Practical Non-monotonic Reasoning

Guido Governatori

Knowledge Techniques Week 2012

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www.nicta.com.au From imagination to impact

Part I

Introduction: Knowledge Representation

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Artificial Intelligence

The design and study of systems that behave intelligentlyFocus on hard problems, often with no, or very ine�cient fullalgorithmic solutionFocus on problems that require “reasoning” (“intelligence”)and a large amount of knowledge about the world

CriticalRepresent knowledge about the worldReason with these representations to obtain meaningfulanswers/solutions

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Symbolic Knowledge Representation

Important objects (collections of objects) and theirrelationships are represented explicitly by internal symbols

Symbolic manipulation of internal symbolic representationsachieves results meaningful in the real world

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Goals of Knowledge Representation

Find representation that are:

Rich enough to express the important knowledge relevant tothe problem at hand

Close to problem at hand: compact, natural, maintainable

Amenable to e�cient computation

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Representational Adequacy

Consider the following facts:Most children believe in Santa.John will have to finish his assignment before he can startworking on his project

Can all be represented as a string! But hard then tomanipulate and draw conclusions

How do we represent these formally in a way that can bemanipulated in a computer program?

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Well-defined Syntax and Semantics

Precise syntax: what can be expressed in the languageFormal language, unlike natural languagePrerequisite for precise manipulation through computation

Precise semantics: formal meaning of expression

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Naturalness of expression

Also helpful if our representation scheme is quite intuitive andnatural for human readers!

Could represent the fact that my car is red using the notation:

“xyzzy ! Zing”where xyzzy refers to redness, Zing refers to by car, and !used in some way to assign properties

But this would not be very helpful. . .

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Inferential Adequacy

Representing knowledge not very interesting unless you canuse it to make inferences:

Draw new conclusions from existing facts.“If its raining John never goes out” + “It is raining today”so. . .Come up with solutions to complex problems, using therepresented knowledge.

Inferential adequacy refers to how easy it is to draw inferencesusing represented knowledge.

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Inferential E�ciency

You may be able, in principle, to make complex deductions, but itmay be just too ine�cient.

The basic tradeo↵ of all KRGenerally the more complex the possible deductions,the less e�cient will be the reasoning process (in theworst case).

The eternal quest of KRNeed representation and inference system su�cientfor the task, without being hopelessly ine�cient

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Inferential Adequacy (2)

Representing everything as natural language strings has goodrepresentational adequacy and naturalness, but very poorinferential adequacy

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Requirements for KR Languages

Representational Adequacy

Clear syntax/semantics

Inferential adequacy

Inferential e�ciency

Naturalness

In practice no one language is perfect, and di↵erent languages aresuitable for di↵erent problems.

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Why Reasoning?

Patient x is allergic to medication m

Anybody allergic to medication m is also allergic tomedication n

Is it ok to prescribe n for x?

Reasoning uncovers implicit knowledge not represented explicitly.Beyond database systems technology

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Syntactic vs Semantic Reasoning

Semantic Reasoning

Sentences P1 . . . ,Pn entail sentence P i↵ thetruth of P is implicit in the truth of P1 . . . ,Pn

Or: if the world satisfies P1 . . . ,Pn then it mustalso satisfy PReasoning usually done by humans

Syntactic Reasoning

Sentences P1 . . . ,Pn infer sentence P i↵ there isa syntactic manipulation of P1 . . . ,Pn thatresults in PReasoning done by humans and machines

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Reasoning: Soundness and Completeness

Sound (syntactic) reasoning:If P is inferred by P1 . . . ,Pn then it is also entailed semanticallyOnly semantically valid conclusions are drawn

Complete (syntactic) reasoningIf P is entailed semantically by P1 . . . ,Pn then it can also beinferredAll semantically valid conclusions can be drawn

Usually interested in sound and complete reasoningBut sometimes we have to give up one for the sake of e�ciency(usually completeness)

