11373_Non Monotonic Reasoning System

8/6/2019 11373_Non Monotonic Reasoning System

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Non Monotonic Reasoning

System


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Limitations of Monotonic System

Logic based systems are monotonic in nature i.e. it remains true under all

circumstances.

A monotonic reasoning system cannot work effectively in real lifeenvironment because

1. information available is always incomplete.

2. as process goes by, situations change and so are the solutions.

3. Default assumptions are made in order to reduce the search time and for

quick arrival of solutions.


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Basic Concepts ofNon Monotonic systems

AI provides solutions for those problems whose facts and rules of inference

are stored in knowledge base. But as mentioned data and knowledge areincomplete in nature.

E.g. we say Rohini is bird. the conclusion that is arrived at is that Rohini can

fly.

B

ut on the other hand, it is not necessary that Rohini should fly because ofvariety of reasons.

--Rohini could be ostrich

--Rohini wings are broken.

--Rohini is too weak to fly.

--Rohini could be caged.

-- Rohini could be a dead bird etc.


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Golden rules for Default reasoning

it is clear that it is not possible for a system to have all the information to

arrive at a decision. Guessing of information is permitted as long as it does

not contradict with the existing ones. Default reasoning assists in generation

of these guesses. The golden rules of default reasoning is

Let X be a piece of information and Y be a conclusion.

Rule 1: if X is not known, then conclude Y.

Rule 2: If X cannot be proved, then conclude Y.

Rule 3: If X cannot be proved in a fixed amount of time, then conclude Y.


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Circumscription

one important kind of default reasoning is circumscription. Consider day to dayactivity. We leave our house on a two wheeler, go to railway station, park our vehicle,

board the train to our destination, reach the destination and attend the office. Here

the system works for the solution of the problem in this case is expected to recognize

certain ground conditions such as

-- there is fuel in two wheeler--two wheeler is in good condition.

-- there is no road blockade.

-- train journey is safe etc.

Consider only those whose existences is required for getting the clear picture

of the solution. This principle of avoiding all unnecessary details and taking into

account only those that are absolutely essential is called circumscription.


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Non Monotonic Reasoning

There is new modal operator M which is interpreted as is consistent orwhose negation is Unprovable.

This formalism helps in representing axiomatically notions like if an animal is a bird,

unless proven it is not so, it can fly

i.e. if q is formula in first order logic then Mq is also a formula in logic. The

modal operator M is consistent with all current beliefs in the sense that the negation

of q cannot be proved from the current information.

E.g.

x: [Bird(x) & M fly(x) p fly(x) ]

it holds until it is possible to prove that x cannot fly

also consider

Bird(Rohini)

from these we have fly (Rohini) because we cannot prove that Rohini cannot fly.


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Lets another axiom

x: [Ostrich(x) p ~fly(x) ]

addition of this does not have any impact and original deduction fly (Rohini) still holdsgood.

If new information is added

Ostrich(Rohini)

then fly(Rohini) can no longer be inferred. A simple addition of new fact makes the

system non monotonic because the previously held inferences are no longer valid.

A non monotonic system evades contradictions. A contradiction occurs when the

system finds that the new state discovered is inconsistent with the existing ones.

Non monotonic reasoning system are more complex than monotonic reasoning

systems. Monotonic reasoning systems generally do not provide facilities for alteringfacts, deleting rules because it will have adverse effect on the reasoning process.


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Bayes TheoremThis theorem provides a method for reasoning about partial beliefs. In this every event is happening or likely to

happen is quantified by pieces of knowledge about he event and the rules of probability dictate how these

numerical values are to be calculated.

To illustrate this , let S stand for the statement the horse challenger with the jockey Abc will win

the race this season.

One can associate with this statement two probability values.

1. Probability of challenger winning the race Probability(challenger- winning)=60%. This is called prior

probability because we do not know the ground situation as on today.

2. once we knowledge about the jockey, ground conditions and other relevant information, the probabilitymight be revised. This probability value is called posterior probability.

