Logics for Data and Knowledge Representation Modeling Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang

Logics for Data and KnowledgeRepresentation

Modeling

Originally by Alessandro Agostini and Fausto GiunchigliaModified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

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Outline Modeling Logical Modeling (formal modeling)

Domain Language Interpretation Theory Model

How to use logical modeling What is a logic? Choosing the right logic and writing the theory

Reasoning services Expressiveness

Expressiveness, Efficiency, Complexity Decidability

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3

Modeling

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World

LanguageL

TheoryT

DomainD

ModelM

DataKnowledge

Meaning

MentalModel

SEMANTICGAP

MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

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Modeling World: the phenomenon we want to describe Domain: the abstract relevant elements in the real world Mental Model: what we have in mind. It is the first

abstraction of the world (subject to the semantic gap) Language: the set of words and rules we use to build

sentences used to express our mental model Model: the formalization of the mental model, i.e. the set

of true facts in the language, in agreement with the theory

Theory: the set of sentences (constraints) about the world expressed in the language that limit the possible models

NOTE: this does not necessarily need to be in formal semantics

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Example of informal Modeling

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World

MentalModel

SEMANTICGAP

ModelM

L: Informal description in NL

D: {monkey, banana, tree}

T: If the monkey climbs on the tree, he can get the banana

M: The monkey actually climbs on the tree and gets the banana

TheoryT

NOTE: a database can be seen as an informal model

LanguageL

DomainD


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Logical Modeling

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Modeling

Realization

World

LanguageL

TheoryT

DomainD

ModelM

DataKnowledge

Meaning

MentalModel

SEMANTICGAP

Inte

rpre

tatio

n

I

En

tailm

en

t⊨

NOTE: the key point is that in logical modeling we have formal semantics


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Logical Modeling Elements Domain: relevant objects Logical Language: the set of formal words and rules we

use to build complex sentences Interpretation: the function that associates elements of

the language to the elements in the domain

Model: the set of true facts in the language describing the mental model, in agreement with the theory

Theory / Knowledge Base (KB): data and knowledge Truth-relation / logical entailment (⊨): deduction,

reasoning, inference

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Example of formal (intentional) modeling

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World

SEMANTICGAP

L = {Monkey, Climbs, GetBanana, , , }

D= {T, F}

T = { (Monkey Climbs) GetBanana}

A possible model M:I(Monkey) = TI(Climbs) = TI(GetBanana) = T

MentalModel

ModelM

TheoryT

LanguageL

DomainD


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Domain Domain (D): the chosen objects from the world

We will deal only with finite domains! Question: what are we leaving out?

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Example: the LDKR class The members of the LDKR class

define a domain D D is a finite set Two “kinds” of the elements in D:

Professor

Student

Both are specializations of Person

Fausto

Mary Paul Jane

HugoSaraRui


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Language Language (L) = a logical language To each logical language we associate an

alphabet of symbols Σ (sigma)

For instance, Σ may contain the logical symbols: ∧ (and)∨ (or)¬ (not)→ (implication)∀ (for all, universal quantifier)∃ (exists, existential quantifier)

L has clear formation rules for formulas

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Logical Language (Syntax) The first step in setting up a logical language is

to list the symbols the alphabet of (formal) symbols Σ formal symbol = a character, or group of characters

taken from some alphabet (it is formal because we specify the meaning)

Symbols in Σ can be divided in: descriptive (non-logical) non-descriptive (logical)

NOTE: English can be restricted to a propositional language, ...but it is not logical (informal syntax)

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Logical language

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The Language L for the LDKR class example with alphabet Σ

Logical symbols:

∧, ∨, ¬, →

Descriptive symbols:Person, Professor, Student

Fausto

Mary Paul Jane

HugoSaraRui


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Formal Syntax The set of rules saying how to construct the

sentences of the language from the alphabet of symbols (i.e. the syntax) is a grammar (i.e., is formal)

Example: ¬A, A ∧ B, A → B¬Professor, Professor ∧ Student, Student → Person

Formal syntax is often called an abstract syntax, in contrast to the concrete syntax used, for instance in implementations.

