Albert Jónsson Forstöðumaður Eignastýringar Lífeyrissjóður starfsmanna ríkisins
The “Logic” of Reachability David E. Smith Ari K. Jónsson
description
Transcript of The “Logic” of Reachability David E. Smith Ari K. Jónsson
The “Logic” of ReachabilityDavid E. Smith
Ari K. Jónsson
Apologies
No resultsideas & formalism
Adverse reactions
“Logic”
Outline
Background & Motivation
Simple Reachability
Mutual Exclusion
“Practical Matters”
Graphplan
Expand plan graph
Derive mutex relationships
If goals are present & consistentsearch for a solution
Graphplan
Expand plan graph
Derive mutex relationships
If goals are present & consistentsearch for a solution
Reachability!(optimistic achivability)
Why Reachability?
Pruning¬reachable ¬achievable
Guidancedistance
TGP
ActionsReal durationConcurrent
Thrust
comlink
Heater
closevalve
TGP Limitations
ActionsPreconditions hold throughoutEffects occur at endAffected propositions undefined during
No exogenous conditions
A eff1
eff2
pre2
pre1
Monotonicity of Reachability
Propositions & actions monotonically increase
¬x
…
x
p
q
¬x
…
x
p
q
¬x
…
A
B
A
B
x
p
q
¬x
r
…
B
A
C
0 1 2 3
Monotonicity of Mutex
Mutex relationships monotonically decrease
x
p
q
¬x
…
x
p
q
¬x
…
A
B
A
B
x
p
q
¬x
r
…
B
A
C
0 1 2 3
¬x
…
Cyclic Plan Graph
x1
p1
q1
¬x0
r3
…
A0
B0
C2
Propositions Actions
Earliest start times
x1
p1
q1
¬x0
r3
…
A0
B0
C2
Cyclic Plan Graph
22
Propositions Actions
Earliest end time
Impact?
ActionsPreconditions hold throughoutEffects occur at endAffected propositions undefined during
Exogenous Conditions
Closed(SJC)t=0600z t=1300z
–5A +5A
A
≥5A
Apre2 cond3
pre1
eff
Windows of Reachability
Propositions Actions
A[0,3],[6,9]
B[11,]
C[…]…
…
p[0,5],[8.1,16]
q[2,17]…
r[3,]…
…
Windows of Mutex
A[0,3],[6,9]
B[11,]
C[…]…
…
p[0,5],[8.1,16]
q[2,17]…
r[3,]…
…
Propositions Actions
[0,3]x[3,4]
[0,3]x[11,]
[3,4]x[11,]
Action Model
Duration
Parallel
(pre) Conditions over intervals
Effects over intervals–5A +5A
A
≥5A
Acond2 cond3
cond1
eff
Acond: r;0, p;[0,2]
eff: r;(0,2), r;2,
e;2r
A
r
p
e
¬ r
Semantics
Acond: r;0, p;[0,2]eff: r;(0,2), r;2,
e;2
A
r
p
e
P stops holding
¬ r r
Semantics
Acond: r;0, p;[0,2]eff: r;(0,2), r;2,
e;2r
A
r
p
e
p stops holding
¬ r
Incomplete
?????????
???
Exogenous Conditions
At(Pkg1, BOS-PO)
At(Truck1, BOS)
Inititial Conditionst=0
Closed(SJC)t=0600z t=1300z
Visible(NGC132)t=0517z t=0642z
Xcond:eff: At(Pkg1, BOS-PO);0
At(Truck1, BOS);0Closed(SJC);[0600,1300]Visible(NGC132);
[0517,0642]…
Outline
Motivation
Simple Reachability
Mutual Exclusion
Practical Matters
Possibility & Reachability
(p;t) p is logically possible at t
∆(p;t) p is reachable at t
(rich;tomorrow)¬∆(rich;tomorrow)
Possibility & Reachability
(p;t) p is logically possible at t
∆(p;t) p is reachable at t
(p;i) t i (p;t)
∆(p;i) t i ∆ (p;t)
Extend to Intervals
Basic Axioms
p;i ∆(p;i)
p;i (p;i)
p;i t i ¬∆(¬p;t)
p;i t i ¬(¬p;t)
Negations are not …
Facts are possible & reachable
∆(p;t) (p;t q;t’) ∆(q;t’)
Transitivity
Basic Axioms
a;t Cond(a;t) Eff(a;t)
X;0
Actions
Exogenous conditions
Closure of X
(Eff(x;0) = ¬p;t) — (p;i)|\ |
Example
0 1 2 3 4 5 6
r
p pX;0
Closure
0 1 2 3 4 5 6
r
p p
p p
r
X;0
closure
Basic
0 1 2 3 4 5 6
r
p p
∆ r
∆p ∆ p
p p
r
X;0
basic
closure
Persistence
