Knowledge BasesKnowledge Baseswith Missing, Uncertain, with Missing, Uncertain,

Fuzzy, or Incomplete Fuzzy, or Incomplete InformationInformation

Based onBased onFuzzy Databases: Principles and ApplicationsFuzzy Databases: Principles and Applications

by Frederick E. Petryby Frederick E. Petry&&

Chapter 7 of Chapter 7 of Aritificial IntelligenceAritificial Intelligence on Reasoning with Uncertain or Incomplete Informationon Reasoning with Uncertain or Incomplete Information

by George F. Luger & William A. Stubblefield by George F. Luger & William A. Stubblefield &&

Querying Disjunctive Databases in Polynomial TimeQuerying Disjunctive Databases in Polynomial Timeby Lars E. Olsonby Lars E. Olson

Outline of TopicsOutline of Topics(A Sampling of Some of the Approaches)(A Sampling of Some of the Approaches)

Imprecision in Conventional DatabasesImprecision in Conventional Databases Null valuesNull values Range valuesRange values

Probabilistic DatabasesProbabilistic Databases Reasoning with Uncertain or Incomplete Reasoning with Uncertain or Incomplete

InformationInformation Fuzzy Set TheoryFuzzy Set Theory

Basic definitionsBasic definitions OperationsOperations

Models of FuzzinessModels of Fuzziness Similarity-based modelsSimilarity-based models Possibility-based modelsPossibility-based models Rough setsRough sets

Disjunctive DatabasesDisjunctive Databases

Missing or Incomplete InformationMissing or Incomplete Information

Missing or incomplete values (i.e. null Missing or incomplete values (i.e. null values) can have many meaningsvalues) can have many meanings Unknown (e.g. salary is not known)Unknown (e.g. salary is not known) Not applicable (e.g. student ID for non-students)Not applicable (e.g. student ID for non-students) Does not exist (e.g. middle name of a person)Does not exist (e.g. middle name of a person) Maybe, possibly, probably, probably not, …Maybe, possibly, probably, probably not, …

Representation (Representation ()) Normally: same symbol, same meaningNormally: same symbol, same meaning Neither Neither nor nor == is reasonable is reasonable Can have marked nulls: Can have marked nulls: ii = = ii

Three Valued Logic for NullsThree Valued Logic for Nulls

Also: x y for x or y null and one of , , , , , .

OK for Cost<10 PartsBut not for Cost<10 Cost10 Parts

Range ValuesRange Values

25 25 AgeAge R, but 35 R, but 35 Age Age R is R is unknown.unknown.

Relations don’t have Relations don’t have duplicates, but both 45-55 duplicates, but both 45-55 elements must remain in elements must remain in AgeAge R.R.

AgeAge Person=P2Person=P2 R < R < AgeAge Person=P4Person=P4 R, but it is unknown whether R, but it is unknown whether AgeAge Person=P2Person=P2 R < R < AgeAge Person=P3 Person=P3 R.R.

For some unknowns,a range may be known.

R = Person Age P1 25 P2 30-35 P3 32-40 P4 45-55 P5 45-55 P6 65

Interesting Consequences:

Probabilistic DatabasesProbabilistic Databases

CleanupCosts:Site Costs…M357 0.4 [$ 50 – 75 K]

0.6 [$ 75 – 100 K]

L121 0.3 [$ 200 – 300 K]0.5 [$ 300 – 400 K]0.2 [* – *]

…

Associate probabilitieswith attributes.

Each group sumsto one. (Missingprobabilities added.)

Example: estimatesof 10 experts ofcleanup costs –4 estimated M357at 50-75K and 6estimated it at75-100K.

Operators extended by allowing conditions on both values and probabilities.Example: SITE COST: 0.5 200K CleanupCosts = {SITE:L121, …}.

