Assumption-Based Truth Maintenance

SystemsMeir Kalech

Outline Last lecture:

1. Consistency-based diagnosis2. GDE – general diagnosis engine3. Conflict generation using ATMS4. Candidate generation

Today’s lecture:1. What is TMS2. TMS architecture3. Justification-based TMS4. Assumption-based TMS

What is TMS? A Truth Maintenance System (TMS) is a

Problem Solver module responsible for:

Enforcing logical relations among beliefs. Generating explanations for conclusions. Finding solutions to search problems Supporting default reasoning. Identifying causes for failure and recover from

inconsistencies.

1. Enforcement of logical relations

AI problem -> search. Search utilizes assumptions. Assumptions change. Changing assumptions -> updating

consequences of beliefs. TMS: mechanism to maintain and update

relations among beliefs.

1. Enforcement of logical relations

Example: If (cs-501) and (math-218) then (cs-570). If (cs-570) and (CIT) then (TMS).

If (TMS) then (AI-experience).The following are relations among beliefs:

(AI-experience) if (TMS).(TMS) if (cs-570), (CIT).(cs-570) if (cs-501), (math-218)

Beliefs are propositional variables TMS is a mechanism for processing large collections

of logical relations on propositional variables.

2. Generation of explanations

Solving problems is what Problem Solvers do. However, often solutions are not enough.

The PS is expected to provide an explanation TMS uses cached inferences for that aim. TMS is efficient:

Generating cached inferences once is more beneficial than running inference rules that have generated these

inferences more than once.

2. Generation of explanations

Example: Q: Shall I have an AI experience after completing the CIT program? A: Yes, because of the TMS course. Q: What do I need to take a TMS course? A: CS-570 and CIT.

There are different types of TMSs that provide different ways of explaining conclusions (JTMS vs ATMS).

In this example, explaining conclusions in terms of their immediate predecessors works much better.

3. Finding solutions to search problems

B

A D

C E

Color the nodes: red (1), green (2) yellow (3). Adjacent nodes are of different colors. The set of constraints describe this problem:

A1 or A2 or A3 not (A1 and B1) not (A3 and C3) not (D2 and E2)B1 or B2 or B3 not (A2 and B2) not (B1 and D1) not (D3 and E3)C1 or C2 or C2 not (A3 and B3) not (B2 and D2) not (C1 and E1)D1 or D2 or D3 not (A1 and C1) not (B3 and D3) not (C2 and E2)E1 or E2 or E2 not (A2 and C2) not (D1 and E1) not (C3 and E3)

To find a solution we can use search:

3. Finding solutions to search problems

A is red A is green A is yellow

B is redB is red B is green B is yellow

C is greenC is green C is redC is yellow

D is red D is yellowD is yellow D is green

E is redE is red E is green E is yellow

4. Default reasoning and TMS

PS must make conclusions based on incomplete information.

“Closed-World Assumption” (CWA) X is true unless there is an evidence to the contrary. CWA helps us limit the underlying search space. The reasoning scheme that supports CWA is called

“default (or non-monotonic) reasoning”.

4. Default reasoning and TMS

Example: Consider the following knowledge base

Bird(tom) and ¬Abnormal(tom) Can_fly(tom) Penguin(tom) Abnormal(tom) Ostrich(tom) Abnormal(tom) Bird(tom) --------------------------------------------- Under the CWA, we assume

¬Abnormal(tom)and therefore we can derive:

Can_fly(tom) Non-monotonic TMS supports this type of reasoning.

5. Identifying causes for failures and recovering from inconsistencies

Inconsistencies among beliefs in the KB are always possible:

wrong data (example: “Outside temperature is 320 degrees.”)

Impossible constraints (example: Big-house and Cheap-house and Nice-house).

TMS maintains help identify the reason for an inconsistency

“dependency-directed backtracking” allows the TMS to recover.

TMS applications Constraint Satisfaction Problems (CSP)

Set of variables Domain over each variable Constraints between variables’ domain Goal: find “solution”: assignments to the

variables that satisfy the constraints Scenario and Planning Problems

Find a path of state transitions lead from initial to final states. (games, strategies).

TMS – identifies of applicable rules.

