cerutti--knowledge representation and reasoning (postgrad seminar @ university of brescia)

University of BresciaDepartment of Information Engineering

Knowledge Engineering and Human-Computer Interaction Research Group

Knowledge Representation andReasoning: an (extremely short)

Overview

Federico Cerutti

Gestione dei Sistemi Informativi AziendaliFriday 1st June, 2012

c© 2012 Federico Cerutti <[email protected]>

The Basic Concepts

Knowledge: some information about the world:

medical information about some particular set of diseases: whatcauses them, how to diagnose them;geographical data: which city is the capital of which country,population statistics, . . . ;common sense physics: bodies cannot go through solid walls, . . .

Representation: how/in which language do we represent thisinformation;

Reasoning: how to extract more information from what isexplicitly represented (because we cannot represent every singlefact explicitly as in a database).

c© 2012 Federico Cerutti <[email protected]> GSIA :: Friday 1st June, 2012 2

The Basic Concepts

We want to be able to talk about some AI programs in terms ofwhat they �know�;

. . . and not just talk about what they know but also havesomething to point to in those systems corresponding to�knowledge� and determining their behaviour, namely explicitlyrepresented symbolic knowledge.


Why Knowledge Representation and Reasoning isSO Important

Expert Systems:

MYCYN (1970s, Stanford University)XCON (1978, Carnegie Mellon University)

Ontologies:

CYC (1980s-today Cycorp, Austin, Texas)WordNetSemantic Web applications

Reasoning about uncertainty:

GoogleAmazonFacebook (ads)


KRR in Classical LogicKRR in Description Logics

KRR in Non-Monotonic LogicsConclusions


Classical Logic: the First Order Logic

A signature is a set of symbols of two kinds: function constantsand predicate constants with a non-negative integer called thearity assigned to each symbol: function constants of arity 0 arecalled object constants, while predicate constants of arity 0 arecalled propositional constants;

Object variables are elements of some �xed in�nite sequence ofsymbols (e.g. x, y, z, x1, x2, . . .);

Terms of a signature σ are formed from object variables and fromfunction constants of σ;

An atomic formula of σ is an expression of the form P (t1, . . . , tn)or t1 = t2 where P is a predicate constant of arity n and each ti isa term of σ;

Formulas are formed from atomic formulas using propositionalconnectives (>,⊥,¬,∧ or &,∨,→ or ⊃,↔ or ≡) and thequanti�ers ∃, ∀.


Variable Scope

Like variables in programming languages, the variables in FOL have ascope determined by the quanti�ers.

P (x) ∧ ∃y[P (y) ∨Q(y)]

x is a free variable, y is a bound variable

A closed formula, or a sentence, is a formula without freevariables;

The universal closure of a formula F is the sentence ∀v1, . . . , vnF ,where v1, . . . , vn are the free variables of F ;

The result of the substitution of a term t for a variable v in aformula F (or F [v/t]) is the formula obtained from F bysimultaneously replacing each free occurrence of v by t.


Semantics

An interpretation or structure of a signature σ consists of:

a nonempty set |I| called the universe (or domain) of I;

for every object constant c of σ, an element cI of |I|;for every function constant f of σ of arity n > 0, a function f I

from |I|n to |I|;for every propositional constant P of σ, an element P I of{FALSE,TRUE};for every predicate constant R of σ of arity n > 0, a function RI

from |I|n to |I|.


Semantics

For any element ξ of its universe |I|, select a new symbol ξ∗ called the name

of ξ. By σI we denote the signature obtained from σ adding all names ξ∗ asobject constants. The interpretation I can be extended to the new signatureσI by de�ning (ξ∗)I = ξ for all ξ ∈ |I|.