Probability(challenger- winning | jockey is Abc)= 65%

when there Is change in the belief about the condition of the horse, the jockey or the ground information, the

probability values are also revised.

The bayesian probability as state , follows the basic axioms of probabilty as

1. Probability of a statement is always greater than zero and less than unity.

2. the probability of a sure proposition is unity.

3. Prob(A or B)= Prob(A) + Prob(B) if A and B are mutually exclusive.

4. Prob(Not A)= 1- Prob(A)


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` in general formula is

Prob(A|B

) = Prob(A) * Prob(B

|A)/ Prob(B

)e.g.

Consider bulb manufacturing unit. Here machines M1,M2 and M3 make 30%, 30% and 40% of

total bulbs. Of their outputs, lets assume that 2%, 3% and 4% are defective. A bulb is drawn at

random and is found defective, what is the probability that the bulb is made by machine M1 or

M2 or M3.

Let E1,E2 And E3 be the events that a bulb selected at random is made by machine M1, M2 and

M3 and let Q denote that it is defective.

Prob(E1)= 0.3, Prob(E2)= 0.3 and Prob(E3)= 0.4

these represent the prior probabilities

Probalibity of drawing a defective bulb made by M1=Prob(Q|E1)= 0.02




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Reasoning about certainty Factor

Probability based reasoning adopted bayes theorem for handling uncertainty. To apply

bayes theorem one needs to estimate prior and conditional probabilities which are

difficult to calculate in many domains. To overcome this problem, the developers of

MYCIN system adopted certainty factors.

A certainty factor (CF) is a numerical estimate of the belief or disbelief on a

conclusion in the presence of set of evidence. various methods of using CF have been

adopted.

1. use a scale from 0 to 1 where 0 represents total disbelief and 1 stands for total

belief. Other values between 0 to 1 represent varying degrees of belief and disbelief.

2. MYCINs CF representation is on scale from -1 to 1. a value of -1 stands for total

disbelief and a value of 1 represents total belief. The value of 0 stands for unknown.

In expert systems, every production rule has certainty factor associated with

it. Herewith we give a typical rule with a CF.

IF there is enough fuel in the vehicle.AND the ignition system is working properly

AND the vehicle does not start

then fault lies in the fuel flow CF=0.75


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The values of the CF are determined by the domain expert who creates the

knowledge base. In typical expert systems involving rules relating to the same

conclusion the rules be structured in such fashion that it helps the user to enhance the

belief or the disbelief. In order to do so, two important components have to be

estimated. They are:

1.Measure ofBelief MB[c,e] representing a value that is an indication on the belief in

conclusion c given the evidence e.

2. Measure of Disbelief MD[c,e] representing a value that is an indication on the

disbelief in conclusion c given the evidence e.

Since at a point of time, a rule should either enhance the belief or disbelief

If MB[c,e]>0 then MD[c,e] =0

If MD[c,e]


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Where:

c- conclusion

CF (composite)[c,e (all)]: composite CF or nett CF on conclusion c after taking account

all evidences.MB[c,e (supp)] : consolidated measure of belief in conclusion c given all evidences

supprting (e supp) it.

MD[c,e (aga)] : consolidated measure of disbelief in conclusion c given all evidences

against (e aga) it.

We have two things, one is evidence e and another the conclusion c. it ispossible for related evidences giving rise to a conclusion. In such situations, we have

the following:

MB[c, s1 &s2] = o when MD[c,s1 &s2] =1

= MB[c,s1] +MB[c,s2] * (1-MB[c,s1]) otherwise

here c is conclusion and, s1 & s2 are the two sources and MB

[c,s1 &s2 ] is the measureof belief on conclusion c based on s1 & s2.

similarly

MD[c, s1 &s2] = o when MB[c,s1 &s2] =1

= MD[c,s1] +MD[c,s2] * (1-MD[c,s1]) otherwise


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CONJUCTIONOF CONCLUSIONS

MB[c1 & c2,e] = Max{ MB[c1,e], MB[c2,e]}

MD[c1 & c2,e] = Min{ MD[c1,e], MD[c2,e]}

DISJUCTIONOF CONCLUSIONS

MB[c1 & c2,e] = Min{ MB[c1,e], MB[c2,e]}

MD[c1 & c2,e] = Max{ MD[c1,e], MD[c2,e]}

where c1 and c2 are the two conclusions and e stands for all evidences.