CSP exampleAllocation problem

Two hosts: {h1,h2} (variables) Three tasks: {t1,t2,t3} (domain) Two constraints:

t1 before t2 on the same host t1 could not be run on the same host of t3

CSP example

t1-h1 t2-h1 t3-h1 t1-h2 t2-h2 t3-h3

t1-h1t2-h1

t2-h1t1-h1

t1-h1t3-h1

t1-h2t2-h2

t2-h2t1-h2

t1-h2t3-h2

t1-h1t2-h1t3-h2

t1-h1t2-h1t3-h1

t1-h2t2-h2t3-h1

t1-h2t2-h2t3-h2… …

48 nodes 6 are solutions

Outline Last lecture:

1. Consistency-based diagnosis2. GDE – general diagnosis engine3. Conflict generation using ATMS4. Candidate generation

Today’s lecture:1. What is TMS2. TMS architecture

3. Justification-based TMS4. Assumption-based TMS

Problem Solver Architecture

Problem Solver

TMS

Justifications, assumptions

Beliefscontradictions

The TMS / PS relationship is the following:

How the TMS and the PS communicate? The PS works with:

assertions (facts, beliefs, conclusions, hypotheses) inference rules procedures

Each one of these is assigned a TMS node. Example: N1: (rule (student ?x) (assert (and (underpaid ?x) (overworked ?x)))) N2: (student Bob)

Given N1 and N2, the PS can infer N3: (and (underpaid Bob) (overworked Bob))

PS threats nodes as logical formulas, While TMS treats nodes as propositional variables.

TMS nodes Different types of TMS support types of nodes:

Premise nodes. These are always true. Contradiction nodes. These are always false. Assumption nodes. PS believes no matter whether or not

they are supported by the existing evidence. Node has a label associated with it. The contents and

the structure of the label depends on the type of TMS. Other properties are node type (premise, assumption,

etc.), node support (justifications, antecedents), node consequences, etc.

TMS justifications If N3, is created by the PS, it reports to the TMS

together with the fact that it follows from N1, N2. justification:(N3 N2 N1)

Here N3 is called the consequent, N1 and N2 are the antecedents of the justification.

Justifications record relations among beliefs or explaining consequents and identifying causes for inconsistencies.

The general format of justifications is the following:(<consequent> <antecedents>)

Propositional specification of a TMS TMS nodes are propositional variables TMS justifications are propositional formulas

N1 & N2 & … & Ni Nj

Here N1, N2, …, Ni, Nj are positive literals, therefore this implication is a Horn formula.

TMS can be viewed as a set of Horn formulas

PS / TMS interactionResponsibilities of the PS:

1. Adds assertions and justifications.

2. Makes premises and assumptions.

3. Retracts assumptions.4. Provides advise on

handling contradictions

Responsibilities of the TMS:

1. Cashes beliefs and consequences and maintains labels.

2. Detects contradictions.3. Performs belief revision.4. Generates explanations.

Outline Last lecture:

1. Consistency-based diagnosis2. GDE – general diagnosis engine3. Conflict generation using ATMS4. Candidate generation

Today’s lecture:1. What is TMS2. TMS architecture3. Justification-based TMS

4. Assumption-based TMS

Justification-based TMSJustifications are used for:

Belief update purpose, when belief state of a node changes.

Handle contradiction: 1. Justification is added to the dependency-

directed backtracking system 2. Then search through the dependency network

for the assumptions of the contradiction3. Contradiction is removed.

Justification-based TMS A justification contains inlist and outlist for a

justified node to be believed: inlist – a set of nodes that must be in outlist – a set of nodes that must be out Syntax: {(inlist),(outlist)}

Premises hold universally: empty in and out Only one context includes the set of

assumptions currently believed.

