Title: Control Rules in Planning
1Control Rules in Planning
- Stephen M. Lee-Urban
- March 26, 2007
2References
- Chapter 10, Control Rules in Planning, in
Automated Planning Theory and Practice - F. Bacchus and F. Kabanza. Using Temporal Logics
to Express Search Control Knowledge for
Planning, in Artificial Intelligence, vol 116,
2000 - Planner TLPlan
- (For)Warning Understanding temporal logic will
be necessary for encoding control rules in your
domains
3Motive
- Classical planning efficiency suffers from
combinatorial complexity (intractable) - Most earlier planners fit in Abstract-Search
Procedure - ND search in node space of set of solution plans
(set of all reachable solutions from n) - Prune function detects and cuts unpromising nodes
- Can improve solving exponential to polynomial
- Same idea used in F.E.A.R.
- Uses domain specific rules to guide forward
chaining algorithm
u is a structured collection of actions and
constraints
4Pruning
- Often involves domain-specific tests
- Identify less desirable solutions below node than
solutions below a different node - Motivating Example
- Forward Search Prune in Container-Stacking
- Consistent ? all containers below c consistent
with g - State Space. Long plans worse than shorter
- If container c position is consistent with goal,
prune what? - Prune states resulting from apply action moving c
- If cs position inconsistent, action a yields
state consistent with goal, and action b moves c
to position inconsistent w/ g - Prune states resulting from applying action b
- Finds near optimal solutions in low-order
polynomial time
How do we express such relationships between
states?
5Simple Temporal Logic ? ? ? ?
- STL extends FOL to include modal operators
- Modal Operators express relationships between
current state and subsequent states - Syntax If L is a func-free FOL, then Lt includes
all of L plus - true, false constant atoms that are always
true/false - Modal Ops What do we need?
- ? (until), ? (always), ? (eventually), ? (next),
GOAL - If F1 and F2 are formulas, then so are
- F1?F2, ?F1, ?F1, ?F1
- If F1 has no modal ops, then GOAL(F1) is a formula
6Semantics of STL
- Interpreting requires triple (S, si, g)
- S lts0, s1, gt is an infinite sequence
- si?? S is the current state
- g is a goal formula
- If Lt is based on L of planning domain, and S is
sequence of states produced by finite plan, isnt
S finite? - Use NOPs. Append infinite final state.
- In coming slides, let F be an STL formula. We now
define whether (S, si, g) ? F
7But First. Entailment
- Aka logical implication
- A B
- A entails B iff every model that makes A true
also makes B true - Example
- A All horses are animals, All stallions
are horses - B All stallions are animals
- A B
8Definitions
? - until ? - always ? - eventually ? - next
- Given ground atom F, (S, si, g) F iff si F
- Eg (S, ((on a b) (on b c) ), g) (on a b)
- Quantifiers and logical connectives have the same
semantic rules as in FOL - Eg (S, si, g) F1?F2 iff (S, si, g) F1 and
(S, si, g) F2 - (S, si, g) ?F iff
- (S, sk, g) F for k?? i
- (S, si, g) ?F iff
- (S, si1, g) F
- (S, si, g) ?F iff
- ?k k ? i, (S, sk, g) F
- (S, si, g) F1?F2 iff
- ?k k ? i, (S, sm, g) F1 for m i, , k-1
and (S, sk, g) F2 - (S, si, g) GOAL(F) iff
- F?g
If F contains no GOAL operators, g is irrelevant.
Write simply (S, si) F
9Simple Examples
? - until ? - always ? - eventually ? - next
- Each means the same for s2
- s2 on(c1,c2) ? on(c2,c3)
- (S,s0) ??(on(c1,c2) ? on(c2,c3))
- (S,s1) ?(on(c1,c2) ? on(c2,c3))
- (S,s2) on(c1,c2) ? on(c2,c3)
- (S,si) ??holding(crane1,c1)
- Holding(crane1,c1) is false for all k ? i
- (S,si) ?x ( on(x,c1) ? ?on(x,c1) )
- The same container is on c1 in all subsequent
states - (S,si) ?x ?( on(x,c1) ? ?on(x,c1) )
10Encoding our Motivating Ex.
? - until ? - always ? - event. ? - next
- Dont move container if position is consistent
with goal - F1(c,d,p) GOAL(in(c,p)) ? ??q GOAL(in(c,q)) ?
