Post on 16-Oct-2020
Answer Set Programming Graphs for Answer Set Programs Constructionof Explanation Graphs from Extended Dependency Graphs ChoosingAssumptions Answer Set Programming Graphs for Answer Set ProgramsConstruction of Explanation Graphs from Extended Dependency GraphsChoosing Assumptions
1
Construction of Explanation Graphs from ExtendedDependency Graphs for Answer Set Programs
Ella Albrecht, Patrick Krumpelmann, Gabriele Kern-Isberner
Lehrstuhl Informatik I, Technische Universitat Dortmund
11. September 2013
1
Overview
Answer Set Programming
Graphs for Answer Set Programs2.1 Extended Dependency Graph2.2 Explanation Graph
Construction of Explanation Graphs from Extended Dependency Graphs
Choosing Assumptions
2
Outline
Answer Set Programming
Graphs for Answer Set Programs2.1 Extended Dependency Graph2.2 Explanation Graph
Construction of Explanation Graphs from Extended Dependency Graphs
Choosing Assumptions
Answer Set Programming 3
Answer Set Programming
Definition (Extended Logic Program)
An extended logic program P is a set of rules r of the formr : h← a1, . . . ,am,not b1, . . . ,not bn.
I body(r) = {a1, . . . ,am,b1, . . . ,bn}I pos(r) = {a1, . . . ,am}, neg(r) = {b1, . . . ,bn}I herbrand basis H (P): set of all grounded literals of P
Definition (Answer Set)
I M ⊆H (P): consistent set of literals
I M is closed under P if head(r) ∈M whenever pos(r)⊆M andneg(r)∩M = /0 for all rules r ∈ P
I M is an answer set of P if M is closed and minimal w.r.t. set inclusionAnswer Set Programming 4
Outline
Answer Set Programming
Graphs for Answer Set Programs2.1 Extended Dependency Graph2.2 Explanation Graph
Construction of Explanation Graphs from Extended Dependency Graphs
Choosing Assumptions
Graphs for Answer Set Programs 5
Extended Dependency Graphs
Definition (Extended Dependency Graph)
The EDG for a logic program P is a directed graph EDG (P) = (V ,E )with nodes V and edges E ⊆ V ×V ×{+,−}.
a1
b2
c3 c4
d5 e6 f 7
++
+
+
++
+
+
− −
−
−
−r1 : a← not b.
r2 : b← not a.
r3 : c ← not a.
r4 : c ← e.
r5 : d ← c .
r6 : e← not d .
r7 : f ← e, not f .
Graphs for Answer Set Programs 6
Extended Dependency Graphs
Definition (Extended Dependency Graph)
The EDG for a logic program P is a directed graph EDG (P) = (V ,E )with nodes V and edges E ⊆ V ×V ×{+,−}.
a1
b2
c3 c4
d5 e6 f 7
a1
b2
c3 c4
d5 e6 f 7
++
+
+
++
+
+
− −
−
−
−r1 : a← not b.
r2 : b← not a.
r3 : c ← not a.
r4 : c ← e.
r5 : d ← c .
r6 : e← not d .
r7 : f ← e, not f .
Graphs for Answer Set Programs 6
Explanation Graph
Definition (Explanation Graph)
An Explanation Graph for a literal a ∈H p ∪H n in a program P w.r.t. an answer setM and an assumption U ⊆H (P) is a directed graph G = (V ,E)
I node set V ⊆H p ∪H n ∪{>,⊥,assume}I edge set E ⊆ V ×V ×{+,−}I H p = {a+ | a ∈H (P)}, H n = {a− | H (P)}.
k+
>
+
l−
⊥
−c−
assume
−
j+
k+
>
l−
⊥
+
−
+ −
Graphs for Answer Set Programs 7
Outline
Answer Set Programming
Graphs for Answer Set Programs2.1 Extended Dependency Graph2.2 Explanation Graph
Construction of Explanation Graphs from Extended Dependency Graphs
Choosing Assumptions
Construction of Explanation Graphs from Extended Dependency Graphs 8
Transformation EDG ⇒ EG
1. Removing irrelevant edges and nodes
2. Marking nodes
3. Construct Explanation Graphs differing five cases of transformation
4. Determination of assumptions
5. Fitting graph to chosen assumption
6. Continue marking and construction process
Construction of Explanation Graphs from Extended Dependency Graphs 9
Transformation EDG ⇒ EG - Removing irrelevant edgesand nodes
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
−
−
f 6−
I Irrelevant edge: An edge(aki ,a
lj ,s) is irrelevant if
ν(aki ) ν(alj ) s
green green −green red +red green +red red −
I Irrelevant node: A node aki isirrelevant if ν(aki ) = red and thereexists l > 0 where ν(ali ) = green.