Certainty TheoryCertainty Theory(Stanford Certainty Factor Algebra)(Stanford Certainty Factor Algebra)

Attempts to formalize the human approach to uncertainty Attempts to formalize the human approach to uncertainty when thinking of facts and rules as being “highly probably,” when thinking of facts and rules as being “highly probably,” “unlikely,” “possible,” …“unlikely,” “possible,” …

Simple assumptions for creating confidence measures and Simple assumptions for creating confidence measures and for combining confidence measures:for combining confidence measures: Evidence is either for or against a particular hypothesis – Evidence is either for or against a particular hypothesis –

ranges from 1 to -1ranges from 1 to -1 MB(H|E): Measure of Belief of hypothesis (H) given evidence (E).MB(H|E): Measure of Belief of hypothesis (H) given evidence (E). MD(H|E): Measure of Disbelief of H given E.MD(H|E): Measure of Disbelief of H given E. Either:Either:

1 > MB(H|E) > 0 while MD(H|E) = 0, or1 > MB(H|E) > 0 while MD(H|E) = 0, or 1 > MD(H|E) > 0 while MB(H|E) = 01 > MD(H|E) > 0 while MB(H|E) = 0

Then: the certainty factor, CF(H|E) = MB(H|E) – MD(H|E).Then: the certainty factor, CF(H|E) = MB(H|E) – MD(H|E). Premises of rules are conjunctions and disjunctions of Premises of rules are conjunctions and disjunctions of

uncertain facts and are modeled respectively by min and max.uncertain facts and are modeled respectively by min and max. Conjunction of Premises: CF(P1 Conjunction of Premises: CF(P1 P2) = min(CF(P1), CF(P2)) P2) = min(CF(P1), CF(P2)) Disjunction of Premises: CF(P1 Disjunction of Premises: CF(P1 P2) = max(CF(P1), CF(P2)) P2) = max(CF(P1), CF(P2))

Certainty TheoryCertainty Theory(Stanford Certainty Factor Algebra)(Stanford Certainty Factor Algebra)

Conclusions of rules have certainty factors based on Conclusions of rules have certainty factors based on assumed complete certainty of premises, but are then assumed complete certainty of premises, but are then tempered multiplicatively by the uncertainty of premises.tempered multiplicatively by the uncertainty of premises.

Example of a rule: (P1 Example of a rule: (P1 P2) P2) P3 P3 R1 (.7) R1 (.7) R2 (.3) R2 (.3) Result 1 (R1) has “estimated certainty” of .7; Result 2 (R2) … .3Result 1 (R1) has “estimated certainty” of .7; Result 2 (R2) … .3 P1, P2, and P3 assumed completely certain for estimating result P1, P2, and P3 assumed completely certain for estimating result

certaintiescertainties Example of certainty calculation assuming P1, P2, and P3 Example of certainty calculation assuming P1, P2, and P3

have certainty .6, .4, and .2 respectivelyhave certainty .6, .4, and .2 respectively CF(P1 CF(P1 P2) = min(.6, .4) = .4 P2) = min(.6, .4) = .4 CF((P1 CF((P1 P2) P2) P3) = max(.4, .2) = .4 P3) = max(.4, .2) = .4 When R1 is added as a derived fact: CF(R1) = CF((P1 When R1 is added as a derived fact: CF(R1) = CF((P1 P2) P2) P3) P3)

.7 = .4 .7 = .4 .7 = .28 .7 = .28 When R2 is added: CF(R1) = .4 When R2 is added: CF(R1) = .4 .3 = .12 .3 = .12

Certainty TheoryCertainty Theory(Stanford Certainty Factor Algebra)(Stanford Certainty Factor Algebra)

Combinations of multiple rules for the same Combinations of multiple rules for the same evidence have rules …evidence have rules …

If CF(R1) and CF(R2) are positive, then If CF(R1) and CF(R2) are positive, then CF(R1) + CF(R2) – (CF(R1) CF(R1) + CF(R2) – (CF(R1) CF(R2)) CF(R2))

If CF(R1) and CF(R2) are negative, then If CF(R1) and CF(R2) are negative, then CF(R1) + CF(R2) + (CF(R1) CF(R1) + CF(R2) + (CF(R1) CF(R2)) CF(R2))

Otherwise (CF(R1) + CF(R2)) / (1 – min(|CF(R1)|, |CF(R2)|))Otherwise (CF(R1) + CF(R2)) / (1 – min(|CF(R1)|, |CF(R2)|)) … … with desirable properties.with desirable properties.

