Lecture Overview. CSE3309 Artificial Intelligence. The lottery paradox. The Bayesian claim. The lottery paradox. A Bayesian model of rationality

Transcription

1 CSE3309 Lecture Overview The lottery paradox A Bayesian model of rationality Lecture 15 Planning Dr. Kevin Korb School of Computer Science and Software Eng. Building 75 (STRIP), Rm 117 korb@csse.monash.edu.au Planning Basic Theoretical Constructs Search-based problem solving From Search to Planning Planning in the Situation Calculus STRIPS planning Situation Space Planner Reading this week s planning lectures Russell & Norvig, Chapter 11 Korb 3 Korb 4 The Bayesian claim Bayesian (probabilistic) reasoning gets it right: Probabilistic Acceptance Rule If and you learn Then infer (accept) What is? Pretty low, so infer Bird from Flies (i.e., modus tollens works). But it depends! If you learn Flies by way of learning Penguin(Tweety), then a different probability applies:!" #%$'&)("$+*,.-0/1 2 The lottery paradox Suppose Reiter runs a lottery, just to confuse me: Generates $ lottery tickets in a fair lottery Set $3-0/546 7/98: Let ;< mean ticket i loses Then.-0/98=/545$3-> ; < So, we can accept each ;5<. 1%1A1 Having accepted ;!?@ ;B, we must accept C < ; <. However, a fair lottery means C < ;<! Oh no! How confused we Bayesians are!! Important NB: The Lottery can stand as a paradigm of inductive reasoning.

2 A Bayesian model of irrationality? First question: how does default logic cope with the Lottery? I.e., Are we Bayesians all alone in Minsky inconsistency, or do we have company? Consider: Answer: Lottery info : Loses(Ticket i) Loses(Ticket i) Default rule says accept so long as Loses(Ticket i) is consistent with the KB Hence, accept Loses(Ticket i) for every ticket but the last! Bayesians might be alone in inconsistency, but is this really preferable? rationality The Bayesian answer to the lottery: Although ;? and ;, C < ; < Kyburg: Conjunctivits is the disease of mind that asserts Whenever and then ensure * Bayesian alternative model of rationality: Reason probabilitistically so long as you can When you can t Find a reasonable Accept and reason qualitatively so long as you can When you can t, give up (revert to probabilistic reasoning, if possible) P.S. A promissory note (or boast): I will give a Bayesian solution to the Frame (tea cup) Problem by the end of semester. Korb 7 Korb 8 Planning: Basic Theoretical Constructs States: how the world is; composed of facts. Goals: how agent would like world to be. More fundamental: Utility function Model of the world: (causal) structure of the world Events: how the world changes. Actions: how the agent can change the world. Intentional Unintended consequences; interaction with world. Planning: deciding upon a sequence of actions to further some goal a likely domain for qualitative reasoning Basic Theoretical Constructs For the qualitative approach we consider next: 1. States 2. Goals 3. Actions/operators

3 Search-based problem solving Given a goal test and an initial state Search the state space by Applying operators (actions) to states Selecting the operator via some heuristic function Returning a sequence of actions (plan), leading to goal Varieties of search: Uninformed (no heuristic function) BFS DFS Informed A A Shopping Problem Problem: Get a quart of milk and a bunch of bananas and a lamp. Initial state: agent is at home, without any desired object Operators: everything the agent can do Heuristic function: Problems? # of things not yet acquired Search complexity From Bayesian perspective: joint probability Korb 11 Korb 12 Search planning Limitations of search-based problem solvers Branching factor huge Heuristic cannot eliminate actions Can t order goals/actions from initial state. Search Another key idea: planning Key ideas behind planning: Opening up representation of states, goals, actions. Representation: Use formal language (e.g. FOL). States and goals - sentences. Actions - logical description of preconditions and effects. Shopping: if goal = have(milk) and buy(x) achieves have(x) then consider plans that buy(milk) don t consider buy(butter) or sleep Divide-and-conquer: use subplans (subgoals) to attack planning problems Assumes subproblems are (relatively) independent Success: buy lamp vs. buy milk Failure: buy milk vs. buy bananas (or, 8-puzzle) When successful, huge reduction in branching factor

4 Planning in the Situation Calculus Need logical sentences to describe: Initial state: description of situation s. at(home,s ) * have(milk,s ) * have(bananas,s ) * have(lamp,s ) * * Goal query asking for suitable situations. s at(home,s) have(milk,s) have(bananas,s) have(lamp,s) Operators: actions satisfying some appropriate successor-state axiom a,s have(milk,result(a,s)) [(a=buy(milk) * at(supermarket,s)) (have(milk,s) * a =drop(milk))] Planning as theorem proving Action sequences: s result([],s) = s a,p,s result([a p],s) = result(p,result(a,s)) A solution is a plan, p, s.t. when applied to S yields a situation satisfying the goal query. Goal query: at(home,result(p,s )) * have(milk,result(p,s )) * have(bananas,result(p,s )) * have(lamp,result(p,s )) Solution p=[go(supermarket),buy(milk),buy(bananas), go(ikea),buy(lamp),go(home)] Korb 15 Korb 16 Practical planning? Problems with planning as theorem proving: Search is inefficient: control/search problems FOL is semidecidable (can t prove there is no plan) Plan may be inefficient: p=[go(market),buy(milk),go(home), go(market),buy(bananas),go(home), go(ikea),buy(lamp),go(home)] Useful to Restrict the language Use special-purpose algorithm, a planner. STRIPS planning Most classical planners describe states and operators in a restricted language: the STRIPS language. STRIPS: STanford Research Institute Problem Solver (after SRI)... States: conjunction of function-free ground literals at(home) * have(milk) * have(bananas) * have(lamp) State description does not need to be complete. (a) set of possible complete states, OR (b) negation as failure : if positive literal not mentioned, then assumed to be false. (aka closed-world assumption )

5 Goals in STRIPS Goals: conjunction of literals, possibly with variables (existentially quantified). at(home) * have(milk) * have(bananas) * have(lamp) STRIPS operators: STRIPS operators 1. Action description: name for a possible action. 2. Precondition: conjunction of atoms (positive literals) what must be true before operator can be applied. x at(x) * sells(x,milk) 3. Effect: conjunction of literals (pos or neg) how the situation changes when operator applied. Korb 19 Korb 20 STRIPS operators (cont.) Syntax for STRIPS operators OP (ACTION: go(there), PRECOND: at(here) * path(here,there), EFFECT: at(there) * at(here)) Operator schema: STRIPS operator with variables. Corresponds to a family of actions, one for each instantiation Usually, only fully instantiated operators can be executed. Operator o is applicable in state s if there is some way to instantiate vars in o so that each precondition is true in s After operator applied, all pos literals in effect(o) hold, all literals in s except those which are neg literals in effect(o). STRIPS assumption: everything literal true in state is true in successor state unless explicitly negated i.e., Frame Problem assumed away.

6 Example State: at(home), path(home,market) go(supermarket) is applicable OP(ACTION: go(market), PRECOND: at(home) * path(home,market), EFFECT: at(market) * at(home)) So, resulting situation contains literals: at(home) at(supermarket) path(home,supermarket) Situation Space Planner Shopping eg: a search space of situations. situation space planner progression planner: High branching factor huge search space. Search backwards, from goal to initial state: regression planner. Desirable when few operators achieve goal(s) Nodes in the search tree = situations. An alternative: search through the space of plans rather than the space of situations.