15381 Artificial Intelligence

1
15-381 Artificial Intelligence

• Means-Ends Analysis and
• Constraint Propagation
• Jaime Carbonell
• 21 January 2003
• Topics Covered
• Homework 1 (generalized state-space search)
• Means-Ends Analysis (back-chaining)
• Search Control Rules in MEA
• Constraint-Based Search

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Search?Planning Parameterized Operations

• Multi-State Transitions
• Instead of Opi,j Si? Sj, We have Opk,l Sk?
  Sl
• Preconditions and Post-Conditions
• Conjunctive set of first-order predicates
• Arguments can be constants or (typed) variables
• Intentional description of subset of all states
• Pre-image Sk states where preconditions are
  true
• Post-image S1 states where post-conditions are
  true
• Requires Consistent variable bindings within and
  across preconditions and post-conditions

3
Search?Planning Parameterized Operations

• First Example
• OPERATOR DRIVE-CAR(ltcargt, ltdrivergt, ltkeysgt,
  ltloc-1gt)
• PRE (AT ltcargt ltloc-1gt)
• (AT ltdrivergt ltloc-1gt)
• (CONTAINS-GAS ltcargt)
• (HAVE ltkeysgt ltdrivergt)
• (CORRESPOND ltkeysgt ltcargt)
• POST (AT ltcargt ltloc-2gt)
• (AT ltdrivergt ltloc-2gt)
• (NOT (AT ltcargt ltloc-1gt))
• (NOT (AT ltdrivergt ltloc-1gt))

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Search?Planning Parameterized Operations

• Second Example
• (Previous example LISP-style, Current one
  PROLOG-style)
• OPERATOR move-robot(r,x,y)
• TYPE ROBOT(r) LOC(x) LOC(y)
• PRE AT(r,x) EMPTY(y) CONNECTED(x,y)
• POST AT(r,y), NOT(AT(r,x))
• OPERATOR pick-up(r,z)
• TYPE ROBOT(r) LOC(x) LOC(y)
• PRE AT(r,x) AT(z,y) NEXT-TO(x,y)
  NOT(holding(r,w,))
• POST HOLDING(r,z)
• NOT(AT(z,y))

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Search?Planning Parameterized Operations

• Interpretation
• A plan is an o-path S0 followed by a sequence of
  instantiated operators which result in the goal
  state.
• Variables match objects in state of specified
  types only for which the preconditions hold at
  plan execution time.
• Planning can proceed by forward or backward (or
  any other) search method.
• More on Planning expected from Veloso (later
  lecture)

6
Means-Ends Analysis

• Backchaining/Subgoaling Search
• Let Scurr S0
• If Scurr SG, then go to next goal (or DONE)
• Let OPSapp match(SG POST(Opi))
• If OPSapp empty, then FAIL
• Else Select OP ? OPSapp, (save alt's)
• If match(PRE(OP), Scurr),
• a. let Scurr apply(OP, Scurr)
• b. Go to step 2
• Else (i.e. if NOT(match(PRE(OP), Scurr)))
• a. MEA(SG) unmatched(PRE(OP)), SI Scurr)
• b. If fail, go to step 4
• c. If succeed, apply OP as above

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Control Rules for MEA

• Choice Points in MEA
• Choose Operator, if several applicable
• Choose Goal, if gt 1 subgoals pending
• Choose Variable Bindings, if gt 1 tuple
• Types of Control Rules
• Select choose an alternative
• and eliminate other contenders
• Reject Reject an alternative
• and retain other contenders
• Prefer Try one alternative first
• and retain others for possible backtracking

8
Control Rules for MEA

• Example
• CONTROL-RULE Carry-before-move
• TYPE SELECT
• PRE Goals(Move(r,x,y), Pick-up(r,z,v)))
• POST Pick-up(r,z)
• CONTROL-RULE Dont polish before machining
• TYPE REJECT
• PRE Goals(Mill(p,f), Drill(p,l,d,s),
  Polish(p))
• POST Polish(p)

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Constraint-Based Search

• Satisfiability problems
• Find consistent bindings to a set of variables
• Consistent satisfy all constraints
• Example (X v Y) (X v Y)
• Example Match applicants to positions
• Two families of search methods apply
• State-space search on bindings
• Satisfiability-search on constraints

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Constraint-Based Search

• Example How fast can you solve this?
• Find a way to fit components (1, 2, 3, 4) into
  slots (A,B,C,D) such that
• Each slot only takes one component
• Slots are in LEFT-RIGHT sequence A, B, C, D
• Slots A and C are T-shaped
• Slots B and D are I-shaped
• Components 1 and 2 are 3-pronged
• Components 3 and 4 are 2-pronged
• 2-pronged fit into T-shaped or I-shaped
• 3-pronged fit only into T-shaped
• Component 3 must be LEFT of component 2

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Constraints

• Least Commitment Method
• For each Variable find all legal
  unary-constrained assignments.
• If no assignments possible, return FAILURE
• Assign most-constrained unassigned variable.
• If all variables assigned, return SUCCESS
• If the assigned variable is a member of a binary
  constraint, propagate instantiation
• Delete all residual un-viable assignments
• Go to 2

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Constraint-Based Search
A 1,2,3,4 B 3,4 C 1,2,3,4 D 3,4
B?4
B?3
A 1,2,4 C 1,2,4 D 4
A 1,2,3 C 1,2,3 D 3 FAILURE
D?4
A 1,2 C 1,2?C 2
D?3
C?2
A 1 SUCCESS
A?1
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(Dis)Advantages of Constraints

• Reduce the search space
• Early failure (upon constraint violation)
• Generate minimal-uncertainty step (least
  commitment strategy)
• Only applicable to satisfiability problems
• Finds an answer, not necessarily optimal
• Not all problems can be cast as constraints to
  satisfy