If t is an object constant, then tI is part of the interpretation I;

For all function constants f of arity n > 0,f(t1, . . . , tn)I = f I(tI1, . . . , t

In);

For any propositional constant P , P I is part of the interpretation I,otherwise we de�ne:

R(t1, . . . , tn) = RI(tI1, . . . , tIn),

⊥I = FALSE,>I = TRUE,(¬F )I = ¬(F I),(F �G)I = �(F I , GI) for every binary connective �,∀wF (w)I = TRUE if F (ξ∗)I = TRUE for all ξ ∈ |I|,∃wF (w)I = TRUE if F (ξ∗)I = TRUE for some ξ ∈ |I|.


Semantics

An interpretation I satis�es a sentence F , or is a model of F(I � F ) if F I = TRUE;

A sentence F is logically valid if every interpretation satis�es F ;

Two sentences are equivalent to each other if they are satis�ed bythe same interpretations;

A formula with free variables is said to be logically valid if itsuniversal closure is logically valid;

Formulas F and G that may contain free variables are equivalentto each other if F ↔ F is logically valid;

A set Γ of sentences is satis�able if there exists an interpretationsatisfying all sentences in Γ;

A set Γ of sentences entails a formula F (F � F ) if everyinterpretation satisfying Γ satis�es the universal closure of F .


Knowledge Representation and Reasoning withFOL

Example

Tony, Mike and John belong to the Alpine Club. Every member of theAlpine Club who is not a skier is a mountain climber. Mountain climbers donot like rain, and anyone who does not like snow is not a skier. Mike dislikeswhatever Tony likes, and likes whatever Tony dislikes. Tony likes rain andsnow.Prove that the given sentences logically entail that there is a member ofAlpine Club who is a mountain climber but not a skier.

KB = {member(tony), member(cmike), member(john),∀ x (member(x) ∧ ¬skier(x))→ climber(x),

∀ x climber(x)→ ¬like(x, rain),∀ x ¬like(x, snow)→ −skier(x),∀ x like(tony, x)→ ¬like(mike, x),∀ x ¬like(tony, x)→ like(mike, x),like(tony, rain), like(tony, snow)}



To prove if KB � ∃ x member(x) ∧ climber(x) ∧ ¬skier(x), let us considerthe Prover9 tool [McCune, 2010] fromhttp://www.cs.unm.edu/~mccune/prover9/.


http://www.cs.unm.edu/~mccune/prover9/

Knowledge Representation and Reasoning withFOLReduction ad absurdum.


KRR in Classical Logic

KRR in Description LogicsKRR in Non-Monotonic Logics

Conclusions


Description Logics

Family of logic-based knowledge representation formalisms well-suitedfor the representation of, and reasoning about:

terminological knowledge;

con�gurations;

ontologies;

database schemata:

schema design, evolution, and query optimisationsource integration in heterogeneous databases/data warehousesconceptual modelling of multidimensional aggregation. . .

. . .


DL Syntax

A description logic is mainly characterised by a set of constructorsthat allow to build complex:

concepts correspond to classes (are interpreted as set of objects);

roles correspond to relations (are interpreted as binary relationson objects).

The DL ALC (Attributive concept Language with Complements):

NC set of concept names, NR set of role names;

>,⊥, and every concept name A ∈ NC is an ALC-conceptdescription;

if C and D are ALC-concept descriptions, and r ∈ NR, thenC uD,C tD,¬C,∀r.C, ∃r.C are ALC-concept descriptions.


DL Syntax

Example (from [Horrocks and Sattler, 2002]).

Man u (∃has-child.Blue) u (∃has-child.Green) u(∀has-child.Happy tRich)


DL Semantics

Semantics given by means of an interpretation I = (∆I , ·I), with∆I 6= ∅ the domain of I, and ·I that maps every ALC-concept to asubset of ∆I , and every role name to a subset of ∆I ×∆I s.t.