Construction of Explanation Graphs from Extended Dependency Graphs 10
Transformation EDG ⇒ EG - Removing irrelevant edgesand nodes
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
−
−
f 6−
I Irrelevant edge: An edge(aki ,a
lj ,s) is irrelevant if
ν(aki ) ν(alj ) s
green green −green red +red green +red red −
I Irrelevant node: A node aki isirrelevant if ν(aki ) = red and thereexists l > 0 where ν(ali ) = green.
Construction of Explanation Graphs from Extended Dependency Graphs 10
Transformation EDG ⇒ EG - Removing irrelevant edgesand nodes
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
−
−
f 6−
I Irrelevant edge: An edge(aki ,a
lj ,s) is irrelevant if
ν(aki ) ν(alj ) s
green green −green red +red green +red red −
I Irrelevant node: A node aki isirrelevant if ν(aki ) = red and thereexists l > 0 where ν(ali ) = green.
Construction of Explanation Graphs from Extended Dependency Graphs 10
Transformation EDG ⇒ EG - Removing irrelevant edgesand nodes
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
f 6−
I Irrelevant edge: An edge(aki ,a
lj ,s) is irrelevant if
ν(aki ) ν(alj ) s
green green −green red +red green +red red −
I Irrelevant node: A node aki isirrelevant if ν(aki ) = red and thereexists l > 0 where ν(ali ) = green.
Construction of Explanation Graphs from Extended Dependency Graphs 10
Transformation EDG ⇒ EG - Removing irrelevant edgesand nodes
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
f 6−
I Irrelevant edge: An edge(aki ,a
lj ,s) is irrelevant if
ν(aki ) ν(alj ) s
green green −green red +red green +red red −
I Irrelevant node: A node aki isirrelevant if ν(aki ) = red and thereexists l > 0 where ν(ali ) = green.
Construction of Explanation Graphs from Extended Dependency Graphs 10
Transformation EDG ⇒ EG - Removing irrelevant edgesand nodes
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
f 6testtt−
I Irrelevant edge: An edge(aki ,a
lj ,s) is irrelevant if
ν(aki ) ν(alj ) s
green green −green red +red green +red red −
I Irrelevant node: A node aki isirrelevant if ν(aki ) = red and thereexists l > 0 where ν(ali ) = green.
Construction of Explanation Graphs from Extended Dependency Graphs 10
Transformation EDG ⇒ EG - Removing irrelevant edgesand nodes
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
I Irrelevant edge: An edge(aki ,a
lj ,s) is irrelevant if
ν(aki ) ν(alj ) s
green green −green red +red green +red red −
I Irrelevant node: A node aki isirrelevant if ν(aki ) = red and thereexists l > 0 where ν(ali ) = green.
