Repetition Prepar Pr a epar t a ion for fo Ex am Ex

Transcription

1 Repetition Preparation for Exam

2 Written Exam Date: Time: 9:30 14:30 No book or other aids Maximal 40 points Grades: 0 19 U (did not pass) (pass) (verygood) (excellent)

3 Exam Topics Basic search (depth-first, breadth-first, best first, hill climbing) Adversarial search (Minimax search, alpha-beta pruning) Logic inference Planning Reasoning under uncertainty ty Paradigm of intelligent robot architectures

4 Search Problem Formulation A search problem is defined by four items: 1. Initial state that agent starts in, e.g., "at Arad 2. Descriptions of possible actions for every state or successor function S(x) = set of action successor pairs e.g., S(Arad) = {<Arad Zerind, Zerind>, <Arad Sibiu, Sibiu>, } 3. Goal test determining whether a given state is a goal state. 4. Path cost function assigning a numeric cost to each path in the state space c(x,a,y) is the step cost of taking action a to go from state x to y. the sum of step costs along a path, e.g., sum of distances, number of actions executed, etc. A solution is a sequence of actions leading from the initial state to a goal state

5 Search Strategies Uninformed search strategies use only the information available in the problem definition Breadth first search Uniform cost search Depth first search.. Informed search strategies use problem specific knowledge to find solutions more efficiently Greedy best first search Best first (A*) search Hill climbing search

6 Uninformed Search Strategies Breadth-first: expand nodes level by level, starting from shallowest level (new nodes at the end of the FRINGE list) Uniform-cost: expand the unexpanded node with the least path cost (arranging the nodes in the FRINGE list in an increasing order of path cost) Depth-first: expand deepest unexpanded node (new nodes in the front of the FRINGE list)

7 Informed Search Strategies Greedy Best-first: expand the unexpanded node that has the best value for the heuristic function h(n) h(n): the estimated cost of the cheapest path from n to goal Best-first (A*): expand the unexpanded node that has the best value for the evaluation function g(n)+h(n) ) g(n): the cost so far to reach n Hill-climbing: It only expands the current node and moves to the child with the best evaluation, terminates when no child improves over the current state. It may use h for evaluation, but different from greedy-best y g y first in selecting among child nodes and no tracing back

8 Avoiding Repeated States Closed list storing all expanded nodes # check if the node is in Closed

9 Minimax Search MINIMAX VALUE( n) = max min Utility ( n ) s Successors( n) s Successors( n) MINIMAX VALUE( s) MINIMAX VALUE( s) if if if n n n is is is a a a terminal state MAX node MIN node Strategy: choose move to position with highest minimax value

10 Alpha Pruning α is the value of the best (i.e., highest-value) choice found so far for any MAX point along the path If v for a later MIN node is worse than α, max will avoid it prune the branches of the MIN node

11 Beta Pruning β is the value of the best (i.e., lowest value) l choice found so far for any MIN point along the path If v for a later MAX node is bigger than β, min will avoid it prune the branches of the MAX node MIN MAX MIN MAX β v

12 General in Logic Logical language supports knowledge representation and it is defined by the following two aspects 1. Syntax describes configurations that can constitute sentences 2 Semantics determines facts in the world to which the 2. Semantics determines facts in the world to which the sentences refer i.e., define truth of a sentence in a world

13 Propositional logic: Syntax * The symbols for propositional logic include - propositional symbols: P, Q, R, S - truth symbols: true, false - connectives:,,,, The propositional symbols P 1, P 2 are sentences. Other p p y 1, 2 sentences can be formed from atomic symbols with the following rules

14 Propositional logic: Semantics Interpretation is the assignment of truth value (either T or F) to the set of propositions The symbol true is always assigned T, and false assigned F The interpretation of truth value for sentences follows the rules below: S is T iff S is F S 1 S 2 is T iff S 1 is T and S 2 is T S 1 S 2 is T iff S 1 is T or S 2 is T S 1 S 2 is T iff S 1 is F or S 2 is T S 1 S 2 is T iff S 1 and S 2 have the same value

15 Rule of Inferences for Propositional Logic Modus Ponens α β, β α And-Elimination i α α 1 2 L α n α i And-Introduction α α α 1 1, α 2, L, α n 2 L α n

16 Inferences Rules Or introduction αi α α L α 1 2 n Double negation elimination A A Unit resolution α β, β α Resolution α β, β γ α β, β γ, or α γ α γ

17 First-order logic first-order logic (like natural language) assumes the world contains Objects: people, houses, numbers, colors, baseball games, wars, Relations:, round, brother of, bigger than, part of, married to; Functions: father of, best friend of, population of

18 Predicates Predicate symbol: referring to a particular relation in the world, e.g. Brother, Bigger than * Atomic sentence =predicate (term 1,...,term n ) Examples: Brother(Richard, John) Married(FatherOf(Richard), MotherOf(John))

