Temporal Graph Plan

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The Logic of ReachabilityDavid E. SmithAri K.
Jónsson
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Apologies
No results ideas formalism Adverse
reactions Logic
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Outline
Background Motivation Simple Reachability Mutual
Exclusion Practical Matters
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Graphplan
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Graphplan
Reachability! (optimistic achivability)
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Mutual Exclusion

• Two actions are mutex if
• one clobbers the others effects or
  preconditions
• they have mutex preconditions
• Two propositions are mutex if
• all ways of achieving them are mutex

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Why Reachability?
Pruning reachable ? achievable Guidance distance
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TGP
Actions Real duration Concurrent
Heater
Thrust
closevalve
comlink
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TGP Limitations
Actions Preconditions hold throughout Effects
occur at end Affected propositions undefined
during No exogenous conditions
pre1
pre2
A
eff1
eff2
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TGP Limitations
Actions Preconditions hold throughout Effects
occur at end Affected propositions undefined
during No exogenous conditions
pre2
cond3
A
Conservative!
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TGP Limitations
Actions Preconditions hold throughout Effects
occur at end Affected propositions undefined
during No exogenous conditions
A
eff
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TGP Limitations
Actions Preconditions hold throughout Effects
occur at end Affected propositions undefined
during Exogenous Conditions
Inititial Conditions
t0
At(Pkg1, BOS-PO)
At(Truck1, BOS)
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Impact?
Actions Preconditions hold throughout Effects
occur at end Affected propositions undefined
during Exogenous Conditions
t0600z
t1300z
Closed(SJC)
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Monotonicity of Reachability
0
1
2
3
x p q x
x p q x
x p q x r
A
A
A
B
B
B
x
C
Propositions actions monotonically increase
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Monotonicity of Mutex
0
1
2
3
x p q x
x p q x
x p q x r
A
A
A
B
B
B
x
C
Mutex relationships monotonically decrease
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Cyclic Plan Graph
Propositions
Actions
x1 p1 q1 x0 r3
A0 B0 C2
Earliest start times
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Cyclic Plan Graph
Propositions
Actions
x1 p1 q1 x0 r3
A0 B0 C2
Earliest end time
2
2
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Impact?
Actions Preconditions hold throughout Effects
occur at end Affected propositions undefined
during Exogenous Conditions
t0600z
t1300z
Closed(SJC)
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Windows of Reachability
Actions
Propositions
A0,3,6,9 B11,? C
p0,5,8.1,16 q2,17 r3,?
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Windows of Mutex
Actions
Propositions
3,4x11,?
A0,3,6,9 B11,? C
p0,5,8.1,16 q2,17 r3,?
0,3x11,?
0,3x3,4
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Action Model
cond1
Duration Parallel (pre) Conditions over intervals
Effects over intervals
cond2
cond3
A
eff
A
5A
5A
A cond r0, p0,2 eff ?r(0,2), r2, e2
p
A
r
r
e
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Semantics
P stops holding
A cond r0, p0,2 eff ?r(0,2), r2, e2
p
A
r
r
e
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Semantics
p stops holding
A cond r0, p0,2 eff ?r(0,2), r2, e2
p
???
A
???
r
r
???
e
???
Incomplete
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Exogenous Conditions
Inititial Conditions
X cond eff At(Pkg1, BOS-PO)0 At(Truck1,
BOS)0 Closed(SJC)0600,1300 Visible(NGC132)
0517,0642
t0
At(Pkg1, BOS-PO)
At(Truck1, BOS)
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Outline
Motivation Simple Reachability Mutual
Exclusion Practical Matters
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Possibility Reachability
?(pt) ? p is logically possible at t
?(pt) ? p is reachable at t
?(richtomorrow) ?(richtomorrow)
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Possibility Reachability
?(pt) ? p is logically possible at t
?(pt) ? p is reachable at t
Extend to Intervals
?(pi) ? ? t? i ?(pt)
?(pi) ? ? t? i ? (pt)
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Basic Axioms
Facts are possible reachable
pi ? ?(pi)
pi ? ?(pi)
Negations are not
pi ? ? t? i ?(pt)
pi ? ? t? i ?(pt)
Transitivity
?(pt) ? (pt ? qt) ? ?(qt)
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Basic Axioms
Actions
at ? Cond(at) ? Eff(at)
Exogenous conditions
X0
Closure of X
(Eff(x0) pt) ?(pi)