Construction of Explanation Graphs from Extended Dependency Graphs 10
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
I Start marking process at nodeswithout incoming edges
I Transformation of fact nodes akia+i
>
+
a+
>
+
e+
>
+
I Transformation of unfoundednodes alja−j
⊥
−
b−
⊥
−
g−
⊥
−
Construction of Explanation Graphs from Extended Dependency Graphs 11
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
I Start marking process at nodeswithout incoming edges
I Transformation of fact nodes akia+i
>
+
a+
>
+
e+
>
+
I Transformation of unfoundednodes alja−j
⊥
−
b−
⊥
−
g−
⊥
−
Construction of Explanation Graphs from Extended Dependency Graphs 11
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
I Start marking process at nodeswithout incoming edges
I Transformation of fact nodes akia+i
>
+
a+
>
+
e+
>
+
I Transformation of unfoundednodes alja−j
⊥
−
b−
⊥
−
g−
⊥
−
Construction of Explanation Graphs from Extended Dependency Graphs 11
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
testtt
testtt
I Start marking process at nodeswithout incoming edges
I Transformation of fact nodes akia+i
>
+
a+
>
+
e+
>
+
I Transformation of unfoundednodes alja−j
⊥
−
b−
⊥
−
g−
⊥
−
Construction of Explanation Graphs from Extended Dependency Graphs 11
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
I Start marking process at nodeswithout incoming edges
I Transformation of fact nodes akia+i
>
+
a+
>
+
e+
>
+
I Transformation of unfoundednodes alja−j
⊥
−
b−
⊥
−
g−
⊥
−
Construction of Explanation Graphs from Extended Dependency Graphs 11
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
testtt
testtt
I Start marking process at nodeswithout incoming edges
I Transformation of fact nodes akia+i
>
+
a+
>
+
e+
>
+
I Transformation of unfoundednodes alja−j
⊥
−
b−
⊥
−
g−
⊥
−
Construction of Explanation Graphs from Extended Dependency Graphs 11
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7
I A green node aki can be marked ifI all predecessor nodes alj marked
I if ν(alj ) = green: there exists n,such that anj marked
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7
I A green node aki can be marked ifI all predecessor nodes alj marked
I if ν(alj ) = green: there exists n,such that anj marked
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7testtt testtt
I A green node aki can be marked ifI all predecessor nodes alj marked
I if ν(alj ) = green: there exists n,such that anj marked
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7testtt testtt
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7testtt testtt
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
a+i
EG (aj1) EG (aj2) ,,, EG (ajn)
s1 s2
sn
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7testtt testtttesttt
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
c+
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7testtt testtttesttt
testtt testtt
+
−
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
c+
EG (a) EG (b)
+
−
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7testtt testtttesttt
testtt testtt
+
−
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
c+
a+ b−
> ⊥
+
−
+ −
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e4
g0
testtt testtt
testtt
testtt
c2 f 7testtt testtt
I Local consistent Explanation(LCE) to a literal ai : minimal setof literals which directly influencetruth value of ai
I LCE for a green node: all directpredecessors
I Transformation of dependentgreen nodes aki
c+
a+ b−
> ⊥
+
−
+ −
f +
g−
⊥
−
−
Construction of Explanation Graphs from Extended Dependency Graphs 12
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e5
g0
c2 f 7
d3 d4
testtt testtt
testtt
testtt
testtt testtt
I A red node aki can be marked if
I at least one predecessor node aljmarked
I if ν(alj ) = red : for all n node anjmarked
Construction of Explanation Graphs from Extended Dependency Graphs 13
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e5
g0
c2 f 7
d3 d4
testtt testtt
testtt
testtt
testtt testtt
testtt testtt
I A red node aki can be marked if
I at least one predecessor node aljmarked
I if ν(alj ) = red : for all n node anjmarked
Construction of Explanation Graphs from Extended Dependency Graphs 13
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e5
g0
c2 f 7
d3 d4
testtt testtt
testtt
testtt
testtt testtt
testtt testtt
I A red node aki can be marked if
I at least one predecessor node aljmarked
I if ν(alj ) = red : for all n node anjmarked
I LCE for a red node akiI considering all nodes representing aiI one predecessor of each node
Construction of Explanation Graphs from Extended Dependency Graphs 13
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e5
g0
c2 f 7
d3 d4
testtt testtt
testtt
testtt
testtt testtt
testtt testtt
I Transformation of dependent red nodesaki
a−i
EG(b1) EG(b2) ,,, EG(bn)
s1 s2
sn
Construction of Explanation Graphs from Extended Dependency Graphs 13
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e5
g0
c2 f 7
d3 d4
testtt testtt
testtt
testtt
testtt testtt
testtt testtttesttt testtt
I pn(d3) ={c2,e5}
pn(d4) ={f 7}
L(d) ={{c, f },{e, f }}
d−
EG(c) EG(f )
Construction of Explanation Graphs from Extended Dependency Graphs 13
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e5
g0
c2 f 7
d3 d4
testtt testtt
testtt
testtt
testtt testtt
testtt testtttesttt testtt
testtt testtt testtt
+
++
I pn(d3) ={c2,e5}
pn(d4) ={f 7}
L(d) ={{c, f },{e, f }}
d−
EG(c) EG(f )
+
−
Construction of Explanation Graphs from Extended Dependency Graphs 13
Transformation EDG ⇒ EG
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
a1 b0
e5
g0
c2 f 7
d3 d4
testtt testtt
testtt
testtt
testtt testtt
testtt testtttesttt testtt
testtt testtt testtt
+
++
I pn(d3) ={c2,e5}
pn(d4) ={f 7}
L(d) ={{c, f },{e, f }}
d−
EG(c) EG(f )
c+
a+ b−
f +
g−
⊥> ⊥
+
+
+
−
+
−
−−
Construction of Explanation Graphs from Extended Dependency Graphs 13
Well-founded model
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
testtt testtt
testtt testtt
testtt
testtt
testtt testtt
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
− − Proposition
All unmarked nodes areundefined by thewell-founded model.
