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Presentation on artificial intelligence planing

Presentation on artificial intelligence planing

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  • The development of general problem solvers has been one of the main goals in Artificial Intelligence. Slide…………….
  • A general problem solver has two main parts a general modeling language for describing problems, and algorithms for solving them.
  • Planning is a form of general problem solving. It is concerned with automatic synthesis of action strategies (plans) from a description of actions, sensors, and goals.AI Planning is general problem solving over a class of models. Models define the scope of a planner, the types of problems it is supposed to handle, the form of the solutions, and the solutions that are best or optimal.
  • When we talk about planning we can identify three main elements.We need a formal representation language to describe the problemModels to understand them and Algorithms to solve them
  • Lets first study models in planning.Why we need a model what can a model do?A model is used to define the scope of a planner. Without a scope it will be impractical to implement them. Also it is used to define what are the solutions to a particular planning problem and what the optimal solution is.
  • Now we understand the importance of models. Our next step is to understand some planning models which are developed to address different categories of planning problems. First is classical planning.Describe the model
  • Here it is shown the definitions for solutions and optimal solutions in classical planning.In simple words a solution is a sequence of applicable action that maps s0 to SG.An optimal solution is a solution that minimizes sum of action costs.In classical planning, it is also assumed that all costs c(a, s) are equal and thus that the optimal plans are the ones with minimal length.
  • Classical planning assumes that the initial state of the system is known and that state transitions are deterministic. But there are situations which we cannot apply these assumptions. In this kind of situations we have to develop a model which includes uncertainty.To do that we have to take sensing and feedback into the account.We need sensing to check what are the states.We need feedback to asses our actions.
  • When we plan with uncertinity we can follow two approches. Those are pure non determinisam and probabilities.In pure non determinism ………………………………….In probabilities……………………………..
  • Now let’s s how the state transitions happens in this model. There are two equations for the two cases. In either case, an action a deterministically maps a belief state b into a new belief state ba.Here a belief state represents set of states deemed possible, and summarizes past actions and observations.
  • If we are going to handle a planning problem with uncertainty without feedback we have to solve it in a deterministic manner.
  • Until now we talked about situations where no additional information available in execution time. But when we plan with sensing in other words feedback we have to deal with the information we receive in execution time.As the slide says………………………..But here there is important fact that is sensing only makes sense in a state of uncertainty; if there is no uncertainty, sensing provides no useful information and can be ignored
  • There are two ctegories of planing with sensing problems. Those areFull-state ObservabilityPartial ObservabilityIn full oberservability…………………..In parialobservability……………………………..
  • As in the earlier situation belief states can be shown as this when planning with sensing.
  • Until this state we does not involve time durations in our planning problems. But some times that is essential. We can use temporal planing in those kind o situations.Slide…..
  • Second bullet pointThis simply says we have to consider each action separately in the set Ai to obtain state transition function.
  • Actually we can see that up to this point this coincide with the model for classical planning if primitive actions are replaced by legal set of actions. When defining the action costs we cannot simply add the cost of each action to get the total cost because the contribution of the actions in Ai depends on the actions taken at the previous steps. So we have to calculate it as this.Total cost to reach the goal can be given as thisInitial cost + sum of transition costs
  • Here parallel planning is temporal planning with actions of unit durations only.SAT and CSP are two approaches which are introduced for parallel planning.
  • As we discussed early we need a formal language to represent complex planning problems. So we can develop languages as described in the slide to address this.Basically planning languages do two major tasks,It specifies the model and reveal useful heuristic information.
  • Stanford Research Institute Problem Solver
  • State Language- a language for describing the world Operator Language - language for describing how the world changes.We consider the Strips language as used currently in planningrather than the original version of Strips that is more complex.
  • Strips has two kinds of symbols. Relational and constant symbols. In the expression given I the slide on is a relational symbol. We call it a relational symbol with arity two because it contains two constant symbols.A main difference between relational and constant symbols in Strips is that the former are used to keep track of aspects of the world that may change as a result of the actions (e.g., the symbol on in on(a, b)), while the latter are used to refer to objects in the domain (e.g., the symbols a and b in on(a, b)).In Strips, there are no functional symbols and the constant symbols are the only terms.
  • There is a construct called atoms in Strips. They acts like boolean variables of the domain. Atom is defined as a ………………………………Each operator has three lists precondition, add and delete.Here a precondition is something like this if we need to by coffee we need to be already in the coffee shop.