GOAL(on(c,d) ? ??e GOAL(on(c,e)) - Holds if acceptable when container c is on item d
in pile p (no goal requiring c in another pile or
on top of something else) - F2(c,p) ok(c,p)?? same(p,pallet) ? ?d
(F1(c,d,p) ? ok(d,p)) - ok(c,p) holds iff c is in pile p and cs position
is consistent with the goal - F3(c) ?p (F2(c,p) ? ok(c,p))
- holds iff cs position is consistent with the
goal - F ?c F3(c) ? ?p?d ?(in(c,p) ? in(c,d))
- holds iff for every container c whose position is
consistent with g, c will never be moved (c
always remains in same pile and on same item in
that pile)
11Break
12Progression
- Computing progression of a control formula F is
essential for pruning - Formula progress(F,si) is true in si1 iff F is
true in si. This is called Fs progression. - progress(F,si) is the formula produced from
progr(F,si) by performing usual simplifications - Replace true ? d with d, ?true with false, etc.
- Both functions can be computed in low-order
polynomial time with algorithms directly
implementing their definitions
(S,si,g) F iff (S,si1,g) progress(F,si)
13Definition of progr(F,si)
? - until ? - always ? - eventually ? - next
- If F contains no modal operators, then
- progr(F,si) true if si F, false if si ? F
- Logical connectives are as usual
- progr(F1?F2,si) progr(F1,si) ? progr(F2,si)
- progr(?F,si) ?progr(F,si)
- Modal operators are as follows
- progr(?F,si) F
- progr(F1?F2,si) ( (F1?F2) ? progr(F1,si) ) ?
progr(F2,si) - progr(?F,si) (?F) ? progr(F,si)
- progr(?F,si) (?F) ? progr(F,si)
14Example of Progr
? - until ? - always ? - eventually ? - next
progr(?F,si) (?F) ? progr(F,si)
- F ?on(c1,c2)
- si on(c1,c2)
- progress(F,si)
- ?on(c1,c2) ? progress(on(c1,c2),si)
- ?on(c1,c2) ? true
- F
- What if si ?on(c1,c2)?
15Using Control Formulas in Planning
- Let S lts0,s1,gt be infinite and F be an STL
formula. - If (S,s0,g) F, then for every finite
truncation S lts0,s1,,sigt of S, progress(F,S)
? false - Let s0 be a state, ? be a plan applicable to s0,
and S lts0,,sngt be the seq. of states produced
by applying ? to s0. - If F is an STL formula and progress(F,S) false,
then S has no extension S lts0,,sn,
sn1,sn2,gt such that (S,s0,g) F
Modify Forward-search to prune any partial plan ?
such that progress(F,S?) false
16Our Handy Example
? - until ? - always ? - eventually ? - next
- Let s0 and g be for a container-stacking problem
with constant symbols c1, , ck. Let F1, F2, F3,
and F be as before. - progress(F,s0)
- progress(?c F3(c) ? ?p?d ?(in(c,p) ?
in(c,d)), s0) - progress(F3(c1) ? ?p?d ?(in(c1,p) ?
in(c1,d)), s0) ? ? progress(F3(ck) ? ?p?d
?(in(ck,p) ? in(ck,d)), s0) - Suppose in s0, c1 is consistent with g and is on
item d1 in pile p1. Then - s0 F3(c1)
- s0 ? F3(ci) for i 2, , k
- Which means progress(F,s0)
- progress( ?p?d ?(in(c1,p) ? in(c1,d)), s0 )
- progress( ?(in(c1,p1) ? in(c1,d1)), s0 )
- ?(in(c1,p1) ? in(c1,d1))
- If an applicable action a to state s0 moves c1
then - ?(s0,a) ? progress(F,s0)
- Thus ?(s0,a) can be pruned.
17Finally, the Planning Procedure!
18Planning Procedure - Comments
- Sound and complete, if the problem is solvable
and STL formula F is entailed for at least one
solution of the problem - Soundness follows from soundness of
Forward-Search - Completeness follows from what?
- the condition on F
Control formulas are like specialized computer
programs and must be debugged.