19 Universal Quantifier <variables> <sentence> Everyone winning Nobel prize is smart: x Win(x,NobelPrize) Win(xNobelPrize) Smart(x) x P is true iff P is true with x being each g possible object in the world

20 Existential quantifier <variables> <sentence> Someone at KTH has won Nobel Prize: x At(xKTH) At(x,KTH) Win(x, Nobel Prize) P i t iff P i t ith bi x P is true iff P is true with x being some possible object in the world

21 Universal Elimination For any variable v and ground term g (without variable) v α SUBST ({ v / g }, α ) E.g., x King(x) Greedy(x) Evil(x) yields King(John) Greedy(John) Evil(John) King(Richard) Greedy(Richard) Evil(Richard) King(Father(John)) Greedy(Father(John)) Evil(Father(John)).

22 Existential Elimination For any sentence α, variable ibl v, and constant symbol lkk that does not appear elsewhere in the knowledge base: v α Subst({v/k}, α) E.g., xcrown(x) OnHead(x,John) yields: Crown(C 1 ) OnHead(C 1,John) provided C 1 is a new constant symbol

23 Generalized Modus Ponens (GMP) For atomic sentences p i, p i, where there is a substitution θ such that SUBST(θ, p i )=SUBST(θ, p i ) for all i p 1 ', p 2 ',, p n ', ( p 1 p 2 p n q) SUBST(θ,q) Example: King(x) Greedy(x) Evil(x) p 1 ' =King(John) p 1 =is King(x) p 2 = Greedy(y) p 2 =Greedy(x) θ ={x/john,y/john q = Evil(x) SUBST(θ,q)= Evil(John)

24 Unification Inorder to applythe generalized modusponens rule, we haveto find substitution to make two involved sentences look the same The job of unification isto find such an substitution given two sentences p and q. Formally we have Unify(p,q) = θ if SUBST(θ, p) = SUBST(θ, q) θ is the unifier of the two sentences p q θ Knows(John,x) Knows(John,Jane) Jane) Knows(John,x) Knows(y,OJ) Knows(John,x) Knows(y,Mother(y)) Knows(John,x) Knows(x,OJ)

25 STRIPS STanford Research Institute Problem Solver Planner used to drive the robot SHAKEY early 1970s. STRIPS addressed the issue of efficient representation and implementation. Successful plans where saved as macro operators. Macros are stored as triplets: Preconditions (P). Add list (A) Delete list (D). Advatage: Specifying everything needed by the frame axiom. Disadvatage: not able to use theorem prover to produce new states. Computers were slow in the 1970s, SHAKEY was thinking for a long time and some plans took half an hour to plan.

26 Operators as Triples Preconditions (P): conditions that must hold before an operator is applied. Add list (A): additions to the state description as a result of applying an operator. Delete list (D): the part of the state description to be removed after the operator is applied. pickup(x): P: gripping() clear(x) ontable(x) A: gripping(x) D: ontable(x) gripping() putdown(x): t k(x Y) stack(x,y) unstack(x,y) P: gripping(x) A: gripping() ontable(x) clear(x) D: gripping(x)

27 Bayes Rule P ( h D ) = P ( h ) P ( D h ) / P ( D ) Based on the evidence or observation D we can revise our prior belief P(h) to the a posteriori probability (posterior) P(h D) The posterior probability is the belief in h after evidence D has been observed.

28 Naïve Bayes Classifier Naïve Bayes Classifier is applied to learning tasks where each instance x is described by a set of attribute values and the target of x takes on a value fromsome finite set V (classification). A set oftraining examples is provided. Now given a new instance described by a tuple of attribute values [a 1, a 2,, a n ], it is required to predict the target value for the new instance. The Bayesian approach is to assign the most probable target value, vmap, given the attribute values, to the new instance

29 Naïve Bayes Classifier Naive Bayes assumption: the attribute values are conditionally independent given the target value: which gives Ni Naive Bayes Classifier: The probabilities P(v ) P(a v ) are estimated in the learning The probabilities P(v j ), P(a i v j ) are estimated in the learning stage based on the frequencies in the training data.

30 Hierarchical Paradigm SENSE PLAN ACT A sequential order of sensing, planning, and acting Sensing is monolithic: all the sensor information is fused into a global world model, which the planar accesses The world model dlcontains ti 1. an a priori (previously acquired) representation of the environment 2. sensing information for what robot sees 3. additional i cognitive ii knowledge which h is relevant

31 Reactive e Structure Functions are independent of each other No planning Behaviors: tight coupling between sense and act Reactive system composed of behaviors

32 Hybrid Paradigms PLAN Modeling and deliberation SENSE ACT Plan: figuring i out how to implement a task and generating a set of behaviors Reactive control: execute the set of behaviors to finish the task