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Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

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Page 1: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

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Page 2: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

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or

B C

C B AÅ

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B

D

B

D

D

D

D

LOF i

LOF −i

(a) Graph Representation of Overlap Constraints.Ó

(b) Graph Representation of LOFs.Ó

D

Interval A

Relationship D

D

Notation:

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{

Page 10: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

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Page 11: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

Relationship:

NC<(A , B)

NC2(A , B)

D(A , B)

B

B

P1

P1 P2

P2 P3

P4 P5Aß

BP1 P2 P3

E = [ { P1, P2 }, { }, { } ]

E = [ {P1, P2}, { }, { P3 } ]

E = [ {P1, P2}, {P4}, {P3,P5} ]

Supporting Probes:à

Evidence Representation:

O(A , B) á P4 P5

B

P1 P2 P3

E = [ {P1, P2}, {P4}, {P3,P5} ]

T�UWVBX��T½�â� 8���Q7Pk_@bG15A:O7VzijOS>�ACBDHD<�eFB®Ed>/g¦BDN/1 8`oJEf?@13>�eF1 V�OSH)Q7>xw�>@BD13HDoWQWbc;=1FbfQ7BCEGOS>�ACN@EG_�-

O(A,B)ã

BAä

BB

C(B,A)å

C(A,B)å

B

D(A,B)

B

NC2(A,B)

Aä BA

äB

ND<(A,B) ND>(A,B)

B

ND1(A,B)NC>(A,B)

B

NC<(A,B)

B

NB(A,B)

B

: fuzzy endinglevel: 0

level: 1

level: 2

level: 3

A. B: DNA fragmentsä

[{},S2,{}][{},{},S3][S1,{},{}]

| : sets of probes

[S1,{},S3]

[S1,S2,S3] [S1,S2,S3] [S1,S2,S3]

[S1,S2,{}]

[S1,S2,S3]

[{},S2,S3]

[{},{},{}]

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Page 13: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

E< < DA�

B C < <A�

B C< <

EC < <D

A�

B C< < EC < <DA�

B C< < EC < <D

LOFLOFiLOF

j

(a) Given LOFi and LOFj.

A�

B C< <A�

B C< <

LOFLOFi�

E C< <D

LOF−j

A�

B C< <

E <<D

A�

B C< <

E C< <D

Known Partial Ordering:

LOFLOFi = LOF−j

(c) Given

LOFLOFi j =(b) Given

six total orderings are possible.

LOF ; one total ordering.

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Page 14: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

LOFLOFi A�

B C�

< <

B C�

< < D

LOFLOFiA�

B C�

< < < D

LOFj�

LOFj�

=

Example:

LOFLOFiLOF

j�=

and

imply�

Rule

(a) For Ordering Relationships between Points.

(b) For Ordering Relationships between Intervals.

imply�

["ND<(A,B)" in LOFi]�

["ND>(A,B)" in LOFj]�and

LOFLOFiLOF

j�=

["O(A,B)" in LOFi,j]�and

A� B

ND>(A,B)

LOFLOFiLOF j

O(A,B)�

A�

B

A�

B

ND<(A,B)

AND

Example 1

imply�and["ND<(A,B)" in LOFi]�

["C(A,B)" in LOFj]�

NO MERGE!!!!!!!!!!!!A�

B

ND<(A,B)

A�

B

C(A,B)�

imply�and["ND<(A,B)" in LOFi]�

["D(A,B)" in LOFj]�

CONFLICT!!!!!!!!!!!!�

A�

B

D(A,B)

A�

B

ND<(A,B)

Rule

Rule

Rule

Example 2

Example 3

["B < C" in LOFi]�

["B < C" in LOFj]�

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Page 15: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

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Page 16: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

NC2(A,C)

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start(B) < start(C)start(B) < end(C)end(B) > start(C)end(B) < end(C)

start(A) < start(C)start(A) < end(C)end(A) ? start(C)end(A) < end(C)

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Page 17: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

NC2(A , B)

AY

BZP1

[P2[

P3[

E = [ {P1, P2}, { }, { P3 } ]

A BZP1

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NC<(A , B) \

E = [ { P1, P2 }, { }, { } ]E = [ { P1, P2 }, { }, { } ]

A BZP3

NC>(A , B) \

E = [ { }, { }, { P3 } ]

