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