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### CS 541: Artificial Intelligence

Breadth first searchDepth first searchDepth first searchDepth first searchDepth first searchDepth first searchDepth first searchDepth first searchDepth first searchDepth first search### Informed search & non-classical search

### Local search algorithms

Lecture II: Problem Solving and Search

Course Information (1)

- CS541 Artificial Intelligence
- Term: Fall 2012
- Instructor: Prof. Gang Hua
- Class time: Wednesday 6:15pm—8:40pm
- Location: Babbio Center Room 210
- Office Hour: Wednesday 4:00pm—5:00pm by appointment
- Office: Lieb/Room305
- Course Assistant: Yizhou Lin
- Course Website:
- http://www.cs.stevens.edu/~ghua/ghweb/ cs541_artificial_intelligence_Fall_2012.htm

Re-cap Lecture I

- Agents interact with environments through actuators and sensors
- The agent function describes what the agent does in all circumstances
- The performance measure evaluates the environment sequence
- A perfectly rational agent maximizes expected performance
- Agent programs implement (some) agent functions
- PEAS descriptions definetaskenvironments
- Environments are categorized along several dimensions:
- Observable? Deterministic? Episodic? Static? Discrete? Single-agent?
- Several basic agent architectures exist:
- Reflex, Reflex with state, goal-based, utility-based

Outline

- Problem-solving agents
- Problem types
- Problem formulation
- Example problems
- Basic search algorithms

Problem solving agent

- Problem-solving agents: restricted form of general agent
- This is offline problem solving—online problem solving involves acting without complete knowledge

Example: Romania

- On holiday in Romania; currently in Arad
- Flight leaves tomorrow from Bucharest
- Formulate goal:
- Be in Bucharest
- Formulate problem:
- States: various cities
- Actions: drive between cities
- Find solution:
- Sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

Problem types

- Deterministic, fully observable single-state problem
- Agent knows exactly which state it will be in; solution is a sequence
- Non-observable conformant problem
- Agent may have no idea where it is; solution (if any) is a sequence
- Nondeterministic and/or partially observable contingency problem
- Percepts provide new information about current state
- Solution is a contingent plan or a policy
- Often interleave search, execution
- Unknown state space exploration problem (“online”)

Example: vacuum world

- Single-state, start in #5. Solution??
- [Right, Suck]
- Conformant, start in [1,2,3,4,5,6,7,8], e.g., right to [2,4,6,8]. Solution??
- [Right, Suck, Left, Suck]
- Contingency, start in #5

Non-deterministic: Suck can dirty a clean carpet

Local sensing: dirt, location only.

Solution??

- [Right, if dirt then Suck]

Single state problem formulation

- A problem is defined by four items
- Initial state
- E.g., “at Arad”
- Action or successorfunctionS(x)= set of action-state pair,
- E.g., S(Arad)={<AradZerind, Zerind>,…}
- Goal test, can be
- Explicit, e.g., x=“at Bucharest”
- Implicit, e.g., NoDirt(x)
- Path cost (additive)
- E.g., sum of distances, number of actions executed, etc.
- c(x, a, y) is the step cost, assumed to be ≥0
- A Solution is a sequence of actions
- Leads from the initial state to a goal state

Selecting a state space

- Real world is absurdly complex
- State space must be abstracted for problem solving
- (Abstract) state = set of real states
- (Abstract) action = complex combination of real actions
- E.g., “AradZerind” represents a complex set of possible routes, detours, rest stops, etc.
- For guaranteed realizability, any real state “in Arad” must get to some real state “in Zerind”
- (Abstract) solution =
- Set of real paths that are solutions in the real world
- Each abstract action should be “easier” than the original problem

Example: vacuum world state space graph

- States??: Integer dirt and robot location (ignore dirt amount, etc..)
- Actions??: Left, Right, Suck, NoOp
- Goal test??: No dirt
- Path cost??: 1 per action (0 for NoOp)

Example: The 8-puzzle

- States??: Integer locations of tiles (ignore intermediate positions)
- Actions??: Move blank left, right, up, down (ignore unjamming etc.)
- Goal test??: =goal state (given)
- Path cost??: 1 per move
- [Note: optimal solution of n-Puzzle family is NP-hard]