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Main KR Approaches

Logic BasedFocus on clean, mathematical semantics: declarativelyExplainability

Frames / Semantic Networks / ObjectsFocus on structure of objects

Rule-based systemsFocus on e�ciencyA) B in logic and rule-based systems

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The Landscape of KR

Predicate logic (first order logic) and its sublanguagesLogic programming, (pure) PrologDescription logicsWeb ontology languages

Predicate logic (first order logic) extensionsModal and epistemic logicsTemporal logicsSpatial logics

Inconsistency-tolerant logics:ParaconsistencyNonmonotonic reasoning

Representing vaguenessProbabilistic logicsBayesian networksMarkov chains

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Part II

Defeasible Reasoning

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Being Lazy: Reasonable Results with Minimum E↵ort

Factual omniscience and (non-)monotonic reasoning

PhD ! Uni

Weekend ! ¬UniPublicHoliday ! ¬Uni

Sick ! ¬UniWeekend ^VICdeadline ! Uni

VICdeadline ^PartnerBirthday ! ¬Uni

Phd ^ (¬Weekend _ (Weekend ^VICdeadline ^¬PartnerBirthday))^¬Sick . . .! Uni

VIC= Very Important Conference

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Inconsistent Information

Classical logics “collapse” in the face of inconsistenciesEverything can be derived

But inconsistencies do happen in real settingsCommon when integrating knowledge from various Websources

Nonmonotonic reasoning is inconsistency tolerant reasoning

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Rules with Exceptions

Natural representation for policies and business rules.

Priority information is often implicitly or explicitly available toresolve conflicts among rules.

Potential applicationsNormative reasoningSecurity policiesBusiness rulesPersonalizationBrokeringBargaining, automated agent negotiations

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Nonmonotonic Reasoning Options

Sceptical vs Credulous

Ambiguity Blocking vs Ambiguity Propagation

Team Defeats vs No Team Defeat

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Basic Reasoning

Suppose you have one pieces of evidence, Evidence A suggestingthat the defendant is responsible.

Given: EvidenceA and the rule

EvidenceA) Responsible

Sceptical: ResponsibleCredulous: Responsible

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Conflict

Suppose that your legal system is based on presumption ofinnocence, and the somebody is guilty if responsibility is proved.

Given the rules

r1 : Responsible ) Guilty

r2 : ) ¬Guilty

Sceptical: ¬GuiltyCredulous: ¬GuiltyWhat about if we have r1 > r2 (same conclusions)

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Sceptical vs Credulous

Suppose you have two pieces of evidence. Evidence A suggestingthat the defendant is responsible, and Evidence A suggesting thatthe defendant is not responsible.

Given: EvidenceA, and EvidenceB and the rules

EvidenceA) Responsible

EvidenceB ) ¬Responsible

Sceptical: no conclusionsCredulous: both conclusions

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Sceptical vs Credulous and preference

Suppose you have two pieces of evidence. Evidence A suggestingthat the defendant is responsible, and Evidence A suggesting thatthe defendant is not responsible. However, Evidence A is morereliable than Evidence B.

Given: EvidenceA, and EvidenceB and the rules

r1 : EvidenceA) Responsible

r2 : EvidenceB ) ¬Responsibler1 > r2

Sceptical: ResponsibleCredulous: Responsible

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Ambiguity Propagation vs Ambiguity Blocking

Suppose you have two pieces of evidence. Evidence A suggestingthat the defendant is responsible, and Evidence A suggesting thatthe defendant is not responsible. If the defendant is responsible,then he is guilty. and we have presupposition of innocence.

Given: EvidenceA, and EvidenceB and the rules

EvidenceA) Responsible

EvidenceB ) ¬ResponsibleResponsible ) Guilty

) ¬GuiltyAmbiguity blocking concludes ¬GuiltyAmbiguity propagation does not concludes ¬Guilty and fails toconclude Guilty .

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Ambiguity Propagation vs Ambiguity Blocking

Suppose you have two pieces of evidence. Evidence A suggesting that thedefendant is responsible, and Evidence A suggesting that the defendant isnot responsible. If the defendant is responsible, then he is guilty. and wehave presupposition of innocence. If the defendant was wrongly accusedthen he is entitled to compensation.