Results are always between 1 and -1.Results are always between 1 and -1. Combining contradictory certainty factors cancel each Combining contradictory certainty factors cancel each

other.other. Combined positive (negative) certainty factors increase Combined positive (negative) certainty factors increase

(decrease) monotonically, as would be expected.(decrease) monotonically, as would be expected.

Certainty TheoryCertainty Theory(Stanford Certainty Factor Algebra)(Stanford Certainty Factor Algebra)

Example:Example:

MotherOfChild(x,y) MotherOfChild(x,z) FullSibling(x,y) (.9)LiveInSameHoushold(x,y) BirthdateWithin15Years(x,y) FullSibling(x,y) (.8)

MotherOfChild(mary,john) (.7)MotherOfChild(mary,alice) (.8)LiveInSameHousehold(john,alice) (.9)BirthDateWithin15Years(john,alice) (1.0)

1st FullSibling Rule: min(.7, .8) .9 = .632nd FullSibling Rule: min(.9, 1.0) .8 = .72Positive Certainty Factor Rule: .63 + .72 – (.63 .72) = 1.35 - .45 = .9

Certainty TheoryCertainty Theory(Stanford Certainty Factor Algebra)(Stanford Certainty Factor Algebra)

Example:Example:

The theory does not attempt to do “correct” reasoning, but The theory does not attempt to do “correct” reasoning, but rather to model the combining of confidence measures in rather to model the combining of confidence measures in an approximate, heuristic, and informal way.an approximate, heuristic, and informal way.

MotherOfChild(x,y) MotherOfChild(x,z) FullSibling(x,y) (.9)LiveInSameHoushold(x,y) BirthdateWithin15Years(x,y) FullSibling(x,y) (.8)

MotherOfChild(mary,john) (.7)MotherOfChild(mary,alice) (.8)LiveInSameHousehold(john,alice) (.9)BirthDateWithin15Years(john,alice) (-.6)

1st FullSibling Rule: min(.7, .8) .9 = .632nd FullSibling Rule: min(.9, -.6) .8 = -.48Mixed Certainty Factor Rule: (.63 + (-.48))/(1 – min(|.63|, |-.48|)) = .15/.52 = .29

Fuzzy Set TheoryFuzzy Set TheoryBasic Idea: extend the notion of a characteristic function:

For an ordinary set S: CharS(x) : U {0, 1} e.g. for U = abcde, 01101 is {b, c, e}

For a fuzzy set F: CharF(x) : U [0, 1] i.e. the degree of set membership ranges from 0 to 1 and represents an opinion, a judgment, or the confidence in membership. The membership function is usually denoted by F(x).

e.g. estimates of high cost of environmental cleanup: HIGH = {0.0/1K, 0.125/5K, 0.5/10K, 0.8/25K, 0.9/50K, 1.0/100K}

Can interpolate to get estimates for other values, e.g. 0.95/75K

Can extend to get other estimates, e.g. HIGH(x) = 1.0 for x 100K

Fuzzy Set OperationsFuzzy Set Operations EqualityEquality

A = B iff A = B iff AA(x) = (x) = BB(x)(x) {0.0/0K, 10/10K} = {0.0/0K, 0.5/5K, 1.0/10K}{0.0/0K, 10/10K} = {0.0/0K, 0.5/5K, 1.0/10K}

ContainmentContainment A A B iff B iff AA(x) (x) BB(x)(x) {0.0/0K, 0.8/10K} {0.0/0K, 0.8/10K} {0.0/0K, 0.5/5K, 0.9/10K} {0.0/0K, 0.5/5K, 0.9/10K}