>I = ∆I , ⊥I = ∅;(C uD)I = CI ∩DI ;

(C tD)I = CI ∪DI ;

(¬C)I = ∆I \ CI ;

(∃r.C)I = {x ∈ ∆I |∃y ∈ ∆I , 〈x, y〉 ∈ rI ∧ y ∈ CI};(∀r.C)I = {x ∈ ∆I |∀y ∈ ∆I , 〈x, y〉 ∈ rI → y ∈ CI};


DL Knowledge Bases: TBoxes

The TBox contains (Terminological Knowledge):

Concept de�nitions, A.= C, where A is a concept name and C is

a ALC-concept (e.g. Father .= Man u ∃has-child.Human);

Axioms, C1 v C2, where Ci are ALC-concepts(∃favourite.Brewery v ∃drinks.Beer).

An interpretation I satis�es:

a concept de�nition A.= C i� AI = CI ;

an axiom C1 v C2 i� CI1 ⊆ CI

2 ;

a TBox T i� I satis�es all de�nitions and axioms in T . In thiscase, I is a model of T (I � T ).


DL Knowledge Bases: ABoxes

The ABox contains (Assertional knowledge):

Concept assertions a : C, where a is an individual name, C aALC-concept (John : Man u ∀has-child.(Male uHappy));

Role assertions 〈a1, a2〉 : r, where ai are individual names, r is arole (〈John,Bill〉 : has-child)

An interpretation I satis�es:

a concept assertion a : C i� aI ∈ CI ;

a role assertion 〈a1, a2〉 : r i� 〈aI1 , aI2 〉 ∈ rI ;an ABox A i� I satis�es all assertions in A. In this case, I is amodel of A (I � A).


DL Entailment

Entailment in DL is de�ned as in FOL.A DL KB entails a concept c (KB � c) i� for every I, if I � KB,then I � c.


Semantic Web

I have a dream for the Web [in which computers] becomecapable of analyzing all the data on the Web � the content,links, and transactions between people and computers. A'Semantic Web', which should make this possible, has yet toemerge, but when it does, the day-to-day mechanisms oftrade, bureaucracy and our daily lives will be handled bymachines talking to machines. The 'intelligent agents' peoplehave touted for ages will �nally materialize.[Berners-Lee and Fischetti, 2000]


Resource Description Framework (RDF): anexample

Title Artist Country Company Price YearEmpire Burlesque Bob Dylan USA Columbia 10.90 1985Hide your heart Bonnie Tyler UK CBS Records 9.90 1988

<rd f :D e s c r i p t i o nrd f : about=" ht tp : //www. recshop . fake /cd/Empire Burlesque "><c d : a r t i s t>Bob Dylan</ c d : a r t i s t><cd:country>USA</ cd :country><cd:company>Columbia</cd:company><cd : p r i c e>10 .90</ cd : p r i c e><cd :yea r>1985</ cd :yea r>

</ rd f :D e s c r i p t i o n>

<rd f :D e s c r i p t i o nrd f : about=" ht tp : //www. recshop . fake /cd/Hide your heart "><c d : a r t i s t>Bonnie Tyler</ c d : a r t i s t><cd:country>UK</ cd:country><cd:company>CBS Records</cd:company><cd : p r i c e>9 .90</ cd : p r i c e><cd :yea r>1988</ cd :yea r>

</ rd f :D e s c r i p t i o n>


Applications May Want More

Complex applications may want more possibilities:

characterisation of properties

identi�cation of objects with di�erent URIs

disjointness or equivalence of classes

construct classes, not only name them

more complex classi�cation schemes

reason about some terms, e.g. if �Person� resources �A� and �B�have the same �foaf:email� property, then �A� and �B� areidentical. . .


OWL (Lite or DL)

A semantic web ontology language developed by the W3C;Design goal: mapping from OWL to an expressive DL (exploitDL results);An OWL ontology can be seen to correspond to a DL TBoxtogether with a role hierarchy, describing the domain in terms ofclasses (corresponding to concepts) and properties (correspondingto roles).

