HO-04 Kecerdasan Buatan

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HO-3 KTL401 Kecerdasan Buatan

Menyelesaikan Masalah

dengan Pencarian

Opim S Sitompul

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Outline

Agen-agen Penyelesaian-masalah

Jenis-jenis masalah

Formulasi masalah

Contoh-contoh masalah

Algoritma pencarian dasar

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Agen-agen Penyelesaian-masalah

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Contoh: Romania

Liburan di Romania; saat ini berada di Arad.

Pesawat berangkat besok dari Bucharest

Formulasi goal:

sampai di Bucharest

Formulasi masalah: states: berbagai kota

actions: mengemudi antar kota

Cari penyelesaian: Deretan kota, mis., Arad, Sibiu, Fagaras, Bucharest

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Contoh: Romania

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Contoh: Romania

Search:

Sebuah agen dengan beberapa pilihan segera dengan nilai yang tidak diketahui dapat memutuskan apa yang dilakukan dengan pertama-tama memeriksa deretan aksi berbeda yang mungkin yang membawa ke state dengan nilai yang diketahui, dan kemudian memilih deretan terbaik.

Algoritma search menerima input sebuah masalah dan mengembalikan sebuah penyelesaian dalam bentuk sederetan aksi.

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Contoh: Romania

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Problem types

Deterministic, fully observable single-state problem

Agent knows exactly which state it will be in; solution is a sequence

Non-observable sensorless problem (conformant problem)

Agent may have no idea where it is; solution is a sequence

Nondeterministic and/or partially observable contingency problem

percepts provide new information about current state

often interleave} search, execution

Unknown state space exploration problem

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Masalah yang terdefinisi baik

dan penyelesaian

Sebuah masalah dapat didefinisikan secara formal oleh empat komponen: State awal, dimana agen itu berada.

Cth.: In(Arad)

Deskripsi tentang kemungkinan aksi yang tersedia bagi si agen. Formulasi yang paling umum menggunakan successor function.

In(Arad) -> initial state

{<Go(Sibiu), In(Sibiu)>, <Go(Timisoara), In(Timisoara)>, <Go(Zerind), In(Zerind)>}

Initial state + successor function -> state space, himpunan semua state yang dapat dicapai dari initial state.

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Masalah yang terdefinisi baik

dan penyelesaian

Goal test, menentukan apakah satu state adalah

goal state.

Path cost, fungsi yang memberikan harga

numerik ke masing-masing path.

Memformulasikan masalah

Abstraksi: proses melenyapkan rincian dari

sebuah representasi.

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Contoh: vacuum world

States: agen ada di salah satu dari dua lokasi,

masing-masing berdebu atau tidak berdebu. Ada 2 x

22 = 8 state

Initial state: sembarang state

Successor function: menghasilkan langkah sah

yang dihasilkan dari mencoba tiga tindakan (Left,

Right, dan Suck)

Goal test: memeriksa apakah semua ruang bersih

Path cost: setiap langkah berharga 1, sehingga path

cost adalah jumlah langkah dalam path.

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Single-state, start in #5.

Solution?

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Single-state, start in #5.

Solution? [Right, Suck]

Sensorless, start in

{1,2,3,4,5,6,7,8} e.g.,

Right goes to {2,4,6,8}

Solution?

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{1,2,3,4,5,6,7,8} e.g.,


Solution?

[Right,Suck,Left,Suck]

Contingency

Nondeterministic: Suck may

dirty a clean carpet

Partially observable: location, dirt at current location.

Percept: [L, Clean], i.e., start in #5 or #7

Solution?

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{1,2,3,4,5,6,7,8} e.g.,


Solution?

[Right,Suck,Left,Suck]

Contingency

Nondeterministic: Suck may

dirty a clean carpet

Partially observable: location, dirt at current location.

Percept: [L, Clean], i.e., start in #5 or #7

Solution? [Right, if dirt then Suck]

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Single-state problem formulation

A problem is defined by four items:

1. initial state e.g., "at Arad“

2. actions or successor function S(x) = set of action–state pairs e.g., S(Arad) = {<Arad Zerind, Zerind>, … }

3. goal test, can be explicit, e.g., x = "at Bucharest"

implicit, e.g., Checkmate(x)

4. 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 leading from the initial state to a goal state

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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., "Arad Zerind" 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

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Vacuum world state space graph

states?

actions?

goal test?

path cost?

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Vacuum world state space graph

states? integer dirt and robot location actions? Left, Right, Suck

goal test? no dirt at all locations

path cost? 1 per action

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Contoh: The 8-puzzle

states?

actions?

goal test?

path cost?