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightPS concludes “nice weather” from A and C

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}

New node in the JTMS

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}PS concludes “swim” from E and G

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}H: Swim {(E,G),()}

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}H: Swim {(E,G),()}

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}H: Swim {(E,G),()}I: Contradiction {(C),()}

Dependency-directed

backtracking system

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}H: Swim {(E,G),()}I: Contradiction {(C),()}X: Handle {(),()} //premiseD: Raining {(X),()}

Context: {(A,D,E),(B,C,F,G,H,I)}

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}H: Swim {(E,G),()}I: Contradiction {(C),()}X: Handle {(),()} //premiseD: Raining {(X),()}J: Read {(D,E),()}K: Contradiction {(J),()} //becomes tired

Dependency-directed

backtracking system

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}H: Swim {(E,G),()}I: Contradiction {(C),()}X: Handle{(),()} //premiseD: Raining {(X),()}J: Read {(D,E),()}K: Contradiction {(J),()} //becomes tiredF: Night {(X),()}

Context: {(A,D,F),(B,C,E,G,H,I,J,K)}

Justification-based TMS – ExamplePropositions: Justifications:A: Temperature>=25 {(),(B)}B: Temperature< 25C: Not raining {(),(D)}D: RainingE: Day {(),(F)}F: NightG: Nice weather {(A,C),()}H: Swim {(E,G),()}I: Contradiction {(C),()}X: Handle {(),()} //premiseD: Raining{(X),()}J: Read {(D,E),()}K: Contradiction {(J),()} //becomes tiredF: Night {(X),()}L: Sleep {(F),()}

Outline Last lecture:

1. Consistency-based diagnosis2. GDE – general diagnosis engine3. Conflict generation using ATMS4. Candidate generation

Today’s lecture:1. What is TMS2. TMS architecture3. Justification-based TMS4. Assumption-based TMS

Assumption-based TMS: Motivation Problem solvers need to explore multiple

contexts at the same time, instead of a single one (the JTMS case)

Alternate diagnoses of a broken system Different design choices Competing theories to explain a set of data

Problem solvers need to compare contexts switching from one context to another.

In JTMS, this can be done by enabling and retracting assumptions.

In ATMS, alternative contexts are explicitly stored.

The idea behind ATMS The assumptions underlying conclusions are

important in problem-solving Solutions can be described as sets of assumptions States of the world can be represented by sets of

assumptions Identify sets of assumptions called here

environments Organize problem solver around manipulating

environments Facilitates reasoning with multiple hypotheses

Assumptions and Justifications

ATMS keeps and manipulates sets of assumptions rather than sets of beliefs

Three types of nodes: Premise nodes. These are always true, but they

are of no special interest for ATMS. Assumption nodes. Once made, assumptions

are never retracted. Contradictions. These are defined by means of

assumptions that originate them. Such sets of assumptions are called nogoods.

ATMS justifications are Horn formulas of the form:Jk: I1, I2, …, In Ck,

where I1, I2, …, In are the antecedents, and Ck is the consequent of justification Jk.

Basic ATMS terminology

ATMS answers queries about whether a node holds in a given set of beliefs.

Definition. A set of assumptions upon which a given node depends is called an environment. Example: {A,B,C}

Definition. A label is a set of environments. Example: {{A,B,C}, … ,{D,F}} That is, the label is the assumptions upon which the node ultimately depends – major difference from JTMS label, where labels are simple, :IN or :OUT.

Definition. An ATMS-node, Nk is a triplet <datum, label(status), justifications>

Basic ATMS terminology

Definition. A node n holds in a given environment E, iff it can be derived from E given the set of justifications J: E,J ⊢ n

An environment is inconsistent if false is derived: E,J ⊢⊥Definition. Let E be a (consistent) environment, and N be

a set of nodes derived from E. Then, E N is called the context of E.

Definition. A characterizing environment is a minimal consistent environment from which a context can be derived.

Each context is completely specified by its characterizing environment.

ATMS efficiency ATMS is provided by a set of assumptions and

justifications. The task of ATMS is efficiently determines the

contexts. Incrementally updating only the changed contexts. Data structure for context-consistency checking and

node inclusion very fast.

Relations between environments

Because environments are monotonic, set inclusion between environments implies logical subsumption of consequences.

Example: E1 = {C} E2 = {C, D} E3 = {D, E}

E1 subsumes E2 E2 is subsumed by E1 E1 neither subsumes or is subsumed by E3

How ATMS answers queriesHow ATMS answers queries about whether a node holds in a given environment?

