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Algorithms of ArtificialIntelligence

Lecture 2: Knowledge

E. Tyugu

Spring 2003

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PrologProlog is a logic-based programming language, i.e. a language forlogic

programming. Itsstatements are Horn clauses.

Examples:

A program: ancestor(X,Z):-parent(X,Z).

ancestor(X,Z):-parent(Y,Z),ancestor(X,Y).

A program:

state(1,0).

state(S,T):-state(S1,pred(T)), nextstate(S,S1,pred(T)).

nextstate(X+1,X,T).

A goal: ?- state(X,3).

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Prolog interpreter

prog- program to be executed;

goals - list of goals, which initially contains the goal

given by a user;

unifier(x,y) - produces the most general unifier ofx

and y, or nil;

apply(x,L) - applies a unifierx to each element of alist L producing a new list.

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Prolog interpreter

A.1.3: exec(prog,goals,success)=

i fempty(goals) thensuccess( )

elsegoal:= head(goals);

goals:= tail(goals);

L: {rest:= prog;

wh i le notempty(rest) doU:=unifier(goal,head(head(rest));i fUn il

thengoals1:=(apply(U,tail(head(rest)));

exec(prog,goals1,success);

i fsuccess thenexit L f if i ;rest:=tail(rest);

od ;failure( )

};exec(prog,goals, succes)

f i

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Semantic networks

Linguists noticed long ago that the structure of a sentence can

be represented as a network. Words of the sentence are nodes,

and they are bound by arcs expressing relations between the

words. The network as a whole represents in this way a

meaning of the sentence in terms of meanings of words and

relations between the words. This meaning is an approximation

of the meaning people can assign to the sentence, analogous in

a way to other approximate representations of the meaning, for

instance, how floating point numbers represent the approximate

meaning of real numbers.

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Example

give

pick up

have a meeting

before

after

after

morning lunch

at the time

who?

John

who?

who?

what?

report

to whom?

me

whos?

his

John must pick up his report in the morning and have

a meeting after lunch. After the meeting he will give the report to me.

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Example continued

Inferences can be made, depending on the properties of the

relations of a semantic network. Let us consider only time

relations of the network in our example, and encode the time

relations by atomic formulas as follows:

before(lunch,morning) = general knowledge

after(morning,lunch)

after(lunch,have a meeting) = specific knowledge

after(have a meeting,give)

at-the-time(morning,pick up)

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Example continuedInference rules:

before(x,y) before(y,z)before(x,z)

after(x,y)before(y,x)

at-the-time(x,z) before(y,z)before(y,x)

Applying these rules, we can infere

after(lunch,have a meeting)before(have a meeting,lunch) and

at-the-time(pick up,morning) before(lunch,morning)before(lunch,pick up) etc.

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Frames

1. The essence of the frame is that it is a module of knowledgeabout something which we can call a concept. This can be asituation, an object, a phenomenon, a relation.

2. Frames contain smaller pieces of knowledge: components,attributes, actions which can be (or must be) taken whenconditions for taking an action occur.

3. Frames contain slots which are places to put pieces ofknowledge in. These pieces may be just concrete values of

attributes, more complicated objects, or even other frames. Aslot is being filled in when a frame is applied to represent aparticular situation, object or phenomenon.

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Inheritance

ideas

even ts states things

actions abstract things

polygons

triangles quadrangles

parallelograms

rectangles rhombuses

An essential idea developed in connection with frames was inheritance. Inheritance is a conveni

way of reusing existing knowledge in describing new frames. Knowing a frame f, one can describ

a new frame as a kind off, meaning that the new frame inherits the properties off, i.e. it will hav

these properties in addition to newly described properties described. Inheritance relation express

very precisely the relation between super- and subconcepts.

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Default theories

A default has the following form

A:B1, ... , Bk--------------

C

where the formulaA is a premise, the formula C is a

conclusion and the formulas B1, ..., Bkarejustifications.

Conclusion of the default can be derived from its premise, if

there is no negation of any justification derived.

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Examples

1. bird(x): flies(x)---------------------

flies(x)

2. Closed world assumption (CWA):

:not F------not F

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Derivation step with a default.

A - premise of a default

C- conclusion of a default

J- justifications of default

A1.4 Default(A,C,J):

fo rB Jdo

ifderrivable(B) thenfailure f i

od;

success( )

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Rules

Rules are a well-known form of knowledge which is easy to use.

A rule is a pair

(condition, action)

which has the meaning: "If the condition is satisfied, then the

action can be taken." Also other modalities for performing the

action are possible - "must be taken", for instance.