Example: robotics assembly

- States??: Real-valued coordinates of robot joint angles, parts of the object to be assembled
- Actions??: Continuous motions of robot joints
- Goal test??: Completely assembly with no robot included!
- Path cost??: time to execute

Tree search algorithm

- Basic idea:
- Offline, simulated exploration of state space by generating successors of explored state (a.k.a., expanding states)

Implementation: states vs. nodes

- A state is a representation of a physical configuration
- A node is a data structure constituting part of a search tree
- Includes parents, children, depth, path cost g(x)
- States do not have parents, children, depth, or path cost
- The EXPAND function creates new nodes, filling in the various fields and using the SUCCESSORSFN of the problem to create the corresponding states

Search strategy

- A search strategy is defined by picking the order of node expansion
- Strategies are evaluated along the following dimensions
- Completeness: does it always find a solution if it exists?
- Time complexity: number of nodes generated/expanded
- Space complexity: number of nodes holds in memory
- Optimality: does it always find a least-cost solution?
- Time and space complexity are measure in terms of
- b: maximum branching factor of the search tree
- d: depth of the least-cost solution
- m: maximum depth of the state space (may be infinity)

Uninformed search strategies

- Uninformed search strategies use only the information available in the problem definition
- List of uninformed search strategies
- Breadth-first search
- Uniform-cost search
- Depth-first search
- Depth-limited search
- Iterative deepening search

Breadth first search

- Expand shallowest unexpanded node
- Implementation:
- fringe is a FIFO queue, i.e., newest successors go at end

Breadth first search

- Expand shallowest unexpanded node
- Implementation:
- fringe is a FIFO queue, i.e., newest successors go at end

Breadth first search

- Expand shallowest unexpanded node
- Implementation:
- fringe is a FIFO queue, i.e., newest successors go at end

- Expand shallowest unexpanded node
- Implementation:
- fringe is a FIFO queue, i.e., newest successors go at end

Properties of breadth-first search

- Complete?? Yes (if b is finite)
- Time?? 1+b +b2+b3+…+bd+b(bd-1)=O(bd+1), i.e., exp. in d
- Space?? O(bd+1), keeps every node in memory
- Optimal?? Yes (if cost=1 per step); not optimal in general
- Space is the big problem; can easily generate nodes at 100M/sec so 24hrs=8640GB

Uniform cost search

- Expand least-cost unexpanded node
- Implementation”
- fringe=priority queue ordered by path cost, lowest first
- Equivalent to breadth-first if step costs all equal
- Complete?? Yes, if step cost ≥ ε
- Time?? # of nodes with g ≤ cost of optimal solution, O(b┌C*/ ε┐) where C* is the cost of the optimal solution
- Space?? # of nodes with g ≤ cost of optimal solution, O(b┌C*/ ε┐)
- Optimal?? Yes – nodes expanded in increasing order of g(n)

Depth first search

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

Depth first search

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

Depth first search

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

- Expand deepest unexpanded node
- Implementation:
- fringe is a LIFO queue, i.e., put successors at front

Properties of depth-first search

- Complete?? No: fails in infinite-depth spaces, spaces with loops
- Modify to avoid repeated states along path
- complete in finite spaces
- Time?? O(bm): terrible if m is much larger than d
- Space?? O(bm), i.e., linear space
- Optimal?? No

Depth limited search

- Depth first search with depth limit l, i.e., node at depth l has no successors

Iterative deepening search

- Iteratively run depth limited search with different depth a range of different depth limit

Properties of iterative deepening search

- Complete?? Yes
- Time?? (d+1)b0+db1+(d-1)b2+…+bd=O(bd)
- Space?? O(bd)
- Optimal?? Yes, if step cost=1
- Can be modified to explore uniform-cost tree
- Comparison for b=10 and d=5, solution at far right leaf:

N(IDS)=6+50+400+3,000+20,000+100,000=123,456

N(BFS)=1+10+100+1,000+10,000+100,000+999,990=1,111,101

- IDS does better because other notes at depth d are not expanded
- BFS can be modified to apply goal test when a node is generated

Repeated states

- Failure to detect repeated states can turn a linear problem into an exponential one!

