Knowledge Representation and Reasoning

University "Politehnica" of Bucharest

Department of Computer Science

Fall 2011

Adina Magda Florea

Master of Science in Artificial Intelligence, 2011-2013

Lecture 4 – NML Supliment

Truth Maintenance Systems TMS Functioning Data structures for a TMS TMS/IE responsibilities TMS/IE interface

TMS

Truth Maintenance Systems

Non monotonic reasoning Increase efficiency of problem solving Good for:

• validate assumptions

• redraw abandoned conclusions NMR dependency directed backtracking (DDBkt)

• control program actions

• explain reasoning3

Dependency Directed Bkt

(1) x {0,1} (2) a = e1(x)

(3) y {0,1} (4) b = e2(x)

(5) z {0,1} (6) c = e3(x)

(7) b c (8) a b

ei(x) = (x+100000)!, i=1,2,3

Find a, b, c to satisfy (1)-(8)

x=0, y=0, z=0 and x=0, y=0, z=1 are rejected because of (7) and (8) – y's value

backtrack to y c = e3(0) and c = e3(1) are lost 4

TMS principles

Each action in the problem solving process has an associated justification

When a contradiction is obtained, find the minimal set of assumptions which generated the contradiction – if we eliminate an element from this set, the justification for the contradiction is not valid any more and the contradiction is removed

Propagate the effects of adding a justification and of eliminating a belief + keep consistency

Select an assumption from the minimal set which generated the contradiction and defeat it

5

TMS Structure

InferenceEngine (IE)

TMS

KnowledgeBase

Justifications

Beliefs

Belief -

expression which can be true or false Justification -

inference rule (step) which lead to

a belief Premise -

true fact (no justification needed)

6

TMS Structure

Set of nodes in the TMS Every node has 2 states

• IN (believed node)

• OUT (node not believed) Set of valid assumptions = set of IN nodes

in the TMS

7

TMS Structure

The nodes may be:

• premises – no justification needed, IN nodes

• belief – may be believed (IN) or not (OUT)

• assumption - with supporting justifications – IN nodes if the justification is true, OUT nodes otherwise

A justification = set of nodes used to infer the justified node, composed of 2 lists:

• IN List – nodes that have to be IN for the justification to be true / to support an IN node

• OUT List – nodes that have to be OUT for the justification to be true / to support an IN node

8

The ABC Murder

Initial set of beliefs

Beneficiar (Alecu) ~Alibi(Alecu)

Beneficiar(Barbu)

Beneficiar(Cezar)

R1: if Beneficiar(x)

and ifnot Alibi(x)

then Suspect(x)

Beneficiar(x) ~ Alibi(x) Suspect(x) : /

Nonmonotonic production rule

9

The ABC Murder

RulesR1: if Beneficiar(x)

and ifnot Alibi(x)then Suspect(x)

R2: if Hotel(x,y)

and Departe(y)

and ifnot Falsificat(y)

then Alibi(x)

R3: if Aparat(x, y)

and ifnot Minte(y)

then Alibi(x)

R4: ifnot ~Spune_adevar(x)

then Alibi(x)

Beneficiar(x) ~ Alibi(x) Suspect(x) : /

10

TMS for the ABC Murder

(Node (IN_List OUT_List))

N1 = Suspect (Alecu) (N1, ( (N2) (N3) ) )N2 = Beneficiar (Alecu)N3 = ~Alibi(Alecu)

Suspect Alecu [IN]

Beneficiar Alecu [IN] Alibi Alecu [OUT]

+ Lista IN Lista OUT

premisã aserþiune nejustificatã

justificare

aserþiune justificatã

justification

justifiedassumption

premise unjustified assumption 11

TMS for the ABC Murder

Suppose new beliefs are addedInregistrat la hotel Alecu, Departe Arad, ~Falsificat registruR2: if Hotel(x,y)

and Departe(y)

and ifnot Falsificat(y)

then Alibi(x) Suspect Alecu [OUT]

Beneficiar Alecu [IN] Alibi Alecu [IN]

+

Înregistrat la hotel Alecu [IN] Departe Arad [IN] Falsificat registru [OUT]

+ +

R1: if Beneficiar(x) and ifnot Alibi(x) then Suspect(x)

12

TMS for the ABC Murder

Suppose another new belief is addedBarbu aparat de CumnatR3: if Aparat(x, y)

and ifnot Minte(y)

then Alibi(x)

Suspect Barbu [OUT]

Beneficiar Barbu [IN] Alibi Barbu [IN]

+

Barbu apãrat de Cumnat [IN] Minte Cumnat [OUT]

+

R1: if Beneficiar(x) and ifnot Alibi(x) then Suspect(x)

13

TMS for the ABC Murder

What about Cezar?R4: ifnot ~Spune_adevar(x)

then Alibi(x)

Suspect Cezar [IN]

Beneficiar Cezar [IN] Alibi Cezar [OUT]

+

Spune adevãrul Cezar [OUT]+

+

R1: if Beneficiar(x) and ifnot Alibi(x) then Suspect(x)

14

TMS for the ABC Murder

Represent contradiction

Contradicþie [OUT]

Suspect Alecu [OUT] Suspect Barbu [OUT] Suspect Cezar [IN] Alþi Suspecþi [OUT]

15

TMS for the ABC Murder

New beliefs are added

Suspect Cezar [OUT]

Beneficiar Cezar [IN] Alibi Cezar [IN]

+

Spune adevãrul Cezar [IN]+

+

Vãzut TV Cezar [IN] TV Falsificat [OUT]

+

16

TMS for the ABC Murder

The new beliefs bring a contradiction

Contradicþie [IN]

Suspect Alecu[OUT] Suspect Barbu[OUT] Suspect Cezar[OUT] Alþi Suspecþi[OUT]