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Using rules

Let us have a set of rules called rules and functions cond(p) and

act(p) which select the condition part and action part of a given

rulep and present them in the executable form. The following

is a simple algorithm for problem solving with rules:

A.1.5

wh i le notgooddo

found := false;

fo rp

rules doi fcond(p) thenact(p); found:=true f i

od ;

i f notfoundthenfailure f i

od

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Decision trees

A simple way to represent rules is decision tree: a tree with

nodes for attributes and arcs for attribute values. Example:

legstwo four

handsno

yes

furryno yes

table animalfurry bird

no

monkey man

yes

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Rete algorithm

Rete algorithm uses a data structure that enables fast search ofapplicable rules. We shall consider it in two parts:

knowledge representation,

knowledge management (i.e. introduction of changes into theknowledge base).

Any rule that is reachable in the Rete graph (see below) via nonemptyrelation nodes can be fired.

Rete algorithm is used in JESS (Java Expert System Shell) and itspredecessor CLIPS (both developed in NASA.)

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Rete algorithm continued

Knowledge includes:

1. facts, e.g.(goal e1 simplify), (goal e2 simplify), (goal e3 simplify),

(expr e1 0 + 3), (expr e2 0 + 5), (expr e3 0 * 2),...

2. patterns, e.g.(goal ?x simplify)

(expr ?y 0 ?op ?a2)(parent ?x ?y)

...

3. and rules, e.g.(R1 (goal ?x simplify) (expr ?x 0 + ?y) => (expr ?x ?y))

(R2 (goal ?x simplify) (expr ?x 0 * ?y) => (expr ?x 0))

...

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Knowledge is represented in the form of an acyclic graph. It is for the

presented example as follows:

root

goal expr

goal expr * expr + ...

*** ***

R2 R1

xe1e2e3

y

2

y

3

5

x ye3 2 x ye1 3

e2 5

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The overall structure of the Rete graph is the following:

root

predicate names layer

patterns layer - alpha nodes (with one input)

rules layer - one node for every rule

beta-nodes(with two inputs)

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Adding facts to Rete graph

When a fact arrives then

1. Select the predicate2. Select the pattern

3. For every relation depending on the selected pattern update

the relation (add a new line to the relation).

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The Rete graph is built, updated and used as follows:

1. One level down from the root are placed all predicate names.

2. The next level down contains alpha-nodes for all patterns of all rules assuccessors of their predicate names.

3. Beta-nodes of the following levels down (with two inputs each) includerelations that unify with the patterns along the path from the root to thenode.

4. The paths lead finally to nodes representing rules.

5. When a new knowledge item arrives, it is placed into the correct places.Finding the places is simple and straightforward, because it is guided

by a relation in every node.6. When a goal is given, the search is simple and straightforward, because

it is guided by a relation in every node.

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Rules with plausibilities

Rules can be extended by adding plausibility values to them.

Let us associate with each rule p a plausibility value c(p) of

application of the rule. These values can be in the range from 0to 1. We shall consider as satisfactory only the results of

application of a sequence of rules p, ..., q for which the

plausibilities c(p) ,..., c(q) satisfy the condition c(p) * ... * c(q) >

cm, where cm is the minimal satisfactory plausibility of the

result. When selecting a new applicable rule, it is reasonablenow to select a rule with the highest value of plausibility

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Plausibilities

A.1.6

c:=1;

whi leno tgooddo

x:=cm;

fo rp rules doifcond(p) andc(p) > x

thena:=act(p);

x:=c(p)

f i

od ;

c:=c*x;i fc > cm thena elsefailure f i

od ;

success

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Using a plausibility function

A.1.7

c:=1;

whi leno tgooddo

x:=cm;

fo rp rules doifcond(p) andplausibility(c(p),c) > x

thena:=act(p);

x:=plausibility(c(p),c)

f i

od ;

c:=x;ifc > cm thena elsefailure( ) f i

od ;

success( )

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Classification of knowledge systems

KNOWLEDGE SYSTEMS

Symbolic

(derivability,

soundness,

completeness)

Rules

(effcient

computabiliy)

Semantic networks

(eloquence,

simplicity)

Frames

(modularity,

inheritance)

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Exercise

Facts: parent(pam,bob).parent(tom,bob).

parent(tom,liz).

parent(bob,ann).

parent(bob,pat).

parent(pat,jim).

Questions and answers:

?-parent(bob,pat). yes

?- parent(liz,pat). no

?- parent(X,liz). X = tom?- parent(bob,X). X = ann

; X = pat

; no

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Bibliography

Bratko, I. (2001)Prolog Programming for Artificial Intelligence.

Addison Wesley.

http://herzberg.ca.sandia.gov/jess/docs/ (Jess ja rete algoritm)

Genesereth, M., Nilsson, N. (1986) Logical Foundations of

Artificial Intelligence. Morgan Kauffmann.
http://herzberg.ca.sandia.gov/jess/docs/http://herzberg.ca.sandia.gov/jess/docs/

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