Artificial Intelligence CS482, CS682, MW 1 2:15, SEM 201, MS 227 Prerequisites: 302, 365 Instructor: Sushil Louis, [email protected], http://www.cse.unr.edu/~sushil Questions Rational agents and performance metrics Suppose that the performance measure is concerned with just the first T time steps of the environment and ignores everything thereafter. Show that a rational agents action may depend not just on the state of the environment but also on the time step it has reached Questions (True or False)

An agent that senses only partial information about the state cannot be perfectly rational There exist task environments in which no pure reflex agent can behave rationally There exists a task environment in which every agent is rational The input to an agent program is the same as the input to the agent function Every agent function is implementable by some program/machine combination Suppose an agent selects its action uniformly at random from the set of possible actions. There exists a deterministic task environment in which this agent is rational True or False

It is possible for a given agent to be perfectly rational in two distinct task environments Every agent is rational in an unobservable environment A perfectly rational poker-playing agent never loses Types of task environments Task Env Soccer Exploring the subsurface oceans of Titan Shopping for used AI books

on the net Playing a tennis match Practicing tennis against a wall Performing a high jump Knitting a sweater Bidding on an item at anauction

Observable Agents Deterministic Episodic Static Discrete Types of task environments Task Env

Observable Agents Deterministic Episodic Static Discrete Soccer

Partial Multi Stochastic Sequential Dynamic Continuous Exploring the

subsurface oceans of Titan Partial Single? Stochastic Sequential Dynamic

Continuous Shopping for used AI books on the net Partial Single ? Deterministic Sequential

Static Discrete Playing a tennis match Fully Multi Stochastic Episodic/Seq

Dynamic Continuous Practicing tennis against a wall Fully Single Stochastic

Episodic/seq Dynamic Continuous Performing a high jump Fully Single

Stochastic Sequential Static Continuous Knitting a sweater Fully Single

Deterministic Sequential Static Continuous Bidding on an item at an auction Fully

Multi Stochastic/ Strategic Sequential Static Discrete Quotes MURPHY'S LAWS

1.Nothing is as easy as it looks. 2.Everything takes longer than you think. 3.Anything that can go wrong will go wrong. 4.If there is a possibility of several things going wrong, the one that will cause the most damage will be the one to go wrong. Corollary: If there is a worse time for something to go wrong, it will happen then. 5.If anything simply cannot go wrong, it will anyway. 6.If you perceive that there are four possible ways in which a procedure can go wrong, and circumvent these, then a fifth way, unprepared for, will promptly develop. 7.Every solution breeds new problems. The Murphy Philosophy Smile . . . tomorrow will be worse.

Arthur C. Clarke Any sufficiently advanced technology is indistinguishable from magic. Outline Problem solving agents Problem types Problem formulation Example Problems

Basic Search Algorithms Problem Solving Agents Restricted form of general agent This is offline problem solving. Search for solution, then execute. During execution we are not using subsequent percepts Problem solving agent example Consider a holiday in Romantic Romania You are an agent, holiday touring in Arad, Romania

What are your performance measures? Improve suntan, look at the sights, check out Transylvania, enjoy the nightlife, become one of the undead, avoid hangovers, The action sequence to do this is long and complicated and you need to read guidebooks, books, talk to people, make tradeoffs Very complex, let us simplify You have a non-refundable ticket to get home from Bucharest tomorrow Now you have a goal: Get to Bucharest in

time to catch your flight tomorrow Romantic Romania Goal: Get to Bucharest Formulate Problem: States: Cities Actions: Drive to city What level of abstraction? Turn wheel or Drive to Bucharest What is a state? What is an action? Goal: Set of states, specifically: {Bucharest}

Solution: Sequence of actions that results in a goal state What type of task environment? Task Env Observable Agents Deterministic Episodic Static

Discrete Task Env Observable Agents Deterministic Episodic Static

Discrete Romantic Romania Yes Single Yes Sequential

Static Discrete Romantic Romania Problem solution A fixed sequence of actions Agent searches for a sequence of actions that will lead to a goal state So we : Formulate the problem, Search for a solution,

Execute the action sequence Execution phase does NOT consider percepts in this simple example. In control theory: Open-Loop system Back to Romanian problem formulation Initial State, S_0 In(Arad) Actions Actions(S) returns set of actions possible in state S {Go(Sibiu), Go(Timisoara), Go(Zerind)} Transition Model: What does an action do?

