Contents Planning vs. problem solving 1 Foundations of Artificial Intelligence Planning in the situation calculus 2 14. Planning STRIPS formalism 3 Solving Logically Specified Problems Step by Step 4 Non-linear planning Wolfram Burgard, Bernhard Nebel, and Martin Riedmiller 5 The POP algorithm Albert-Ludwigs-Universit¨ at Freiburg Graphplan 6 August 2, 2011 Heuristic search planning 7 Outlook: Extensions & non-classical planning 8 (University of Freiburg) Foundations of AI August 2, 2011 2 / 54 Planning Planning vs. Problem-Solving Basic difference: Explicit, logic-based representation States/Situations: Through descriptions of the world by logical formulae Given a logical description of the initial situation, vs. data structures → The agent can explicitly think about it and communicate. a logical description of the goal conditions, and Goal conditions as logical formulae vs. goal test (black box) a logical description of a set of possible actions, → The agent can also reflect on its goals. → find a sequence of actions (a plan) that brings us from the initial Operators: Axioms or transformation on formulae vs. modification of situation to a situation in which the goal conditions hold. data structures by programs → The agent can gain information about the effects of actions by inspecting the operators. (University of Freiburg) Foundations of AI August 2, 2011 3 / 54 (University of Freiburg) Foundations of AI August 2, 2011 4 / 54
Planning vs. Automatic Programming Planning as Logical Inference (1) Planning can be elegantly formalized with the help of the situation calculus . Initial state : Difference between planning and automatic programming (generating At ( Home , s 0 ) ∧ ¬ Have ( Milk , s 0 ) ∧ ¬ Have ( Banana , s 0 ) ∧ ¬ Have ( Drill , s 0 ) programs): Operators (successor-state axioms): In planning, one uses a logic-based description of the environment. ∀ a , s Have ( Milk , Do ( a , s )) ⇐ ⇒ { a = Buy ( Milk ) ∧ Poss ( Buy ( Milk ) , s ) ∨ Have ( Milk , s ) ∧ a � = ¬ Drop ( Milk ) } Plans are usually only linear programs (no control structures). Goal conditions (query): ∃ s At ( Home , s ) ∧ Have ( Milk , s ) ∧ Have ( Banana , s ) ∧ Have ( Drill , s ) When the initial state, all prerequisites and all successor-state axioms are given, the constructive proof of the existential query delivers a plan that does what is desired. (University of Freiburg) Foundations of AI August 2, 2011 5 / 54 (University of Freiburg) Foundations of AI August 2, 2011 6 / 54 Planning as Logical Inference (2) The STRIPS Formalism The variable bindings for s could be as follows: STRIPS: STanford Research Institute Problem Solver (early 70s) Do ( Go ( Home ) , Do ( Buy ( Drill ) , Do ( Go ( Hardware store ) , The system is obsolete, but the formalism is still used. Usually, a simplified Do ( Buy ( Banana ) , Do ( Buy ( Milk ) , Do ( Go ( Supermarket ) , s 0 )))))) version is used: I.e., the plan (term) would be World state (including initial state): Set of ground atoms (called fluents), � Go ( Supermarket ) , Buy ( Milk ) , . . . � no function symbols except for constants, interpreted under closed world However, the following plan is also correct: assumption (CWA). Sometimes also standard interpretation, i.e., negative � Go ( Supermarket ) , Buy ( Milk ) , Drop ( Milk ) , Buy ( Milk ) , . . . � facts must be explicitly given In general, planning by theorem proving is very inefficient. Goal conditions : Set of ground atoms Alternative: Specialized inference system for a limited representation Note : No explicit state variables as in sitation calculus. Only the current world state is accessible. → Planning algorithm (University of Freiburg) Foundations of AI August 2, 2011 7 / 54 (University of Freiburg) Foundations of AI August 2, 2011 8 / 54
STRIPS Operators Actions and Executions Operators are triples, consisting of Action Description : Function name with parameters (as in situation An action is an operator, where all variables have been instantiated: calculus) Op ( Action: Go ( Home , Supermarket ) , Preconditions : Conjunction of positive literals; must be true before the Precond: At ( Home ) , Path ( Home , Supermarket ) , operator can be applied (after variables are instantiated) Effect: At ( Supermarket ) , ¬ At ( Home ) ) Effects : Conjunction of positive and negative literals; positive literals are An action can be executed in a state, if its precondition is satisfied. It added (ADD list), negative literals deleted (DEL list) (no frame problem!). will then bring about its effects. Op ( Action: Go ( here , there ) , Precond: At ( here ) , Path ( here , there ) , Effect: At ( there ) , ¬ At ( here ) ) (University of Freiburg) Foundations of AI August 2, 2011 9 / 54 (University of Freiburg) Foundations of AI August 2, 2011 10 / 54 Linear Plans Searching in the State Space We can now search through the state space (the set of all A sequence of actions is a plan states formed by truth assignments to For a given initial state I and goal conditions G , such a plan P can be successfully executed in I iff there exists a sequence of states fluents) – and in this s 0 , s 1 , . . . , s n such that way reduce planning - the i -th action in P can be executed in s i − 1 and results in s i to searching. - s 0 = I and s n satisfies G We can search forward ( progression P is called a solution to the planning problem specified by the planning ): operators, I and G Or alternatively, we can start at the goal and work backwards ( regression planning ). Possible since the operators provide enough information (University of Freiburg) Foundations of AI August 2, 2011 11 / 54 (University of Freiburg) Foundations of AI August 2, 2011 12 / 54
Searching in the Plan Space Plan = Sequence of Actions? Instead of searching in the state space, we can search in the space of all Often, however, it is neither meaningful nor possible to commit to a plans . specific order early-on (put on socks and shoes). The initial state is a partial plan containing only start and goal states: → Non-linear or partially-ordered plans (least-commitment planning) Partial Order Plan: Total Order Plans: Start Start Start Start Start Start Start The goal state is a complete plan that solves the given problem: Right Right Left Left Right Left Sock Sock Sock Sock Sock Sock Left Right Sock Sock Left Left Right Right Right Left Sock Shoe Shoe Sock Sock Sock LeftSockOn RightSockOn Right Left Right Left Left Right Left Right Shoe Shoe Shoe Sock Sock Shoe Shoe Shoe Operators in the plan space: Left Right Left Right Left Right Shoe Shoe Shoe Shoe Shoe Shoe Refinement operators make the plan more complete (more steps etc.) LeftShoeOn, RightShoeOn Finish Finish Finish Finish Finish Finish Finish Modification operators modify the plan (in the following, we use only refinement operators) (University of Freiburg) Foundations of AI August 2, 2011 13 / 54 (University of Freiburg) Foundations of AI August 2, 2011 14 / 54 Representation of Non-linear Plans Completeness and Consistency Complete Plan : A plan step = STRIPS operator (or action in the final plan) Every precondition of a step is fulfilled: A plan consists of ∀ S j ∀ c ∈ Precond ( S j ) : A set of plan steps with partial ordering ( < ), where S i < S j implies S i ∃ S i with S i < S j and c ∈ Effects ( S i ) and must be executed before S j . for every linearization of the plan: A set of variable assignments x = t , where x is a variable and t is a ∀ S k with S i < S k < S j , ¬ c �∈ Effect ( S k ) . constant or a variable. Consistent Plan : A set of causal relationships S i → S j means “ S i produces the if S i < S j , then S j � < S i and if x = A , then x � = B for distinct A and B precondition c for S j ” (implies S i < S j ). for a variable x . ( unique name assumption = UNA) Solutions to planning problems must be complete and consistent . A complete, consistent plan is called a solution to a planning problem (all linearizations are executable linear plans ) (University of Freiburg) Foundations of AI August 2, 2011 15 / 54 (University of Freiburg) Foundations of AI August 2, 2011 16 / 54
Example Plan Refinement (1) Regression Planning: Fulfils the Have predicates: . . . after instantiation of the Actions : variables: Op ( Op ( Action: Go ( here , there ) , Action: Buy ( store , x ) , Precond: At ( here ) ∧ Path ( here , there ) , Precond: At ( store ) ∧ Sells ( store , x ) , Effect: At ( there ) ∧ ¬ At ( here ) ) Effect: Have ( x ) ) Note: there , here , x , store are variables. Note: In figures, we may just write Buy ( Banana ) instead of Thin arrow = < , thick arrow = causal relationship + < Buy ( SM , Banana ) (University of Freiburg) Foundations of AI August 2, 2011 17 / 54 (University of Freiburg) Foundations of AI August 2, 2011 18 / 54 Plan Refinement (2) Plan Refinement (3) First, you have to go there . . . Shop at the right store . . . Note : So far no searching, only simple backward chaining. Now : Conflict! If we have done Go ( HWS ) , we are no longer At ( Home ) . Likewise for Go ( SM ) . (University of Freiburg) Foundations of AI August 2, 2011 19 / 54 (University of Freiburg) Foundations of AI August 2, 2011 20 / 54
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