Given: EvidenceA, and EvidenceB and the rules

EvidenceA) Responsible

EvidenceB ) ¬ResponsibleResponsible ) Guilty

) ¬Guilty¬Guilty ) Innocent

Innocent ) Compensation

Ambiguity blocking concludes Compensation

Ambiguity propagation does not conclude Compensation

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Team Defeat vs No Team Defeat

r1 :General ) Attack

r2 :Captain) ¬Attackr1 > r2

r3 :Bishop ) Attack

r4 :Priest ) ¬Attackr3 > r4

Team Defeat concludes AttackNo Team Defeat does not conclude Attack

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Weak and Strong Support

Suppose that a drunk person testify that the accused (not known to him)was in location di↵erent from the crime scene at the time of the crime.Secure footage from high definition camera shows the accused at thecrime scene at the time of the crime.

r1 :drunk ) ¬CrimeScene

r2 :camera) CrimeScene

r3 :¬CrimeScene ) Alibi

r2 > r1

Do we have scintilla of evidence to claim that the accuse was at thecrime scene at the time of the crime?

Is it reasonable to say that we have substantial evidence supporting forthe same claim?

Is it reasonable to claim that beyond any reasonable doubts the accused

has an alibi?

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Why Defeasible Logic?

Rule-based non-monotonic formalism

Flexible

E�cient (linear complexity)

Directly skeptic semantics

Argumentation semantics

Constructive proof theory

Optimised/e�cient implementations (1.700.000 rules)

Extensible

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Defeasible Logic: Strength of Conclusions

Derive (plausible) conclusions with the minimum amount ofinformation.

Definite conclusionsDefeasible conclusions

Defeasible TheoryFactsStrict rules (A! B)Defeasible rules (A) B)Defeaters (A; B)Superiority relation over rules

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Conclusions in Defeasible Logic

A proof is a finite sequence P = (P(1), . . . ,P(n)) of tagged literalssatisfying four conditions

+∂p (-∂p): p is (not) derivable using ambiguity blocking,team defeat

+∂ ⇤p: p is derivable using ambiguity blocking, no team defeat

+dp: p is derivable using ambiguity propagation, team defeat

+d ⇤p p is derivable using ambiguity propagation, no teamdefeat

+sp: p is a credulous conclusion using team defeat

+sp: p is a credulous conclusion using no team defeat

+s�p: p is a credulous weak conclusion

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Proving Conclusions in Defeasible Logic

1 Give an argument for the conclusion you want to prove

2 Consider all possible counterarguments to it3 Rebut all counterarguments

Defeat the argument by a stronger oneUndercut the argument by showing that some of the premisesdo not hold

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Example

Facts: A1, A2, B1, B2

Rules: r1:A1 ) Cr2:A2 ) Cr3:B1 ) ¬Cr4:B2 ) ¬Cr5:B3 ) ¬C

Superiority relation:r1 > r3r2 > r4r5 > r1

Phase 1: Argument for CA1 (Fact), r1 : A1 ) CPhase 2: Possible counterargumentsr3 : B1 ) ¬Cr4 : B2 ) ¬Cr5 : B3 ) ¬CPhase 3: Rebut the counterargumentsr3 weaker than r1r4 weaker than r2r5 is not applicable

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Derivations in Defeasible Logics: Ambiguity blocking

+∂p

1) 9 an applicable rule r pro p2) 8 rule t con p either:

2.1) t is not applicable2.2) t is defeated by an applicable rule s pro p stronger than t

+∂ ⇤p

1) 9 an applicable rule r pro p2) 8 rule t con p either:

2.1) t is not applicable2.2) t is defeated by r where r is stronger than t

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Derivations in Defeasible Logics: Ambiguity propagation

+∂p

1) 9 an applicable rule r pro p2) 8 rule t con p either:

2.1) t is not applicable2.2) t is defeated by an applicable rule s pro p stronger than t

+dp

1) 9 an applicable rule r pro p2) 8 rule t con p either:

2.1) t is not applicable2.2) t is defeated by a supported rule s pro p stronger than s

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Derivations in Defeasible Logics: Support

+sp

1) 9 a supported rule r pro p2) 8 rule s con p either

2.1) s is not applicable using ambiguity propagation (i.e., �d ,�d ⇤)2.2) s is not stronger than r

+s�p

9 a supported rule r pro p

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Properties of Defeasible Logic

Theorem

Defeasible logic is consistent. +∂a and +∂¬a cannot be bothderived, unless they are already known as certain knowledge (facts)

Theorem

Defeasible logic is coherent. +#a and �#a cannot be derivedfrom the same knowledge base.