AND]

IMPLY

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E = [ { P1, P2 }, {P3 }, {P4 } ] E = [ {P5 }, { }, { P6 } ]

P6`P5

`B C

a

A_ P1`

P2`

CP6`

P5`

P3

NC2(B,C) b

D(A , C)

D(A , B)

E = [{P1,P2}, {P3,P5}, {P6}]

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Page 18: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

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Page 24: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

NC>(A,B)

Ak B

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Ak

B

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Ak B

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C(B,A)lcontains

Ak

B

C(A,B)lcontains

O(A,B)m

Ak

B

overlaps

Ak

B

D(A,B)disjoint

Ak

B

not−disjoint 1ND1(A,B)

Anot−disjoint>ND>(B,A)

B

A

not−disjoint<ND<(B,A)

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B A

not−contained>NC>(B,A)

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NC2(B,A)

A

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B

overlaps

Anot−disjoint 1

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ND1(B,A)

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B

disjoint

Ak

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NA(A,B)not−after

AB

NA(B,A)not−after

Ak

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C(A,B)lcontains

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A. Table of Fram Definition and Inversionk

LOFiLOF−i LOF−iLOF−i

NA(B<A)A is not after B B is not after A B is not disjoint with A

ND>(A,B) ND>(B,A)A is not disjoing with B

(1 endpoint relationship)

(3 endpoint relationships, and one is redundant)

(3 endpoint relationships, and one is redundant)

(4 endpoint relationships, and 2 are redundant)

(4 endpoint relationships, and 2 are redundant)

(4 endpoint relationships, and 1 is redundant)

(2 endpoint relationships, and both are essential)

(2 endpoint relationships, and 1 is redundant)

NA(A, B)

ND<(A, B)A is not disjoint with B B is not disjoint with A

ND<(B,A)

NA(B,A)NC2(A, B)A is not contained in B B is not contained in A

and B is nontained in Aand B is not contained in AC(B,A) C(B,A)B contains A B contains A

C(A,B)A contains B

C(A,B)A contains B

O(B,A)B overlaps with A

O(A,B)A overlaps with B

ND1(B,A)B is not disjoint with A

ND1(A, B)A is not disjoint with B

NC>(A, B) NC<(B,A)B is not contained in AB is not contained in A

NC<(A, B)A is not contained in B A is not contained in B

NC>(B,A) D(A,B) D(B,A)B is disjoint with AA is disjoing with B

(3 endpoint relationships, and 1 is redundant)

(2 endpoint relationships, and 1 is redundant) (4 endpoint relationships, and 3 are redundant)

s(A) ? s(B)s(A) < e(B)e(A) ? s(B)e(A) ? e(B)

s(A) > s(B)s(A) < e(B)e(A) > s(B)e(A) < e(B)

s(A) < s(B)s(A) < e(B)e(A) > s(B)e(A) > e(B)

s(A) < s(B)s(A) < e(B)e(A) > s(B)e(A) < e(B)

s(A) < s(B)s(A) < e(B)e(A) > s(B)e(A) ? e(B)

s(A) < s(B)s(A) < e(B)e(A) ? s(B)e(A) < e(B)

s(A) ? s(B)s(A) < e(B)e(A) > s(B)e(A) ? e(B)

s(A) ? s(B)s(A) < e(B)e(A) ? s(B)e(A) < e(B)

s(A) < s(B)s(A) < e(B)e(A) ? s(B)e(A) ? e(B)

s(A) < s(B)s(A) < e(B)e(A) < s(B)e(A) < e(B)

s(A) ? s(B)s(A) < e(B)e(A) > s(B)e(A) < e(B)

LOFi

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Page 25: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

No

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Oo No LOFi=LOFj

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ND1(A,B)

NC2(A,B)

ND<(A,B)

ND>(A,B)

D(A,B)

O(A,B)q

C(A,B)r

C(B,A)r

LOFiLOFj

NoND> −> C

NoNC< −> C

NoND1 −> CND1 −> C

ND< −> C

NC< −> C

NC> −> ND>

NC< −> ND<

NC< −> ND<

NC> −> ND>

If this is the case, the upgraded relationship is shown in the second entry.Note: For some cells even though LOFi and LOFj cannot be combined, one of the relationships can be upgraded to a more precise relationship.