State Space of our problem Result (In(Arad), Go(Zerind)) = In(Zerind) State space is a directed graph A Path in the state space is a sequence of states connected by a sequence of actions Goal State(s) In(Bucharest) Path COST function Some agents are better than others lower cost

Path costs are non-negative (>= 0) A solution is a sequence of actions leading from the initial state to a goal state Abstraction The real world is absurdly complex so state space must be abstracted for problem solving In(Arad) means somewhere in Arad but where Result(In (Arad), Go(Zerind)) = In (Zerind). Yay but how do you find the highway out and what side do you drive on and wheres the gas station, and .. In a more expressive, less abstract representation of the world, In(Arad) must correspond to some real location in Arad (Hotel Phoenix perhaps) Similarly a solution, a sequence of actions, must correspond to real

actions in the less abstract real-world. A Solution Path must correspond to a real path Our abstraction should make the original problem easier while at the same time enabling a correspondence with a more expressive representation Vacuum world. States and transitions Vacuum world States ? Actions ?

Transition model (see figure) Goal test ? Path cost ? Vacuum world States Dirt location (0, 1), Robot location (0, 1) Initial state can be any state Actions

Left, Right, Suck, NoOp Transition model Goal test No Dirt. All squares are clean Path cost 1 per action, 0 for NoOp 8 puzzle States Location of every tile and blank

Initial state Any state Actions Movement of blank Up, down, left, right Transition model New state after blank move Goal Test Test if configuration matches figure Path cost

1 per blank move 8 Queens States ? Initial State ? Actions ? Transition model

? Goal Test ? Path cost ? Real world problems

Route finding TSP VLSI Robot Navigation Automatic assembly sequencing Solving Romania Romanian problem formulation Initial State, S_0 In(Arad) Actions

Actions(S) returns set of actions possible in state S {Go(Sibiu), Go(Timisoara), Go(Zerind)} Transition Model: What does an action do? State Space of our problem Result (In(Arad), Go(Zerind)) = In(Zerind) State space is a directed graph A Path in the state space is a sequence of states

connected by a sequence of actions Goal State(s) In(Bucharest) Path COST function Some agents are better than others lower cost Path costs are non-negative (>= 0) A solution is a sequence of actions leading from the initial state to a goal state Frontier Frontier Frontier Frontier

Arad is loopy Why should we ignore loopy (redundant) paths? 1. DynProg 2. PathCost Should we always ignore redundant paths? Graph search avoids redundant paths And, very importantly, getting rid of redundant paths reduces the number of tree nodes from pow(b, d) to approximately 2 d^2 !!!!! b = branching factor d = tree depth Graph search makes a state tree

Graph search frontier separates explored and unexplored states Implementing graph search Node != problem state (states do not have parent, action, path-cost, ) Parent Action

State Path-cost function ChildNode(problem, parent, action) returns Node return a Node with State = problem.Result(parent.State, action) Parent = parent Action = action Path-cost = parent.Path-cost + problem.Step-cost(parent.State, action)

If node contains goal state, then you have to construct the solution a path by following the parent chain to the root Implementing graph search Frontier: Queue FIFO LIFO Priority Path-Cost? Explored-Set: Hash table

Ready for Search Different search strategies are defined by the order in which we choose nodes from the frontier to expand Lifo, fifo, We compare search strategies along the following dimensions Completeness: Does it always find a solution if one exists? Time Complexity: Number of nodes expanded/generated

Space Complexity: Max number of nodes in Memory Optimality: Does it always find least-cost solution Time and space complexity are measured in terms of b maximum branching factor of search tree d depth of least cost solution m maximum depth of the tree (may be infinite!) Uninformed Search

Breadth-first Uniform-cost Depth-first Depth-limited Iterative deepening Breadth-first search FIFO Q BFS Complete: Yes shallowest goal node Time == Number of nodes expanded assume b constant O(b^d) if you check for goal state upon generation of node or O(b^(d+1)) if you check when you pick node for expansion

Space == Space for nodes = number of nodes in explored set + number of nodes in frontier O(b^(d-1)) in explored + O(b^d) in frontier Uh-oh! Can generate nodes at the rate of 100MB/sec so 24 hours means 8640GB Look at figure 3.13 in the book With b = 10, d = 16, and 1M nodes/sec, 350 Years and 10 exabytes of storage needed Optimality: Optimal if path cost is non-decreasing function of depth Do BFS

