1. Data and Knowledge Representation
Mathematical Logic
Data and Knowledge Representation
Originally by Alessandro Agostini and Fausto Giunchiglia
Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese 1

2. Outline Modeling and logical modeling Languages
Domain
Language
Theory
Model
Languages
Logic: formal languages
Using Logic
Specification
Automation
Expressiveness
Efficiency VS. Complexity
Decidability

3. Modeling the world A model is an abstraction of a part of the world
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
A model is an abstraction of a part of the world
INDIVIDUAL
SET
RELATION
(e.g. Near, Sister)
DOMAIN
(e.g. the girls in foreground)

4. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Theory T
Data
Knowledge
World
Mental
Model
Domain D
Model M
Meaning
SEMANTIC
GAP

5. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Modeling: explained
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
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
Theory: the set of sentences (constraints) about the world expressed in the language that limit the possible models
Model: the formalization of the mental model, i.e. the set of true facts in the language, in agreement with the theory
Semantic gap: the impossibility to capture the entire reality
NOTE: this does not necessarily need to be in formal semantics

6. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Mental models
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
There is a monkey in a laboratory with some bananas hanging out of reach from the ceiling. A box is available that will enable the monkey to reach the bananas if he climbs on it. The monkey and box have height Low, …
… but if the monkey climbs onto the box he will have height High, the same as the bananas. [...]”
NOTE: each picture is a different model 6

7. Example of informal Modeling
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
World
Mental
Model
Language L
Domain D
Theory T
Model M
L: natural language
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”
SEMANTIC
GAP

8. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Theory T
Data
Knowledge
Modeling
Interpretation
Entailment
World
Mental
Model ⊨ I
Realization
Domain D
Model M
Meaning
SEMANTIC
GAP
NOTE: the key point is that in logical modeling we have formal semantics

9. Logical Modeling: explained
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
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
Theory / Knowledge Base (KB): the set of facts which are always true (in data and knowledge)
Model: the set of true facts in the language describing the mental model (the part of the world observed), in agreement with the theory
Truth-relation / logical entailment (⊨): deduction, reasoning, inference

10. Intensional vs Extensional semantics
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Intensional: I can only express the fact that a given proposition is true or false
“Bananas are yellow” “Bananas have a curve shape”
“All men are mortal” man  mortal
GirlsPlaying = True
Extensional: I provides the objects of the domain corresponding to the proposition
“The author of Romeo and Juliet is Shakespeare”
author(R&G, Shakespeare)
GirlsPlaying = {Mary, Sara, Julie}

11. Example of formal (intentional) modeling
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
World
Mental
Model
Language L
Domain D
Theory T
Model M
L = {MClimbs, MGetBanana, , , }
D= {T, F}
T = { MClimbs MGetBanana}
A possible model M:
I(MClimbs) = T
I(MGetBanana) = T
SEMANTIC
GAP

12. Example of formal (extensional) modeling
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
World
Mental
Model
Language L
Domain D
Theory T
Model M
L = {Monkey, Climbs, GetBanana, , , }
D= {Cita, Banana#1, Tree#1}
T = { (Monkey  Climbs)  GetBanana}
A possible model M:
I(Monkey) = {Cita}
I(Climbs) = {(Cita, Tree#1)}
I(GetBanana) = {(Cita, Banana#1)}
SEMANTIC
GAP

13. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
A (usually finite) set of words (the alphabet) and rules to compose them to build “correct sentences”
e.g. Monkey and GetBanana are words
e.g. Monkey  GetBanana is a sentence (rule: A  B)
A tool for codifying our (mental) model (what we have in mind):
Sentences (syntax) with an intended meaning (semantics)
e.g. with the word Monkey we mean

14. Language and correctness
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
We need an algorithm for checking correctness of the sentences in a language
PARSER
s, L
Yes, correct! No
We say that a language is decidable if it is possible to create such a tool that in finite time can take the decision

15. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Syntax and Semantics
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Syntax: the way a language is written:
Syntax is determined by a set of rules saying how to construct the expressions of the language from the set of atomic tokens (i.e., terms, characters, symbols).
The set of atomic tokens is called alphabet of symbols, or simply the alphabet).
Semantics: the way a language is interpreted:
It determines the meaning of the syntactic constructs (expressions), that is, the relationship between syntactic constructs and the elements of some universe of meanings (the intended model).
Such relationship is called interpretation.

16. Example of Syntax and Semantics
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Suppose we want to represent the fact that Mary and Sara are near each other.
ENGLISH
Mary is near Sara.
formal syntax, informal semantics
‘SYMBOLIZED’ ENGLISH
near(M,S)
near(Mary, Sara)
LOGICS with an interpretation function I
“near” by I it means near as spatial relation
“M” by I it means Mary the girl
“S” by I it means Sara the girl
formal syntax, formal semantics

17. Formal vs. Informal languages
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Language = Syntax (what we write) + Semantics (what we mean)
Formal syntax
Infinite/finite (always recognizable) alphabet
Finite set of formal constructors and building rules for phrase construction
Algorithm for correctness checking (a phrase in a language)
Formal Semantics
The relationship between syntactic constructs in a language L and the elements of an universe of meanings D is a (mathematical) function I: L  pow(D)
Informal syntax/semantics
the opposite of formal, namely the absence of the elements above.