rel1

rel2

NC<

NC>

ND1

NC2

ND<

ND>

D

O

no infono infono infono info

no infono infono info

NC<

no info

no info

NC>s

ND1s

NC2s

ND<s

ND> D O

no infono infono info

no info

no infono infono info

NC<(A,C)sno info

no infono infoNC<(C,A)s

no infono infono info

NC<(A,C)no infono infoNC<(C,A)

no info

no infono info

no info

no infono info no info

no infoNC>(C,A)

NC>(A,C)

no info

no infono infono infoNC>(C,A)

NC>(A,C)

no infono info NB(A,C)

no infono info

no infono info

NC<(A,C)sno infono infoNC<(C,A)s

no infono infoNC>(C,A)

NC>(A,C) NB(A,C)sNB(C,A)s no info

no info

NC2(A,C)

NC2(C,A)

no info

no infono info

no infono info

NC<(A,C)no infono info no info

Ct

no info

no info

NC<(A,C)sno infoNC<(C,A)s no info

no info

NC<(A,C)no infono infoD(C,A)

NC<(A,C)sno infoNC<(C,A)s

NC<(A,C)sno infoNC<(C,A)s

no infono info

NC>(C,A)

NC>(A,C)no infono infoNC>(C,A) NC>(C,A)

sNC>(A,C)s

no infoNC>(A,C)sNC>(C,A)s

no infono info

NC<(A,C)NC>(C,A)

no infoND1(A,C)s

NC<(A,C)s

NC<(C,A)s

NC>(C,A)

NC>(A,C)no infono infoD(C,A)

D(A,C) NC2(A,C)s

NC2(C,A)s

NC<(A,C)

NC<(C,A)

NC>(A,C)NC>(C,A)

no infono infono info

NC<(A,C)sno infoNC<(C,A)s

no info

no info

NB(A,C)sno info

NC<(C,A)s

no info

no infono info

no info

no info

NC>(C,A)sNC>(A,C)sno info

NB(C,A)s

NC>(C,A)sNC>(A,C)s

no infono info

NC>(A,C)

NC<(C,A)

no infono infoNC<(C,A)sD(A,C)

no infono info

NC>(A,C)s

D(C,A)

NC>(A,C)sNC<(C,A)s no info

no infoD(C,A)

D(A,C)

no infono info

no infoNC<(C,A)s

NC>(C,A)sNC>(A,C)sno info

NC2(A,C)s

NC2(C,A)s

no infono info

NC<(A,C)s

NC<(C,A)sno info

NC>(C,A)sNC>(A,C)sno info

ND1(A,C)s

no info

NC<(A,C)sNC>(C,A)sNC<(C,A)s

NC<(A,C)sno info

NC<(C,A)sND1(A,C)s NC<(A,C)

NC<(A,C)NC>(C,A)

D(C,A)

NC<(A,C)sND1(A,C)sNC<(C,A)

NC<(A,C)

no infoND1(A,C)

ND<(C,A)no info no info

NC>(C,A)sNC>(A,C)sno infoND1(A,C)s

D(A,C)NC>(C,A)sNC>(C,A)sNC<(A,C)s

NC<(A,C)s

NC>(C,A)

NC>(A,C)

ND1(A,C)

no infoND>(A,C)NC>(C,A)ND1(A,C)

D(A,C)NC>(A,C)sNC<(C,A)sNC<(C,A)s

NC>(A,C)sNC>(A,C)sNC<(C,A)sD(C,A)

D(A,C)no infono infoD(C,A)

D(A,C)

D(C,A)

NC>(A,C)NC<(C,A)

D(A,C)

D(C,A)NC>(A,C)

NC<(C,A)

NC<(A,C)s

NC<(C,A)sND1(A,C)s

NC>(C,A)sNC>(A,C)sND1(A,C)s

D(A,C)

D(C,A)

NC>(C,A)s NC2(A,C)

NC2(C,A)

ND1(A,C)ND1(A,C)

NC<(A,C)ND>(A,C)

NC<(A,C)s

NC>(C,A)ND<(C,A)

ND<(A,C)sND1(A,C)sno infoNC<(C,A)u

ND1(A,C)sND>(C,A)s

NC>(A,C) NC>(A,C)sno info

NC<(A,C)sNC>(C,A)sD(A,C)D(C,A)