Construction of Explanation Graphs from Extended Dependency Graphs 14
Outline
Answer Set Programming
Graphs for Answer Set Programs2.1 Extended Dependency Graph2.2 Explanation Graph
Construction of Explanation Graphs from Extended Dependency Graphs
Choosing Assumptions
Choosing Assumptions 15
Assumptions
I Tentative assumptions in an EDG:T A (G ′) = {ai | aki ∈V ′,ν(aki ) = red ,(aki ,a
lj ,−)∈ E ,aki not marked}
I G = (V ,E ): validly colored EDGI G ′ = (V ′,E ′): EDG after removing irrelevant edges and nodes and
marking nodes
I Assumption: Subset U ⊆T A (G ′) such that all nodes in the EDGcan be marked.
I Indeterminateness arises from cycles ⇒ consideration ofdependencies within and between cycles
Lemma
In an EDG without cycles all nodes can be marked.
Choosing Assumptions 16
Choosing Assumptions - Approach 1
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
testtt testtt
testtt testtt
testtt
testtt
testtt testtt
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
I One tentative assumption from eachminimal cycle
I C1 ={h8,p16,n14,o15}
C2 ={j10,k11, l12,p16}
C3 ={h8, i9, j10,k11, l12,m13,n14,o15}
C4 ={r18,s19}
I Assumptions(G ′) ={{p,k, r},{p, i , r},{p,m, r},{p,o, r},{o,k, r}}
Choosing Assumptions 17
Choosing Assumptions - Approach 2
a1
c2
b0
d3
e5 f 6
d4
f 7
g0
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
testtt testtt
testtt testtt
testtt
testtt
testtt testtt
+
−
−
−−
− −
−
−
− −
+
−
−−
−−
−
−
−−
+
−−
I Linked cycles: strongly connected components
I 1. Start at linked cycles without externalincoming edges
2. Find critical nodes3. Determine pre-conditions of each
critical node4. Find successful combinations of
pre-conditions5. Determine pre-condition paths and path
assumptions6. Built assumptions for the linked cycle
Choosing Assumptions 18
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−
−−
+
−−
I Linked cycles: strongly connected components
I 1. Start at linked cycles without externalincoming edges
2. Find critical nodes3. Determine pre-conditions of each
critical node4. Find successful combinations of
pre-conditions5. Determine pre-condition paths and path
assumptions6. Built assumptions for the linked cycle
Choosing Assumptions 18
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−
−−
+
−−
I Linked cycles: strongly connected components
I 1. Start at linked cycles without externalincoming edges
2. Find critical nodes3. Determine pre-conditions of each
critical node4. Find successful combinations of
pre-conditions5. Determine pre-condition paths and path
assumptions6. Built assumptions for the linked cycle
Choosing Assumptions 18
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−
−−
+
−−
I Linked cycles: strongly connected components
I 1. Start at linked cycles without externalincoming edges
2. Find critical nodes3. Determine pre-conditions of each
critical node4. Find successful combinations of
pre-conditions5. Determine pre-condition paths and path
assumptions6. Built assumptions for the linked cycle
Choosing Assumptions 18
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−
−−
+
−−
I Critical nodes: {j10,n14}I Successfull combination: {i9,p16}, {m13,p16}I PA (path(i9)) ={i ,o}
PA (path(p16)) ={p,k,o}
PA (path(m13)) ={m,k}