  • We can represent a problem in Strips as a tuple like this.Here P is the problem……………………………
  • This mapping defines the semantics of a Strips planning problem P, whose solution is given by the solution of the state model S(P).GPT (Bonet & Geffner 2000) and MBP (Bertoli et al. 2001),
  • Heuristic not admissible (not lower bounded ) but informative and fastUsefull for optimal planningThe heuristic h+(s) if a state is h+(G) where G is the goal, and is obtained by solving the above first equation with single shortest path algorithms.
  • Atoms are independent – the cost of achieving a set of atoms corresponds to the sum of the costs of achieving each atom in the set
  • Heuristic admissible but not very informative
  • To find the heuristic functionRecent idea of relaxationConsidering a set of possible values as a same value
  • Every element has one main branch, which represents the principal line of development, and may have multiple subbranches, each of which represents a separate line of developmentParallel development- we can consider branches separatelyThis is different from branching factor
  • Other – especial cases no need to describe here
  • Now I am going to classify different planners according to how they operateIn AI planning classification is slightly differentthe estimated cost of the remaining plan h(s) that depends only on the state s obtained by progression or regression and which summarizes the partial plan p completelybranching factor in temporal planning, where the set of parallel macro actions is exponential in the number of primitive actions, and in a number of sequential domains like Sokoban (Junghanns & Schaeffer 1999), where the number of applicable actions is just too large.
  • Search a partially developed planIf we have already identified heads and tails we can use this
  • Huristic , branching combined
  • Heuristic and constraint-based approaches, so powerful in the deterministic setting, are not directly applicable to problems involving non-determinism and feedback, as the solution of these problems is not a sequence of actions but a function mapping states into actions.Dynamic programming (DP) methods compute a value function over all states, and use this function to define the policy
  • The greedy policy 𝜋V (s) relative to a given value function V corresponds to the function that maps states s into actions a that minimize the worst cost or the expected cost of reaching the goal from s according to whether state transitions are modeled nondeterministically or probabilistically.The greedy policy 𝜋V (s) relative to a given value function V corresponds to the function that maps states s into actions a that minimize the worst cost or the expected cost of reaching the goal from s according to whether state transitions are modeled nondeterministically or probabilistically.
  • Dynamic programming (DP) methods compute a value function over all states, and use this function to define the policyThe greedy policy 𝜋V (s) relative to a given value function V corresponds to the function that maps states s into actions a that minimize the worst cost or the expected cost of reaching the goal from s according to whether state transitions are modeled nondeterministically or probabilistically.Greedy policy is optimal when V is the optimal cost functionany heuristic function h determines a greedy policy 𝜋V for V = hDynamic programming (DP) methods compute a value function over all states, and use this function to define the policyThe greedy policy 𝜋V (s) relative to a given value function V corresponds to the function that maps states s into actions a that minimize the worst cost or the expected cost of reaching the goal from s according to whether state transitions are modeled nondeterministically or probabilistically.Greedy policy is optimal when V is the optimal cost functionany heuristic function h determines a greedy policy 𝜋V for V = h
  • The optimal cost function is the solution of the Bellman equation for the non-deterministic and stochastic cases respectively,In both cases with V (s) = 0 for all goal states. Value iteration solves the Bellman equation by plugging an estimate Vi function on the right hand side, and obtaining a new value function i+1 on the left hand side. This process is iterated until a fixed point is reached (in the probabilistic case, the convergence is defined in a slightly different way
  • ..Heuristic function is to limit the no of avoid consideration of most states.
  • In the last few years, promising strategies that integrate DP and heuristic search methods have been proposed. Real time dynamic programmingMore precisely, in every non-goal state s, the best action a according to the heuristic is selected (i.e., a = πh(s)) and the heuristic value h(s) of the state s is updated using Bellman equation.Then a random successor state of s and a is selected using the transition function or transition probabilities, and this process is repeated until the goal is reached.

Planing presentation Presentation Transcript

  • 1. Perspectives on ArtificialIntelligence PlanningBased on the research paperby,Professor H´ector Geffner
  • 2. General problem solversA general problem solver is a program that accepts high-level descriptions of problems and automatically computes their solution
  • 3. Problem solvers con. Problem Solver General modeling Algorithms language
  • 4. What is planning?Planning is a key area in Artificial Intelligence. In its general form, planning is concerned with the automatic synthesis of action strategies (plans) from a description of actions, sensors, and goals.