17

Removing the contradiction

• Identify the minimal set of beliefs that brought the contradiction

• OUT some belief to remove contradiction• Select an assumption (node) N and add a valid

justification to a node N’ OUT List of NOR

• OUT a node N’ IN List of N by adding a valid justification to a node N” OUT List of N’

18

Removing the contradiction

• Justification to remove the contradiction• {Suspect Alecu, Suspect Barbu, Suspect Cezar, Alti

Suspecti}

Contradicþie [OUT]

Suspect Alecu[OUT] Suspect Barbu[OUT] Suspect Cezar[OUT] Alþi Suspecþi[IN]

Alþi Suspecþi [IN]

Suspect Alecu [OUT] Suspect Barbu [OUT] Suspect Cezar [OUT]

19

Data structures for a TMS

Data structures

(1)Node. Contains the following slots: Value – the representation of the associated fact; a unique

value, which is identical with the representation in the KB Label – state of the node - IN or OUT. NodeJustification – list of justifications which justify a given

node. Note that a node may have several justifications IsConsequence – list of justifications in which the node take

part. It is formed of 2 lists:

• ConseqIN – list of justifications in which the node appears in the IN list (N.ConseqIN)

• ConseqOUT - list of justifications in which the node appears in the OUT list (N.ConseqOUT)

Contradiction – a flag indicating if the node is a contradiction 20

Data structures for a TMS

(2) Justification. Contains the following slots: Type – represents the inference type of a justification,

namely premise, Modus Ponens, rule, inheritance, etc. Depends on the Inference Engine and it is given by the IE to the TMS

Consequence – the node the justification justifies Premises – list of nodes that participated in the

inference, formed of the INList of nodes and the OUTList of nodes

(N.Justification)(3) An indexing structure and mechanism to allow fast

search of nodes in the TMS

21

Types of nodes

(Node (IN_List OUT_List))

N1 = Suspect (Alecu) (N1, ( (N2) (N3) ) )N2 = Beneficiar (Alecu)N3 = ~Alibi(Alecu)

Suspect Alecu [IN]

Beneficiar Alecu [IN] Alibi Alecu [OUT]

+ Lista IN Lista OUT

premisã aserþiune nejustificatã

justificare

aserþiune justificatã

Justification

JustifiedAssumption(Regular node)

Premise Unjustified assumption(Assumption)

Rau Alecu [IN]

Unjustified assumption(Assumption)

+

22

Types of nodes

Premise nodes – always true (IN)

Unjustified assumptions nodes (Assumptions) – nodes which IE wants to believe (IN) or not (OUT), although they are not supported by the existing evidence

Justified assumptions nodes (Regular nodes) - nodes which are believed only if there is a valid reason for that (a valid justification). An assumptions that has a valid justification is treated as a Regular node and is IN.

Contradiction nodes – always false (OUT)

23

TMS/IE responsibilities

Responsibilities of the IE:

Adds premises and makes assumptions

Apply rules and adds justifications

Retracts assumptions Provides advise on

handling contradictions

Responsibilities of the TMS:

Cashes beliefs and maintains node labels

Detects contradictions Performs belief revision Generates explanations

24

TMS/IE interface IE action

A. Provides a mapping between IE and TMS data structures:

1. If a new node is needed, IE must inform TMS about it, and give the connection between this node and the corresponding assertions.

2. To use TMS nodes, IE must be able to retrieve TMS nodes associated with assertions that IE works with.

3. IE must provide a way for defining nodes as premises, and for enabling/retracting assumptions.

TMS action

A. Creates nodes with specified properties

25

TMS/IE interface IE action

B. Provides facilities for representing justifications.

C. Provides facilities for inspecting TMS beliefs, i.e. accessing label information.

TMS actionB. Accepts records of IE

deductions (as justifications)

C. Computes the correct label for nodes and supplies them on request.

1. Derives consequences of assumptions & premises based on the dependency network

2. When assumptions are retracted, their consequences are retracted

3. Provides explanations for beliefs, e.g., inferences

26

TMS/IE interface

IE action

D. Provides facilities for contradiction handling.

TMS action

D. Detects contradictions, based on contradiction nodes and explicit dependencies.

Ensuring a proper mapping between IE and TMS structures

Assertion i Cash assertion i Node I

TMS network Belief status for Node i

27

TMS – Computes the correct label

Algorithm: Add a node justification

1. Create justification J for node N

2. Update the links to referred nodes for connecting J in the TMS

3. if justified node N is IN

then STOP /* justification does not modify N state*/

4. if justified node N is OUT

then

4.1. Evaluate justification J

4.2. if J is valid

then

4.2.1. Change N state in IN

/* node changes from OUT to IN */

4.2.2. Propagate(N)

end 28

TMS – Computes the correct label

Propagate(N)

1. for all nodes Ni in N.ConseqIN do

1.1. if Ni is OUT

then

1.1.1. Evaluate new state of Ni based on Ni.Justification

/* considering new state IN of N */

1.1.2. if new sate of Ni is IN

then Propagate(Ni)

/* if Ni is IN no modification required */

29

TMS – Computes the correct label

2. for all nodes Ni in N.ConseqOUT do

2.1. if Ni is IN

then

2.1.1. Build list LN with all nodes recursively justified by N

2.1.2. for all nodes Nj in LN do

i. Nj.Label OUT

/* mark Nj with OUT */

2.1.3. for all nodes Nj in LN do

i. Evaluate Nj.State

/* Nj.State may be IN or OUT */

ii. Nj.Label Nj.State

/* update Nj state */

end30

Network update when changing a node state in OUT

N2 [IN] N3 [OUT]

N1 [IN] N4 [IN]

N

OUT

OUT

OUT

[OUT]

[IN][OUT]

[OUT]

+

+