ComplementComplement AA = {(1- = {(1-AA(x)/x}(x)/x} {0.7/10K}{0.7/10K} = {0.3/10K} i.e. the judgment that 10K is not in the set is 0.3 = {0.3/10K} i.e. the judgment that 10K is not in the set is 0.3

(when the judgment that it is in the set is 0.7)(when the judgment that it is in the set is 0.7) Set UnionSet Union

AABB(x) = Max((x) = Max(AA(x), (x), BB(x))(x)) {0.0/0K, 0.8/10K} {0.0/0K, 0.8/10K} {0.0/0K, 0.5/5K, 0.9/10K} = {0.0/0K, 0.5/5K, {0.0/0K, 0.5/5K, 0.9/10K} = {0.0/0K, 0.5/5K,

0.9/10K}0.9/10K} Set IntersectionSet Intersection

AABB(x) = Min((x) = Min(AA(x), (x), BB(x))(x)) {0.0/0K, 0.8/10K} {0.0/0K, 0.8/10K} {0.0/0K, 0.5/5K, 0.9/10K} = {0.0/0K, 0.4/5K, {0.0/0K, 0.5/5K, 0.9/10K} = {0.0/0K, 0.4/5K,

0.8/10K}0.8/10K} Concentration (CON(A))Concentration (CON(A))

CON(A)CON(A)(x) = ((x) = (AA(x))(x))22

Concentrates fuzzy elements by reducing the membership gradeConcentrates fuzzy elements by reducing the membership grade Dilation (DIL(A))Dilation (DIL(A))

DIL(A)DIL(A)(x) = ((x) = (AA(x))(x))1/21/2

Dilates fuzzy elements by increasing the membership gradeDilates fuzzy elements by increasing the membership grade

Similarity-Based ModelsSimilarity-Based Models

Allows us to measure nearness of domain Allows us to measure nearness of domain elementselements

Uses a similarity relationUses a similarity relationSim(x, y) A B C D E

A 1.0 0.8 0.4 0.5 0.8 B 0.8 1.0 0.4 0.5 0.9 C 0.4 0.4 1.0 0.4 0.4 D 0.5 0.5 0.4 1.0 0.5 E 0.8 0.9 0.4 0.5 1.0

The similarity of A and E is 0.8.

Properties of Similarity RelationsProperties of Similarity Relations

Reflexive: Sim(x,x) = 1.0Reflexive: Sim(x,x) = 1.0 Symmetric: Sim(x,y) = Sim(y,x)Symmetric: Sim(x,y) = Sim(y,x) Transitive: Sim(x,z) = MaxTransitive: Sim(x,z) = Max{y}{y}(Min[Sim(x,y), Sim(y,z)])(Min[Sim(x,y), Sim(y,z)])

Reflexive, Symmetric, Transitive: implies an equivalence Reflexive, Symmetric, Transitive: implies an equivalence relationrelation

Sim(x, y) A B C D E

A 1.0 0.8 0.4 0.5 0.8 B 0.8 1.0 0.4 0.5 0.9 C 0.4 0.4 1.0 0.4 0.4 D 0.5 0.5 0.4 1.0 0.5 E 0.8 0.9 0.4 0.5 1.0

e.g. Sim(A,C) = Max(Min[Sim(A,A), Sim(A,C)], Min[Sim(A,B), Sim(B,C)], Min[Sim(A,C), Sim(C,C)],e.g. Sim(A,C) = Max(Min[Sim(A,A), Sim(A,C)], Min[Sim(A,B), Sim(B,C)], Min[Sim(A,C), Sim(C,C)],Min[Sim(A,D), Sim(D,C)], Min[Sim(A,E), Sim(E,C)]) = Max(Min[1.0, 0.4], Min[0.8, 0.4], Min[0.4, 1.0],Min[Sim(A,D), Sim(D,C)], Min[Sim(A,E), Sim(E,C)]) = Max(Min[1.0, 0.4], Min[0.8, 0.4], Min[0.4, 1.0],Min[0.5, 0.4], Min[0.8, 0.4]) = Max(0.4, 0.4, 0.4, 0.4, 0.4) = 0.4.Min[0.5, 0.4], Min[0.8, 0.4]) = Max(0.4, 0.4, 0.4, 0.4, 0.4) = 0.4.

