Using Plausible Reasoning Theory

Claudiu PoznaUniversity Transilvania of Brasov,

Departament of Product Design and Robotics, Bd. Eroilor 28, 500036 Brasov, Romania,E-mail: cpunitbvro

Abstract: The paper intends to continue the research program of are presented and also the related works of Cox and E.T.human knowledge. We will present a theory of commune sense Jayne. We will mention also the work of E. Yudkowsky [3]which is related to plausible reasoning. In order to do this we where an epistemology based on Thomas Bayesian result iswill establish the principles and some theoretical results which presented and also J. Pearl work on causal reasoning [4]. Thehave been obtained from the mentioned principles: the Bayesian bridge between the Bayesian plausible reasoning and mobiletheorem and the Bayesian filter. The next step was to use these robots has been inspired by the work of C. Pradalier, wheretheoretical results in models creation. The paper presents such tsnasibeen ofia bile robot is Prolled usingan example. the navigation of a mobile robot is controlled using

Bayesian's filters [5].Our intention is to establish a scientific knowledge

structure: impose the principles, develop (by analysis) theprinciples in theoretical results and use these results in modelsbuilding. For this we have transformed the rules of "Laplacemodel of commune sense" into principles, and presentedsome theoretical results which are obtained from these

Present paper intends to continue the research program of principles. These examples are resumed to the Bayesianhuman knowledge process which was started in [1]. We theorem and Bayesian filter.remember that the results of the phenomenological researches In the end we will use the mentioned theoretical results andon artificial intelligent (Al) sytagma (collocation) were seven build a model. This is an example which is connected to thequestions which allowed the possibility to deep the mobile robots locomotion. This will give the possibility tounderstanding of the Al and which can drive to intelligent analyze results of the plausible reasoning theory.product construction. These seven questions are:

II THE THEORY OF COMUNE SENS, THE PLAUSIBLE1. Are they known theories that have as object the human REASONING

knowledge?2. How can we use them in order to develop a human

knowledge model? In [1] we have presented a model of human scientific3. How can we simulate this model and how can we knowledge which can be briefly described by the following

improve it? sentences:4. What is the technology - the methods and the tools -

which can be used in order to copy the model?5. What are the properties of the object that can be * the reality is divided in several region, physic, biology,

transformed in intelligent object? human knowledge etc.6. How can we experiment the intelligent object? * for each region a set of fundamentals troughs are

proposed;* mathematical language is involve in order to analyze

One possible answer to the first question could be "The ths rnil n sals hoeia aeLaplce mdelf cmmun sene". he amedmode is * using these theoretical base, phenomena from the

based on reverend Thomas Bayesian and Laplace results andaetoe elt einca etasoe nmdlwas developed in [2].The backgrounds of the present work are E.T Jayne's

probability theory [2] where the rules of the mentioned model

1-4244-1 234-X/O1/$25.OO ©2001 IEEE. 1 17

SACI 2007 - 4th International Symposium on Applied Computational Intelligence and Informatics

Using the same epistemological tree we can transform the The theoretical results:"Laplace commune sense model" presented in [2] into atheory of plausible reasoning. The background of the theory Analyzing the mentioned principles theoretical results canconsists on principles or axioms. The difference between be deduced. From the beginning we will mention that becausethese two concepts consists on the fact that the axioms are the probability function has the same properties (1 ... 5) it canself evident fundamental and the principles are accepted be accepted that the plausibility function is synonymous withfundamental reason. This is the reason that we have choose to the probability function. This is the only reasons thatname the next fundamental reasons principles theoretical results from probability theory can be transferred

to the theory of plausible reasoning.It is obvious that we do not intend to present exhaustive

The principles of plausible reasoning: theoretical results. We will resume presenting the Bayesiantheorem which can easily deduced from (1).

1. The representation for the degree of true (the plausibility) The degree of trough for sentence A in condition ofis given by the plausibility function: knowing 0, is proportionally with degree of trough for the

sentence A and with the degree of true for the sentence 0 inp: (D -X [O 1]; p(A X) = y (1) condition that A is trough. The degree of trough for sentence

A in condition of knowing 0 is inverse proportional with thewhere: degree of trough of sentence 0o is a set of sentencesp(A IX) is a continuous and monotonic function which p(A 0) = p()p(IA) (6)associates a particularly degree of trough for the p(°)sentence A in the condition that sentence X is true;

Proof: p(AO) = p(A)p(O IA) = p(O)p(A 0) => (5)