Human uMale

<owl :C la s s><ow l : i n t e r s e c t i o nO f rd f :parseType=" Co l l e c t i on ">

<owl :C la s s rd f : abou t="#Human"/><owl :C la s s rd f : abou t="#Male"/>

</ ow l : i n t e r s e c t i o nO f></ owl :C la s s>


OWL Constructors and Axioms (partial)

Constructor DL syntax

instersectionOf C1 u · · · u Cn

unionOF C1 t · · · t Cn

complementOf ¬CallValuesFrom ∀P.CsomeValuesFrom ∃P.C

Axiom DL syntax

subClassOf C1 v C2

equivalentClass C1.= C2

disjointWith C1 v ¬C2


Protege Ontology Editor

People ontology fromhttp://owl.man.ac.uk/2005/07/sssw/people.owl

An old lady is an eldery, female person, who has some animalsbut only cats.

A cat is an animal.

Minnie is an eldery female, and she has a pet, Tom.

Who has a pet, is a person.

Since Minnie has a pet, she is a person.

Since Minnie is an eldery, female person, she is a old lady.

Since Minnie is an old lady, and she has a pet, Tom, and since oldladies have only cats, Tom is a cat.


http://owl.man.ac.uk/2005/07/sssw/people.owl

DL and Other Formalisms

Most DLs are decidable fragments of FOL (ALC-concepts can bemapped into �rst order formulae);

DLs far more expressive than ALC:number restrictions: �people having at most 2 cats and exactly 1dog�complex roles:

inverse (�has-child�, �child-of �);transitive closure (�offspring�, �has-child�);role inclusion (�has-daughter�, �has-child�);. . .


A More Expressive Logic: the Second-Order Logic

Richer syntax than FOL because, along with object variables, weassume now an in�nite sequence of function variables of arity n > 0,and an in�nite sequence of predicate variables of arity n ≥ 0. Objectvariables are viewed as function variables of arity 0.

∀α, β ∃γ ∀x (γ(x) = α(β(x)))Sentence expressing the possibility of composing any two functions

∀x, y (Q(x, y)↔ ∀q (F (q)→ q(x, y)))where F (q) stands for

∀x1, y1 (P (x1, y1)→ q(x1, y1))∧∀x1, y1, z1 ((q(x1, y1) ∧ q(y1, z1))→ q(x1, z1))

Q is the intersection of all transitive relations containing P


Cyc

The Cyc Knowledge Server is a very large knowledge base andinference engine

Developed by Cycorp http://www.cyc.com/

It aims to provide a deep layer of �common sense knowledge� tobe used by other knowledge-intensive programs

Contains terms and assertions in formal language CycL, basedsecond order logic

Knowledge base contains classi�cation of things, facts, rules ofthumb, heuristics for reasoning about everyday objects.


http://www.cyc.com/

Cyc Example of Opposite Relationships

c r e a t e constant k1c r e a t e constant k2(#$ i s a #$k1 #$Sca l a r I n t e r v a l )(#$ i s a #$k2 #$Sca l a r I n t e r v a l )(#$muchLessThan #$k1 #$k2 )

Query : (#$muchGreaterThan ?X ?Y)



KRR in Non-MonotonicLogicsConclusions


Non Monotonic Logics

Classical logic is monotonic: whenever a sentence A is a logicalconsequence of a set of sentences T (T � A), then A is also aconsequence of an arbitrary superset of T ;

Commonsense reasoning is di�erent: we often draw plausibleconclusions based on the assumption that the world is normaland as expected;

This is farm from being irrational: it is the best we can do insituations in which we have only incomplete information;

It can happen that our normality assumptions turn out to bewrong: in this case we may have to revise our conclusions.


The Tweety Example

Birds �ies, in Prolog: assert((flies(X) :- bird(X))).

Tweety is a bird. . . (assert(bird(tweety)).)

. . . then Tweety �ies.

Tux is a penguin (penguin(tux).)

Every penguin is a bird (bird(X) :- penguin(X).)

Then Tux �ies (?!?!)