Example: context-free grammars 13


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Interpretation Interpretation (I) = a mapping of L into D

I must be effective (i.e., computable)

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Intensional interpretation

I(Professor) = T

I(Student) = T

I(Person) = T

Extensional interpretation

I(Professor) = {Fausto}

I(Student) = {Mary, Paul, Jane, Rui, Sara, Hugo}

I(Person) = {Fausto, Mary, Paul, Jane, Rui, Sara, Hugo}

Fausto

Mary Paul Jane

HugoSaraRui


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Theory Theory T (also L-Theory) = set of facts of L

A fact defines a piece of knowledge (about D), namely something true

A theory is a way to put constraints on the intended models

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Theory A finite theory T is called:

An ontology if it contains knowledge only (T-BOX) A knowledge base (KB) if it contains knowledge (T-

BOX) and data (A-BOX) A database (DB) if it only contains data. A DB + its schema is the simplest kind of knowledge

base

NOTE: Sometimes the terms Ontology and Knowledge Base

are used interchangeably

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Theory

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The set of (true) facts T over L

Student → Person

Professor → Person

Student → ¬ Professor

Professor → ¬ Student

Fausto

Mary Paul Jane

HugoSaraRui


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Model Model (M) = the abstract (in mathematical

sense) representation of the intended truths via interpretation I of the language L

M is called L-model of D: M ⊨ TM satisfies T

T holds in M T is TRUE in MM yields T

with T set of arbitrary complex formulas

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Model

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Intensional interpretation

I(Professor) = T

I(Student) = T

I(Person) = T

The I is a model for the theories below:

M ⊨ {Person}

M ⊨ {Professor ∨ Student}

M ⊨ {Person, Professor ∨ Student, Student → Person}

…

Fausto

Mary Paul Jane

HugoSaraRui


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Theory and Model We have:

M ⊨ T iff M ⊨ A, for each formula A in T

A model M of a theory T is any interpretation function that satisfies all the facts in T

There can be many models satisfying the theory T. They are a subset of all possible interpretation functions over L.

In case there are no models for T, we say that the theory T is unsatisfiable.

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Theory and Model

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T

Student → Person

Professor → Person

Student → ¬ Professor

Professor → ¬ Student

Fausto

Mary Paul Jane

HugoSaraRuiM




M is a model for T (M ⊨ T )

M’

I(Professor) = {Fausto, Mary}



M’ is not, because Mary is both a student and a professor


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Same theory, different models (MODEL#1) L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana,

, , }

T = { (MonkeyLow BananaHigh MonkeyGetBanana)

(MonkeyLow MonkeyClimbBox)

( MonkeyLow BananaHigh MonkeyGetBanana)}

Informal Semantics:

“If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “

Formal Semantics:

I(MonkeyLow) = T

I(BananaHigh) = T I(MonkeyClimbBox) = F

I(MonkeyGetBanana) = F

MODEL#1


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Same theory, different models (MODEL#2) L = {MonkeyLow, BananaHigh, MonkeyClimbBox,

MonkeyGetBanana, , , }




Informal Semantics:

“If the monkey climbs onto the box, he becomes high in position (not low anymore) and can get the banana.”

Formal Semantics:

I(MonkeyLow) = F

I(BananaHigh) = T I(MonkeyClimbBox)

= T

I(MonkeyGetBanana) = T

MODEL#2


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Same theory, different models (MODEL#3) L = {MonkeyLow, BananaHigh, MonkeyClimbBox,





Informal Semantics:

“If the monkey is low and the banana is also low, then the monkey can get the banana. “

Formal Semantics:

I(MonkeyLow) = T

I(BananaHigh) = F I(MonkeyClimbBox)

= F


MODEL#3


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How to use logical modeling

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1. Define a logic most often by reseachers once for all (not a trivial task!)

2. Choose the right logic for the problem Given a problem the computer scientist must choose

the right logic, most often one of the many available

3. Write the theory The computer scientist writes a theory T

4. Use reasoning services The computer scientist uses reasoning services to

solve her programs

MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: EXPRESSIVENESSMODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

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What is a Logic? Logic = <L, I, ⊨ > where

L LanguageSet of phrases/sentences/formulas (alphabet + formation rules)

I Interpretation functionWhat phrases mean in a chosen domain Dwith I: L -> D

⊨ Satisfiability relation (over M)How to compute the fact that a formula A

is TRUE in M, notationally M ⊨ A

A ⊨ B in M iff M ⊨ A implies M ⊨ B

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Choose the right logic for the problem

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Problem: the LDKR class

Specification:

“In the LDKR class Fausto is the professor. There are 6 students. They are Mary, Paul, Jane, Rui, Hugo and Sara”.

Formalization: We want to represent both classes

of objects (Professor, Student) and individuals (Mary, Paul…)

Choose a logic that allows for them, e.g. ClassL with Individuals

Fausto

Mary Paul Jane

HugoSaraRui

L = {Professor, Student , ∧, ∨, ¬, →}

D = {Fausto, Mary, Paul, Jane, Rui, Sara, Hugo}


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Define the theory for the problem

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L = {Professor, Student , ∧, ∨, ¬, →}

D = {Fausto, Mary, Paul, Jane, Rui, Sara, Hugo}

In the theory we want to model the fact that the set of professors is always disjoint from the set of students…

Fausto

Mary Paul Jane

HugoSaraRui

T = {Student → ¬ Professor; Professor → ¬ Student}

M



M ⊨T

Then we can use reasoning services to handle our problem


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Reasoning Services: EVALModel Checking (EVAL)Is a sentence ψ true in model M?