ND<(A,C)ND>(C,A)NC>(A,C)NC<(C,A)

ND1(A,C)no infoC(C,A)

C(A,C)

C

NAs

NA

NA(A,C)

no infoNA(C,A)

NA(A,C)

NA(C,A)

NA(A,C)

NA(C,A)

NA(A,C)

NA(C,A)NA(A,C)NA(C,A) NA(C,A)

s

NA(C,A)s

NA(C,A)sNA(C,A)s

NA(C,A)s

NA(C,A)sNA(C,A)s

NA(A,C)sNA(A,C)s

NA(A,C)s

NA(A,C)s

NA(A,C)s

NA(C,A)sNA(A,C)s

NA(C,A)sNA(A,C)s

NA(A,C)sNA(C,A)s

NA(C,A)sNA(A,C)s

NA(C,A)NA(A,C)

NA(C,A)s

NA(C,A)sNA(C,A)sNA(C,A)s

NA(A,C)sNA(C,A)s

NA(C,A)sNA(C,A)sNA(C,A)s

NA(C,A)s

NA(A,C)s

NA(A,C)s

NA(A,C)sNA(A,C)sno info

NA(A,C)

NA(A,C)NA(C,A)

NA(A,C)

NA(A,C)NA(C,A)

NA(A,C)

NA(C,A)

NA(A,C)s

NA(A,C)sNA(C,A)s

NA(A,C)sNA(C,A)sNA(C,A)sNA(C,A)s

NA(A,C)s

NA(A,C)s

NA(C,A)s

NA(A,C)sNA(C,A)s

no infoNA(A,C)s

NA(C,A)s

no infoNA(A,C)

NA(C,A)s

NA(A,C)

NC<(A,C)s

NC>(C,A)

D(A,C)

rel1(B,A) and rel2(B,C)entry 3 =rel1(A,B) and rel2(B,C)entry 1 =

In each cell,

rel1(A,B) and rel2(C,B)entry 2 = rel1(B,A) and rel2(C,B)entry 4 =

NA(A,C)

NA(C,A)NA(A,C)sNA(C,A)

sNA(A,C)

NA(A,C)

B. Table of Frame Composition Inference Rules

C. Table of Transitivity Rules

ë$ß

Page 26: Electrical Engineering and Computer Science · n s. \[ ] `_ \_]] \]] \,]]], @] \

rel1

rel2

NC

<

NC

>

ND

1

NC

2

ND

<

ND

>

D O

NC

<N

C>

ND

1N

C2

ND

<

C

NA

NA

− − − −

− − − −− −−−

− −

− −− −

− −[s1,

{},{

}]

[s4,

{},{

}]

− − −−− −

− − − −− −

− − − −− −− −− −[{

},{}

,s6]

[{},

{},s

3]

− − −

− − − − − −− − − − − −

− −− −

[s1,

{},{

}]

[s4,

{},{

}]

[{},

{},s

6]

[{},

{},s

3]

− − − − − −

− −− − − − − −− −− − − − −

− −−−−

− −− −

− − − −− −

− − − −

− −

− [s4,

{},{

}]

[s1,

{},{

}]

[s4,

{},{

}][{

},{}

,s6]

[{},

{},s

3]

[s1,

{},{

}]

[s4,

{},{

}]

[{},

{},s

3][{

},{}

,s6]

[{},

s5,{

}]

ND

>

− − − − − − − − − −[{},

{},s

6]

[{},

{},s

3]

− −[{},

{},s

6]

[{},

{},s

3]

−− −−− − − −− [{},

s5,{

}]

[{},

{},s

6][{

},s5

,s3U

s6]

[{},

{},s

6]

− [{},

{},s

3][{

},{}

,s5U

s6]

[{},

{},s

5][s

5,{}

,{}]

[s5,

s2U

s6,s

3][{

},{}

,s6]

− [{},

{},s

3]

− − − − [s1,

{},{

}]

[s4,

{},{

}]−− − − − −− − − −[s

1,{}

,{}]

[s4,

{},{

}]

− − − −− − −[s1,

{},{

}]− [s

4,{}

,{}]

[{},

s5,{

}]

[s4,

{},{

}]

[s1U

s4,s

5,{}

]

[s1,

s2U

s4,s

5][{

},{}

,s5]