Theorem

Defeasible logic has linear complexity.

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Part III

Modal Defeasible Logic

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Modal Logic

Guido gives a tutorial KTW 2012

Normal Modal Logic1 propositional logic2 2(A! B)! (2A!2B)3 ` A/ `2A or A ` B/2A `2B4 2A! A (2A ` A)5 2A! ¬2¬A (2A ` ¬2¬A)6 2A!22A (2A `22A)7 2A! ¬2¬2A (2A ` ¬2¬2A)

1 + 2 + 3 = Logical omniscience (and expected side-e↵ects)1 = monotonic

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What is a rule?

A rule is a binary relationship between a set of ‘expressions’ and an‘expression’

What’s the strength of the relationship?

What’s the type of the relationship?

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Modal Defeasible Logic: Mode and Strength

1 The strength describes how strong is the relationshipsbetween the antecedent and the consequent of a rule.

A1, . . . ,An ! B (B is an indisputable consequence ofA1, . . . ,An)A1, . . . ,An ) B (normally B if A1, . . . ,An)

2 The mode qualifies the conclusion of a rule.A1, . . . ,An )BEL B (an agent forms the belief B whenA1, . . . ,An are the case)A1, . . . ,An )INT B (an agent has the intention B whenA1, . . . ,An are the case)A1, . . . ,An )OBL B (an agent has the obligation B whenA1, . . . ,An are the case)

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Conclusions in Basic Modal Defeasible Logic

+�2iq, which is intended to mean that q is definitelyprovable (i.e., using only facts and strict rules of mode 2i );

��2iq, which is intended to mean that we have proved thatq is not definitely provable in D;

+∂2iq, which is intended to mean that q is defeasiblyprovable in D using rules of mode 2i ;

�∂2iq which is intended to mean that we have proved that qis not defeasibly provable in D using rules of mode 2i .

We obtain 2ip i↵ +∂2ip.

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Recipe for Modal Defeasible Logics

Choose the appropriate modalitiesCreate a defeasible consequence relation for each modalityIdentify relationships between modalities:

inclusion21f !22f

conflicts21f ,22¬f !?

conversions from one modality to another modality

A1, . . . ,An )21 B

22A1, . . . ,22An `22B

Put in a mixer and shake well!

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Proofs for Modal Defeasible Logic

Conflict 21 ! ¬22¬1 Give an argument for the conclusion you want to prove

2 Consider all possible counterarguments to it using rules forboth 21 and 22

3 Rebut all counterargumentsDefeat the argument by a stronger oneUndercut the argument by showing that some of the premisesdo not hold

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Proofs for Modal Defeasible Logic

Conversion 21 to 22

1 Give an argument for the conclusion you want to prove usingrules for either 22 or rules of mode 21 st all premises areprovable with mode 22

2 Consider all possible counterarguments to it3 Rebut all counterarguments

Defeat the argument by a stronger one (same as 1)Undercut the argument by showing that some of the premisesdo not hold (for rules of mode 21 show that the premises arenot provable with mode 22)

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DL for cognitive agents

D = (F ,RBEL,RDES,R INT,ROBL,>)

RBEL rules for belief !BEL, )BEL, ;BEL

RDES rules for desire !DES, )DES, ;DES

R INT rules for intention !INT, )INT, ;INT

ROBL rules for obligation !OBL, )OBL, ;OBL

For X 2 {INT,DES,OBL}D ` XA i↵ D `+∂XAD ` A i↵ D `+∂BELA

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Interactions

OBLa! ¬INT¬a social

INTa! ¬DES¬a stable

(INTa^OBL(a! b))! INTb conversion

(OBLa^ INT(a! b))! OBLb conversion

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Conversions

What do we conclude from

A1,A2 )OBL C

and INTA1 and INTA2?What about

A1, INTA2 )OBL C

and INTA1 and INTA2?

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Social Agents

Definition (Social agent)

An agent is social if in case of a conflict between an obligationand an intention, the agent prefers the obligation to her intention

IJCAIdeadline )OBL Uni

SoccerWorldCup )INT ¬Uni

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BIO Logical Agents

A set of rules for beliefs:

a1, . . . ,an )BEL c

The agent derives BELc from a1, . . . ,anA set of rules for intentions:

a1, . . . ,an )INT c

The agent derives INTc from a1, . . . ,anA set of rules for :

a1, . . . ,an )OBL c

The agent derives OBLc from a1, . . . ,anBelief rules are stronger than obligation rules which in turns arestronger than intention rules

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From Beliefs to Intentions

If we want to model realistic agents, the model must conform withthe real world. According to current legal theories:If an agent knows/beliefs that B is a consequence of A, and theagent intends A, then the agent intends B (unless she has somejustifications for not intending it).

From a1, . . . ,an )BEL c , and

INTa1, . . . , INTan derive

INTc

If an agent believes that dropping a glass will break it, and sheintends to drop the glass, she intends to break it.

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Good News

Why should we use BIO Logical agents

Theorem

The complexity of defeasible logic for BIO logical agents is linear

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Part IV

Adding Time

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Temporalised Defeasible Logic

Temporalised Defeasible Logic is an umbrella expression for a zooof variants of logics.

time points: A : t (A holds at time t)

intervals: A[ts , te ] (A holds from ts to te)

durations: A : d (A holds for d time units)

. . .

A temporalised defeasible theory

(F ,R ,>,T )

T discrete total order of instants

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Persistent and Transient Conclusions

linear discrete time line with a fixed granularitypropositions (literals) are associated with instants of time

C : t is persistent at time t, if C continues to hold after tunless some event occurs to terminate it.C : t is transient at time t, if C is guaranteed to hold at time tonly.

partition the rules into persistent rules and transient rules

ClapHands : t !t MakeSomeNoise : t

TearPaper : t !p ShreddedPaper : t

A1 : t1, . . .An : tn )x C : t

no constraints over t1, . . . , tn and t.

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Proving Persistence

1 Generate an argument for the persistent conclusion now usingpersistent rules.

Take a rule for the conclusion that is applicable now orShow the there is a time in the past where the persistentconclusion obtains.

2 Consider all possible counterarguments for the conclusionTake all rules for its negation that obtain nowTake all rules for its negation that have obtained since thetime in the past.

3 rebut the counterargumentsshow that the rules have been discarded (not applicable ordefeated).

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Example

Facts: A : t0, B : t2,C : t2, D : t3

Rules: r1 : A : t )p E : tr2 : B : t )p ¬E : tr3 : C : t ;p E : tr4 : D : t )t ¬E : t

Superiority relation:r3 > r2r1 > r4

Conclusions at time t0A, E using r1 (E is persistent)Conclusions at time t1EConclusions at time t2B , C , EConclusions at time t3D, ¬E

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Linear Time

Theorem

The extension of a temporalised defeasible theory D can becomputed in O(|R |⇤ |H|⇤ |T |)

R is the set of rules in D

H is the Herbrandt base of D, i.e., the set of distinctpropositional atoms

T is the set of distinct instant of time explicitly occurring inD.

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Running out of time: Deadlines

Many kinds of deadlines; di↵erent functions

Research ProblemHow to represent deadlines (in contracts)?What happens after the deadline?Characterise types of deadlines

Approach:Identify key parameters; template formulasTemporalised Defeasible Deontic Logic

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Modelling Intervals

interval (set of instants) [ts , te ]

) A[ts , te ] shorthand for

)p A : ts

;t ¬A : te

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Basic Deadline

+ Sanction

Basic Deadline

+ Sanction

Customers must pay within 30 days after receiving the invoice.

