3. Knowledge Representation, Reasoning, and Planning

Transcription

1 3. Knowledge Representation, Reasoning, and Planning 3.1 Common Sense Knowledge 3.2 Knowledge Representation Networks 3.3 Reasoning Propositional Logic Predicate Logic: PROLOG 3.4 Planning Planning vs. Problem Solving Planning Using Situation Calculus Planning Examples Summary Origins Problem solving using state-space searching Theorem proving Robotics Planners are like problem solving agents states goals actions (2) Planners Differences: Use logic to represent goals, states, and actions. Searches are executed differently. Use of partial plans and planning algorithms to achieve a solution. In some ways more intelligent than a problem solving agent. Planning vs. Problem Solving Planning Using Situation Calculus Planning Examples Summary A planning agent uses its percepts of the world state to form a model of the world. The agent first generates a goal to achieve. Then the agent constructs a plan to achieve it from the current state. Once it has a plan, it keeps executing it until the plan is finished. Then the agent begins again with a new goal. Planners must be able to distinguish two special cases: infeasible goals goals that are already achieved

2 Plans: A Simple Example Suppose we are at home and we want to go to school, and we must take the bus to do so. A simple plan would likely be: state: We are at home. Actions required to get to school: Go to the bus stop. Get on the bus. Ride the bus to school. Get off of the bus. Walk from the bus stop to the school. Goal State: We are at school. Planning vs. Problem Solving Planning Using Situation Calculus Planning Examples Summary From Problem Solving to Planning From Problem Solving to Planning (2) Basic elements of a search-based problemsolver: Representation of actions. Actions are described by programs that generate successor state descriptions. Representation of states. All state representations are complete. State representations are used for successor generation, heuristic function evaluation, and goal testing. Representation of goals. The goal test and heuristic function are the only way to test for a goal state. Black box decisions on their desirability Representation of plans. A solution is an unbroken sequence of actions from an initial state to goal states. A planning agent needs a more flexible way of structuring its deliberations: work on whichever part of the problem is most likely to be solvable, given the current information. Video Rental - Search Searching vs. Planning Goal: Rent a video, buy some bread, and come home Go to Restaurant Go to Dry Cleaners Go to Rogers Go to store Eat Pick up clothes Rent video Buy Milk... Goal Get a quart of milk and a bunch of bananas and a variable-speed cordless drill. Sit in a chair Go to Sleep Buy Bread Read book...

3 Key Ideas Behind Planning Key Ideas Behind Planning (2) Open up the representation of states, goals, and actions. First-order logic States and goals : sets of sentences Actions: logical descriptions of preconditions and effects Consequently, we get direct connections between states and actions. Example: Goal: Have(Milk) Database: Buy(x) -> Have(x) Chosen action: Buy(Milk) The planner is free to add actions to the plan wherever they are needed, rather than in an incremental sequence at the initial state. Example: choose Buy(Milk) even before knowing where and how to get Milk. There is no necessary connection between the order of planning and the order of execution. Representing states as sets of logical sentences is essential in making this freedom possible. Example: At(Supermarket) represents an entire class of states. Key Ideas Behind Planning (3) Video Rental - Search (revisited) Most parts of the world are independent of most other parts. This makes it feasible to solve conjunctive goals in a divide-and-conquer strategy (generate subplans). Example: divide the Supermarket subplan into a Milk and Bananas subplan. Goal: Rent a video, buy some bread, and come home Go to Restaurant Go to Dry Cleaners Go to Rogers Go to store Sit in a chair Go to Sleep Eat Pick up clothes Rent video Buy Milk Buy Bread Read book... Goal... Video Rental - Planning A simple progressive plan might be state: At Home without Video and Bread Actions: Go to Store Buy Bread Go to Rogers Rent Video Go Home Goal state: At Home with Video and Bread Planning vs. Problem Solving Planning Using Situation Calculus Planning Examples Summary