2. The consistence of the commune sense requires the Some comments are necessary:following property for the functionp * Equation (6) has been proposed for the first time by

Thomas Bayes in* For many scientific works this equation consist the

a. p(AB IX)= p(A X)p(B AX) (2) background for probabilistic and epistemologicalb. p(A X)+p(-,A B) =1 (3) theory [3];c. p(A+B X)=p(AJX)+p(BJX)-p(ABJX) (4) Theories have like objective the knowledge improvement.d. p(A1 X) = - I In science the theories becomes operational by constricting

n models. In order to converge to the model construction wewhere {Ai=1 n is a complete set of mutual excusive will link this theoretical result to the Bayesian filter [6]:sentence

A Bayesian filter allows to estimate the state Xt for aMarkovian system in condition of knowing the observation

Some comments are necessaries: Zl,..Zt. In order to solve this problem several steps arenecessary:

* by consistence we mean:* every possible way of reasoning a sentence must lead * variable definition:

to the same result;* the equivalent sentences have an equal degree of true {X1 }O.i.t the system states; {Z1 }O.i.t observations;

- the same plausibility;* in order to obtain the plausibility for a sentence we * decomposition

must take into account all the available evidence; t* p(AB lX) means the plausibility of sentence A and B p(XO -..X,, Z0-.Z,) = J7J p(X1 X11l )p(Z1 X1) (7)

in the condition that sentence X is true; .inta knwede* -A means non A;* p(A + B lX) means the plausibility of sentence A or B th intastt. itiuin

in the condition that sentence X is true;p(X0) (8)

1 18

C. Pozna Using Plausible Reasoning Theory

* the transition model from state i-Ito state expK'(Xi -( + 0.5))2p(X1 X1.1) (9) p(X1 Xi-,) (13)

* the sensor model; ze2 (,-( +0.5))2

p(Z1 Xi) (10) In order to determine this distribution experimentations are

performed. More exactly we measure the transition from statei-i to state i and we compare this results with the theoretical

* the question results obtained with (12). After a statistical analyze of thisp(Xt IZt...Z (11) comparison we can obtain (13).

For the sensor model (10) we have proposed the normalizeddistribution (14)

Some comments are necessary: ep((Z, -(Xi +0.5))2* The Bayesian filter concept systematizes a plausible e 2P.2(I +O Z )2

reasoning problem construction; p(z1 Xi)= /Z (i+0 ) (14)* The Bayesian filter suppose two levels: the problem y,exp(ZX+0))

description and the question; z- 2.(1+o.1z1 )2)* The first level consists also from two parts: specification

of the model and identification of the parameters. Once again we must do experiments. In this case the sensor

In he extsecionwe illusetheBaysia fiterin rde tois experimented. More precisely we measure with the sensor

Inotel anex secutio e wilxamlue.teBysafiernodrto and we compare the results with the measurement obtainedmodel and simulatean example. ~with a more precisely measurement system. The result of this

comparison is the distribution (14)IIIACASESTUDY ~~~~For the initial position we have use the distribution (15), see

IIIACASESTUDY ~~~~figure 2.

In order to exemplify the mentioned theoretical results we expr (xo )2will consider the case of a mobile robot which modifies his y2.0.52state (position) and - from time to time- make observations p(X0)=X 2(15)(measure his position), see figure 1. Iexp( (x0)52

yO 2.05

p(Xt Z) Even the initial data are not crispy because we must admit

K ~~~~~~~~~~thatwe don't know with precision this data.

I -- ~~~~~~~~~~~~~~~~~~~~~0.2

A -Vt I X'-) ~~~~~0.16 --- ------- --- - - - ~ - --

0.14

Figure 1. The mobile robot 0.2 - J--- I--L

-- --0.1- --The dynamic model of the mobile robot is very simple, the

0.2 - -

SACI 2007 - 4th International Symposium on Applied Computational Intelligence and Informatics

Using these models we have imagined and simulate the The second situation:following situations:

The robot performs several observations without

The first situation: performing any transition. This situation is computed withequation (17)

The robot has several state transition and no observations aremade during this transitions. This situation is computed withequation (16).

ZP(X1)P(Zi Xi)P(X1) = (17)

ZZPIXI)PIZ I X1) (7

Z P(Xil)P(Xi Xi-,) Xi ZiP(X) =El (16)

Z Z P(X11, )P(X1 X'1)Xxi -i

In figure 4 and 5 where we have presented the results ofSimulation results are presented in figure 3. If we this simulation we can see that the degree of plausibilityanalyze this result the main conclusion is that even the increases continuously and converges to value 1translation value - according to (12) - remains constant, (absolute trust).the degree of plausibility has decreased continuouslyfrom translation to translation. This means that thedegree of trust decrees continuously.