  • 5. Elements of Planning1. Representation languages for describing problems conveniently.2. Mathematical models for making the different planning tasks precise3. Algorithms for solving these models effectively
  • 6. ModelsModels needed to define scope of a planner• What is a planning problem• What is a solution(plan)• What is an optimal solution
  • 7. Classical PlanningClassical planning can be understood in terms ofdeterministic state models characterized by the followingelements• A finite and discrete state space S,• An initial situation given by a state s0 ∈ S,• A goal situation given by a non empty set SG ⊆ S,• Actions a(s) ⊆ A applicable in each state s ∈ S,• A deterministic state transition function f(a, s) for a ∈ a(s)• Positive action costs c(a, s) for doing action a in s.
  • 8. Classical Planning con.
  • 9. Planning with Uncertainty Unlike in classical planning here states of the system and that state transitions are nondeterministic so we have to define how uncertainty is modeled. Also we have to take sensing and feedback in to the account.
  • 10. Modeling Uncertainty• Pure non determinismUncertainty about the state of the world isrepresented by the set of states S’ ⊆ S that aredeemed possible• ProbabilityRepresented by a probability distribution over S.
  • 11. Modeling uncertainty in statetransitions
  • 12. Planning under uncertaintywithout feedbackIn both above cases the problem of planning under uncertaintywithout feedback reduces to a deterministic search problem in beliefspace, a space which can be characterized by the followingelements.• A space B of belief states over S,• An initial situation given by a belief state b0 ∈ B,• A goal situation given by a set of target beliefs BG• Actions a(b) ⊆ A applicable in each belief state b• Deterministic transitions b to ba for a ∈ a(b) givenby (1) and (2) above• Positive action costs c(a, b).
  • 13. Planning with SensingWith the ability to sense the world, the choice ofthe actions depends on the observation gathered and thus the form of the plans changes.
  • 14. Planning with Sensing con.Full-state ObservabilityIn the presence of sensing, the choice of the action ai at timei depends on all observations o0, o1, . . . , oi−1 gathered up tothat point.Partial Observabilityobservations reveal partial information about the true stateof the world and it is necessary to model how the two arerelated. The solution then takes the form of functionsmapping belief states into actions, as states are no longerknown and belief states summarize all the information fromprevious belief states and partial observations
  • 15. Planning with Sensing con.
  • 16. Temporal Planning Temporal models extend classical planning in a different direction. This is a simple but general model where actions have durations and their execution can overlap in time.• we assume a duration d(a) > 0 for each action a, and a predicate comp(A) that defines when a set of actions A can be executed concurrently.
  • 17. Model for temporal planning• Need to replace the single actions a in that model by sets of legal actions. {A0,A1,A2,…..}• each set Ai start their execution at the same time ti. The end or completion time of an action a in Ai is thus ti + d(a) where d(a) is the duration of a.• t0 = 0 and ti+1 is given by the end time of the first action in A0, . . . , Ai that completes after ti.
  • 18. Model for temporal planningcon.• The initial state s0 is given, while si+1 is a function of the state si at time ti and the set of actions Ai in the plan that complete exactly at time t + 1; i.e., si+1 = fT (Ai, si).• The state transition function fT is obtained from the representation of the individual actions• A valid temporal plan is a sequence of legal sets of actions mapping the initial state into a goal state.
  • 19. Model for temporal planningcon.
  • 20. Temporal planning vs.sequential planning• Though the model for sequential planning and the model for temporal planning both appear to be close from a mathematical point of view they are quite different from a computational point of view.• Heuristic search is probably the best current approach for optimal and non-optimal sequential planning, it does not represent the best approach for parallel planning
  • 21. Languages In large problems, the state space and state transitions need to be represented implicitly in alogical action language, normally through a set of (state) variables and action rules. A good actionlanguage is one that supports compact encodings of the models of interest.
  • 22. Strips• In AI Planning, the standard language for many years has been the Strips language introduced in 1971 by Fikes & Nilsson.• While from a logical point of view, Strips is a very limited language, Strips is well known and helps to illustrate the relationship between planning languages and planning models, and to motivate some of the extensions that have been proposed.
  • 23. Strips con. Strips State language Operator language
  • 24. Elements of Strips• The Strips language L is a simple logical language made up of two types of symbols: relational and constant symbols. E.g. on(a, b) relational constant symbol symbols• In Strips, there are no functional symbols and the constant symbols are the only terms.
  • 25. Elements of Strips con.• Atomscombination p(t1, . . . , tk) of a relational symbol pand a tuple of terms ti of the same arity as p.• Operatorsdefined over the set of atoms in L. Each operatorop has a precondition, add, and delete listsPrec(op), Add(op), and Del(op) given by sets ofatoms.