Easiest way: associate with each node all of the environments

Better way: we can record only those environments which satisfy the following four properties:

1. Soundness: a node holds in any of the environments associated with it.

2. Consistency: no environment is a nogood.3. Completeness: every consistent environment is either

associated with it or with a superset of it.4. Minimality: no environment is a subset of any other.

Example, dependency network:

Is H believed? Yes, because its label is non-empty.

Is H believed under {B, C, D, Z, X}? Yes, because {B, C, D} {B, C, D, Z, X}

Is H believed under {C, D}? No.

ATMS labels

D

C

F

G

{{B, C}}

{{C, D}}

{{A},{B,C,D}}

Certain nodes can be declared as contradictions: Every environment which allows a contradiction is

inconsistent. Inconsistent environments are called nogoods.

Example:

Contradictions

F

G

{B,C}

{A,B,C}

Special labels in ATMSCase 1: Label = { } (empty label)

This means that there is no known consistent environment in which the node is believed, i.e. either there is no path from assumptions to it, or all environments for it are inconsistent.

Case 2: Label = {{}} (empty environment)

This means that the node is believed in every consistent environment, i.e. the node is either a premise or can be derived strictly from premises.

Label propagationR

C

D

G

L

Label propagation: enable A

R

C

D

G

L

Label propagation: enable B

R

C

D

G

L

Label propagation: enable C

R

D

G

L

C

{{C

C}}

C}}

C}, {B,C}}

Label propagation: enable D

R

G

L

C

{{C

C},{D}}

C}}

C}, {B,C}}

D

Example:

datum justificationsenvironments

For example see Franz Wotawa’s slides page 7-8

Properties of ATMS

Environment Lattice

Comments to lattice If an environment is nogood, then all of its

superset environments are nogood as well. All nogoods are the result of the nogood {A, B, E}.

The ATMS associates every datum with its contexts. If a datum is in a context, then it is in every superset as well (the inconsistent supersets are ignored).

Comments to lattice The circled nodes indicate all the contexts of

The square nodes indicate all the contexts of

If PS infers y=0 from x+y=1 and x=1:

Then the context for y=0 is the intersection of the contexts of the above:

Comments to lattice One sound and complete label for the consequent is the

set whose elements are the union of all possible combinations of picking one environment from each antecedent node label. Thus one sound and complete label is:

The environment {A, B, C, D} is removed because it is subsumed by {A, B, C}.

The environment {A, B, D, E} is not included because it is a superset of the inconsistent {A, B, E}.

ATMS algorithms Logical specification of the ATMS:

ATMS does propositional reasoning over nodes. ATMS justifications are Horn clauses. Contradictions are characterized by nogoods.

Every ATMS operation which changes a node label can be viewed as adding a justification, i.e. this is the only operation we have to be concerned here is label update as a result of adding a justification.

ATMS algorithms Step 1: Compute a tentative new (locally

correct) label for the affected node as followsGiven Jik the label of the i’th node of k’th justification

for consequent node n, a complete label for node n:

Lnew = k {x | x = i xi , where xi Jik} Step 2: All nogoods and subsumed

environments are removed from Lnew to achieve global correctness.

Propagating label changes

To update node Ni, compute its new label as described If the label has not changed

DONE. Else

If Ni is a contradiction node do Mark all environments in its label as nogoods. For every node in the network,

check its label for environments marked as nogoods and remove from every node label.

Else recursively update all Ni’s consequences (other

nodes having justifications which mention Ni).

ExampleAssumptions:#: Proposition: Label:

Justifications:A: Temperature >= 25 {{A}} {(A)}B: Temperature < 25 {{B}} {(B)}C: Not Raining {{C}} {(C)}D: Day {{D}} {(D)}

Derived facts:E: Nice weatherF: SwimG: ReadH: Sleep

Rules:1. A and C E2. E and D F3. D and out(C)

G4. Out(D)

H5. A and B

⊥

Example – empty environment

Assumptions:#: Proposition: Label:

Justifications:A: Temperature >= 25 {{A}} {(A)}B: Temperature < 25 {{B}} {(B)}C: Not Raining {{C}} {(C)}D: Day {{D}} {(D)}H: Sleep {{Out(D)}} {(Out(D))}

The problem solver applies the breadth first search strategy, to provide the assumptions: The empty environment is provided first, then A, B…, {A,B},{A,C}…,{A,B,C}..etc. The rules are fired by the PS as justifications to the assumptions.