Summary

- Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored
- Variety of uninformed search strategies
- Iterative deepening search uses only linear space and not much more time than other uninformed algorithms
- Graph search can be exponentially more efficient than tree search

Lecture II: Part II

Outline

- Informed search
- Best first search
- A* search
- Heuristics
- Non-classical search
- Hill climbing
- Simulated annealing
- Genetic algorithms (briefly)
- Local search in continuous spaces (very briefly)

Review of tree search

- A strategy is defined by picking the order of node expansion

Best first search

- Idea: using an evaluation function for each node
- Estimate of desirability
- Expand the most desirable unexpanded node
- Implementation:
- fringe is a priority queue sorted in decreasing order of desirability
- Special case:
- Greedy search
- A* search

Greedy search

- Evaluation function h(n) (heuristic)
- Estimate of cost from n to the closest goal
- E.g., hSLD(n)=straight-line distance from n to Bucharest
- Greedy search expands the node that appears to be closest to the goal

Properties of greedy search

- Complete?? No: can get stuck in loops
- LasiNeamtLasiNeamt (when Oradea is the goal)
- Time?? O(bm): but a good heuristic can give dramatic improvement
- Space?? O(bm), i.e., keep everything in memory
- Optimal?? No

A* search

- Idea: avoid expanding paths that are already expensive
- Evaluation function f(n)=g(n)+h(n)
- g(n)=cost so far to reach n
- h(n)=estimated cost to goal from n
- f(n)= estimated total cost of path through n to goal
- A* search uses an admissible heuristic
- i.e., h(n)≤h*(n) where h*(n) is the true cost from n
- also require h(n)≥0, so h(G)=0 for any goal G.
- E.g., hSLD(n) never overestimates the actual road distance
- Theorem: A* search is optimal (given h(n) is admissible in tree search, and consistent in graph search

Optimality of A* (standard proof)

- Suppose some suboptimal goal G2 has been generated and is in the queue.
- Let n be an unexpanded node on a shortest path to an optimal goal G1.
- f(G2) = g(G2) since h(G2)=0

> g(G1) since G2 is suboptimal

≥ f(n) since h is admissible

- Since f(G2)>f(n), A* will never select G2 for expansion before selecting n

Optimality of A* (more useful)

- Lemma: A* expands nodes in order of increasing f value
- Gradually addes“f-contour”of nodes (c.f., breadth-first adds layer)
- Contour i has all nodes f=fi, where fi<fi+1

Properties of A* search

- Complete?? Yes
- Unless there are infinitely many nodes with f<f(G)
- Time?? Exponential in [relative error in hxlength of solution.]
- Space?? Keep all nodes in memory
- Optimal?? Yes – cannot expand fi+1 until fi is finished
- A* expands all nodes with f(n)<C*
- A* expands some nodes with f(n)=C*
- A* expands no nodes with f(n)>C*

Proof of Lemma: Consistency

- A heuristic is consistent if

h(n)≤c(n, a, n')+h(n')

- If h is consistent, then

f(n')=g(n')+h(n')

=g(n)+c(n, a, n')+ h(n')

≥g(n)+h(n)

=f(n)

- f(n) is non-increasing along any path

Admissible heuristics

- E.g., 8 puzzle problem
- h1(n): number of misplaced tiles
- h2(n): total Manhattan distance
- h1(s) =?? 6
- h1(s) =?? 4+0+3+3+1+0+2+1=14
- Why they are admissible?

Dominance

- If h2(n)≥h1(n) for all n (both admissbile), then h2 dominates h1 and is better for search
- Typical search cost:
- D=14: IDS=3,473,941 nodes
- A*(h1)=539 nodes
- A*(h2)=113 nodes
- D=24: IDS≈54,000,000,000 nodes
- A*(h1)=39,135 nodes
- A*(h2)=1,641 nodes
- Given any admissible heurisitcsha and hb
- h(n)=max(ha, hb) is also admissible and dominate ha and hb

Relaxed problem

- Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem
- If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution
- If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution
- Key point: the optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem

Relaxed problem

- Well-known example: traveling sales-person problem (TSP)
- Find the shortest tour visiting all cities exactly once
- Minimum spanning tree can be computed in O(n2) and is a lower bound on the shortest (open) tool

Summary

- Heuristic functions estimate costs of shortest paths
- Good heuristics can dramatically reduce search cost
- Greedy best-first search expands lowest h
- Incomplete and not always optimal
- A* search expands lowest g + h
- complete and optimal
- also optimally efficient (up to tie-breaks, for forward search)
- Admissible heuristics can be derived from exact solution of relaxed problems