I Assumptions(LCred ) ={{i ,p},{i ,k},{i ,o},{o,p},{o,k},{o}}∪{{p,m},{p,k},{k,m},{k},{o,m},{o,k}}
I µAssumptions(LCred ) = {{o},{k},{i ,p},{p,m}}
Choosing Assumptions 19
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−
−−
+
−−
I Critical nodes: {j10,n14}I Successfull combination: {i9,p16}, {m13,p16}I PA (path(i9)) ={i ,o}
PA (path(p16)) ={p,k,o}
PA (path(m13)) ={m,k}
I Assumptions(LCred ) ={{i ,p},{i ,k},{i ,o},{o,p},{o,k},{o}}∪{{p,m},{p,k},{k,m},{k},{o,m},{o,k}}
I µAssumptions(LCred ) = {{o},{k},{i ,p},{p,m}}
Choosing Assumptions 19
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−−
+
−−
I Assumption: o
o−
assume
−
h+
o−
assume
−
−
I Two cases:
1. All nodes of the EDG marked ⇒ DONE2. There are still unmarked nodes in the
EDG ⇒ Remove marked nodes andstart again with the determination ofassumptions
Choosing Assumptions 20
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−−
+
−−
testtt−
− −
+
−
−−
−−
−
−−
+
−−
I Assumption: o
o−
assume
−
h+
o−
assume
−
−
I Two cases:
1. All nodes of the EDG marked ⇒ DONE2. There are still unmarked nodes in the
EDG ⇒ Remove marked nodes andstart again with the determination ofassumptions
Choosing Assumptions 20
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−−
+
−−
testtt−
− −
+
−
−−
−−
−
−−
+
−−−
I Assumption: o
o−
assume
−
h+
o−
assume
−
−
I Two cases:
1. All nodes of the EDG marked ⇒ DONE2. There are still unmarked nodes in the
EDG ⇒ Remove marked nodes andstart again with the determination ofassumptions
Choosing Assumptions 20
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−−
+
−−
testtt−
− −
+
−
−−
−−
−
−−
+
−−
I Assumption: o
o−
assume
−
h+
o−
assume
−
−
I Two cases:
1. All nodes of the EDG marked ⇒ DONE2. There are still unmarked nodes in the
EDG ⇒ Remove marked nodes andstart again with the determination ofassumptions
Choosing Assumptions 20
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−−
+
−−
testtt−
− −
+
−
−−
−−
−
−−
+
−−
I Assumption: o
o−
assume
−
h+
o−
assume
−
−
I Two cases:
1. All nodes of the EDG marked ⇒ DONE2. There are still unmarked nodes in the
EDG ⇒ Remove marked nodes andstart again with the determination ofassumptions
Choosing Assumptions 20
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−−
+
−−
testtt−
− −
+
−
−−
−−
−
−−
+
−−
testtt
I Assumption: o
o−
assume
−
h+
o−
assume
−
−
I Two cases:
1. All nodes of the EDG marked ⇒ DONE2. There are still unmarked nodes in the
EDG ⇒ Remove marked nodes andstart again with the determination ofassumptions
Choosing Assumptions 20
Choosing Assumptions - Approach 2
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
h8 i9
j10
k11l12m13
n14
o15
p16
q17 r18
s19
−
− −
+
−
−−
−−
−
−−
+
−−
testtt−
− −
+
−
−−
−−
−
−−
+
−−
testtt testtt
testtt
testtttesttttesttt
testtt
testtt
testtt
testtt testtt
testtt
−
− −
+
−
−−
−−
−
−−
+
−−
I Assumption: o
o−
assume
−
h+
o−
assume
−
−
I Two cases:
1. All nodes of the EDG marked ⇒ DONE2. There are still unmarked nodes in the
EDG ⇒ Remove marked nodes andstart again with the determination ofassumptionsChoosing Assumptions 20
Summary
I Construction of Explanation Graphs from validly colored ExtendedDependency Graphs
I Marking nodesI Five types of transformation
I Choosing assumptionsI Approach 1: linear, but non-minimalI Approach 2: exponential, but minimal
Choosing Assumptions 21