Equivalence ClassesEquivalence Classesfor Similarity Relationsfor Similarity Relations

Sim(x, y) A B C D E

A 1.0 0.8 0.4 0.5 0.8 B 0.8 1.0 0.4 0.5 0.9 C 0.4 0.4 1.0 0.4 0.4 D 0.5 0.5 0.4 1.0 0.5 E 0.8 0.9 0.4 0.5 1.0

The -level set S ={(x,y)|Sim(x,y) } isan equivalence relation.

e.g. S0.8 = {{A,B,E}, {C}, {D}}

Partion Tree:

{A, B, C, D, E} = 0.4

{A, B, D, E} {C} = 0.5

{A, B, E} {D} {C} = 0.8

{A} {B, E} {D} {C} = 0.9

{A} {B} {E} {D} {C} = 1.0

Fuzzy RelationsFuzzy Relations

Fuzzy Relation: (essentially) subset of the cross product(D1) (D2) … (Dn), where (D) is the power setof the domain D.

Fuzzy Tuple: any member of of a fuzzy relation.

Interpretation (of a fuzzy tuple): (essentially) select anyone element from each set of the tuple. The space ofinterpretations is the cross product D1 D2 … Dn.

Fuzzy QueriesFuzzy Queries

Relational AlgebraRelational Algebra Expressions plus a clause defining minimum Expressions plus a clause defining minimum

thresholds for attributes over similarity relationsthresholds for attributes over similarity relations Evaluate and merge tuples (forming sets in the Evaluate and merge tuples (forming sets in the

powerset) so long as there are no minimum powerset) so long as there are no minimum threshold value violationsthreshold value violations

Relational CalculusRelational Calculus An augmentation similar to the augmentation An augmentation similar to the augmentation

for relational algebrafor relational algebra Predicate-calculus statements with minimum Predicate-calculus statements with minimum

level values specified for relations and/or level values specified for relations and/or comparison expressionscomparison expressions

Query ExampleQuery Example(Find Consensus in Opinion Survey)(Find Consensus in Opinion Survey)

How well to the experts and residents agree about the effects of pollutants?

Similarity Relation for EFFECTSimilarity Relation for EFFECT

Then if Thres(EFFECT) 0.85, for example, then the valuesin the same blocks of the partition determined by = 0.85are conisdered to be equal. Thus, for this example,{Minimal, Limited, Tolerable, Moderate} are equal, as are{Major, Extreme}.

Consensus of ExpertsConsensus of Experts

R1 = POLLUTANT,NAME,EFFECTTYPE=ExpertSurvey with Thres(EFFECT) 0.85 and Thres(NAME) 0.0

Consensus of ResidentsConsensus of Residents

R2 = POLLUTANT,NAME,EFFECTTYPE=ResidentSurvey with Thres(EFFECT) 0.85 and Thres(NAME) 0.0

Consensus in SurveyConsensus in SurveyR1

R2

R1 POLLUTANT,EFFECT R2

with Thres(EFFECT) 0.85 and Thres(NAME) 0.0

Possibility-Based ModelsPossibility-Based Models

Similarity-based modelsSimilarity-based models Capture imprecision in distinction of elements Capture imprecision in distinction of elements

of domainsof domains Example: is “limited” different from Example: is “limited” different from

“moderate”?“moderate”? Possibility-based modelsPossibility-based models

Capture uncertainty about the elements Capture uncertainty about the elements themselves (or about relationships among the themselves (or about relationships among the elements)elements)

Example: Is the salary high? How much is Example: Is the salary high? How much is “likes(person, movie)” certain?“likes(person, movie)” certain?