  • 26. Planning problems in Strips P = <A,O, I,G>• A stands for the set of all atoms in the domain• O is the set of operators• I and G are sets of atoms defining the initial and goal situations.
  • 27. Planning problems in Strips con.The problem P defines a deterministic state modelS(P)like below• The states s are sets of atoms from A• The initial state s0 is I• The goal states are the states s such that G ⊆ s• A(s) is the set of operators o ∈ O s.t. Prec(o) ⊆ s• The state transition function f is such that f(a, s) =s + add(a) − del(a) for a ∈ a(s)• Costs c(a, s) are all equal to 1
  • 28. Advanced languages• Domain-independent planners with expressive state and action languages like GPT (Bonet & Geffner2000) and MBP (Bertoli et al. 2001) has been introduced. both of them provide additional constructs for expressing non determinism and sensing.• Knowledge-based planners has been introduced which provide very rich modeling languages, often including facilities for representing time and resources
  • 29. ComputationThis relates to the algorithms and techniques use to solve planning problems.
  • 30. Heuristic Search• Simple , powerful and explains success of recent approaches• Maps planning problems into search problems• Explicitly searches state space with heuristic h(s) that estimates cost from s to Goal• Heuristic h extracted automatically from problem representation• Uses A*, IDA*…… etc to plan
  • 31. Heuristic Functions• Heuristic search depends on the choice of heuristic function• Heuristics derived as optimal cost function of relaxed problems • Simple relaxation used in planning • Ignore delete lists • Reduce large goal sets to subsets • Ignore certain atoms
  • 32. Additive Heuristic
  • 33. Additive Heuristic• Assume atoms are independent• Heuristic not admissible (not lower bounded )• Informative and fast• Useful for optimal planning• The heuristic h+(s) of a state is h+(G) where G is the goal, and is obtained by solving the above first equation with single shortest path algorithms.
  • 34. Fast Forward Heuristic• Does not assume that atoms are independent• Solve the relaxation suboptimally• Extracts useful information for guiding hill climbing search
  • 35. Max Heuristic
  • 36. Generalisation: hm heuristics• For fixed m=1,2…. Assume cost of achieving set C given by cost of most costly subset of size m – For m=1 , hm = hmax – For m=2 , hm = hG (Graphplan) – For any m , hm admisible and polinomial
  • 37. Pattern databases• Project the state space S of a problem into a smaller state space S’• S’ can be solved optimally or exhaustively• Heuristic h(s) for the original state space is obtained from the solution cost h’∗(s’) of the projected state s’ in the relaxed space.
  • 38. Branching Scheme• A branch is an object that specifies a linear sequence of versions of an element• Enable parallel development• Usually seldom mentioned in texts• But has very strong influence on performance
  • 39. Types of branching• Forward Branching – Start from the initial state and go forward till the goal state is found• Backward Branching – start from the goal state and go backward till the initial state• Other – When both above methods do not workout
  • 40. Classification in AI planners• State space planners – Progression and regression planning – Search in the space of states – Build plans from head or tail only – Estimated cost consists of two parts • the accumulated cost of the plan g(p) that depends on p • the estimated cost of the remaining plan h(s) that depends only on the state s obtained by progression or regression – High branching factor
  • 41. Classification in AI planning• Partial –order planners – Search in the space of plans – The partial plan heads or tails can be suitably summarized – Useful for computing the estimated cost f(p) of the best complete plans
  • 42. Heuristics and branchingconvergenceit should be possible to combine informative lowerbounds and effective branching rules, thus allowingus to prune partial solutions p whose estimatedcompletion cost f(p) exceeds a bound B.
  • 43. Search in Non-Deterministic Spaces• Heuristic and constraint-based approaches are not directly applicable to problems involving non-determinism and feedback• The solution of these problems is not a sequence of actions but a function mapping states into actions.• Dynamic programming is used
  • 44. Dynamic programming
  • 45. Mathematics…… 
  • 46. More Mathematics …….  Bellman equations
  • 47. DP features• Works well for spaces containing large number of states• For larger spaces, the time and space requirements of pure DP methods is expensive• In comparison heuristic search methods for deterministic problems that can deal with huge state spaces provided a good heuristic function
  • 48. Converging DP and HeuristicmethodsNew strategies that integrate DP and heuristicsearch methods have been proposed. • Real time dynamic programming • In every non-goal state s, the best action a according to the heuristic is selected • Heuristic value h(s) of the state s is updated using Bellman equation. • Then a random successor state is selected • This process is repeated until the goal is reached