4 Planning in Situation Calculus Formulation of planning as a logical inference problem, using situation calculus: Initial state: An arbitrary logical sentence about a situation S 0. At(Home, S 0 ) Have(Milk, S 0 ) Have(Bananas, S 0 ) Have(Drill, S 0 ) Goal state: A logical query asking for suitable situations. s At(Home,s) Have(Milk,s) Have(Bananas,s) Have(Drill,s) Planning in Situation Calculus (2) Operators: A set of descriptions of actions. a,s Have(Milk, Result(a,s)) [(a = Buy(Milk) At(Supermarket,s) (Have(Milk,s) a Drop(Milk)) ] Result(a,s): the situation resulting from executing action a in situation s. Planning in Situation Calculus (3): Handling Action Sequences ResultSeq(k, s): the situation resulting from executing the sequence of actions k starting in situation s. s ResultSeq( [ ], s) = s a, p, s ResultSeq( [a p], s) = ResultSeq(p, Result(a, s)) Plan for Shopping Example A solution to the shopping problem is a plan p that when applied to the start state S 0 yields a situation satisfying the goal query. Therefore, we search a p such that At(Home, ResultSeq(p, S 0 )) Have(Milk, ResultSeq(p, S 0 )) Have(Bananas, ResultSeq(p, S 0 )) Have(Drill, ResultSeq(p, S 0 )) Plan for Shopping Example (2) We search a p such that At(Home, ResultSeq(p, S 0 )) Have(Milk, ResultSeq(p, S 0 )) Have(Bananas, ResultSeq(p, S 0 )) Have(Drill, ResultSeq(p, S 0 )) So why don t we just use a theorem prover? and we are done! Solution: p = [ Go(SuperMarket), Buy(Milk), Buy(Bananas), Go(HardwareStore), Buy(Drill), Go(Home) ]

5 Planner vs. Theorem Prover The fundamental difference between a planner and a theorem prover is that the planner requests a sequence of steps that, if executed, will result in the goal state being true. The theorem prover evaluates the sentences for validity. Theorem Provers as Planners? The major drawback of using a theorem-prover for a planner is that a theoretical solution and a practical solution are often not one and the same. In fact, the only guarantee we have for the solution is that we have reached the goal state, but no information on the means. (DoNothing, DoNothing,, Plan) (DoSomething, Plan, UndoSomething) Theorem Provers as Planners? (2) The method of logical inference is semidecidable. This affects planning if the method of inference is unguided by making it extremely inefficient. The solution to the problem of semidecidability is to restrict the language understood by the prover - i.e. create a planner rather than a general purpose theorem prover. Planning vs. Problem Solving Planning Using Situation Calculus Planning Examples Summary The planner will have an algorithm to handle its language efficiently. The most basic representation used for creating planners was developed in a language known as STRIPS STanford Research Institute Problem Solver (Today: SRI International; planner rather than problem solver) Extensions of the STRIPS language are often used in planners today. STRIPS blends efficient planning with the expressive situation calculus representation. Representations: States and Goals States are represented as conjunctions of literals, though planners will often not store entire sets of literals describing the current state. At(Home, S 0 ) Have(Milk, S 0 ) Have(Bananas, S 0 ) Have(Drill, S 0 ) In fact, incomplete states often assume that the literals not indicated are to be treated as false. Goals are conjunctions of literals and (often existentially) quantified variables. At(Home,s) Have(Milk,s) Have(Bananas,s) Have(Drill,s)

