the Sudoku puzzle. First, by the

formalization of Sudoku game as a graph. Then, a solving algorithm based on heuristic reasoning on the graph may

be proposed.

In order to evaluate the difficulty levels of puzzles, a quantitative measurement of the complexity level of

Sudoku puzzles based on the graph structure and information theory may be proposed. Experimental results show

that all the puzzles can be solved fast using heuristic reasoning, and that the proposed game complexity metrics can

discriminate between difficulty levels of puzzles.

Origami is the art of paper folding. It also does have many educational benefits. Its connection with geometry is

clear. But also the study of Origami and Mathematics may be considered into the field of Topology, although it may

be more related with Combinatorics, or Graph Theory.

2. About Artificial Intelligence

Among the things that AI needs to implement a representation are Categories, Objects, Properties, Relations and

so on. All them are connected to Mathematics, as well as being very adequate illustrative examples. For instance,

showing Fuzzy Sets together with the usual Crisp or Classical Sets, which are a particular case of the previous; or

introducing concepts and strategies from Discrete Mathem atics, as the convenient use of Graph Theory tools on

many fields.

The problems in AI can be classified in two general types, Search Problems and Representation Problems . Then,

we have Logics, Rules, Frames, Nets, as interconnected models and tools. All them are very mathematical topics.

The origin of the ideas about thinking machines, the mechanism through work the human brain, the possibility of

mimic its behavior, if we produce some computational structure similar to neuron, or to neural system, their synapsis

or connections between neurons, to produce Neural Networks... All this can appear with resonances of a Science

Fiction history, or perhaps a movie, but it is a real subject of study, and it is so from many years ago, and more in

the last times.

The basic purpose of the A I will be to create an admissible model for the human knowledge. Its subject is,

therefore, "pure form". We try to emulate the way of reasoning of a human brain. This must be in successive,

approximating steps, but the attempts proceed always in this sense.

3. Searching strategies

Between the Nets, the more recent studies to deal with Bayesian Nets, or Networks. Before than its apparition, the

purpose was to obtain useful systems for me dical diagnosis, by classical statistical techniques, such as the Bayes

Rule. A Bayesian Net is represented as a pair (G, D), where G is a directed, acyclic and connected graph, and D will

be a probability distribution, associated with random variables. Such distribution verifies the Property of Directional

Separation, according to which the probability of a variable does not depends of their not descendant nodes.

The Inference in BNs consists in establish on the Net, for the known variables, their values, and for the unknown

variables, their respective probabilities. The objective of BNs in Medicine is to find the probability of success with

we can to give determined diagnosis, known certain sympto ms. We need to work with the subsequent Hypotheses:

Exclusivity, Exhaustivity and Conditional Independence. According the Exclusivity, two different diagnoses cannot

be right at time. With the Exhaustivity, we suppose at our disposition all the possible diagnosis. And by the C I

(acronym of Conditional Independence), the discoveries found must be mutually independents, to a certain

diagnosis. The usual problem with such hypotheses will be their inadequacy to the real world. For this, it will be

necessary to introduce Bayesian Networks.

In the searching process, we have two options: without information of the domain (Blind Search); and with

information about of the domain (Heuristic Search). In the first case, we can elect, according the type of problem,

between Search in extent and Search in depth.

There are other methods, obtained from the previous, such as Searching in Progressive Depth and Bidirectional

Searching, both with names sufficiently allusive to its nature. Also we can found another method, in this case not

derived, the General Search in Graphs. In such procedure, it is obvious the possibility of immediate translation to

matrix expression, through their incidence matrices. All these methods joined to their algorithms.

Blind Search, or search without inform

ation of the dom

ain, appears with the initial attempts to solve, by

idealizations of the real world, playing problems, or the obtaining of automatic proofs.

Angel Garrido / Procedia Social and Behavioral Sciences 2 (2010) 1133–1136

1135

Searching in extent. We advance in the graph through levels. So, we obtain the lesser cost solution, if exists.

Whereas, in the Depth Searching, we expand one link each time, from the root - node. If we reach a blind alley

into the graph, we back until the nearest node and from this, we take one ramification in the graph. It is usual

establish an exploration limit, or depth limit, fi xing the maximal length of the path, from the root.

Heuristic Search, i.e. searching with knowledge of the domain. Initially, were usual to think that all the paths can

be explored by the computer. But it is too optimist. Such exploration will be very difficult, because of phenomenon

as "combinatorial explosion" of branching, when we expand. Its spatial and temporal complexity can advise us

against its realization. For this, we need to select the more promising trajectories. In this way, we cannot obtain the

best solution (optima), but an efficient approach to her.

4. Conclusions

Not only may be these techniques very useful into the class-room, because through them Mathematics obtain a

support on the aforementioned Games (Chess, Checkers, Stratego, Sudoku, …), but also our students can be

introduced in more subtle analyses, as may be the Prissoner´s Dilemma. Also it will be disposable any information

about games as Chess, Go, …, its Rules, Tricks, Hints, and so, in the Web pages, being as well possible to play with

them. And we can obtain information of papers, web explanations, etc., about the history of such games, which will

be very illustrative and motivating for the students. All these techniques has been implemented in the class-room

with students of secondary level, increasing with them its interest in Mathematics and simultaneously, in TIC new

technologies and its fundamental basis. Furthermore, with students of undergraduate university level, in studies of

Mathematics and Computer Science, reaching a very positive reaction, which increments their interest and results.

As an example of educational practice for the classroom, we may dispose a group of people of reasonable size

(from twenty to thirty students), of secondary level, or instead of undergraduate university level, in Mathematics or

Computer Science.

