Assignment 1 Info
Artificial Intelligence: Methods and Applications Lecture 4: FO Logic and Knowledge Representation
Info about Assignment 1
Henrik Björklund Umeå University
21 november 2011
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21 november 2011
1 / 26
Björklund (UmU)
Assignment 1 Info
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21 november 2011
2 / 26
Assignment 1 Info
Lab sessions
What to hand in
You will have to hand in the following Java files: �
A class definition that implements the interface OthelloAlgorithm.
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Assignment 1 is due at 12:00 (noon) on December 5th.
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There are lab sessions on November 23d, November 30th, and December 1st.
A class definition that implements the interface OthelloEvaluator, and that works together with your algorightm class.
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The session on November 23d (tomorrow) will only last until 11:15.
A class definition that extends the abstract class AbstractOthelloPosition and that works together with your algorithm and evaluator classes.
You also have to hand in a report that describes the structure of your solution. Notice that the class OthelloAction has been updated. (If you do not use Java, contact me about what to hand in.)
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21 november 2011
3 / 26
Björklund (UmU)
Assignment 1 Info
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21 november 2011
4 / 26
21 november 2011
6 / 26
Minesweeper
Requirements
With appropriate parameters set (e.g., depth), your solution should not use more than 10 seconds per move on average over a game. It should never use more than 30 seconds for any single move.
Minesweeper
The algorithms will be run on a computer from room MA426 or MA436. These computers are HP DC7900 with the following specs: �
OS: Debian GNU/Linux 6.0 64bit.
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Processor: Intel Core 2 Quad, 4x266MHz
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Memory: 4GB
Your solutions should beat an Alpha-Beta algorithm using the board representation from AbstractOthelloPosition and a very simple heuristic.
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21 november 2011
5 / 26
Björklund (UmU)
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Minesweeper
Minesweeper
How it should work
Basics
We assume that the system already has vocabulary and axioms for natural numbers, i.e. a relation Num, a function +, relations < and ≤, etc. We use the two unary relations MaxRow and MaxCol to keep track of the number of rows and columns. These numbers have to be unique:
Assume that we want to play a game of Minesweeper on an n × m board. We should be able to dothe following: 1. Tell our KRS that the bord has n rows and m columns. 2. Click on some square in the Minesweeper application. 3. Enter all information we gain into the KRS (e.g., ”square (5,3) has the number 2 on it”). 4. Ask the KRS for a list of safe squares to click on, if there are any. 5. Repeat from (2).
Björklund (UmU)
AIM
21 november 2011
7 / 26
Björklund (UmU)
AIM
Minesweeper
21 november 2011
8 / 26
Minesweeper
Basics
Rows, columns, and squares
We assume that the system already has vocabulary and axioms for natural numbers, i.e. a relation Num, a function +, relations < and ≤, etc. We use the two unary relations MaxRow and MaxCol to keep track of the number of rows and columns. These numbers have to be unique:
We define rows and colums: ∀x (Row(x) ⇔ Num(x) ∧ 0 < x ∧ ∃y (MaxRow(y ) ∧ x ≤ y )) ∀x (Col(x) ⇔ Num(x) ∧ 0 < x ∧ ∃y (MaxCol(y ) ∧ x ≤ y ))
¬∃ x, y (x �= y ∧ MaxRow(x) ∧ MaxRow(x)) ¬∃ x, y (x �= y ∧ MaxCol(x) ∧ MaxCol(x))
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21 november 2011
8 / 26
Björklund (UmU)
Minesweeper
AIM
21 november 2011
9 / 26
Minesweeper
Rows, columns, and squares
Neighbors
Next, we define the neighbor relation between squares: We define rows and colums:
∀x1 , y1 , x2 , y2 (Neighbor (x1 , y1 , x2 , y2 ) ⇔
∀x (Row(x) ⇔ Num(x) ∧ 0 < x ∧ ∃y (MaxRow(y ) ∧ x ≤ y ))
Square(x1 , y1 ) ∧ Square(x2 , y2 ) ∧ (x1 �= x2 ∨ y1 �= y2 )∧
x1 ≤ x2 + 1 ∧ x2 ≤ x1 + 1 ∧ y1 ≤ y2 + 1 ∧ y2 ≤ y1 + 1)
∀x (Col(x) ⇔ Num(x) ∧ 0 < x ∧ ∃y (MaxCol(y ) ∧ x ≤ y )) And squares:
Björklund (UmU)
∀x, y (Square(x, y ) ⇔ Row(x) ∧ Col(y ))
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21 november 2011
9 / 26
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21 november 2011
10 / 26
Minesweeper
Minesweeper
Neighbors
Values
Next, we define the neighbor relation between squares:
Finally, we need to define what the numbers on the squares mean. We use one formula for each of the values 0, 1, 2, 3, 4, 5, 6, 7, 8.