19Extensions
- Function Symbols
- Axioms (Horn-clauses)
- Restrict axioms to Horn-clauses and use a
Horn-clause theorem prover - Attached Procedures
- Allow some func./predicate symbols to be
evalutated as attached procedures - Time
- Actions with time durations and overlapping
- Extended Goals
- Add control rules like F ??at(r1,bad-loc)
- Reach goal in 2 actions or fewer F g V (?g)
20TLPlan
- On vega.cc.lehigh.edu /home/sml3/planning/sys/tlp
lan - http//www.cs.toronto.edu/fbacchus/tlplan-manual.
html - Requires one input script file referring to two
others - (load-file "BlocksWorld.tlp")
- (load-file "BlocksProblems.tlp")
- (set-statistics-file "BlocksProblems.csv")
- (set-goal (goal0))
- (set-initial-world (state0))
- (set-plan-name "Problem0")
- (plan) try and solve the
problem - (select-final-world) needed for next line
- (print-pddl-plan)
- (exit) remove this line for interactive mode
- Running from the directory containing your
domain - ../runtlplan script.tlp
Domain File
Problem File
Defined in Problem File
Also in tlplan.log
or make a symbolic link in your domain directory
via ln s ltpathgt/runtlplan
21Sample Domain File
(clear-world-symbols) Remove old dom symb
WORLD SYMBOLS (declare-described-symbols
(predicate on 2) and so on (predicate
ontable 1)) (declare-defined-symbols
(predicate goodtower 1) (predicate
goodtowerabove 1) and so on (function
depth 1)) DEFINED PREDICATES
(def-defined-predicate (goodtower ?x) (and
(clear ?x) (goodtowerbelow ?x)))
TEMPORAL CONTROL FORMULA (define (bw-control1)
(always (forall (?x) (clear ?x)
(implies (goodtower ?x) (next
(goodtowerabove ?x))) )))
OPERATORS (def-strips-operator (pickup ?x)
(pre (handempty) (clear ?x) (ontable ?x))
(add (holding ?x)) (del (handempty)
(clear ?x) (ontable ?x))) PRINT ROUTINES and
FUNCTIONS
Problem File
(define (state0) (clear a) (clear b)
(clear c) (ontable a) (ontable b)
(ontable c) (handempty)) (define
(goal0) (on a b) (on b c) (ontable c))
22Defining a Domain Init/Def
- (clear-world-symbols)
- Must call first in a new domain definition file
- Clears the prev. domain's language definition
- Resets temporal control formula and the print
world command to their defaults - (declare-described-symbols (functionpredicate
name arity no-cycle-checkrewritable) ...) - must be declared prior to any other symbols
- (declare-defined-symbols (functionpredicategener
ator name arity) ...) - must be declared after the described symbols
- (def-defined-predicate (name parameters)
(local-vars declarations) formula) - Includes a new predicate, defined in terms of a
FO formula involving other predicates - Can be recursive
- (define name list)
- name is an abbreviation for list. Allows macro
subst. in domain definition files. - Use to define temporal control formulas
- ( ?variable value)
- assigns value to ?variable
23Formula Syntax
Termsconstant symbol, number, or
string?variable any symbol starting with
?(?array index ) dimension must agree with
args(function term ) built-in or declared w/
init. decl. Atomic Formulas (predicate term )
predicate declared w/ init. decl.( term1 term2)
predefined equality binary predicate (TRUE)
const. atomic formula always true (FALSE)
const. atomic formula always false First-Order
Formulas atomic-formula (and formula ) --
?? (or formula ) ? (xor formula ) ?? (not
formula ) ? (implies formula1 formula2) --
? (if-then-else formula1 formula2
formula3) (forall var-gen formula) --
?? (exists var-gen formula) ? (exists!
var-gen formula) unique ? (exactly one)
Temporal Logic Formulas (TF)first-order-formula(
next tf) -- ? (eventually tf) ?(t-eventually
ispec tf) ispec is an interval of
states(always tf) ?(t-always ispec tf)
ispec is again an interval(t-until ispec tf1
tf2) ? Modalities(goal formula tf
generator) eval in goal world(previous formula
tf generator) eval in prev. state(current
formula tf generator) eval in cur. state
24Defining Operators
STRIPS Operator(def-strips-operator name pre del
cost duration priority) where(name v )
Declares op name and params(pre formula)
Precondition list(add p ) Add list(del p )
Delete list(cost n) Cost of action, default
is 1(duration n) Duration of action, default
1(priority n) Used in search to order
successor states, default 0
(def-strips-operator (pickup ?x) (pre
(handempty) (clear ?x) (ontable ?x)) (add
(holding ?x)) (del (handempty) (clear ?x)
(ontable ?x)))
25Defining TL Control Formulas
(define (bw-control1) (always (forall (?x)
(clear ?x) (implies (goodtower ?x)
(next (goodtowerabove ?x))) )))
26Contact Me
- Start early, defining control rules is tricky,
REQUIRES DEBUGGING and tuning - sml3_at_lehigh.edu