6 Representations: Actions Actions are described by STRIPS operators, each of which consists of three components Action description: what an agent returns to the environment in order to do something. Within a planner it serves only as a name for a possible action. Precondition: a conjunction of atoms (positive literals) stating what must be true before the operator can be applied Effect of an operator: a conjunction of literals (positive or negative) that describes how the situation changes when the operator is applied Operator Example Op( ACTION: Go(there), PRECOND: At(here) Path(here, there), EFFECT: At(there) At(here) ) Operator: Go(there) Situation and Plan Spaces Situations are nodes of the search tree between the initial and goal states. Planners can search through the situation space progressively or regressively. Regressive searches are possible because the operators have enough knowledge about the resulting state and the preconditions to decide if an operator is actually applicable. Regressive searching is desirable as goals are not as complex as initial states (fewer conjuncts), but it is very complicated in application, and inefficient to do attempt to overcome this. Situation and Plan Spaces (2) Plan spaces are made up of partial plans - basically frameworks for plans. Plan modification involves inserting, removing, and ordering steps, instantiating variables, etc. The solution is the final plan. The steps taken to achieve this are irrelevant. Example: Video Rental At(Home) Rents(Rogers, Video) Sells(Store, Bread) At(Home) Go(Rogers) At(Rogers) Rents(Rogers, Video) Rent(Video) At(Rogers) Go(Store) At(Store) Sells(Store, Bread) Buy(Bread) At(Store) Go(Home) Situation and Plan Spaces (3) Two operators are generally used on plans: Refinement: constrain or eliminate plans from the space Modification: any other type of plan modification Plan frameworks will all contain a start state and a goal descriptor. The start state sets the preconditions for the first action, and the goal state matches the results of the last action.

7 Plans: Totally and Partially Ordered Least Commitment - don t make a decision until you have to. If a decision is not important at a given stage, leaving it undecided allows for greater flexibility later on and results in less backtracking This leads to a partial order within the plan. Any given (total order) plan will be a linearization of this partial ordering. Plans: Totally and Partially Ordered (2) Example: Putting on a Pair of Shoes Op(ACTION: RightShoe, PRECOND: RightSockOn, EFFECT: RightShoeOn) Op(ACTION: RightSock, EFFECT: RightSockOn) Op(ACTION: LeftShoe, PRECOND: LeftSockOn, EFFECT: LeftShoeOn) Op(ACTION: LeftSock, EFFECT: LeftSockOn) Plans: Totally and Partially Ordered (3) Example : Putting on a Pair of Shoes Data Structure for Plans A set of plan steps. Each step is one of the operators. A set of step ordering constraints. Each constraint is of the form S i < S k Step S i must occur sometime before S k. A set of variable binding constraints, of the form v = x, where v is a variable and x is a constant or a variable. A set of causal links. Data Structure for Plans: Shoe Example Plan ( STEPS: { S 1 : Op(ACTION: ), S 2 : Op(ACTION:, PRECOND: RightShoeOn LeftShoeOn) }, ORDERINGS: {S 1 < S 2 }, BINDINGS: {}, LINKS: {} ) Causal Links Causal links are formed based on precondition fulfilling. A causal link is created when a given state S 1 contains a precondition c for state S 2. S 1 c S 2 S 1 achieves c for S 2. A purpose of S 1 is to achieve the precondition c of S 2. Causal links are protected.

8 Protecting Threatened Causal Links Protecting Threatened Causal Links S 1 c S 3 S 2 c When linearizing a plan, threats that may delete a precondition are ordered either - before the first state (demoted) or - after the second (promoted) when linearizing the plan. This prevents removing the link. S 1 c S 3 S 2 c demote S 3 S 1 S 2 c c promote S 1 S 2 S 3 c c S 3 threatens a condition c established by S 1 and protected by the causal link from S 1 to S 2. S 3 threatens a condition c established by S 1 and protected by the causal link from S 1 to S 2. Solutions Solutions are executable plans. Complete Solutions If S c 1 S 2 then S 1 < S 2 and there is no S 3 such that c results from it, where S 1 < S 3 < S 2 Consistent Solutions There are no orderings such that S 1 < S 2 and S 2 < S 1, or where a variable v = A and v = B (or v = x = B) Possible Threats When a state threatens to remove a precondition necessary to reach another state, it is called a threat. Often when variables are not yet instantiated, they pose possible threats, depending on what they are later assigned. There are 3 methods of dealing with the possible threat. Note the transitivity of the < and = operators Possible Threat Solutions Resolving with an equality constraint Immediately instantiate the variable to something that is nonthreatening Resolving with an inequality constraint Immediately mark the variable as not equal to the threatening value. This has lower commitment. Resolve the threat later This has the lowest commitment, but may not result in an actual solution plan. Planning vs. Problem Solving Planning Using Situation Calculus Planning Examples Summary