Uniform-cost search Expand node with lowest path-cost Goal test on expansion Replace frontier node if you find better path to same node.State Uniform cost search Draw the Uniform-cost search tree for getting from Sibiu to Bucharest Uniform cost search Complete if every step cost is > 0 Optimal

Time/Space Strictly more than BFS Do UCS Depth-first search LIFO Q DFS Often easy to implement recursively Completeness: Graph search version is complete in finite spaces Tree search version can be infinitely loopy Not-optimal

Time: If d is depth of shallowest optimal solution, and m is max depth of tree, DFS may generate O(b^m) >> O(b^d) Space: O(bm) ! Not bad and we can go lower to O(m) with some fancy housekeeping (backtracking search) Some kind of DFS used a lot in AI because space requirements are low What kinds? Depth-limited search DFS with depth limit, l (el) If l < d you will never find solution (incomplete) If l > d non-optimal DFS = DLS with l = infinity

Romanian problem depth is 20 == number of states Actually 9. The diameter of the state space (max steps between any pair of states) DLS (or DFS) Remove limit to make DFS Iterative deepening DFS DLS but keep increasing limit Why? Space efficient like DFS and complete and optimal like BFS Not much extra work since the number of nodes at depth d is b^d And number of interior nodes = b^d -1

Most nodes are leaves Numerical comparison for b = 10 and d = 5, solution at far right leaf: N(IDS) = 50 + 400 + 3; 000 + 20; 000 + 100; 000 = 123; 450 N(BFS) = 10 + 100 + 1; 000 + 10; 000 + 100; 000 + 999; 990 = 1; 111; 100 Iterative deepening Do DFS Iterative lengthening Expand all nodes with cost less than < C Increase C if goal not found Min of all node costs explored in prior step

Check textbook Bidirectional Search b^(d/2) + b^(d/2) << b^d Search forwards from start and backwards from goal Check for frontier intersection One search must be BFS for good check on frontier intersection

How do you search backwards for Romania Vacuum cleaner 8-queens Comparison of uninformed search Informed Search Best First Search A* Heuristics Basic idea Order nodes for expansion using a specific search strategy

Remember uniform cost search? Nodes ordered by path length = path cost and we expand least cost This function was called g(n) Order nodes, n, using an evaluation function f(n) Most evaluation functions include a heuristic h(n) For example: Estimated cost of the cheapest path from the state at node n to a goal state Heuristics provide domain information to guide informed search Romania with straight line distance heuristic h(n) = straight line distance to Bucharest

Greedy search F(n) = h(n) = straight line distance to goal Draw the search tree and list nodes in order of expansion (5 minutes) Time? Space? Complete? Optimal? Greedy search Greedy analayis Optimal? Path through Rimniu Velcea is shorter

Complete? Consider Iasi to Fagaras Tree search no, but graph search with no repeated states version yes In finite spaces Time and Space Worst case where m is the maximum depth of the search space Good heuristic can reduce complexity

f(n) = g(n) + h(n) = cost to state + estimated cost to goal = estimated cost of cheapest solution through n Draw the search tree and list the nodes and their associated cities in order of expansion for going from Arad to Bucharest 5 minutes

A* f(n) = g(n) + h(n) = cost to state + estimated cost to goal = estimated cost of cheapest solution through n Seem reasonable?

If heuristic is admissible, is optimal and complete for Tree search Admissible heuristics underestimate cost to goal If heuristic is consistent, is optimal and complete for graph search Consistent heuristics follow the triangle inequality If n is successor of n, then h(n) c(n, a, n) + h(n) Is less than cost of going from n to n + estimated cost from n to goal Otherwise you should have expanded n before n and you need a different heuristic f costs are always non-decreasing along any path contours Non decreasing f implies We can draw contours

Inside the 400 contour All nodes have f(n) 400 Contour shape Circular if h(n) = 0 Elliptical towards goal for h(n) If C* is optimal path cost A* expands all nodes with f(n) < C* A* may expand some nodes with f(n) = C* before getting to a goal state If b is finite and all step costs > e, then A* is complete since There will only be a finite number of nodes with f(n) < C* Because b is finite and all step costs > e

Pruning A* does not expand nodes with f(n) > C* The sub-tree rooted at Timisoara is pruned Search Problem solving by searching for a solution in a space of possible solutions Uninformed versus Informed search Atomic representation of state Solutions are fixed sequences of actions