Heuristic Search Planners

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Heuristic Search Planners
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Planning as heuristic search

• Use standard search techniques, e.g. A,
  best-first, hill-climbing etc.
• Attempt to extract heuristic state evaluator
  automatically from the Strips encoding of the
  domain
• Here, generate relaxed problem by assuming action
  preconditions are independent

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Recap A search

• Best-first search using node evaluation
• f(n) g(n) h(n)
• where
• g(n) accumulated cost
• h(n) estimate of future cost
• For A, h(.) should never overestimate the cost.
  In this case, the solution will be optimal. Then
  h is called an admissible heuristic.

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Derive cost estimate from a relaxed planning
problem

• Ignore the deletes on actions
• BUT still NP-hard, so approximate
• For individual propositions p
• d(s, p) 0 if p is true in s
• 1 min(d(s, pre(a))) otherwise
• min over actions a that add p

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Cost of a conjunction

• How to compute d(s,pre(a)) or d(s,G) ?
• Different options
• Additive d(s, P) sum d(s, p) over p in P
• Max d(s, P) max d(s, p)
• Then h(s) d(s, G)
• Can compute d(.,.) in polynomial time

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Admissibility and information

• Is h (additive version) admissible? How about
  h-max?

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Admissibility and information II

• If h is not admissible, why would we use it
  rather than h-max?

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HSP algorithm overview

• Hill-climbing search restarts if plateau for
  too long
• Some ad hoc choices for the planning competition
• Hill-climbing search is not complete

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HSP2 overview

• Best-first search, using h
• Based on WA - weighted A
• f(n) g(n) W h(n).
• If W 1, its A (with admissible h).
• If W gt 1, its a little greedy generally finds
  solutions faster, but not optimal.
• In HSP2, W 5

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Experiments

• Does ok compared with IPP (Graphplan derivative)
  and Blackbox.

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Regression search

• Motivation for HSPr
• HSP and HSP2 spend up to 80 of their time
  computing the evaluation function.
• Slow to generate nodes compared to other
  heuristic search systems.
• Search backwards from goal, then re-use cost
  estimates from s0 to the goal, since we always
  have a single start state s0.
• Common wisdom regression planning is good
  because the branching factor is much lower

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HSPr problem space

• States are sets of atoms (correspond to sets of
  states in original space)
• initial state s0 is the goal G
• Goal states are those that are true in s0
• Still use h. h(s) sum g(s0, p)

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Mutexes in HSPr

• Problem many of the regressed goal states are
  impossible prune them with mutexes
• E.g in blocksworld (on(c,d), on(a,d), ..) is
  probably unreachable.

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Mutexes in HSPr

• First definition
• A set M of pairs R p, q is a mutex set if
• (1) R is not true in s0
• (2) for every op o that adds p,
• o deletes q
• Sound, but too weak.

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Mutexes in HSPr, take 2

• Better definition
• A set M of pairs R p, q is a mutex set if
• (1) R is not true in s0
• (2) for every op o that adds p,
• either o deletes q
• or o does not add q, and for some precond r of
  o,
• r, q is in M.
• Recursive definition allows for some interaction
  of the operators

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Computing mutex sets

• Start with some set of potential mutex pairs
• Delete any that dont satisfy (1) and (2) above
• Keep going until you dont delete any more
• Initial set? could be all pairs (usually too
  expensive)

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Initial set of potential mutexes

• Ma p, q some action adds p and deletes q
• Mb r, q p, q is in Ma, some action adds
  p,
• and has r in the precondition
• Initial set Ma u Mb
• Mutex set derived from Ma u Mb is M

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HSPr algorithm

• WA search using h(s0) and M
• W 5 as before
• Prune states that contain pairs in M

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Experiments comparing HSP2 and HSPr

• Sometimes HSPr does better, sometimes HSP2 does
  better. Why?
• Two reasons (per B G)
• Still have spurious states
• Since HSP2 recomputes the estimate in each state,
  it actually has more information

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Evidence for spurious states

• Re-run HSPr using mutex set derived from all
  possible pairs.
• No difference in most domains
• Improvement in tire-world domain (with complex
  interactions)
• Slows down in logistics domain

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Branching factor

• Varies widely from instance to instance. (Always
  seems greater in forward chaining though)
• Performance of HSP2 vs HSPr doesnt seem to
  correlate with branching factor
• Other factors dominate, e.g. informedness of
  heuristic

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Derivation of heuristics

• h has problems when there are positive or
  negative interactions
• Can efficient heuristics better capture the
  interactions?
• H2 use the cost of the most expensive pair of
  goals
• Still admissible, more informative than hmax,
  still cheap
• Room for domain-dependent options?

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Comparing HSPr and Graphplan

• Both search forwards in relaxed space, then
  backwards
• Planning graph encodes an admissible heuristic
  hg(s) j if j is the first level where s appears
  without a mutex
• Graphplan encodes IDA efficiently as solution
  extraction but this makes it hard to use other
  search algorithms.

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Overall

• Planning as heuristic search HSP family are
  elegant, quite efficient for domain-independent,
  and use clear principles of search
• Simple algorithms and relatively thorough
  analysis make it easy to consider lots of
  extensions

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Ways to extend

• Improving automatically generated heuristics
• More flexible action representations
• Probably easier to encode in forwards than
  backwards search
• Principles and format for encoding
  domain-dependent heuristics
• Both the estimate function and other control