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Contoh: The 8-puzzle

states? locations of tiles

actions? move blank left, right, up, down

goal test? = goal state (given)

path cost? 1 per move

[Note: optimal solution of n-Puzzle family is NP-hard]

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Contoh: robotic assembly

states?: real-valued coordinates of robot joint angles parts of the object to be assembled

actions?: continuous motions of robot joints

goal test?: complete assembly

path cost?: time to execute

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Tree search algorithms

Basic idea: offline, simulated exploration of state space by generating

successors of already-explored states (a.k.a.~expanding

states)

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Tree search example

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Tree search example

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Tree search example

Fringe: kumpulan node yang telah

dibuat tetapi belum di-expand. Masing-

masing elemen fringe adalah leaf

node.

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Implementation: general tree search

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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 state, parent node, action, path cost g(x), depth

The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states.

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Search strategies

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 one exists?

time complexity: number of nodes generated

space complexity: maximum number of nodes in memory

optimality: does it always find a least-cost solution?

Time and space complexity are measured 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 ∞)

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Uninformed search strategies

Uninformed search strategies use only the

information available in the problem definition

Breadth-first search

Uniform-cost search

Depth-first search

Depth-limited search

Iterative deepening search

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Expand shallowest unexpanded node

Implementation:

fringe is a FIFO queue, i.e., new successors go at

end

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Implementation:

fringe is a FIFO queue, i.e., new successors go

at end

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Implementation:


end

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Implementation:


end

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Properties of breadth-first search

Complete? Yes (if b is finite)

Time? 1+b+b2+b3+… +bd + b(bd-1) = O(bd+1)

Space? O(bd+1) (keeps every node in memory)

Optimal? Yes (if cost = 1 per step)

Space is the bigger problem (more than time)

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Uniform-cost search

Expand least-cost unexpanded node

Implementation:

fringe = queue ordered by path cost

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(bceiling(C*/ ε)) where C* is the cost of the optimal solution

Space? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε))

Optimal? Yes – nodes expanded in increasing order of g(n)

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Depth-first search

Expand deepest unexpanded node

Implementation:

fringe = LIFO queue, i.e., put successors at front

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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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Depth-first search


Implementation:


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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

but if solutions are dense, may be much faster than

breadth-first

Space? O(bm), i.e., linear space!

Optimal? No

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Depth-limited search

= depth-first search with depth limit l,

i.e., nodes at depth l have no successors

Recursive implementation:

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Iterative deepening search l =0

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Number of nodes generated in a depth-limited search to depth d with branching factor b:

NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bd

Number of nodes generated in an iterative deepening search to depth d with branching factor b:

NIDS = (d+1)b0 + d b^1 + (d-1)b^2 + … + 3bd-2 +2bd-1 + 1bd

For b = 10, d = 5,

NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111

NIDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456

Overhead = (123,456 - 111,111)/111,111 = 11%

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Properties of iterative

deepening search

Complete? Yes

Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd =

O(bd)

Space? O(bd)

Optimal? Yes, if step cost = 1

Breadth-first search Procedure breadth-first-serach;

Begin

open = [start];

closed = [];

while open ≠ []

begin

remove leftmost state from open, call it X;

if X is a goal then return(success)

else begin

generate children of X;

put X on closed;

eliminate children of X on open or closed;

put remaining children on right-end of open;

end;

return(failure)

End. 58

Depth-first search Procedure breadth-first-serach;

Begin

open = [start];

closed = [];

while open ≠ []

begin

remove leftmost state from open, call it X;

if X is a goal then return(success)

else begin

generate children of X;

put X on closed;

eliminate children of X on open or closed;

put remaining children on left-end of open;

end;

return(failure)

End. 59

Adjacency List

Arad Zerind(75) Sibiu(140) Timisoara(118)

Zerind Oradea(71)

Sibiu Fagaras(99) RimnicuVilcea(80)

Timisoara Lugoj(111)

Oradea Sibiu(151)

Lugoj Mehadia(70)

Fagaras Bucharest(211)

RimnicuVilcea Pitesti(97) Craiova(146)

Pitesti Bucharest(101)

Craiova Pitesti(138)

Mehadia Drobeta(75)

Drobeta Craiova(120)

Bucharest Giurgiu(90) Urziceni(85)

Urziceni Hirsova(98) Vaslui(142)

Hirsova Eforie(86)

Vaslui Lasi(92)

Lasi Neamt(87)

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Summary of algorithms

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Repeated states

Failure to detect repeated states can turn a

linear problem into an exponential one!

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Graph search

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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

Documents

HO-04 Kecerdasan Buatan