Planning I: Total Order Planners

Sections 11.1 - 11.4

ProblemMechanically find a sequence of actions that

take us from the start state to the goal state now we want a mechanical way of doing the

“proof” efficientlyDefine a start (initial) state, a goal state, &

some operatorsFind a plan

sequence of operators/actions that leads from the start to the goal state

AssumptionsGoals are conjunctions of atoms

to achieve a goal, you achieve a set of subgoalsActions are deterministic

no uncertainty in performing an actionThe agent is the only “operator” in the worldWorld is accessible [the agent is omniscient]Closed world assumption (sometimes is adopted)

negation by failure

Approaches to planning

Situational calculusState-space search

Situational calculus [Ch. 7]FOL plus a “timestamp” added to most predicatesresult(action, state) function returns the next state

after applying action in stateFrame problem

effect axioms describes what changes, but doesn’t specify what stays the same

frame axioms describe what stays the same, but many are required (representation problem) and you have to reason with them (inferential problem).

Successor-state axioms: combine frame axioms with effect axioms

Good representation; slow inference procedure: too fine grained

State-space search [Ch. 3-4]Initial state, goal test, operators [actions]Doesn’t allow reasoning about the states &

operators operators are just used to generate the next state

• can’t reason about which operators should occur before which other operators which would greatly reduce the number of states needed to test

goal test is a black box that doesn’t allow us to reason about how to get to the goal

• heuristic is simply a numeric score of distance from the goal

Weak in terms of representation & ability to reason about the world

Solution

Combine the two approaches simplify the representation language

• allows us to reason about how to achieve the goal• inference procedure is faster than resolution

“open up” the representation of state, operators & goal test in the search

• rather than blindly applying operators, reason about which ones are most important

• reduce the number of nodes that are considered

STRIPSFacts: conjunction of ground atomsGoal: conjunction of atoms

variables allowed: assume all variables in a goal description are existential

Operators/Actions [e.g., StackOnto] precondition: conjunction of atoms action effects: add (positive literals) & delete (negative literals)

list assumption: everything stays the same unless explicitly

on the delete list [avoids the frame problem]

Approaches to planningSituation-space search

search the space of all possible situations initial state is one node goal node is all of the sub-goals solved plan is the sequence of actions from start -> goal

Plan-space search search the space of all (partial) plans node is a partial plan start node is an initial node goal node contains the complete plan arcs (typically) add to or modify a previous node’s plan

Situation-space plannersProgression: forward chaining

similar to state-space search except for representation

inefficient due to large state space to exploreRegression: backward chaining

start from the goal state & solve its sub-goals [preconditions]

more efficient & goal-directed than progression [fewer applicable operators]

STRIPS: goal-stack based regression planner

Goal stack: what to do nextCurrent state: facts that are truePick order of achieving (sub-)goals

find operator that achieves the goal push the operator on to the stack push its preconditions (in some order) on to the stack check that when we eventually get back to the

original goal, that all of the preconditions that we needed to satisfy are still satisfied

Key assumptionSub-goals are independent of each other

divide & conquer the problem without worrying about other parts of the “equation”

• e.g., buying items: the order doesn’t matter; buying milk before eggs doesn’t usually make it impossible to then buy eggs

• whole plan = sum of all sub-plans STRIPS makes this assumption

• thus, incomplete

Sussman anomaly sub-goals interfere with each other