∆(p;i) meets(i,j) (p;j) ∆(p;i||j)
0 1 2 3 4 5 6
r
p p
∆ r
∆p ∆ p
p p
r
X;0
basic
closure
Persistence
∆(p;i) meets(i,j) (p;j) ∆(p;i||j)
0 1 2 3 4 5 6
r
p p
p p
r
X;0
closure
∆p ∆p
∆ rbasic &persist
Actions
∆Cond(a;t) Eff(a;t) ∆(a;t)
Reachability
∆p1;i1 … ∆pn;in ∆(p1;i1 … pn;in)
Conjunctive optimism
Action Application
0 1 2 3 4 5 6
∆p ∆p
∆ r
∆A
r
A
r
p
e
¬ r
Acond: r;0, p;[0,2]
eff: r;(0,2), r;2,
e;2
∆Cond(a;t) Eff(a;t) ∆(a;t)
Action Application
0 1 2 3 4 5 6
∆p ∆p
∆ r
∆A
∆ ¬ r
r
A
r
p
e
¬ r
∆ e
Acond: r;0, p;[0,2]
eff: r;(0,2), r;2,
e;2
∆Cond(a;t) Eff(a;t) ∆(a;t)
Persistence Again
∆(p;i) meets(i,j) (p;i) ∆(p;i||j)
0 1 2 3 4 5 6
∆p ∆p
∆ r
∆A
∆ ¬ r
r
A
r
p
e
¬ r
∆ e
Persistence (revised)
∆(p;i) meets(i,j) (p;i) ∆(p;i||j)
a;t ∆(a;t) p;i PersistEff(a;t) meets(i,j) (p;i) ∆(p;i||j)
r
A
rp
e
¬ r
Persistence
0 1 2 3 4 5 6
∆p ∆p
∆ r
∆A
∆ ¬ r
∆ e
a;t ∆(a;t) p;i PersistEff(a;t) meets(i,j) (p;i) ∆(p;i||j)
Outline
Motivation
Simple Reachability
Mutual Exclusion
Practical Matters
Mutual Exclusion
M(p1;t1, …, pn;tn)
M(p1;i1, …, pn;nn)
t1 i1, …, tn in M(p1;t1, …, pn;tn)
(∆p1;i1 … ∆pn;in ) ¬M(p1;i1, …, pn;nn) ∆(p1;i1 … pn;in)
Conjunctive optimism
Intervals
Logical Mutex
M(p;t, ¬p;t)
Consequences
¬(1 … n) M(1, …, n)
Consequences
M(A;t, ¬p;t+)
Consequences
Acond: p; …
eff: e;
…
A;t p;t+
A;t e;t+e
M(A;t, ¬e;t+)
¬(1 … n) M(1, …, n)
Consequences
¬(1 … n) M(1, …, n)
M(A;t, B;t+–)
Consequences
Acond: p; …
A;t p;t+
B;t ¬p;t+eBcond: ¬p; …
Implication Mutex
M(1, …, n) ( 1) M(, …, n)
Implication Mutex Example
M(1, …, n) (1 1) M(1, …, n)
Example
B cond:q;0eff:f;1
M(1, …, n) ( 1) M(, …, n)
M(p;1,q;1)
A cond:p;0eff:e;1
p;1
q;1
A;1
B;1
e;2
f;2
Implication Mutex Example
M(1, …, n) (1 1) M(1, …, n)
Example
B cond:q;0eff:f;1
p;1
q;1
A;1
B;1
e;2
f;2
M(1, …, n) ( 1) M(, …, n)
M(p;1,q;1)
A cond:p;0eff:e;1
A;t p;t
B;t q;t
Implication Mutex Example
M(1, …, n) (1 1) M(1, …, n)
Example
B cond:q;0eff:f;1
p;1
q;1
A;1
B;1
e;2
f;2
M(1, …, n) ( 1) M(, …, n)
M(p;1,q;1)
A cond:p;0eff:e;1
A;t p;t
B;t q;t
M(A;1,q;1)
M(p;1,B;1)
Implication Mutex Example
M(1, …, n) (1 1) M(1, …, n)
Example
B cond:q;0eff:f;1
p;1
q;1
A;1
B;1
e;2
f;2
M(1, …, n) ( 1) M(, …, n)
M(p;1,q;1)
A cond:p;0eff:e;1
A;t p;t
B;t q;t
M(A;1,q;1)
M(p;1,B;1)
M(A;1,B;1)
Implication Mutex for Intervals
M(1, …, n) ( 1) M(, …, n)
M(1;i1, …, n;in) j= {t: ;t t1 i1 1;t1}
M(;j, …, n;in)
p;[1,3)
q;[2,3)
A;[1,3)
B;[2,3)
e;…
f;…
Explanatory Mutex
{( 1) M(, …, n)} M(1, …, n)
If “all ways of proving” 1 are mutex with 2, …, n M(1, …, n)
p;1
q;1
A;1
B;1
e;2
f;2
A
Bp
A p
Outline
Motivation
Simple Reachability
Mutual Exclusion
Practical Matters
Limiting Mutex
Reachable propositions
Time spread
p
A
q
M(p;2, q;238)[0,2] [236,240]
Mutex spread theorem ?
CSP?
A[0,3],[6,9]
B[11,]
C[…]…
…
p[0,5],[8.1,16]
q[2,17]…
r[3,]…
…
Propositions Actions
Initial Domains
A[0, )
B[0, )
C[0, )
…
p[0, )
q[0, )
r[0, )
…
Propositions Actions
Interval Elimination
A[0, )
B[0, )
C[0, )
…
p[0,5],[8.1, )
q[0, )
r[0, )
…
Propositions Actions
Reachability? Mutex
Mutex Representation
M(A;t, B;[t+2,t+10])p
B
[0,4]
¬p
A
[6,10]
B
A
M(A, B, [2,10])
M(A, B, , I)
Final Remarks
Reachabilitysimple
Mutexsurprisingly simplecomplex realization
Questionslimiting mutexCSP implementation?mutex representationTGP