0.04

'9 i--- -T- --- -- --i- --- i-- --i ---i--- --i- --- V-- --i 0,' --------------- ----------- ---------------------

_ !fl.06i ~~~~~~~~~~~~~~ -!,- - -1 1 | | | | | | | i rx2i slX ' \~.0J A r - rI T

I' J, D ~~~~~~~~~~~ ~ ~ ~~ ~~~~~~~~~~~~~~~~0 326 ------- ----------------

- D A- r- - - - - D0.02 - -

----------~ ~ ~ ~ ~ ~ 006 I- -----------------------

MA[r r/-n{ - - -n D s

t11 J J X / W1A/SAI \ X X \\\ -- P -SS ~ ~ -------E --3-------

00 2 s3 i e 7 9 lo Figure 4. Two observation ( and ---

which starts from the same state

Figure 3.Transition without observation

This is an obvious situation, because a scientist has afreadythe feeling that using repeatedly a model the degree ofcofdec wil derae In thi cas th beneit sthtw In figure 4 two particular situations are compared. There arecan ~copt thi derasn an of corew cntk two observations which start from the same state. In the first

deiin afe ths reut. case, when the observation reproduces the value of the state,we will obtain a bigger rising. In contrary, in the second case

120

C. Pozna Using Plausible Reasoning Theory

there is a difference between the observation and the state. are below of this value. This is a more realistic strategy whichThis difference will rule to a smaller degree of true. is presented in figure 6.

0.06r 025

I ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~012......5.....

I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~V

05 10 15~~~~~~~~~~~~~~~~~~~~~~~~~00

Figure 5. x[nsIncreasing the plausibility by

several observations Figure 6Transitions (-) followed by observations

If we realize several observations which have the same valuethe degree of true will increase continuously to one. In figure 6 the minimum trust value is 0. 1. We have started

from 0.18 and after five transition we a below this value. In

The third situation: this moment we have performed an observation whichincreased the confidence value to 0.22.

The robot performs transitions and after each transitionperforms observations. We have presented in figure 5 twosituations. The first involves two observations after eachtransition, and the second only one observation after eachtransition. It can be observed that the first strategy increases IV CONCLUSIONSthe degree of plausibility for the current state of the robot.

Starting from the seven questions of the performedphenomenological analysis [1] this paper presents a possible

----------------- GM theory of human knowledge. Elements of this theory are------- -----------prop--osed in [2], but we have structured them from a new

-- ------------------------point of view. This point of view corresponds to anepistemological tree which starts with principles, developtheories and construct models. The answer was the theory of

aO3-------DEQ --------I------commune sense. We consider that the main advantage of thisI--------theory consist in fact that it allows epistemological model

which contains both inductive and deductive process. Thepresented simulations underline this aspect. hincreasing the

Figure 5 plausibility of a sentence by performing observation means toTransitions (-) followed by two or perform the induction. We will underline also two aspects

one observations (-.) whlichiliihaveiben obtainedc from simulation. We wvill me-ntion

SACI 2007 - 4th International Symposium on Applied Computational Intelligence and Informatics

reach this level. An observation will be made after the trustvalue is below the minimum value.The main drawback of this theory is the time consuming

computation. This means that if we choose if for real time REFERENCESapplications we must find new formalisms in order tominimize this time.

Several researches directions have been opened by this [1] Pozna, C., A phenomenological analysis trial of the Al syntagma, Inpaper: Pr. of the 7th. International symposium of Hungarian researchers,

Budapest 2006, 159-165;[2] Jaynes, E.T., Probability theory with application in science and

* The first is to focus on plausible reasoning and answer to engineering, Washington University 1974;the remained questions proposed in [1]; [3] http://yu_dkowskynet/bayes/bayes.htmI Yudkowsky, E., An Intuitive

* The second is to use the plausible reasoning to a more Explanation of Bayesian Reasoning[4] J. Pearl. Causality: Models, Reasoning, and Inference, Cambridge

complicated driving situation such as obstacle avoiding. Univ. Press, New York, 2000.* Imagine new formalism which will minimize the [5] Praladier, C., Navigation intentionnelle d'un robot mobile, Doctoral

computing time. Thesis of L'INPG 2004

122