\
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Example
r
X0
p
?p
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Closure
r
X0
p
?p
? p
? p
closure
? r
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Basic
? r
basic
?p
? ?p
r
X0
p
?p
? p
? p
closure
? r
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Persistence
?(pi) ? meets(i,j) ? ?(pj) ? ?(pij)
? r
basic
?p
? ?p
r
X0
p
?p
? p
? p
closure
? r
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Persistence
?(pi) ? meets(i,j) ? ?(pj) ? ?(pij)
? r
basic persist
?p
??p
r
X0
p
?p
? p
? p
closure
? r
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Actions
Reachability
?Cond(at) ? ?Eff(at) ? ?(at)
Conjunctive optimism
?p1i1 ? ? ?pnin ? ?(p1i1 ? ? pnin)
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Action Application
?Cond(at) ? ?Eff(at) ? ?(at)
A cond r0, p0,2 eff ?r(0,2), r2, e2
?A
? r
?p
??p
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Action Application
?Cond(at) ? ?Eff(at) ? ?(at)
A cond r0, p0,2 eff ?r(0,2), r2, e2
? e
? r
?A
? r
?p
??p
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Persistence Again
?(pi) ? meets(i,j) ? ?(pi) ? ?(pij)
? e
? r
?A
? r
?p
??p
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Persistence (revised)
?(pi) ? meets(i,j) ? ?(pi) ? ?(pij)
?at ?(at) ? pi ? PersistEff(at) ? meets(i,j)
? ?(pi) ? ?(pij)
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Persistence
?at ?(at) ? pi ? PersistEff(at) ? meets(i,j)
? ?(pi) ? ?(pij)
? e
? r
?A
? r
?p
??p
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Outline
Motivation Simple Reachability Mutual
Exclusion Practical Matters
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Mutual Exclusion
M(p1t1, , pntn)
Intervals
M(p1i1, , pnnn) ? ? t1? i1, , tn? in
M(p1t1, , pntn)
Conjunctive optimism
(?p1i1 ? ? ?pnin ) ? M(p1i1, , pnnn) ?
?(p1i1 ? ? pnin)
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Logical Mutex
(?1 ? ? ?n) ? M(?1, , ?n)
Consequences
M(pt, pt)
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Consequences
(?1 ? ? ?n) ? M(?1, , ?n)
Consequences
A cond p? eff e?
At ? pt?
At ? ete
M(At, pt?)
M(At, et?)
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Consequences
(?1 ? ? ?n) ? M(?1, , ?n)
Consequences
A cond p?
At ? pt?
Bt ? pte
B cond p?
M(At, Bt??)
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Implication Mutex
M(?1, , ?n) ? (? ? ?1) ? M(?, , ?n)
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Implication Mutex Example
M(?1, , ?n) ? (?1 ? ?1) ? M(?1, , ?n)
M(?1, , ?n) ? (? ? ?1) ? M(?, , ?n)
p1 q1
A1 B1
e2 f2
Example
M(p1,q1)
A cond p0 eff e1
B cond q0 eff f1
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Implication Mutex Example
M(?1, , ?n) ? (?1 ? ?1) ? M(?1, , ?n)
M(?1, , ?n) ? (? ? ?1) ? M(?, , ?n)
p1 q1
A1 B1
e2 f2
Example
M(p1,q1)
A cond p0 eff e1
At ? pt
B cond q0 eff f1
Bt ? qt
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Implication Mutex Example
M(?1, , ?n) ? (?1 ? ?1) ? M(?1, , ?n)
M(?1, , ?n) ? (? ? ?1) ? M(?, , ?n)
p1 q1
A1 B1
e2 f2
Example
M(p1,q1)
A cond p0 eff e1
At ? pt
M(A1,q1)
B cond q0 eff f1
Bt ? qt
M(p1,B1)
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Implication Mutex Example
M(?1, , ?n) ? (?1 ? ?1) ? M(?1, , ?n)
M(?1, , ?n) ? (? ? ?1) ? M(?, , ?n)
p1 q1
A1 B1
e2 f2
Example
M(p1,q1)
A cond p0 eff e1
At ? pt
M(A1,q1)
B cond q0 eff f1
Bt ? qt
M(p1,B1)
M(A1,B1)
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Implication Mutex for Intervals
M(?1, , ?n) ? (? ? ?1) ? M(?, , ?n)
M(?1i1, , ?nin) ? j t ?t ? ? t1? i1 ?1t1
? M(?j, , ?nin)
p1,3) q2,3)
A1,3) B2,3)
e f
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Explanatory Mutex
?? (? ? ?1) ? M(?, , ?n) ? M(?1, , ?n)
If all ways of proving ?1 are mutex with ?2, ,
?n ? M(?1, , ?n)
A
p
B
p1 q1
A1 B1
e2 f2
A
p
?
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Outline
Motivation Simple Reachability Mutual
Exclusion Practical Matters
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Limiting Mutex
Reachable propositions Time spread
M(p2, q238)
0,2
236,240
p
q
A
Mutex spread theorem ?
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CSP?
Actions
Propositions
A0,3,6,9 B11,? C
p0,5,8.1,16 q2,17 r3,?
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Initial Domains
Actions
Propositions
A0, ?) B0, ?) C0, ?)
p0, ?) q0, ?) r0, ?)
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Interval Elimination
Actions
Propositions
A0, ?) B0, ?) C0, ?)
p0,5,8.1, ?) q0, ?) r0, ?)
?Reachability ? Mutex
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Mutex Representation
M(At, Bt2,t10)
M(A, B, 2,10)
M(A, B, ?, I)
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Final Remarks
Reachability simple Mutex surprisingly
simple complex realization Questions limiting
mutex CSP implementation? mutex representation TGP