We will give previously a basic introduction to the foundations to graphs and probability. For instance, we may

shown this case of study: Bayesian Networks, as a directed, acyclic, connected graph, jointly with a probability

distribution associated with each node, that express the mutual relationship between nodes representing states, by

directed edges. It is possible to expose some very classical and illustrative examples, as may be the net ASIA, on

infectious diseases, as tuberculosis. Or more simply, the well-known case of the wet grass, and if it is due to which

today is raining, or perhaps it would be due to a water-spinkler. Such examples are motivating not only to learn with

more interest and motivation the graph theory tools, but also permits an elemental survey on the current research in

AI, or more generally, in Computer Science.

By this new approach we defend, Computer Science occupies, partially and in a natural way, the role Physics and

its problems have played as support of mathematical reasoning, a fact in the past two centuries (although Physics do

not disappear from the view, being a necessary aid). We propose showing such Methods through the parallel study

of Mathematics and Computer Science foundations. Other Co mputer Science subfields could be carriers of this

method too, but perhaps AI is the current better choice, given its characteristics, which practically coincide with

many mathematical techniques and objectives.

The creative learning permits to understand the development and practice of creativity. The possibility of

founding new solutions is one specific characteristic of the creative process. It may consists in the art of formulate

questions to obtain ideas, increasing capacities, defying the current conventionalism in the educative world. So, the

benefits of such an innovative educative method must consist in a more progressive regard of Mathematical

Education in modern times, with the final purpose of produc ing adaptive and creative minds, capable of solving new

problems and challenges.

References

Courant, R. (1996), What Is Mathematics? An Elementary Approach to Ideas and Methods. Oxford UP.

Chen, Z. (2009), "Heuristic Reasoning on Graph and Game Complexity of Sudoku". The Smithsonian/ NASA Astrophysics System .

McCorduck, P. (2004), Machines Who Think (2nd ed.), Natick, MA: A. K. Peters, Ltd.

Mitchell, D. (2009): Complete Origami. Firefly Books.

Pólya, G. (2009): How to Solve It: A New Aspect of Mathematical Method. Ishi Press. First ed. 1945.

1136 Angel Garrido / Procedia Social and Behavioral Sciences 2 (2010) 1133–1136

Pólya, G. (1990): Mathematics and Plausible Reasoning. Vol. I: Induction and Analogy in Mathematics. Vol. II: Patterns of Plausible Inference.

Both in Princeton University Press.

Pólya, G. (1981): Mathematical Discovery: On Understanding, Learning and Teaching Problem Solving. Wiley.

Russell, S., and Norvig, P. (2003), Artificial Intelligence: A Modern Approach (2nd ed.), Prentice Hall.

Samuel, A. L. (1959), "Some studies in machine learning using the game of checkers", IBM Journal ofResearch and Development 3 (3):

210í219.

Searle, J. (1980), "Minds, Brains, and Programs", Behavioral and Brain Sciences 3 (3): 417–457.

Turing A. (1950), "Computing Machinery and Intelligence", Mind, LIX (236): 433–460.

... Una RNA es un modelo matemático computacional formado por una capa de entrada, capas de salidas y capas ocultas, estás a su vez están constituidas por neuronas artificiales e interconectadas mediante sinapsis, cada sinapsis con su respectivo peso, el funcionamiento de una RNA se asemeja al comportamiento del cerebro humano , capaz de aprender mediante el aprendizaje, precedido de un entrenamiento repetitivo (Amershi et al., 2005;Brazdil y Jorge, 2001;Brusilovsky y Peylo, 2003;Garrido, 2010;Vila y Penín, 2007). ...

Fernando ALAIN Incio Flores
Dulce Lucero Capuñay Sanchez
Ronald Omar Estela Urbina
Segundo Edilberto Vergara Medrano

Predecir los resultados académicos de los estudiantes permite al docente buscar técnicas y estrategias en el tiempo indicado durante el proceso de enseñanza y aprendizaje con el fin de mejorar el logro de competencias en sus estudiantes. En esta investigación se implementó una red neuronal artificial (RNA) para predecir los resultados académicos del curso de física de los estudiantes del II ciclo de la carrera profesional de Ingeniería Civil de la universidad Nacional Intercultural Fabiola Salazar Leguía de Bagua-Perú en función de datos históricos. La RNA se diseñó e implemento en el Software MATLAB, su arquitectura está formada por una capa de entrada, una capa oculta y una capa de salida, para el entrenamiento de la RNA se utilizó dos algoritmos que posee la Toolbox de MATLAB: el Scaled Conjugate Gradient logrando un porcentaje de predicción del 70% y el Levenberg-Marquardt logrando un porcentaje de predicción 86%.

John R. Searle

This article can be viewed as an attempt to explore the consequences of two propositions. (I) Intentionality in human beings (and animals) is a product of causal features of the brain. I assume this is an empirical fact about the actual causal relations between mental processes and brains. It says simply that certain bran processes are sufficient for intentionality. (2) Instantiating a computer program is never by itself a sufficient condition of intentionality. The main argument of this paper is directed at establishing this claim. The form of the argument is to show how a human agent could instantiate the program and still not have the relevant intentionality. These two propositions have the following consequences: (3) The explanation of how the brain produces intentionality cannot be that it does it by instantiating a computer program. This is a strict logical consequence of 1 and 2. (4) Any mechanism capable of producing intentionality must have causal powers equal to those of the brain. This is meant to be a trivial consequence of 1. (5) Any attempt literally to create intentionality artificially (strong AI) could not succeed just by designing programs but would have to duplicate the causal powers of the human brain. This follows from 2 and 4. 'Could a machine think?' On the argument advanced here only a machine could think, and only very special kinds of machines, namely brains and machines with internal causal powers equivalent to those of brains. And that is why strong AI has little to tell us about thinking, since it is not about machines but about programs, and no program by itself is sufficient for thinking.