The empty environment caused the PS to provide out(D) as justification to H

Example – {D}Assumptions:#: Proposition: Label:

Justifications:A: Temperature >= 25 {{A}} {(A)}B: Temperature < 25 {{B}} {(B)}C: Not Raining {{C}} {(C)}D: Day {{D}} {(D)}H: Sleep {{Out(D)}} {(Out(D))}G: Read {{D,Out(C)}} {{D,Out(C)}}

Environments {A},{B} and {C} have not changed

Example – {A,B}Assumptions:#: Proposition: Label:

Justifications:A: Temperature >= 25 {{A}} {(A)}B: Temperature < 25 {{B}} {(B)}C: Not Raining {{C}} {(C)}D: Day {{D}} {(D)}H: Sleep {{Out(D)}} {(Out(D))}G: Read {{D,Out(C)}} {(D,Out(C))}

⊥ {{A,B}} {(A,B)}PS will fire the fifth rule (A and B ⊥)

ATMS will add this environment to the nogood DB

Example – {A,C}Assumptions:#: Proposition: Label:

Justifications:A: Temperature >= 25 {{A}} {(A)}B: Temperature < 25 {{B}} {(B)}C: Not Raining {{C}} {(C)}D: Day {{D}} {(D)}H: Sleep {{Out(D)}} {(Out(D))}G: Read {{D,Out(C)}} {(D,Out(C))}E: Nice weather {{A,C}} {(A,C)}

PS will fire the first rule (A and C E){A,D} and {B,D} can fire the third rule, but it has

already been fired

Example – {A,C,D}Assumptions:#: Proposition: Label:

Justifications:A: Temperature >= 25 {{A}} {(A)}B: Temperature < 25 {{B}} {(B)}C: Not Raining {{C}} {(C)}D: Day {{D}} {(D)}H: Sleep {{Out(D)}} {(Out(D))}G: Read {{D,Out(C)}} {(D,Out(C))}E: Nice weather {{A,C}} {(A,C)}F: Swim {{A,C,D}} {(E,D)}•PS will fire the second rule (E and D F)•{A,B,C},{A,B,D} and {A,B,C,D} are superset of {A,B} and so are not fired•If: Label(E)={{A,C},{X,Y}} then Label(F)={{A,C,D},{X,Y,D}}

Example – environment latticeWhen sleep (H) holds?

Square – where H holdsCircle – where G holdsRhombus - where F holds

Back to Diagnosis…

M1

M2

M3

*

*

*

A1

A2

+

+G=12,{{A2,M2,M3}}

B=2,{{}}

E=3,{{}}

C=2,{{}}

D=3,{{}}

A=3,{{}}

z=6,{{M3}}

y=6,{{M2}}

x=6,{{M1}}

F=12,{{A1,M1,M2}}F=10,{{}}

NOGOODS: {A1,M1,M2}x=4,{{A1,M2}}

y=4,{{A1,M1}}

G=10,{{A1,M1,M3,A2}}G=12,{{}}

{A1,M1,M3,A2}

z=6,{{M3},{A2,M2}}

y=6,{{M2},{A2,M3}}

z=8,{{A1,A2,M1}}

x=4,{{A1,M2},{A1,A2,M3}}

Bibliography 1. Kenneth D. Forbus and Johan de Kleer, Building Problem

Solvers, The MIT Press, 1993.2. Johan de Kleer, An assumption-based truth maintenance

system, Artificial Intelligence 28, 127-162, 1986.3. Johan de Kleer, Problem Solving with the ATMS, Artificial

Intelligence 28, 197-224, 1986.4. Johan de Kleer, Extending the ATMS , Artificial Intelligence 28,

163-196, 1986.5. Mladen Stanojevic and Sanja Vranes and Dusan Velasevic,

Using Truth Maintenance Systems: A Tutorial, IEEE Expert: Intelligent Systems and Their Applications 9(6), 45-56, 1994.