Beyond classical search

Iterative improvement algorithms

- In many optimization problems, path is irrelevant;
- The goal state itself is the solution
- Then state space = set of "complete" configurations;
- Find optimal configuration, e.g., TSP
- Or, find configuration satisfying constraints, e.g., timetable
- In such cases, can use iterative improvement algorithms;
- keep a single "current" state, try to improve it
- Constant space, suitable for online as well as online search

Example: traveling salesperson problem

- Start with any complete tour, perform pair-wise exchanges
- Variants of this approach get within 1% of optimal very quickly with thousands of cities

Example: n-queens

- Goal: put n queens on an n x n board with no two queens on the same row, column, or diagonal
- Move a queen to reduce number of conflicts
- Almost always solves n-queens problems almost instantaneously for very large n, e.g., n=1million

Hill-climbing (or gradient ascent/descent)

- "Like climbing Everest in thick fog with amnesia”

Hill climbing

- Useful to consider state space landscape
- Random-restart hill climbing overcomes local maxima -- trivially complete
- Random sideways moves escape from shoulders loop on flat maxima

Simulated annealing

- Idea: escape local maxima by allowing some "bad" moves, but gradually decrease their size and frequency

Properties of simulated annealing

- At fixed "temperature" T, state occupation probability reaches Boltzman distribution:
- p(x) = exp{E(x)/kT}
- T decreased slowly enough always reach best state x
- because exp{E(x*)/kT}/exp{E(x)/kT} = exp{(E(x*)-E(x))/kT}>>1 for small T
- Is this necessarily an interesting guarantee??
- Devised by Metropolis et al., 1953, for physical process modeling
- Widely used in VLSI layout, airline scheduling, etc.

Local beam search

- Idea: keep k states instead of 1; choose top k of all their successors
- Not the same as k searches run in parallel!
- Searches that find good states recruit other searches to join them
- Problem: quite often, all k states end up on same local hill
- Idea: choose k successors randomly, biased towards good ones
- Observe the close analogy to natural selection!
- Particle swarm optimization

Genetic algorithm

- Idea: stochastic local beam search + generate successors from pairs of states

Genetic algorithm

- GAs require states encoded as strings (GPs use programs)
- Crossover helps i substrings are meaningful components
- GAs 6= evolution: e.g., real genes encode replication machinery!

Continuous state space

- Suppose we want to site three airports in Romania:
- 6-D state space dened by (x1; y2), (x2; y2), (x3; y3)
- Objective function f(x1; y2; x2; y2; x3; y3) = sum of squared distances from each city to nearest airport
- Discretization methods turn continuous space into discrete space,
- e.g., empirical gradient considers ±ᵟ change in each coordinate
- Gradient methods compute
- To reduce f, e.g., by
- Sometimes can solve for exactly (e.g., with one city).
- Newton{Raphson (1664, 1690) iterates
- to solve , where

Assignment

- Reading Chapter 1&2
- Reading Chapter 3&4
- Required:
- Program Outcome 31.5: Given a search problem, I am able to analyze and formalize the problem (as a state space, graph, etc.) and select the appropriate search method
- Problem 1: Russel & Norvig Book Problem 3.9 (programming needed) (4 points)
- Program Outcome 31.11: I am able to find appropriate idealizations for converting real world problems into AI search problems formulated using the appropriate search algorithm.
- Problem 2: Russel & Norvig Book Problem 3.2 (2 points)
- Program outcome 31.12: I am able to implement A* and iterative deepening search. I am able to derive heuristic functions for A* search that are appropriate for a given problem.
- Problem 3: Russel & Norvig Book Problem 3.3 (2 points)
- Program outcome 31.9: I am able to explain important search concepts, such as the difference between informed and uninformed search, the definitions of admissible and consistent heuristics and completeness and optimality. I am able to give the time and space complexities for standard search algorithms, and write the algorithm for it.
- Problem 4: Russel & Norvig Book Problem 3.18 (2 points)
- Optional bonus you can gain:
- Russel & Norvig Book Problem 4.3
- Outlining algorithm, 1 piont
- Real programming, 3 point

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