9 Planning Examples Partial-order Planner (POP) Example Blocks World Shakey s World Video Rental Problem Partial Order Planner (POP) POP uses regressive planning to create a solution. 1. s with a minimal partial plan and extends the plan at each step by achieving the precondition of a state S. operators are chosen from any steps currently in the plan or the operator pool 2. It then sets up causal links and eliminates threats. 3. If it cannot find an operator at a given point, it backtracks to the previous choice point. The POP algorithm is sound (truth-preserving) and complete - it always finds a solution if one is possible. POP Example: Shopping POP Example: Shopping (2) Initial state: Op(ACTION:, EFFECT: At(Home) Sells(HWS, Drill) Sells(SM, Milk) Sells(SM, Banana)) Goal state: Op(ACTION:, PRECOND: Have(Drill) Have(Milk) Have(Banana) At(Home)) Actions: Op(ACTION: Go(there), PRECOND: At(here), EFFECT: At(there) At(here) Op(ACTION: Buy(x), PRECOND: At(store) Sells(store, x), EFFECT: Have(x) ) POP Example: Shopping (3) POP Example: Shopping (3) Op(ACTION: Buy(x), PRECOND: At(store) Sells(store, x), EFFECT: Have(x) ) Op(ACTION: Buy(x), PRECOND: At(store) Sells(store, x), EFFECT: Have(x) )

10 POP Example: Shopping (3) POP Example: Shopping (3) Causal links Op(ACTION: Buy(x), PRECOND: At(store) Sells(store, x), EFFECT: Have(x) ) POP Example: Shopping (3) POP Example: Shopping (4) Causal links Op(ACTION:, EFFECT: At(Home) Sells(HWS, Drill) Sells(SM, Milk) Sells(SM, Banana)) Op(ACTION:, EFFECT: At(Home) Sells(HWS, Drill) Sells(SM, Milk) Sells(SM, Banana)) POP Example: Shopping (5) POP Example: Shopping (6) A partial plan achieving At preconditions of the three Buy actions. A flawed plan, getting the agent to the HWS and SM.

11 POP Example: Shopping (7) POP Example: Shopping (8) Causal link protection. POP Example: Shopping (8) POP Example: Shopping (9) Causal link protection. A solution to the shopping problem Blocks World Builds stacks of blocks on a table top. Uses two state descriptors and an operator: On(x, y) : indicates x is on top of y Clear(x) : indicates x has nothing on top of it Move(x, y, z) : Preconditions: Clear(x) /\ Clear(z) /\ On(x, y) Effect: A C A B B C Goal Blocks World (2) On(x, z) /\ Clear(y) /\ On(x, y) /\ Clear(z) Solution: Change the meaning of Clear(x) to mean there is room on x for a block Add a new rule MoveToTable(x, y) Preconditions: On(x, y) /\ Clear(x) /\ Table(y) Effect: On(x, Table) /\ Clear(y) /\ On(x, y) Modify the Move(x, y, z) rule: Preconditions: Effect: C A B A B C On(x,y) /\ Clear(x) /\ Clear(z) /\ Table(z) On(x, z) /\ Clear(y) /\ On(x, y) /\ Clear(z) Goal Note: Move(x, y, Table) and Move(x, Table, z) have some concerns with Clear(Table). In the former, the table must be clear to move the block and in the later, the table is then marked as clear.

