Transcript class notes (powerpoint)

Complexity of domain-independent planning
Jim Blythe
USC INFORMATION SCIENCES INSTITUTE
CS 541: Planning complexiity
1
Reviewing decidability
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A decision problem is a set of related problem instances,
e.g. “is 1 prime?”, “is 2 prime?” etc.
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A decision problem is decidable if there is a program
that takes any instance and correctly halts with the
answer “yes” or “no”.
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A problem is semi-decidable if a program guarantees to
halt and give the correct answer in one of the cases
[either “yes” or “no”] but not in the other case.

Otherwise, any program will sometimes run forever in
either case - this problem is undecidable.
USC INFORMATION SCIENCES INSTITUTE
CS 541: Planning complexiity
2
Decidability results from Erol et al.
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Consider plan existence, given operators, objects,
predicates an initial state and goal description.
Based on transformations to logic programming.
Can transform a planning problem without delete lists
or negative preconditions to a logic program (and
vice versa) in polynomial time.

function symbols => undecidable unless have
acyclicity and boundedness conditions.
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No function symbols, and finite initial state =>
decidable.
USC INFORMATION SCIENCES INSTITUTE
CS 541: Planning complexiity
3
Acyclicity
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A “level mapping” maps propositions to natural
numbers. A “predicate level mapping” works on
predicate symbols, e.g. on, at-truck,..
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If there is a level mapping such that for every
operator, l(add) > l(pre) for every literal in the add list
or preconditions, the P is atomically acyclic
USC INFORMATION SCIENCES INSTITUTE
CS 541: Planning complexiity
4
Worst-case complexity of problems.

If a problem is decidable, we might ask how many
resources a program requires to compute the answer
(in the worst case).

We measure the resources a program takes in terms
of time or memory (space), as a function of the size
of the input.

If a problem is known to be in some complexity class,
then we know there is a program that solves it using
resources bound by that class.
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CS 541: Planning complexiity
5
Complexity classes

If a problem is in P, there exists a program to decide
it taking polynomial time in the size of the input.
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PSPACE - polynomial space, EXPTIME and
EXPSPACE - exponential.

If a problem is in NP, there exists a nondeterministic
program that can solve it in polynomial time. You can
think of this program as one that guesses the correct
answer and then must verify it is correct.

Also have NEXPTIME, etc.
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CS 541: Planning complexiity
6
Hierarchy of complexity

We don’t know if there are any problems in NP but
not in P. But clearly NP contains P.
EXPSPACE
NEXPTIME
EXPTIME
PSPACE
NP
P
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CS 541: Planning complexiity
7
States, operators, plans.
How many, how big?
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If there are n objects, m predicates with arity r and o
operators (with s variables each):
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There are A = m * n^r possible atoms. It takes this much
space to describe a state.

So there are 2^A possible states (double exponential).

There are o * n^s possible ground operators.

In general plans will be bounded by the number of states.
(Why?)
USC INFORMATION SCIENCES INSTITUTE
CS 541: Planning complexiity
8
Bounds on complexity

Now we’ll assume no function symbols and only
finitely many constant symbols, so the plan existence
problem is decidable.

With no restrictions, the problem is in EXPSPACE.
(What’s an algorithm that shows this?)
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CS 541: Planning complexiity
9
Some special cases - no delete lists.
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If there are no delete lists, then operators only need
to appear once.
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So it’s in NEXPTIME. (What’s an algorithm that
shows this?)
USC INFORMATION SCIENCES INSTITUTE
CS 541: Planning complexiity
10
No negative preconditions
and no delete lists
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Now plans for different subgoals won’t negatively
interfere with each other.
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Now it’s in EXPTIME.
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If each operator has only one precondition, the
number of subgoals does not increase in a
backward-chaining search - now it’s in PSPACE.
USC INFORMATION SCIENCES INSTITUTE
CS 541: Planning complexiity
11
Propositional case
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If the predicates have no arguments, so nor do operators,
the number of possible operators and atoms is polynomial.
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This reduces the complexity of the algorithms we have
considered until now:
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General case -> PSPACE
No delete lists -> NP
No negative preconds -> P
Not more than one precondition -> NLOGSPACE
If you know the operators in advance, this in effect bounds
the arity of predicates and operators, with the same result.
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CS 541: Planning complexiity
12
What does all this mean?
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Domain-independent planning in general is very
hard.
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The search for special cases seems fruitful.
e.g. Backstrom and Klein: each op changes at most
one proposition, no more than one op to make a
proposition true (or false). -> P.
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CS 541: Planning complexiity
13
Other approaches
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Try to reuse past experience, with machine learning
or case-based reasoning. Even if a general problem
domain is hard, perhaps the subset we typically
encounter can be solved more easily and we can
cache problem parts that are reusable.
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There are many possible approaches to this:
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Learning search control knowledge
Learning macro-operators
Saving and reusing cases
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CS 541: Planning complexiity
14
We still need to think about each domain

These results paint a general landscape.

Even if a problem is in P, it can still be prohibitively
expensive to solve.

Many special cases exist that aren’t covered in this
framework.

What is the complexity of plan existence for the
blocksworld? PSPACE?
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CS 541: Planning complexiity
15