Possibility TheoryPossibility Theory(for Single Values)(for Single Values)

The available information about the value of a single-The available information about the value of a single-valued attribute A for a tuple t is represented by a valued attribute A for a tuple t is represented by a possibility distribution possibility distribution A(t)A(t) on D on D {e}, where D is the {e}, where D is the domain of attribute A and e is an extra element which domain of attribute A and e is an extra element which stands for the case when the attribute does not apply to stands for the case when the attribute does not apply to t.t.

The possibility distribution The possibility distribution A(t)A(t) defines a mapping from defines a mapping from D D {e} to [0, 1] {e} to [0, 1]

Example: middle-aged might have a possibility Example: middle-aged might have a possibility distribution defined over the domain age = [0, 100], for distribution defined over the domain age = [0, 100], for every tuple t that contains age, as follows:every tuple t that contains age, as follows:

middle-aged(t) =

0, for age(t) 35(45 – 35)/10, for 35 age(t) 451, for 45 age(t) 50(55 – 50)/5, for 50 age(t) 550, for 55 age(t)

{(e) = 0

Possibility DistributionsPossibility Distributionsfor Usual Situationsfor Usual Situations

Possibility DistributionsPossibility Distributionsfor Ill-known Valuesfor Ill-known Values

ComputingComputingPossibile and Certain ConditionsPossibile and Certain Conditions

(from the possibility distribution (from the possibility distribution A(t)A(t)

and a set S – ordinary or fuzzy)and a set S – ordinary or fuzzy)

Possible(d) = minPossible(d) = minddDD((A(t)A(t)(d), (d), SS(d))(d)) PossibilityDegree(d) = supPossibilityDegree(d) = supddDDPossible(d)Possible(d) Certain(d) = maxCertain(d) = maxddDD(1 - (1 - A(t)A(t)(d), (d), SS(d))(d)) CertaintyDegree(d) = infCertaintyDegree(d) = infddDDCertain(d)Certain(d)

Possibility-Based QueriesPossibility-Based Queries

Note: Unlike earlier, where middle-aged was the possibility distribution, here middle-aged is a fuzzy set and John’s age has a possibility distribution (e.g. we are assuming that someone has an idea what the possibilities are for John’s age).

If John’s age and the fuzzy predicate P(middle-aged) are represented as:

Then we can evaluate the condition: John’s age = middle-aged as:

Possible

PossibilityDegree

Certain

CertaintyDegree

Rough SetsRough Sets

Captures indiscernibility among set valuesCaptures indiscernibility among set values Formalized as follows:Formalized as follows:

R is an indiscernibility relation or equivalence R is an indiscernibility relation or equivalence relation among indiscernible values.relation among indiscernible values.

[x][x]RR denotes the equivalence class of R containing denotes the equivalence class of R containing x.x.

The elementary sets are the equivalence classes of The elementary sets are the equivalence classes of R.R.

The definable sets are unions of elementary sets.The definable sets are unions of elementary sets. Lower and upper approximations of a set X over the Lower and upper approximations of a set X over the

elements of the universe U are respectivelyelements of the universe U are respectively RX = { xRX = { xU | [x]U | [x]RR X } X } RX = { xRX = { xU | [x]U | [x]RRX X } }

Examples for Rough SetsExamples for Rough SetsIndiscernibility Relation, R

An equivalence class:[SAND]R = {BEACH, COAST, SAND}

Elementary sets

A definable set: [URBAN]R [SAND]R

= {BUILDING, URBAN, BEACH, COAST, SAND}

For X = {BUILDING, URBAN, SAND},lower approximation: RX = {BUILDING, URBAN}upper approximation: RX = {BUILDING, URBAN, BEACH COAST, SAND}

Rough Relational ModelRough Relational Model

A rough relation R is (essentially) a subset of A rough relation R is (essentially) a subset of the set cross product the set cross product (D(D11) ) (D(D22) … ) … (D(Dnn).).

A rough tuple tA rough tuple tii for R is (d for R is (di1i1, d, di2i2, …, d, …, dinin) where ) where ddijij D Djj..

An interpretation (aAn interpretation (a11, a, a22, …, a, …, ann) of a rough ) of a rough tuple ttuple tii = (d = (di1i1, d, di2i2, …, d, …, dinin) is any value ) is any value assignment such that aassignment such that ajj d dijij for all j. for all j.

Tuples tTuples tii = (d = (di1i1, d, di2i2, …, d, …, dinin) and t) and tkk = (d = (dk1k1, d, dk2k2, , …, d…, dknkn) are redundant if [d) are redundant if [dijij] = [d] = [dkjkj] for all j.] for all j.

Sample Rough RelationSample Rough RelationRough Relation X:

Tuple i

FEATURE equivalence classes:

COUNTRY equivalence classes:

Interpretations of Tuple i: (U157, US, SAND) … (U157, UNITED-STATES, SAND) …

Tuple i and the tuple(U157, {USA, MEXICO}, {ROAD, BEACH})are redundant.

Rough SelectionRough SelectionA=aX is a rough relation Y where RY = { tX | i[ai] = j[bj] }, aia, bjt(A) RY = { tX | i[ai] j[bj] }, aia, bjt(A)

X

FEATURE={BEACH, ROAD}X

ID COUNTRY FEATURE U157 {USA, MEXICO} {SAND, ROAD} M007 MEXICO {COAST, AIRPORT}( U147 US {BEACH, ROAD, URBAN} )

Rough ProjectionRough ProjectionB(X) is the rough relation { t(B) | t X }.Rough projection must maintain lower and upper approximations withequivalent tuples, one from the lower and one from the upper beingmaintained as the lower approximation.

COUNTRYXUS{US, MEXICO}MEXICOBELIZE{BELIZE, INT}

X

FEATUREFEATURE={BEACH, ROAD}X {SAND, ROAD}( {SAND, ROAD, URBAN} )

Rough JoinRough JoinThe specification of is lengthy, but straightforward: for testing whether tuples join, lowerapproximation uses equality (as usual) and upper approximation uses rough set subset (asusual), but in either direction. Both lower and upper tuples are compared.

(COUNTRY,FEATUREFEATURE={SAND,ROAD}X)COUNTRY

(COUNTRY,FEATUREFEATURE=MARSHX) US {SAND, ROAD, URBAN} MARSH US {SAND, ROAD, URBAN} {MARSH, LAKE} US {SAND, ROAD, URBAN} {MARSH, PASTURE, RIVER}( {US, MEXICO} {SAND, ROAD} MARSH )( {US, MEXICO} {SAND, ROAD} {MARSH, LAKE} )

( {US, MEXICO} {SAND, ROAD} {MARSH, PASTURE, RIVER} )

(COUNTRY,FEATUREFEATURE={SAND,ROAD}X) {US, MEXICO} {SAND, ROAD} MEXICO {SAND, ROAD}( US {SAND, ROAD, URBAN} )

(COUNTRY,FEATUREFEATURE=MARSHX) US MARSH( US {MARSH, LAKE} )( USA {MARSH, PASTURE, RIVER} )

Rough Relational Set OperatorsRough Relational Set Operators Rough difference, X – Y:Rough difference, X – Y:

RT = { tRT = { tRX | tRX | tRY } andRY } and RT = { tRT = { tRX | tRX | tRY }RY }

Rough union, X Rough union, X Y: Y: RT = { tRT = { tRXRXRY } andRY } and RT = { tRT = { tRXRXRY }RY }

Rough intersection, X Rough intersection, X Y: Y: RT = { tRT = { tRXRXRY } andRY } and RT = { tRT = { tRXRXRY }RY }

FEATURECOUNTRY=USX {MARSH, LAKE} MARSH {MARSH, PASTURE, RIVER} {FOREST, RIVER} {SAND, ROAD, URBAN}( {SAND, ROAD} )

X

- FEATURECOUNTRY=MEXICOX {SAND, ROAD} BEACH SAND

Note that ( {SAND, ROAD} ), which is an upperapproximation after the selection, disappears inthe projection, and is thus not removed in thedifference operation.