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RuleML Telecon – Nov.27th, 2009

Degrees (of freedom)Degrees (of freedom)in Uncertain Reasoningin Uncertain Reasoning

Davide SottaraDEIS, Faculty of Engineering, University of Bologna

40100 Bologna (BO), Italy

davide.sottara2@unibo.it

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Imperfect InformationDifferent types of “afflictions”:Different types of “afflictions”:

ReliabilityReliability (Confidence)(Confidence)• E.g.: Temperature = 25° E.g.: Temperature = 25° (( = = 70/100)70/100)

Measures the “strength” of the estimate (Conflicting) alternatives may be considered

Uncertainty, properUncertainty, proper (Probability, “prior”)(Probability, “prior”)Both subjective (bayesian) and objective (frequentist)Both subjective (bayesian) and objective (frequentist)

• E.g.: TemperatureE.g.: Temperature = 25° = 25° (( = = 70%) 70%) Temperature is expected to be (precisely) 25° May be completely different

VaguenessVagueness (Fuzzy, “posterior”)(Fuzzy, “posterior”)• E.g.: TemperatureE.g.: Temperature ≈ 25° ≈ 25° (( = 0.7) = 0.7)

The value, e.g. 22°, is known to be “similar” to 25°. Data known precisely, but expressed vaguely.

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Simple Examples

rulerule "Bet - Fuzzy""Bet - Fuzzy"

whenwhen

$b: Bet( $a : amount $b: Bet( $a : amount isis highhigh, $n : number ), $n : number )

Extraction( outcome == $n ) Extraction( outcome == $n )

thenthen

insertinsert((newnew Win($a Win($a),”),”very highvery high”); ”);

endend   

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Simple Examples

rulerule "Bet - Fuzzy""Bet - Fuzzy"

whenwhen

$b: Bet( $a : amount $b: Bet( $a : amount isis highhigh, $n : number ), $n : number )

Extraction( outcome == $n ) Extraction( outcome == $n )

thenthen

insertinsert((newnew Win($a Win($a),”),”very highvery high”); ”);

endend   

rulerule "Bet - Probabilistic""Bet - Probabilistic"

whenwhen

$b: Bet( $a : amount, $n : number )$b: Bet( $a : amount, $n : number )

Extraction( outcome Extraction( outcome ==== $n ) $n )

thenthen

gain = $a / gain = $a /

endend   

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Simple Examples

rulerule "Bet - Fuzzy""Bet - Fuzzy"

whenwhen

$b: Bet( $a : amount $b: Bet( $a : amount isis highhigh, $n : number ), $n : number )

Extraction( outcome == $n ) Extraction( outcome == $n )

thenthen

insertinsert((newnew Win($a Win($a),”),”very highvery high”); ”);

endend   

rulerule "Bet - Probabilistic""Bet - Probabilistic"

whenwhen

$b: Bet( $a : amount, $n : number )$b: Bet( $a : amount, $n : number )

Extraction( outcome Extraction( outcome ==== $n ) $n )

thenthen

gain = $a / gain = $a /

endend   

rulerule "Bet - Mixed""Bet - Mixed"

whenwhen

$b: Bet( $a : amount $b: Bet( $a : amount isis highhigh, $n : number ), $n : number )

negneg Extraction( outcome Extraction( outcome ==== $n ) $n )

thenthen

ExpectedRisk r = ...ExpectedRisk r = ...

endend   

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vs

BenefitsBenefits

RobustnessRobustness Handle Inaccurate inputsHandle Inaccurate inputs Avoid arbitrary thresholdsAvoid arbitrary thresholds

ConvenienceConvenience Knowledge is rarelyKnowledge is rarely preciseprecise

ConcisenessConciseness Compact expressions Compact expressions

DrawbacksDrawbacks

ComplexityComplexity In writing rulesIn writing rules In evaluating rulesIn evaluating rules

CoherenceCoherence MaintainMaintain soundness soundness

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Expressing Imperfection

Damasio, Pan, Stoilos & Straccia (2008)Damasio, Pan, Stoilos & Straccia (2008)

Different Different non-booleannon-boolean rules can be encoded rules can be encoded Generalize the idea of (truth) Generalize the idea of (truth) degreedegree Generalize the idea of logic Generalize the idea of logic operatoroperator

Their work has led to Their work has led to fuzzyRuleMLfuzzyRuleML

What about Evaluation?What about Evaluation?

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(Truth) Degrees Different Different modelsmodels generalize {T,F} generalize {T,F}

High order models may combine different typesHigh order models may combine different types

0“false”

1“true”

0“false”

1“true”

0“false”

1“true”

0“false”

1“true”

Real Value [0,1]

Interval [0,1]2

Distribution [0,1] → [0,1] Type II fuzzy set

Imprecise Distribution [0,1] → [0,1]3

Type III fuzzy set

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Evaluators Predicates are no longer “just” true (or false)Predicates are no longer “just” true (or false)

Information may come from Information may come from differentdifferent sources: sources:

FactsFacts : “a priori” information : “a priori” information Embedded EvaluatorEmbedded Evaluator

• Possibly externalPossibly external RulesRules : chaining : chaining

⇒

Age > 18

⇒

Degrees have to be Degrees have to be mergedmerged ( (∩∩ )) May be May be missingmissing** ( (⊘⊘)) May be May be discounteddiscounted** ( () by confidence) by confidence May May overrideoverride** others ( others ())

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Operators

α

α

isA

Bet

high

amount

isA

Extract

$n

number

= 3

value

⊗ ∧

⊗

LiteralLiteral and and LogicalLogical Evaluators (Operators) are Evaluators (Operators) are configurableconfigurable Implementation Implementation kindkind** chosen individually chosen individually

• With With defaultdefault**• Refined using Refined using argumentsarguments**

Bet( $a : amount Bet( $a : amount isis highhigh, $n : number , $n : number Extraction( value Extraction( value ==== 3 ) 3 )

  

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Full Entailment

GeneralizedGeneralizedPropagationPropagation** policy policy ““Propagate Propagate iffiff true” not suitable true” not suitable

ImplicationImplication is evaluated is evaluated

Modus PonensModus Ponens is evaluated is evaluated Using Premise and ImplicationUsing Premise and Implication

α

α

⊗

β

β

⇒

→

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Engine extensionCore component : Core component : FactoryFactory

Builds Builds DegreesDegrees Builds (coherent) Builds (coherent) OperatorsOperators

Provides Provides PropagationPropagation Policy Policy• PASSPASS, , HOLDHOLD, , DROPDROP

Provides Provides MergeMerge Strategy Strategy• Handles missing valuesHandles missing values• Discounts and OverridesDiscounts and Overrides

AttributesAttributes are passed to the factory are passed to the factory

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f-RuleML Attributes / 1@@degreedegree

AppliesApplies--toto : : Evaluators, Operators, RulesEvaluators, Operators, Rules ValueValue : : any any RoleRole : : prior, constant value (may be merged withprior, constant value (may be merged with

other contributions)other contributions) FactoryFactory:: parses the value to return a Degreeparses the value to return a Degree

@@kindkind AppliesApplies--toto : : Operators, (Evaluators), RulesOperators, (Evaluators), Rules ValueValue : : depends on logic familydepends on logic family Role Role : : select the type of operatorselect the type of operator FactoryFactory:: chooses the actual implementation chooses the actual implementation

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f-RuleML Attributes / 2@@argsargs

AppliesApplies--toto : : Evaluators, (Operators), (Rules)Evaluators, (Operators), (Rules) ValueValue : : String String Role Role : : additional initialization parameters for the additional initialization parameters for the

evaluator / operatorevaluator / operator FactoryFactory:: passes the String to the constructorpasses the String to the constructor

@@defaultdefault AppliesApplies--toto : : Operators, Evaluators, RulesOperators, Evaluators, Rules ValueValue : : n.a.n.a. Role Role : : forces the default optionsforces the default options FactoryFactory:: ignores other initialization parametersignores other initialization parameters

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Examples / 1

rulerule "Attr1""Attr1"

whenwhen

Toss ( side == Toss ( side == @[ degree=0.5 ]@[ degree=0.5 ] “heads”) “heads”)

thenthen

// ...// ...

endend    rulerule "Attr2""Attr2"

whenwhen

Coin( weight Coin( weight << @[args=”tol=0.2,unit=g”]@[args=”tol=0.2,unit=g”]

10 ])

oror @[ kind=”Max” ]@[ kind=”Max” ]

Dice( faces == 4)Dice( faces == 4)

thenthen ... ... endend   

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f-RuleML Attributes / 3

@@idid AppliesApplies--toto : : Evaluators, OperatorsEvaluators, Operators ValueValue : : IDID Role Role : : unique identifier. unique identifier.

May be used to chain rules May be used to chain rules (i.e. a rule entails the truth of a constraint)(i.e. a rule entails the truth of a constraint)

FactoryFactory:: n.a.n.a.

@@filterfilter AppliesApplies--toto : : Operators, Evaluators, RulesOperators, Evaluators, Rules ValueValue : : ** Role Role : : selects the rule-propagation strategyselects the rule-propagation strategy FactoryFactory:: chooses the actual implementationchooses the actual implementation

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f-RuleML Attributes / 4@@booleanboolean

AppliesApplies--toto : : Evaluators, Operators, RulesEvaluators, Operators, Rules ValueValue : : “true” | “false” (optional)“true” | “false” (optional) Role Role : : the result is approximated with a booleanthe result is approximated with a boolean FactoryFactory:: casts the actual Degree to a Degreecasts the actual Degree to a Degree

modelling T (resp. F)modelling T (resp. F)

@@crispcrisp AppliesApplies--toto : : Operators, Evaluators, RulesOperators, Evaluators, Rules ValueValue : : “true” | “false” (optional)“true” | “false” (optional) Role Role : : forces the canonical evaluationforces the canonical evaluation FactoryFactory:: the result is cast to boolean andthe result is cast to boolean and

propagation is halted on falsepropagation is halted on false

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Examples / 2rulerule "Attr3""Attr3"

whenwhen

$c : Coin( )$c : Coin( )

Toss ( coin ==Toss ( coin == @[ crisp] @[ crisp] $c ) $c )

thenthen

insertinsert($c,”id_lucky”)($c,”id_lucky”)

endend   

rulerule "Attr4""Attr4"

whenwhen

$c : Coin( this $c : Coin( this is @[ id=”id_lucky”] “lucky”)

thenthen

......

endend   

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f-RuleML Attributes / 5@@mergemerge

AppliesApplies--toto : : Evaluators, Operators, RulesEvaluators, Operators, Rules ValueValue : : ** Role Role : : Degree fusion strategyDegree fusion strategy FactoryFactory:: chooses the actual implementationchooses the actual implementation

@@missingmissing AppliesApplies--toto : : Evaluators, Operators, RulesEvaluators, Operators, Rules ValueValue : : ** Role Role : : Missing value completion strategyMissing value completion strategy FactoryFactory:: chooses the actual implementationchooses the actual implementation

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f-RuleML Attributes / 5@@overrideoverride

AppliesApplies--toto : : Evaluators, Operators, RulesEvaluators, Operators, Rules ValueValue : : ** Role Role : : ““Defeat” degree strategyDefeat” degree strategy FactoryFactory:: chooses the actual implementationchooses the actual implementation

@@discountdiscount AppliesApplies--toto : : Evaluators, Operators, RulesEvaluators, Operators, Rules ValueValue : : ** Role Role : : Confidence-based strategyConfidence-based strategy FactoryFactory:: chooses the actual implementationchooses the actual implementation

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Examples / 3

rulerule "Attr5""Attr5"

whenwhen

Toss( side == “heads”,Toss( side == “heads”,this is @[ degree=”[0.4,0.6]”,

merge=”Intersect”, missing=”OWA” ]

“probable” )

thenthen

......

endend   

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Work–in–progressExtending RETE: Extending RETE:

@id

@filter

Evaluator(@kind,@args)

Degree[ ]

∩,,⊘

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Conclusionsf-RuleMLf-RuleML supports Imperfect Rules: supports Imperfect Rules:

Rules Rules definedefine the the abstractabstract constraints constraints Attributes Attributes specifyspecify the the concreteconcrete semantics semantics

Evaluation requires more customizationsEvaluation requires more customizations Additional Additional attributesattributes can be used can be used

Different logics can be used by setting the Different logics can be used by setting the defaultdefault values appropriately values appropriately Individual rules can override themIndividual rules can override them