∀x1 , y1 , x2 , y2 (Neighbor (x1 , y1 , x2 , y2 ) ⇔
Square(x1 , y1 ) ∧ Square(x2 , y2 ) ∧ (x1 �= x2 ∨ y1 �= y2 )∧
x1 ≤ x2 + 1 ∧ x2 ≤ x1 + 1 ∧ y1 ≤ y2 + 1 ∧ y2 ≤ y1 + 1)
Neighbors with mines on them: ∀x1 , y1 , x2 , y2 (MNeighbor (x1 , y1 , x2 , y2 ) ⇔
Neighbor (x1 , y1 , x2 , y2 ) ∧ Mine(x2 , y2 )
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21 november 2011
10 / 26
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Minesweeper
21 november 2011
11 / 26
Minesweeper
Values
Values
Finally, we need to define what the numbers on the squares mean. We use one formula for each of the values 0, 1, 2, 3, 4, 5, 6, 7, 8. For Value(x, y ) = 0 we get
Finally, we need to define what the numbers on the squares mean. We use one formula for each of the values 0, 1, 2, 3, 4, 5, 6, 7, 8. For Value(x, y ) = 0 we get
∀x, y (Value(x, y ) = 0 ⇔ ¬∃x1 , y1 (MNeighbor (x, y , x1 , y2 ))).
∀x, y (Value(x, y ) = 0 ⇔ ¬∃x1 , y1 (MNeighbor (x, y , x1 , y2 ))). For Value(x, y ) = 1: ∀x, y (Value(x, y ) = 1 ⇔ ∃x1 , y1 (MNeighbor (x, y , x1 , y1 )∧
¬∃x2 , y2 ((x1 �= x2 ∨ y1 �= y2 ) ∧ MNeighbor (x, y , x2 , y2 ))))
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21 november 2011
11 / 26
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Minesweeper
21 november 2011
11 / 26
Minesweeper
Value(x, y ) = 3
Vocabulary
In summary, we have the following vocabulary (on top of the one for natural numbers):
∀x, y (Value(x, y ) = 3 ⇔ ∃x1 , y1 , x2 , y2 , x3 , y3 ((x1 �= x2 ∨ y1 �= y2 ) ∧ (x1 �= x3 ∨ x1 �= y3 )∧ (x2 �= x3 ∨ y2 �= y3 )∧
MNeighbor (x, y , x1 , y1 ) ∧ MNeighbor (x, y , x2 , y2 ) ∧ MNeighbor (x, y , x3 , y3 )∧
¬∃x4 , y4 (x1 �= x4 ∨ y1 �= y4 ) ∧ (x2 �= x4 ∨ y2 �= y4 )∧
(x3 �= x4 ∨ y3 �= y4 ) ∧ MNeighbor (x, y , x4 , y4 )
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21 november 2011
12 / 26
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No constant names (except for numbers)
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The unary relation names Row and Col
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The binary relation names Square and Mine
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The four-ary relation names Neighbor and MNeighbor
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The two-ary function name Value
Björklund (UmU)
AIM
21 november 2011
13 / 26
Minesweeper
Minesweeper
Playing the game
Playing the game
To initialize a game on an n times m board, we simply have to give our KRS the following two assertions:
To initialize a game on an n times m board, we simply have to give our KRS the following two assertions:
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MaxRow(n)
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MaxRow(n)
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MaxCol(m)
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MaxCol(m)
Once we click on some squares, we gain informations and add more assertions, e.g.
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21 november 2011
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Value(1, 1) = 1
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Value(2, 2) = 2
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Minesweeper
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14 / 26
21 november 2011
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Ontologies for Knowledge Representation
Playing the game
To initialize a game on an n times m board, we simply have to give our KRS the following two assertions: �
MaxRow(n)
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MaxCol(m)
Ontologies
Once we click on some squares, we gain informations and add more assertions, e.g. �
Value(1, 1) = 1
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Value(2, 2) = 2
Before the next move, we query the KRS for safe squares: �
? ∃x, y (¬Mine(x, y ))
Björklund (UmU)
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21 november 2011
14 / 26
Björklund (UmU)
Ontologies for Knowledge Representation
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Ontologies for Knowledge Representation
Specialized vs. general KRSs
What is an ontology?