Check M ⊨ ψ

EVALψ , MYes, M ⊨ ψ

No, M ⊭ ψ


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EVAL L = {MonkeyLow, BananaHigh, MonkeyClimbBox,





I(MonkeyLow) = T


= F


Evaluate ψ1 and ψ2 in M

ψ1 = MonkeyClimbBox is true in M (YES)

ψ2 = MonkeyGetBanana is false in M (NO)

M


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Reasoning Services: SATSatisfiability (SAT)Is there a model M where ψ is true?

find M such that M ⊨ ψ

SATψM

No


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SAT L = {MonkeyLow, BananaHigh, MonkeyClimbBox,





I(MonkeyLow) = F


= T


ψ = MonkeyGetBanana

Is there a model M where ψ is true? (YES, the model on the left)

SAT is a search problem (find a model M)


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Reasoning Services: VALValidity (VAL) Is ψ true according to all possible models?

Check whether for all M, M ⊨ ψ

VALψYes

No


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VAL L = {MonkeyLow, BananaHigh, MonkeyClimbBox,





I(MonkeyLow) = F

I(BananaHigh) = T I(MonkeyClimbBox) = T


ψ = MonkeyLow MonkeyGetBananaIs ψ true according to all possible model M? (No, the monkey can be high without getting the banana)

VAL is a search problem (check ψ in all M)NOTE: It is enough to find a counterexample to conclude for non validity


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Reasoning Services: ENTEntailment (ENT) ψ1 true in M (all models) implies ψ2 true in M (all models)

check A ⊨ B in M by checking M ⊨ A implies M ⊨ B

ENTψ1 , ψ2, MYes, ψ1 ⊨ ψ2

No


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ENT L = {MonkeyLow, BananaHigh, MonkeyClimbBox,





ψ1 = MonkeyClimbBoxψ2 = MonkeyLowψ1 true in M (all models) implies ψ2 is true in M (all models). (Yes)

NOTE: ψ1 entails ψ2 in all models


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An Important Trade-Off There is a trade-off between:

expressive power (expressiveness) and computational efficiency provided by a (logical)

language

This trade-off is a measure of the tension between specification and automation

To use logic for modeling, the modeler must find the right trade off between expressiveness in the language for more tractable forms of reasoning services.

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Examples of Expressiveness

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LANGUAGE NL SENTENCE FORMULA

Propositional logic

Fausto likes skiing

I like skiing

Fausto-likes-skiing

I-like-skiing

Modal logic I believe I like skiing B(I-like-skiing)

First-order logic Every person likes skiing

I like skiing

Fausto likes skiing

∀ person.like-skiing(person)

like-skiing(I)

like-skiing(Fausto)

Description Logic

Every person likes cars person ⊑ ∃ likes.Car

…


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Efficiency VS. Complexity Efficiency

Performing in the best possible manner; satisfactory and economical to use [Webster]

In modeling it applies to reasoning We use the more specific term computational

complexity (time, space,...)

Complexity (or computational complexity) of reasoning It is the difficulty to compute a reasoning task

expressed by using a logic With degrees of expressiveness, we may classify the

logical languages according to some “degrees of complexity”

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Degrees of Complexity

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The more you specify, the more cost grows

CO

ST

& P

RE

CIS

ION

OF

TH

E

SP

EC

IFIC

AT

ION

Natural Language Diagram Logics

FORMALITY

0

1000000

2000000

3000000

4000000

5000000

6000000

7000000

8000000

9000000

10000000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


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When is the use of logics appropriate?

When logic is used we always pay a performance price

We therefore use it when it is cost-effectiveTo prove correctness (offline use)To draw conclusions (online use)


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Examples of offline use (specification)For data and knowledge representation

In safety-critical applicationsTrainsPlanes…

In security critical applicationsbank transactions…


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Examples of online use (reasoning) To let programs interoperate

Web 1.0 Web 2.0

LET PEOPLE INTEROPERATE

(informal semantics)

Web 3.0

LET PROGRAMS INTEROPERATE

(formal semantics)


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Decidability The existence of an effective method to

determine the validity of formulas in a logical language

A logic is decidable if there is an effective method to determine whether arbitrary formulas are included in a theory

A decision procedure is an algorithm that, given a decision problem, terminates with the correct yes/no answer.

In this course we focus on logics that are expressive enough to model real problems but are still decidable

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Documents

Logics for Data and Knowledge Representation Modeling Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang