Practical Non-monotonic Reasoning

Practical Non-monotonic Reasoning Guido Governatori

Knowledge Techniques Week 2012 NICTA Members

NICTA Partners

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 intelligently Focus on hard problems, often with no, or very inefficient full algorithmic solution Focus on problems that require “reasoning” (“intelligence”) and a large amount of knowledge about the world

Critical Represent knowledge about the world Reason with these representations to obtain meaningful answers/solutions

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Symbolic Knowledge Representation

Important objects (collections of objects) and their relationships are represented explicitly by internal symbols Symbolic manipulation of internal symbolic representations achieves 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 to the problem at hand Close to problem at hand: compact, natural, maintainable Amenable to efficient 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 start working on his project

Can all be represented as a string! But hard then to manipulate and draw conclusions How do we represent these formally in a way that can be manipulated in a computer program?

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Well-defined Syntax and Semantics

Precise syntax: what can be expressed in the language Formal language, unlike natural language Prerequisite 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 and natural 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 can use 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 the represented knowledge.

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

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Inferential Efficiency

You may be able, in principle, to make complex deductions, but it may be just too inefficient. The basic tradeo↵ of all KR Generally the more complex the possible deductions, the less efficient will be the reasoning process (in the worst case). The eternal quest of KR Need representation and inference system sufficient for the task, without being hopelessly inefficient

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Inferential Adequacy (2)

Representing everything as natural language strings has good representational adequacy and naturalness, but very poor inferential adequacy

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Requirements for KR Languages

Representational Adequacy Clear syntax/semantics Inferential adequacy Inferential efficiency Naturalness In practice no one language is perfect, and di↵erent languages are suitable 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 to medication 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↵ the truth of P is implicit in the truth of P1 . . . , Pn Or: if the world satisfies P1 . . . , Pn then it must also satisfy P Reasoning usually done by humans Syntactic Reasoning Sentences P1 . . . , Pn infer sentence P i↵ there is a syntactic manipulation of P1 . . . , Pn that results in P Reasoning 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 semantically Only semantically valid conclusions are drawn

Complete (syntactic) reasoning If P is entailed semantically by P1 . . . , Pn then it can also be inferred All semantically valid conclusions can be drawn

Usually interested in sound and complete reasoning But sometimes we have to give up one for the sake of efficiency (usually completeness)

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Main KR Approaches

Logic Based Focus on clean, mathematical semantics: declaratively Explainability

Frames / Semantic Networks / Objects Focus on structure of objects

Rule-based systems Focus on efficiency A ) B in logic and rule-based systems

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The Landscape of KR

Predicate logic (first order logic) and its sublanguages Logic programming, (pure) Prolog Description logics Web ontology languages

Predicate logic (first order logic) extensions Modal and epistemic logics Temporal logics Spatial logics

Inconsistency-tolerant logics: Paraconsistency Nonmonotonic reasoning

Representing vagueness Probabilistic logics Bayesian networks Markov chains 16/67

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 ! ¬Uni

PublicHoliday ! ¬Uni Sick ! ¬Uni

Weekend ^ VICdeadline ! Uni

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

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Inconsistent Information

Classical logics “collapse” in the face of inconsistencies Everything can be derived But inconsistencies do happen in real settings Common when integrating knowledge from various Web sources 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 to resolve conflicts among rules. Potential applications Normative reasoning Security policies Business rules Personalization Brokering Bargaining, 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 suggesting that the defendant is responsible. Given: EvidenceA and the rule EvidenceA ) Responsible Sceptical: Responsible Credulous: Responsible

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Conflict

Suppose that your legal system is based on presumption of innocence, and the somebody is guilty if responsibility is proved. Given the rules r1 : Responsible ) Guilty

r2 :

) ¬Guilty

Sceptical: ¬Guilty Credulous: ¬Guilty What about if we have r1 > r2 (same conclusions)

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Sceptical vs Credulous

Suppose you have two pieces of evidence. Evidence A suggesting that the defendant is responsible, and Evidence A suggesting that the defendant is not responsible. Given: EvidenceA, and EvidenceB and the rules EvidenceA ) Responsible

EvidenceB ) ¬Responsible Sceptical: no conclusions Credulous: both conclusions

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Sceptical vs Credulous and preference Suppose you have two pieces of evidence. Evidence A suggesting that the defendant is responsible, and Evidence A suggesting that the defendant is not responsible. However, Evidence A is more reliable than Evidence B. Given: EvidenceA, and EvidenceB and the rules r1 : EvidenceA ) Responsible

r2 : EvidenceB ) ¬Responsible

r1 > r2

Sceptical: Responsible Credulous: Responsible

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Ambiguity Propagation vs Ambiguity Blocking Suppose you have two pieces of evidence. Evidence A suggesting that the defendant is responsible, and Evidence A suggesting that the 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 ) ¬Responsible Responsible ) Guilty ) ¬Guilty

Ambiguity blocking concludes ¬Guilty Ambiguity propagation does not concludes ¬Guilty and fails to conclude Guilty . 25/67

Ambiguity Propagation vs Ambiguity Blocking Suppose you have two pieces of evidence. Evidence A suggesting that the defendant is responsible, and Evidence A suggesting that the defendant is not responsible. If the defendant is responsible, then he is guilty. and we have presupposition of innocence. If the defendant was wrongly accused then he is entitled to compensation. Given: EvidenceA, and EvidenceB and the rules EvidenceA ) Responsible

EvidenceB ) ¬Responsible Responsible ) Guilty ) ¬Guilty

¬Guilty ) Innocent

Innocent ) Compensation

Ambiguity blocking concludes Compensation Ambiguity propagation does not conclude Compensation 26/67

Team Defeat vs No Team Defeat

r1 :General ) Attack

r2 :Captain ) ¬Attack r1 > r2

r3 :Bishop ) Attack

r4 :Priest ) ¬Attack r3 > r4

Team Defeat concludes Attack No 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 the crime scene at the time of the crime. r1 :drunk ) ¬CrimeScene r2 :camera ) CrimeScene r2 > r1

r3 :¬CrimeScene ) Alibi

Do we have scintilla of evidence to claim that the accuse was at the crime scene at the time of the crime? Is it reasonable to say that we have substantial evidence supporting for the same claim? Is it reasonable to claim that beyond any reasonable doubts the accused has an alibi? 28/67

Why Defeasible Logic?

Rule-based non-monotonic formalism Flexible Efficient (linear complexity) Directly skeptic semantics Argumentation semantics Constructive proof theory Optimised/efficient implementations (1.700.000 rules) Extensible

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Defeasible Logic: Strength of Conclusions

Derive (plausible) conclusions with the minimum amount of information. Definite conclusions Defeasible conclusions

Defeasible Theory Facts Strict 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 literals satisfying four conditions +∂ p (-∂ p): p is (not) derivable using ambiguity blocking, team defeat +∂ ⇤ p: p is derivable using ambiguity blocking, no team defeat +d p: p is derivable using ambiguity propagation, team defeat +d ⇤ p p is derivable using ambiguity propagation, no team defeat +s p: p is a credulous conclusion using team defeat +s p: 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 it Rebut all counterarguments

3

Defeat the argument by a stronger one Undercut the argument by showing that some of the premises do not hold

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Example Facts: A1 , A2 , B1 , B2 Rules: r1 :A1 ) C r2 :A2 ) C r3 :B1 ) ¬C r4 :B2 ) ¬C r5 :B3 ) ¬C Superiority relation: r1 > r3 r2 > r4 r5 > r1

Phase 1: Argument for C A1 (Fact), r1 : A1 ) C Phase 2: Possible counterarguments r3 : B1 ) ¬C r4 : B2 ) ¬C r5 : B3 ) ¬C Phase 3: Rebut the counterarguments r3 weaker than r1 r4 weaker than r2 r5 is not applicable

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Derivations in Defeasible Logics: Ambiguity blocking +∂ p 1) 9 an applicable rule r pro p 2) 8 rule t con p either: 2.1) t is not applicable 2.2) t is defeated by an applicable rule s pro p stronger than t +∂ ⇤ p 1) 9 an applicable rule r pro p 2) 8 rule t con p either: 2.1) t is not applicable 2.2) 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 p 2) 8 rule t con p either: 2.1) t is not applicable 2.2) t is defeated by an applicable rule s pro p stronger than t +d p 1) 9 an applicable rule r pro p 2) 8 rule t con p either: 2.1) t is not applicable 2.2) t is defeated by a supported rule s pro p stronger than s

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Derivations in Defeasible Logics: Support

+s p 1) 9 a supported rule r pro p 2) 8 rule s con p either 2.1) s is not applicable using ambiguity propagation (i.e., 2.2) s is not stronger than r

d , d ⇤)

+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 both derived, unless they are already known as certain knowledge (facts)

Theorem Defeasible logic is coherent. +#a and from the same knowledge base.

#a cannot be derived

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 Logic 1 propositional logic 2 2(A ! B) ! (2A ! 2B) 3 ` A/ ` 2A or A ` B/2A ` 2B 4 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

2

The strength describes how strong is the relationships between the antecedent and the consequent of a rule. A1 , . . . , An ! B (B is an indisputable consequence of A1 , . . . , An ) A1 , . . . , An ) B (normally B if A1 , . . . , An )

The mode qualifies the conclusion of a rule.

A1 , . . . , An )BEL B (an agent forms the belief B when A1 , . . . , An are the case) A1 , . . . , An )INT B (an agent has the intention B when A1 , . . . , An are the case) A1 , . . . , An )OBL B (an agent has the obligation B when A1 , . . . , An are the case)

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Conclusions in Basic Modal Defeasible Logic

+ 2i q, which is intended to mean that q is definitely provable (i.e., using only facts and strict rules of mode 2i ); 2i q, which is intended to mean that we have proved that q is not definitely provable in D;

+∂2i q, which is intended to mean that q is defeasibly provable in D using rules of mode 2i ; ∂2i q which is intended to mean that we have proved that q is not defeasibly provable in D using rules of mode 2i . We obtain 2i p i↵ +∂2i p.

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Recipe for Modal Defeasible Logics

Choose the appropriate modalities Create a defeasible consequence relation for each modality Identify relationships between modalities: inclusion conflicts

21 f ! 22 f 21 f , 22 ¬f ! ?

conversions from one modality to another modality A1 , . . . , An )21 B 2 2 A 1 , . . . , 22 A n ` 2 2 B

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 for both 21 and 22 Rebut all counterarguments

3

Defeat the argument by a stronger one Undercut the argument by showing that some of the premises do 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 using rules for either 22 or rules of mode 21 st all premises are provable with mode 22

2

Consider all possible counterarguments to it Rebut all counterarguments

3

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

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DL for cognitive agents

D = (F , R BEL , R DES , R INT , R OBL , >) R BEL rules for belief !BEL , )BEL , ;BEL R DES rules for desire !DES , )DES , ;DES R INT rules for intention !INT , )INT , ;INT R OBL rules for obligation !OBL , )OBL , ;OBL For X 2 {INT, DES, OBL} D ` XA i↵ D ` +∂X A

D ` A i↵ D ` +∂BEL A

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Interactions

OBL a ! ¬INT¬a INTa ! ¬DES¬a (INTa ^ OBL(a ! b)) ! INTb (OBL a ^ INT(a ! b)) ! OBL b

social stable conversion 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 obligation and 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 , . . . , an A set of rules for intentions:

a1 , . . . , an )INT c

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

a1 , . . . , an )OBL c

The agent derives OBLc from a1 , . . . , an Belief rules are stronger than obligation rules which in turns are stronger than intention rules 50/67

From Beliefs to Intentions

If we want to model realistic agents, the model must conform with the real world. According to current legal theories: If an agent knows/beliefs that B is a consequence of A, and the agent intends A, then the agent intends B (unless she has some justifications 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 she intends 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 zoo of 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) ...

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Persistent and Transient Conclusions linear discrete time line with a fixed granularity propositions (literals) are associated with instants of time C : t is persistent at time t, if C continues to hold after t unless some event occurs to terminate it. C : t is transient at time t, if C is guaranteed to hold at time t only.

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 using persistent rules. Take a rule for the conclusion that is applicable now or Show the there is a time in the past where the persistent conclusion obtains.

2

Consider all possible counterarguments for the conclusion Take all rules for its negation that obtain now Take all rules for its negation that have obtained since the time in the past.

3

rebut the counterarguments show that the rules have been discarded (not applicable or defeated).

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Example

Facts: A : t0 , B : t2 , C : t2 , D : t3 Rules: r1 : A : t )p E : t r2 : B : t )p ¬E : t r3 : C : t ;p E : t r4 : D : t )t ¬E : t Superiority relation: r3 > r2 r1 > r4

Conclusions at time t0 A, E using r1 (E is persistent) Conclusions at time t1 E Conclusions at time t2 B, C , E Conclusions at time t3 D, ¬E

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Linear Time

Theorem The extension of a temporalised defeasible theory D can be computed 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 distinct propositional atoms T is the set of distinct instant of time explicitly occurring in D.

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Running out of time: Deadlines

Many kinds of deadlines; di↵erent functions Research Problem How to represent deadlines (in contracts)? What happens after the deadline? Characterise types of deadlines

Approach: Identify key parameters; template formulas Temporalised 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

Basic Deadline Customers must pay within 30 days after receiving the invoice. t1 invoice

OBL pay ¬pay

t1 + 31

viol(inv )

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

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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. t1 invoice invinit invterm invviol invsanc

OBL pay ¬pay

t1 + 31

viol(inv )

OBL fine

invoice : t1 )OBL pay : [t1 , max] OBL pay : t2 , pay : t2 ;OBL ¬pay : t2 + 1 invoice : t1 , OBL pay : t1 + 31 ) viol(inv ) : t1 + 31 viol(inv ) : t )OBL fine : [t, max] 60/67

Maintenance

Maintenance Deadlines Customers must keep a positive balance, for 30 days after opening an bank account. t1 openAccount

t2 vio(pos)

OBL positive ¬positive

t1 + 30

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

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Persistent

Persistent Obligation after Deadline Customers must pay within 30 days after receiving the invoice. t1 invoice

OBL pay ¬pay

t1 + 31

vio(pos)

invinit invoice : t1 )OBL pay : [t1 , max] invterm OBL pay : t2 , pay : t2 ;OBL ¬pay : t2 + 1 invviol invoice : t1 , OBL pay : 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. wedding

OBL cake order

t1

¬cake

t2

viol(wed)

wedinit order : t1 , wedding : t2 )OBL cake : [t1,t2 ] wedterm OBL cake : t3 , cake : t3 ;OBL ¬cake : t3 + 1 wedviol wedding : t2 , OBL cake : 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, November 2006. 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 defeasible logic. 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 6–11 2005.

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