The Tweety Example Revisited

Birds �ies unless they are �abnormal�(flies(X) :- bird(X), \+ abnormalbird(X).)\+ means negation as failure, equivalent to:not(P) :- call(P), !, fail. if P, then not(P) failsnot(P). else not(P) holds

We know that penguin are �abnormal� birds(abnormalbird(X) :- penguin(X). andbird(X) :- penguin(X).)

Tweety is a bird (we do not know if normal or abnormal). . . (bird(tweety).)

Tux is a penguin. . . (penguin(tux).)

. . . Tweety �ies, Tux does not.

What would happen if we know that also Tweety is a penguin?(penguin(tweety).)

Tweety does not �y.


A Logic for Default Reasoning (sketched)[Reiter, 1980]

A default theory is a pair (D,W ) where W is a set of sentences in�rst order logic, and D is a set of defaults.

A default is an expression:

A : B1, . . . , Bn

C

Recalling Tweety example:

Bird(x) : MFly(x)

Fly(x)MFly ' �it is consistent to assume that �ies�

∀x, Penguin(x)→ ¬Fly(x)


Answer Set Programming

Answer Set Programming is a recent problem solving approach;

It has roots in KR, logic programming, and nonmonotonicreasoning;

The idea: stop trying to prove something, represent solutions, ormodels (Answer Sets)!

Normal logic program P is a �nite set of rules of the form:

a← b1, . . . , bm, not c1, . . . , not cn

where a, bi, cj are literals of the form p or ¬p (strong negation,also written as �-�) where p is a �rst-order atom from a classicalFOL signature.

An answer set is a set of ground atoms that are �collectivelyacceptable�


Answer Set Programming: the Tweety Example

f l i e s (X) :− bi rd (X) , not abnormal (X) .abnormal (X) :− penguin (X) .b i rd (X) :− penguin (X) .b i rd ( tweety ) .penguin ( tux ) .

Resulting Answer Sets:

{penguin ( tux ) , f l i e s ( tweety ) , b i rd ( tweety ) ,b i rd ( tux ) , abnormal ( tux )}


Answer Set Programming: the Nixon Diamond

Usually, Quakers are paci�st

Usually, Republicans are not paci�st

Richard Nixon is both a Quaker and a Republican

quaker ( nixon ) .r epub l i can ( nixon ) .p a c i f i s t (X) :− quaker (X) , not −p a c i f i s t (X) .−p a c i f i s t (X) :− r epub l i can (X) , not p a c i f i s t (X) .

Resulting Answer Sets:

{ quaker ( nixon ) , r epub l i can ( nixon ) , p a c i f i s t ( nixon )}{quaker ( nixon ) , r epub l i can ( nixon ) , −p a c i f i s t ( nixon )}


Argumentation: an Informal Example (Courtesyof Prof. Massimiliano Giacomin)

The reason

The conclusion

We are justified in believing that we should run LHC

We should run Large Hadron Collider

LHC allows us to understand the Laws

of the Universe

Understandingthe Laws of the Universe is good



The reason

The conclusion

We are justified in believing that we should run LHC



of the Universe


In Argumentation (and in real life as well):

- reasons are not necessary “conclusive”

(they don’t logically entail conclusions)

- arguments and conclusions can be “retracted”

in front of new information, i.e. counterarguments

BUT





of the Universe


We should not run LHC

LHC will generate black holes

destroying Earth

Destroying Earth is bad

Now we are justified in believing that we should not run LHC





of the Universe




destroying Earth


Black holes will not destroy Earth

Black holes will evaporate because

of Hawking radiation

Now we are again justified in believing that we should run LHC





of the Universe




destroying Earth





Hawking radiationdoes not exist

Dr Azzeccagarbuglisays so

Now we are again justified in believing that we should not run LHC





of the Universe




destroying Earth





Hawking radiationdoes not exist

Dr Azzeccagarbuglisays so

Dr Azzeccagarbugliis not expert in physics

He is a lawyer

Now we are again justified in believing that we should

run LHC


What is Argumentation?

[Prakken, 2011] Argumentation is the process of supportingclaims with grounds and defending them against attack.