The above methodology is used by MYCIN for certainty factor calculations.

the following steps are adopted by MYCIN

1. the CF of the conjunction of several facts is taken to be minimum of the CFs of the

individual facts.

2. . the CF for the conclusion is obtained by multiplying the CF of the rule with the

minimum CF of IF part.

3. the CF for the fact produced as the conclusions of one or more rules is the

maximum of the CFs produced.

To illustrate, consider the following rule.

Rule 1. IF p & q & r then z (CF= 0.65)

Rule 2. IF u & v & w then z (CF= 0.7)


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assume that the user is certain about p, q, r, u, v & w to 0.6, 0.45, 0.3, 0.7, 0.5 and 0.6

respectively. Then the CF value of z is

from Rule 1: CF(p) = 0.6CF(q) = 0.45

CF(r) = 0.3

Min of these is CF(r) = 0.3

CF(z) based on p, q and r is 0.3* 0.65 = 0.195

from Rule 2: CF(u) = 0.7

CF(v) = 0.5

CF(w) = 0.6

Min of these is CF(v) = 0.5

CF(z) based on u, v and w is 0.5 *0.7 = 0.35

Since Rule 1 and Rule 2 have different set of antecedents, CF of z based on these is

given by maximum of CF(z)

= Max (0.195,0.35)

= 0.35


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Lets take real life example. In MYCIN, there is production rule which states

IF the organism is showing gram positive

AND the organism is coccus

AND the organism is growing in chainsTHEN the organism is streptococcus CF= 0.7

the interpretation is

if Rule 1,2 &3 are 100% certain, then the organism is only 70% certain to be

streptococcus

assume that the user is 100 % certain for Rule 1 &2. the system them poses the

question

MYCIN: did the organism grows in clumps, chains or pairs?

User: chains(8), clumps(4), pairs(-5)

the answer implies that the user is certain to 0.8 for chains, 0.4 for clumps and -0.5

from pairs

Since MB[chains, e] = 0.8 MD[chains, e] = 0

MB[pairs, e] = 0 MD[pairs, e] = -0.5

MB[clumps, e] = 0.4 MD[clumps, e] = 0


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The new CF is

CF[h1,s1 &s2 &s3]= MB[h1,s1&s2&s3] MD[h1,s1&s2&s3]

since rule 3 is not 100% certain but only to 80%, the new value of

MB[h1,s1&s2&s3] = 0.7 * 0.8= 0.56

since all the values of MB are greater than zero, MD values are equal to zero.

Hence the new CF = 0.56 0 = 0.56

the strength of rule is reduced to reflect the uncertainty in rule 3.

MYCIN proved to the massive success in its certainty factor calculation.


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Fuzzy Logic

Fuzzy set theory allows us to represent set membership as a possibility distribution.

The motivation for fuzzy sets is provided by the need to represent such propositions

as:

John is very tall.Mary is slightly ill.

Exceptions to the rule are nearly impossible.

Once set membership has been redefined , it is possible to define a reasoning system

based on techniques for combining distributions. Such reasoners have been applied in

control systems for devices as diverse of trains and washing machines.


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Dempster Shafer Theory

this considers sets of propositions and assigns to each of them an interval

[Belief, Plausibility]

in which degree of belief must lie. Belief (denoted by Bel) measures the strength of

the evidence in favor of a set of propositions. It range from 0 (no evidence) to1(denoting certainty).

Plausibility (PI) is denned to be

PI(s) = 1- Bel(~s)

it also ranges from 0 to 1 and measures the extent to which evidence in

favor of ~s, then Bel(~s) will be 1 and PI(s) will be 0. thus possible value ofBel(s) is also

0.

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11373_Non Monotonic Reasoning System