As we have seen, we can handle things like arithmetic and minesweeping with rather small, specialized knowledge representation systems.
In philosophy, ontology is the study of that which is.
When do we need larger ones?
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onto-: being, that which is
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logia: science, study
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Robotics
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Natural language processing
In knowledge representation, an ontology is a systematic way of representing knowledge about the world.
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Etc.
This means that there can be many different ontologies.
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21 november 2011
16 / 26
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21 november 2011
17 / 26
Ontologies for Knowledge Representation
Ontologies for Knowledge Representation
Biological ontologies
DAG ontologies
Classification of human beings: Kingdom: Phylum: Class: Order: Family Tribe: Genus: Species:
Most ontologies, however, are not pure trees.
Animalia Chordata Mammalia Primates Hominidae Hominini Homo Homo sapiens
Consider adding the category ”Things that walk on two legs” to a biological ontology. A human and a wolf are closer to each other in the biological ontology than either are to an ostrich, which - ironically - belongs to the class aves. Yet both humans and ostriches are ”Things that walk on two legs”, while wolfs are not.
The dominating feature of most ontologies is a hierarchical, tree-like structure.
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21 november 2011
18 / 26
Björklund (UmU)
Ontologies for Knowledge Representation
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21 november 2011
Ontologies for Knowledge Representation
Building a general purpose ontology
Categories
Challenges:
To be able to do any useful reasoning, we have to be able to reason about categories, rather than only about single objects.
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Generalizations are, generally speaking, not absolute
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A general ontology should be applicable to all special-purpose domains Different domains must be handled in a uniform way
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A general ontology is a social contract: it works only as long as people agree on it
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21 november 2011
20 / 26
Compare the following: �
Fruit is healthy
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Apples are healthy and oranges are healthy and bananas are healthy and peaches are healthy and ...
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Healthy (Apple1 ) ∧ Healthy (Apple2 ) ∧ Healthy (Apple3 ) ∧ ...
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Ontologies for Knowledge Representation
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21 november 2011
21 / 26
Ontologies for Knowledge Representation
Categories
Reification Rather than treating all categories as relations (predicates) in FO, we can use an object for each category. Let Apples be constant name that refers to such an object.
Categories can be seen as sets � �
We can then use a membership relation to say that another object x belongs to the category Apples:
Each apple is a member of the category of all apples: Apple1 ∈ Apples All apples are fruits: Apples ⊂ Fruits
This makes it possible to state properties that are inherited by sub-categories � �
∀x (x ∈ Fruits ⇒ Healthy (x))
It follows that all apples are healthy.
Member (x, Apples) Which we abbreviate as: x ∈ Apples The benefit is that we will need fewer relations and fewer axioms.
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21 november 2011
22 / 26
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21 november 2011
23 / 26
Ontologies for Knowledge Representation
Ontologies for Knowledge Representation
Objects with objects as parts
Representing time
PartOf (CS, NT ) PartOf (NT , UmU) PartOf (CS, UmU)
The starting point we choose and the unit we measure in are arbitrary: any choice will do. The objects we work with are intervals. These objects belong to the category Intervals.
The PartOf relation is transitive and reflexive. Notice that this is neither the same as the subset relation nor the Member relation. � � �
The subset relation relates two categories: Apples ⊂ Fruits
The Member relation relates an object and a category: Apple1 ∈ Apples
The PartOf relation relate two objects: PartOf (MyLeftHand, MyLeftArm).
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21 november 2011
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Ontologies for Knowledge Representation
Time interval relations [Allen, 1983]
Meet(i, j) Before(i, j) Overlap(i, j) After (i, j) During(i, j) Begins(i, j) Finishes(i, j) Equals(i, j)
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⇔ End(i) = Begin(j) ⇔ End(i) < Begin(j)
⇔ Begin(i) < Begin(j) < End(i) < End(j) ⇔ Before(j, i)
⇔ Begin(j) < Begin(i) < End(i) < End(j) ⇔ Begin(i) = Begin(j) ⇔ End(i) = End(j)
⇔ Begin(i) = Begin(j) ∧ End(i) = End(j)
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The Begin funktion gives the moment when an interval begins
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The End funktion gives the moment when an interval ends
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The Time function maps moments to our time scale
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