[van Eemeren et al., 1996] Argumentation is a verbal and socialactivity of reason aimed at increasing (or decreasing) theacceptability of a controversial standpoint for the listener orreader, by putting forward a constellation of propositionsintended to justify (or refute) the standpoint before a rationaljudge.

A framework for practical and uncertain reasoning able to copewith partial and inconsistent knowledge.


The Elements of an Argumentation System[Prakken and Vreeswijk, 2001]

1 The de�nition of an argument (possibly including an underlyinglogical language + a notion of logical consequence)

2 The notion of attack and defeat (successful attack) betweenarguments;

3 An argumentation semantics selecting acceptable (justi�ed)arguments


Arguments and Attacks: Assumption-BasedArgumentation [Bondarenko et al., 1993]

An Assumption-Based Argumentation (ABA) system is a tuple〈L,R,A, ·〉 s.t.

L is a set of sentences;R is a set of rules of the form s1 ← s2, . . . sn where each si is asentence;A ⊆ L is a set of candidate assumptions, and each assumptioncannot be the head of any rule;a is the contrary of assumption a;

An argument is a deduction supported by a set of assumptions;

An argument A attacks another argument A′ if the conclusion ofA is the contrary of one of the assumptions supporting A′.


Arguments and Attacks: ABA Example (from[Gaertner and Toni, 2008])

Assumptions: {all_likes(adrian),mom_hates(adrian)}Rules:{acceptable(adrian)←all_like(adrian), easy_to_remember(adrian)easy_to_remember(adrian)← short(adrian)some_dislike(adrian)← mom_hates(adrian)some_dislike(adrian)← dad_hates(adrian)dad_hates(adrian)← too_common(adrian)dad_hates(adrian)← uncle_has(adrian)mom_not_hate(adrian)← mom_said_ok(adrian)mom_said_ok(adrian)short(adrian)}Contraries: all_like(adrian) = some_dislike(adrian),mom_hates(adrian) = mom_not_hate(adrian).


Arguments and Attacks: Argument Schemes[Walton, 1996]

An argument scheme is a reasoning pattern giving us thepresumption in favour of its conclusion.

A critical question is a question that can be posed by an opponentin order to undermine the validity of the stated argument.

There are several argument schemes in literature.

Expert testimony

Premise 1: E is expert on DPremise 2: E says PPremise 3: P is in DConclusion: P is the case

Critical questions:

1 Is E biased?

2 Is P consistent with what other experts say?

3 Is P consistent with known evidence?c© 2012 Federico Cerutti <[email protected]> GSIA :: Friday 1st June, 2012 39

Argumentation Semantics: AbstractArgumentation [Dung, 1995]

1 The de�nition of an argument (possibly including an underlyinglogical language + a notion of logical consequence)

2 The notion of attack and defeat (successful attack) betweenarguments;

3 An argumentation semantics selecting acceptable (justi�ed)arguments

Abstract argumentation focuses on the third aspect.

An abstract argumentation framework AF is a tuple 〈A,R〉, where Ais a set of argument (whose origin and structure is not speci�ed), andR ⊆ A×A is a set of attack (or defeat) relations.

Argument evaluation: given an argumentation framework, determinethe justi�cation state (defeat status) of arguments. In particular, whatargument emerge undefeated from the con�ict, i.e. are acceptable?


Nixon Diamond

AFN = 〈AN , RN 〉, where AN = {A1, A2}, RN = {〈A1, A2〉, 〈A2, A1〉},and

A1: since Nixon is a quaker, then he is also a paci�st;

A2: since Nixon is a republican, he is not a paci�st.


Argumentation Semantics (Courtesy of Prof.Massimiliano Giacomin)

• Specification of a method for argument evaluation, or of

criteria to determine, given a set of arguments, their “defeat status”

Argumentation Framework

Semantics

Defeat status

Defeat status

Undefeated

Defeated

Provisionally Defeated


Extension-based Semantics (Courtesy of Prof.Massimiliano Giacomin)

Set of extensions ℰS(AF) Argumentation framework AF

Semantics S

Defeat/Justification Status

Skeptically justi�ed argument: belongs to all the extensions;Credulously justi�ed argument: belongs to at least one;Indefensible argument: does not belong to any extension.