18. Levels of Formalization
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Both Syntax and Semantics can be formal or informal.
Programming Languages
Diagrams NL
Logics
Level1
Leveln
English Italian Russian Hindi ...
ER UML ...
SQL ...
PL FOL DL ...
FORMALITY
informal
semi-formal
formal 18 18

19. Informal languages: natural language
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Let us try to recognize relevant entities, relations and properties in the NL text below
The Monkey-Bananas (MB) problem by McCarthy, 1969
“There is a monkey in a laboratory with some bananas hanging out of reach from the ceiling. A box is available that will enable the monkey to reach the bananas if he climbs on it. The monkey and box have height Low, but if the monkey climbs onto the box he will have height High, the same as the bananas. [...]”
Question:
How shall the monkey reach the bananas?

20. Languages with formal syntax
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
In the Entity-Relationship (ER) Model [Chen 1976] the alphabet is a set of graphical objects, that are used to construct schemas (the sentences).
Relation
Attribute
Entity
Examples of ER sentences:
Monkey
Box
Climb
0..1
0..n
Banana
Height

21. Languages with formal syntax and semantics
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Logics has two fundamental components:
L is a formal language (in syntax and semantics)
I is an interpretation function which maps sentences into a formal model M (over all possible ones) with a domain D
Domain D = {T, F} or D’ = {o1, …, on}
Language L = {A,,,}
Interpretation I: L  D is intensional or I: L  pow(D’) is extensional
Theory: a set of sentences which are always true in the language (facts)
Model: the set of true facts in the language describing the mental model (the part of the world observed), in agreement with the theory

22. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Theory
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
A theory T is set of facts:
A fact written in the language L defines a piece of knowledge (something true) about the domain D
A theory can be seen as a set of constraints on possible models to filter out all undesired ones

23. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Data and Knowledge
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Knowledge is factual information about what is true in general (axioms and theorems) e.g. singers sing songs
Data is factual information about specific individuals (observed knowledge) e.g. Michael Jackson is a singer
A finite theory T is called:
An ontology if it contains knowledge only
A knowledge base (KB) if it contains knowledge and data
A database (DB) if it only contains data.
A database + its schema is the simplest kind of knowledge base
NOTE: Sometimes the terms Ontology and KB are used interchangeably

24. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
A model M is an abstract (in mathematical sense) representation of the intended truths via interpretation I of the language L
e.g. Madonna is a singer, Sting is a singer, …
M ⊨ T (M entails 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 (M satisfies 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.

25. Usages of Representation Languages
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
The two purposes in modeling
Specification:
Representation of the problem at the proper level of abstraction
Allow informal/formal syntax and informal/formal semantics
Automation (Automated Reasoning)
Computing consequences or properties of the chosen specifications
It requires formal semantics.
Logics have formal syntax and formal semantics!

26. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Why Natural Languages?
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Used for
Advantages
Disadvantages
For informal specification
Often used at the very beginning of problem solving, when we need a direct, “flexible”, well- understood language and the problem is still largely unclear
Useful to interact with users
Semantics is informal, largely ambiguous
Pragmatically inefficient for automation

27. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Why Diagrams?
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Used for
Advantages
Disadvantages
To provide more structured and organized specification than natural languages
Informal/formal syntax (depends on the kind of diagram)
Largely structured and organized; usually used in representation with unified languages when things are non-trivial or more precision is required w.r.t. Natural Language
Useful to interact with users
Semantics is informal, largely ambiguous
Pragmatically inefficient for automation 27 27

28. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Why Logic?
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Used for
Advantages
Disadvantages
Formal specification
Automation
Well-understood with formal syntax and formal semantics: we can better specify and prove correctness
Pragmatically efficient for automation exploiting the explicitly codified semantics: reasoning services
It can be hardly used to interact with users
An exponential grow in cost (computational, man power) 28 28

29. Logics for Automation: reasoning
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Logics provides a notion of deduction
axioms, deductive machinery, theorem
Deduction can be used to implement reasoners
Reasoners allow inferring conclusions from known facts (i.e., a set of “premises”, premises can be axioms or theorems).
From implicit knowledge to explicit knowledge
Reasoning services:
Model Checking (EVAL) Is a sentence ψ true in model M?
Satisfiability (SAT) Is there a model M where ψ is true?
Validity (VAL) Is ψ true according to all possible models?
Entailment (ENT) ψ1 true implies ψ2 true (in all models) 29 29

30. How to use logical modeling in practice
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Define a logic
most often by reseachers
once for all (not a trivial task!)
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
Write the theory
The computer scientist writes a theory T
Use reasoning services
Computer scientists use reasoning services to solve their programs 30

31. An Important Trade-Off
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
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.

32. Examples of Expressiveness
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
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
∀ person.like-skiing(person)
like-skiing(I)
like-skiing(Fausto)
Description Logic
Every person likes cars
person ⊑ ∃ likes.Car …

33. Efficiency VS. Complexity
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Efficiency
Performing in the best possible manner; satisfactory and economical to use [Webster]
In modeling it applies to reasoning
In time, space, consumption of resources
Complexity (or computational complexity) of reasoning
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”
NOTE: When logic is used we always pay a performance price and therefore we use it when it is cost-effective

34. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
Decidability
MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC
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

35. Appendix: Greek lettersΑα
Alpha Νν Nu Ββ
Beta Ξξ Xi Γγ
Gamma Οο
Omicron Δδ
Delta Ππ Pi Εε
Epsilon Ρρ
Rho Ζζ
Zeta
Σσς
Sigma Ηη
Eta Ττ
Tau Θθ
Theta Υυ
Upsilon Ιι
Iota Φφ
Phi Κκ
Kappa Χχ
Chi Λλ
Lamdba Ψψ
Psi Μμ Mu Ωω
Omega 35