12 Blocks World Example C A B A B C Goal State: On(A, Table) /\ On(C, A) /\ Clear(C) /\ On(B, Table) /\ Clear(B) MoveToTable(C, A) - PRECOND: On(C, A), Clear(C) EFFECT: On(C, Table) /\ Clear(A) /\ On(C, A) Move(B, Table, C) - PRECOND: On(B, Table), Clear(C) EFFECT: On(B, C) /\ Clear(Table) /\ On(B, Table) /\ Clear(C) Move(A, Table, B) - PRECOND: On(A, Table), Clear(B) EFFECT: On(A, B) /\ Clear(Table) /\ On(A, Table) /\ Clear(B) Goal State: On(C, Table) /\ On(B, C) /\ On(A, B) /\ Clear(A) Shakey s World A planner designed to allow a robot to move between rooms and perform some restricted actions. Uses actions such as Go(loc), Push(obj, loc1, loc2), Climb(box) and Down(box) TurnOn(light_switch) and TurnOff(light_switch) Predicates such as Pushable(object), On(Shakey, level), At(Shakey, loc), Climbable(box), In(obj, loc) Constant - Floor Shakey s World (2) Video Rental - Search Goal: Rent a video, buy some bread, and come home Go to Restaurant Eat Go to Dry Cleaners Go to Rogers Go to store Sit in a chair Go to Sleep Pick up clothes Rent video Buy Milk Buy Bread Read book... Goal... Video Rental - Planning At(Home) Rents(Rogers, Video) Sells(Store, Bread) Video Rental - Planning At(Home) Rents(Rogers, Video) Sells(Store, Bread) At(s) Rents(s, Video) Rent(Video) At(s) Sells(s, Bread) Buy(Bread)

13 Video Rental - Planning At(Home) Rents(Rogers, Video) Sells(Store, Bread) Video Rental - Planning At(Home) Rents(Rogers, Video) Sells(Store, Bread) At(Rogers) Rents(Rogers, Video) At(Store) Sells(Store, Bread) At(x) Go(Rogers) At(x) Go(Store) Rent(Video) Buy(Bread) At(Rogers) Rents(Rogers, Video) Rent(Video) At(Store) Sells(Store, Bread) Buy(Bread) Video Rental - Planning At(Home) Rents(Rogers, Video) Sells(Store, Bread) Video Rental - Planning At(Home) Rents(Rogers, Video) Sells(Store, Bread) At(Home) Go(Rogers) At(Home) Go(Store) At(Home) Go(Rogers) At(Rogers) Go(Store) At(Rogers) Rents(Rogers, Video) Rent(Video) At(Store) Sells(Store, Bread) Buy(Bread) At(Rogers) Rents(Rogers, Video) Rent(Video) At(Store) Sells(Store, Bread) Buy(Bread) At(Store) Go(Home) Video Rental - Planning At(Home) Go(Rogers) At(Rogers) Rents(Rogers, Video) Rent(Video) At(Rogers) Go(Store) At(Store) Sells(Store, Bread) Buy(Bread) At(Store) Go(Home) Planning vs. Problem Solving Planning Using Situation Calculus Planning Examples Summary

14 Summary Planners use formal representations of states, goals, and actions to allow greater flexibility in determining paths from the start state to the finish. Planners often take one of two approaches to solving a problem - progressive or regressive plan building. Least Commitment is making decisions only when it is necessary, thus reducing backtracking. Partial Plans are often a better solution mechanism than searching as they reduce the complexity of the method. Causal links are used to determine and resolve conflicts when working with partial-order plans. References Kasabov, N. (1996). Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering. Cambridge, MA, MIT Press. Nilsson, N. (1998). Artificial Intelligence A New Synthesis. San Francisco, CA, Morgan Kaufmann. Russell, S. and Norvig, P. (1995). Artificial Intelligence A Modern Approach. Upper Saddle River, NJ, Prentice Hall.