Complete Semantics (Courtesy of Prof.Massimiliano Giacomin)

Acceptability

α acceptable w.r.t. (“defended by”) S

• all attackers of α are attacked by S

Admissible set S

• conflict-free

• every element acceptable w.r.t. S

(defends all of its elements)

α

S

IF

also includes allacceptable elementsw.r.t. itself

Completeextension

Complete semantics

All traditional semanticsselect complete extensions


Complete Semantics Examples (Courtesy of Prof.Massimiliano Giacomin)

α β γ

ChainAdmissible sets:

ø, {α}, {α, γ}

Only one complete extension:

ℰCO(AF) = {{α, γ}}

Nixon Diamond

βα

βα

βα

βα

All admissible sets

are complete

ℰCO(AF) =

{ ø, {α}, {β} }


Complete Semantics Examples (Courtesy of Prof.Massimiliano Giacomin)

Nixon Diamond + node

βα

βα

βα

Admissible sets:

ø, {α}, {β}, {α, γ}

ℰCO(AF) = {

ø

{α, γ},

{β} }

βα γ

ℰCO(AF)

γ

γ

γ


Grounded Semantics (Courtesy of Prof.Massimiliano Giacomin)

Undefeated

Defeated

Provisionally Defeated

Grounded extension GE(AF):

Least complete extension

Defeat status

included in all extensions

of any traditional semantics

Grounded semantics is

the “most skeptical” one


Grounded Semantics Examples (Courtesy of Prof.Massimiliano Giacomin)

α β γ

Chain

GE(AF) = {α, γ}

Nixon Diamond

βα GE(AF) = ø

Nixon Diamond + node

βα γ GE(AF) = ø


Grounded Semantics Problem (Courtesy of Prof.Massimiliano Giacomin)

β

α

γ δ

β

α

γ δ VS

What we (may) wantGrounded Semantics

• Actually, grounded semantics is polynomially computable

• But sometimes a more discriminative behavior is desirable

THE CASE OF FLOATING ARGUMENTS

• A problem for all possible unique status approaches

Let us consider multiple status approaches!


Preferred Semantics (Courtesy of Prof.Massimiliano Giacomin)

Preferred semantics

Preferred extension

Maximal complete extension = max Set:

• is conflict-free

• defends all of its elements

[P.M. Dung, ’95]

Stable extensions are maximal complete extensions

• conflict-free: by definition• admissible: every argument attacking an extension is outside

⇒ attacked by the extension itself• maximal: no argument can be included!


Preferred Semantics Examples (Courtesy of Prof.Massimiliano Giacomin)

β

α

γ δ

β

α

γ δ

β

α

γ δ

β

α

γ δ

β

α

γ δ

Grounded semantics:

ℰPR(AF) = ℰST(AF) = { {α, δ}, {β, δ} } ⇒ δ is justified



KRR in Non-Monotonic Logics

Conclusions


Conclusions

Knowledge: some information about the world;

Representation: how to represent this information:

Classical Logic;Description Logics;Nonmonotonic Logics;

Reasoning: how to extract more information from what isexplicitly represented:

logic inference;representing solutions (or models).


What is Not Covered in This Presentation(among the others)

SAT Solver;

CSP;

Conceptual Graphs;

Autoepistemic Reasoning;

Belief revision;

Modal logic;

Deontic logic;

Temporal reasonnig;

Spatial reasoning;

Physical reasoning;

Event calculus;

Temporal action logic;

Multi-agent systems;

Bayesian Networks;

Neural Networks;

Markovian Chains;

. . .


cerutti--knowledge representation and reasoning (postgrad seminar @ university of brescia)

Technology