[s5,

{},{

}][s

4Us5

,{},

{}]

[s1,

{},{

}]− [s

4,{}

,{}]

− − − − − −[s1,

{},{

}]

[s4,

{},{

}]

− −−−[{},

{},s

6]

[{},

{},s

3]

− −[s1,

{},s

6]

[s4,

{},s

3]

− − − − − − − −[s1,

{},{

}]

[s4,

{},{

}]

[s1,

{},{

}][{

},{}

,s3]

[{},

{},s

6][s

4,{}

,{}]

[{},

{},s

6]

[{},

{},s

3][s

1,s2

Us4

,s6]

[s4,

s2U

s6,s

3][s

1,{}

,s6]

[s4,

{},s

3]

[{},

{},s

5]

[s5,

{},{

}]

[{},

{},s

6]

[{},

s2U

s6,s

3]

[{},

{},s

6]− − [{

},{}

,s3]

DO

− −[s1,

{},{

}]

−− [{},

{},s

3]−− − − − − [s

1Us2

Us4

,{},

{}]

[{},

{},s

2Us3

Us6

][s

2,s3

Us5

,s6]

[s4,

s5U

s1,s

2]

C

[s1,

{},{

}]

[s4,

{},{

}]

− −− − − − −− [{},

{},s

6][{

},{}

,s3]

− [{},

s2,{

}]

[s1,

{},{

}][{

},{}

,s6]

[{},

{},s

3][s

4,{}

,{}]

− − −[s1,

{},{

}]

[{},

{},s

3]

[s1U

s4,s

5,s3

Us6

][{

},s2

Us5

,{}]

[s1U

s4,s

2,s3

Us6

]

[{},

s2,s

3Us6

]

[s1,

s2U

s4,s

5][{

},{}

,s5U

s6]

[s5,

s2U

s6,s

3][s

4Us5

,{},

{}]

[{},

s2,s

3Us6

][{

},{}

,s3]

[{},

s2,{

}]

[s1,

{},{

}][{

},s2

,{}]

[s1U

s4,s

2,{}

]

[s1,

s2U

s4,s

5Us6

]

[s1,

{},{

}]

[s4,

{},{

}]

− −−−− − − −[{},

{},s

6]

[{},

{},s

3]

[s1,

{},s

6]

− − −− [s4,

{},s

3]

− − [{},

{},s

6][s

4,{}

,{}]

[s1U

s4,s

5,{}

][{

},s5

,s3U

s6]

[s1,

{},s

6]

[s4,

{},s

3]

− −[s1,

{},{

}]− [s

4,{}

,{}]

[{},

{},s

6]

− [{},

{},s

3]

[{},

{},s

5][s

5,{}

,{}]

[s4U

s5,s

2Us6

,s3]

[s1,

{},{

}][{

},{}

,s2]

[s2,

{},{

}]

[s2,

s3U

s5,s

6][s

4,s5

Us1

,s2]

[{},

{},s

2][s

2,{}

,{}]

[{},

{},s

2Us3

][s

1,s2

Us3

Us4

Us5

,s6]

[s4,

s1U

s2U

s5U

s6,s

3][s

1Us2

,s3U

s5,s

6][{

},{}

,s2]

[s2,

{},{

}][s

4,s1

Us5

,s2U

s3]

[s2,

{},{

}][{

},{}

,s2]

[s1,

s3U

s5,s

6]

[{},

s3U

s5,s

6][s

4,s5

Us1

,{}]

[s1,

{},{

}]

[s4,

{},{

}]

[s1,

s2U

s4,{

}]

In e

ach

cell,

entr

y 2

=en

try

1 =

entr

y 3

=en

try

4 =

evid

ence

of (

rel1

(A,B

) an

d re

l2(B

,C))

evid

ence

of (

rel1

(A,B

) an

d re

l2(C

,B))

evid

ence

of (

rel1

(B,A

) an

d re

l2(C

,B))

evid

ence

of (

rel1

(B,A

) an

d re

l2(B

,C))

[s1,

s2,s

3]

[s4,

s5,s

6]

− − [s4,

s5U

s1,s

3]

[s1U

s4,s

2,{}

]

D. T

able

of E

vide

nce−

Com

bina

tion

Rul

es fo

r T

rans

itivi

ty

ë9á