¬payinvoice

viol(inv)

OBLfine

t1 t1+31OBLpay

invinit invoice : t1 )OBL pay : [t1,max]invterm OBLpay : t2,pay : t2 ;OBL ¬pay : t2+1invviol invoice : t1,OBLpay : t1+31) viol(inv) : t1+31

invsanc viol(inv) : t )OBL fine : [t,max]

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Basic Deadline + Sanction

Basic Deadline + Sanction

Customers must pay within 30 days after receiving the invoice.Otherwise, a fine must be paid.

¬payinvoice

viol(inv)OBLfine

t1 t1+31OBLpay

invinit invoice : t1 )OBL pay : [t1,max]invterm OBLpay : t2,pay : t2 ;OBL ¬pay : t2+1invviol invoice : t1,OBLpay : t1+31) viol(inv) : t1+31invsanc viol(inv) : t )OBL fine : [t,max]

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Maintenance

Maintenance Deadlines

Customers must keep a positive balance, for 30 days after openingan bank account.

¬positiveopenAccount

vio(pos)

t1 t1+30

t2

OBLpositive

posinit openAccount : t1 )OBL positive : [t1, t1+30]postermposviol ¬positive : t2,OBLpositive : t2 ) viol(pos) : t2

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Persistent

Persistent Obligation after Deadline

Customers must pay within 30 days after receiving the invoice.

¬payinvoice

vio(pos)

t1 t1+31OBLpay

invinit invoice : t1 )OBL pay : [t1,max]invterm OBLpay : t2,pay : t2 ;OBL ¬pay : t2+1invviol invoice : t1,OBLpay : t1+31) viol(inv) : t1+31

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Non-persistent

Non-persistent Obligation after Deadline

A wedding cake must be delivered, before the wedding party.

¬cakeorder

wedding

viol(wed)

OBLcake

t1 t2

wedinit order : t1,wedding : t2 )OBL cake : [t1,t2 ]wedterm OBLcake : t3,cake : t3 ;OBL ¬cake : t3+1wedviol wedding : t2,OBLcake : t2 ) viol(wed) : t2

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References for Defeasible Logic

Grigoris Antoniou, David Billington, Guido Governatori, and Michael J.Maher.Representation results for defeasible logic.ACM Transactions on Computational Logic, 2(2):255–287, April 2001.

Grigoris Antoniou, David Billington, Guido Governatori, and Michael J.Maher.Embedding defeasible logic into logic programming.Theory and Practice of Logic Programming, 6(6):703–735, November2006.

David Billington, Grigoris Antoniou, Guido Governatori, and Michael J.Maher.An inclusion theorem for defeasible logic.ACM Transactions in Computational Logic, 12(1):article 6.

Ho-Pun Lam and Guido Governatori.The making of SPINdle.In Guido Governatori, John Hall, and Adrian Paschke, editors, RuleML

2009, pages 315–322, Springer, 2009.65/67

References for Modal Defeasible Logic

Guido Governatori and Antonino Rotolo.BIO logical agents: Norms, beliefs, intentions in defeasible logic.Journal of Autonomous Agents and Multi Agent Systems, 17(1):36–69,2008.

Duy Hoang Pham, Guido Governatori, Simon Raboczi, Andrew Newman,and Subhasis Thakur.On extending RuleML for modal defeasible logic.In Nick Bassiliades, Guido Governatori, and Adrian Paschke, editors,RuleML 2008, pages 89–103, Springer, 2008.

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References for Temporal Defeasible Logic

Guido Governatori and Antonino Rotolo.Changing legal systems: legal abrogations and annulments in defeasiblelogic.Logic Journal of IGPL, 18(1):157–194, 2010.

Guido Governatori, Antonino Rotolo, and Giovanni Sartor.Temporalised normative positions in defeasible logic.In Anne Gardner, editor, 10th International Conference on Artificial

Intelligence and Law (ICAIL05), pages 25–34. ACM Press, June 6–112005.

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