1. Logics for Data and Knowledge Representation
ClassL (part 1): syntax and semantics

2. Outline Introduction Syntax Semantics Reasoning Alphabet
Formation rules
Semantics
Class-valuation
Venn diagrams
Satisfiability
Validity
Reasoning
Comparing PL and ClassL
ClassL reasoning using DPLL

3. Introduction: ClassL, the logic of classes
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It is a propositional logic
Sentences expressing propositions (something true or false)
It is also called Propositional Description Logic (DL) or ALC DL
Different alphabet and semantics w.r.t. PL (notational variant)
The logical constants (“operators”) are: ⊓ (“and, intersection”), ⊔ (“or, disjunction”),  (“not”)
Meta-logical symbols: ⊥, ⊤
Extensional interpretation
The domain is a set of objects. Propositions are interpreted using an extensional interpretation.

4. Intensional vs Extensional interpretation
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Intentional interpretation: D = {T, F}
Lion
Monkey
Tree . T F .
Lion1
Lion2
The World
The Mental Model
The Formal Model 4

5. Intensional vs Extensional interpretation
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Extensional interpretation: D = {Cita, Kimba, Simba}
Lion
Monkey
Tree
Kimba .
Cita . .
Simba
Lion1
Lion2
The World
The Mental Model
The Formal Model 5

6. they can be substituted by any proposition or formula
Language (Syntax)
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The syntax of ClassL is similar to PL
Alphabet of symbols Σ0 Σ0
Descriptive
Logical
⊓, ⊔, 
Constants
one proposition only
A, B, C …
Variables
they can be substituted by any proposition or formula
P, Q, ψ …
NOTE: not only characters but also words (composed by several characters) like “monkey” are descriptive symbols 6

7. Additional Symbols Auxiliary symbols Parentheses: ( )
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Auxiliary symbols
Parentheses: ( )
Additional logical constants:
Logical constants are, for all propositions P:
⊥ (falsehood symbol, false, bottom) ⊥  P ⊓ ¬P
T (truth symbol, true, top) T  ¬ ⊥
Note that differently from PL, in ClassL they are not defined symbols but they are logical facts, i.e. theorems 7

8. Formation Rules (FR): well formed formulas
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Well formed formulas (wff) in ClassL can be described by the following BNF grammar (codifying the rules):
<Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤
<wff> ::= <Atomic Formula> | ¬<wff> | <wff> ⊓ <wff> | <wff> ⊔ <wff>
Atomic formulas are also called atomic propositions
Wff are class-propositional formulas (or just propositions)
A formula is correct if and only if it is a wff
Σ0 + FR define a propositional language
PARSER
ψ, ClassL
Yes, ψ is correct! No 8

9. Semantics means providing an interpretation
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So far the elements of our propositional language are simply strings of symbols
without formal meaning
The meanings which are intended to be attached to the symbols and propositions form the intended interpretation σ (sigma) of the language 9

10. Extensional Semantics: Extensions
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The semantics of a propositional language of classes L are extensional (semantics)
The extensional semantics of L is based on the notion of “extension” of a formula (proposition) in L
The extension of a proposition is the totality, or class, or set of all objects D (domain elements) to which the proposition applies

11. Extensions - Remarks
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If a proposition applies to an individual object, its extension is simply the one object designated (denoted) by the proposition.
If a proposition applies to a group of objects, its extension is the class consisting of all the objects, if any, to which it applies.
In ClassL, a proposition is also called a concept

12. Examples Take the proposition lion:
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Take the proposition lion:
its extension may include (according to the modeler) not only living lions, but also all the lions of the past, and those of the future
Take the proposition Rome:
its extension can be simply the singleton set whose element is the city of Rome (notice that several cities may have the same name, so we need to specify which Rome)
Take the proposition red ⊓ apple:
its extension can the class containing all the red apples

13. Extensional Interpretation
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Given a domain (or universe) of interpretation U, the extensional interpretation I of a proposition P, denoted by I(P) or PI is a subset of U
This is fundamental to make the language formal.
NOTE: By assuming one world, i.e. one domain, the extension of a proposition is unique.
Take P = ‘airplane’.
I(airplane) = {Boeing , Boeing n, piper1, piperk, ...}
= … all airplanes occurring in the part of the world being modeled

14. Class-valuation σ
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In extensional semantics, the first central semantic notion is that of class-valuation (the interpretation function)
Given a Class Language L
Given a domain of interpretation U
A class valuation σ of a propositional language of classes L is a mapping (function) assigning to each formula ψ of L a set σ(ψ) of “objects” (truth-set) in U:
σ: L  U

15. Class-valuation σ σ(⊥) = ∅ σ(⊤) = U (Universal Class, or Universe)
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σ(⊥) = ∅
σ(⊤) = U (Universal Class, or Universe)
σ(P)  U, as defined by σ
σ(¬P) = {a  U | a ∉ σ(P)} = comp(σ(P)) (Complement)
σ(P ⊓ Q) = σ(P) ∩ σ(Q) (Intersection)
σ(P ⊔ Q) = σ(P) ∪ σ(Q) (Union)

16. Example
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Suppose Person and Female are atomic formulas (also called concepts)
Person ⊓ Female
denotes those persons that are female
Person ⊓ Female
denotes those that are not female
Person ⊔ Person
is the concept describing the whole world (⊤)

17. Venn Diagrams and Class-Values
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By regarding propositions as classes, it is very convenient to use Venn diagrams
Venn diagrams are used to represent extensional semantics of propositions in analogy of how truth-tables are used to represent intentional semantics
Venn diagrams allow to compute a class valuation σ’s value in polynomial time
In Venn diagrams we use intersecting circles to represent the extension of a proposition, in particular of each atomic proposition
The key idea is to use Venn diagrams to symbolize the extension of a proposition P by the device of shading the region corresponding to the proposition, as to indicate that P has a meaning (i.e., the extension of P is not empty).

18. Venn Diagram of P, ⊥ σ(P) σ(⊥)
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σ(P)
Venn diagrams are built starting from a “main box” which is used to represent the Universe U. P
σ(⊥)
The falsehood symbol corresponds to the empty set. ⊥

19. Venn Diagram of ¬P,⊤ σ(¬P) σ(⊤)
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σ(¬P)
¬P corresponds to the complement of P w.r.t. the universe U. P
σ(⊤)
The truth symbol corresponds to the universe U. 19

20. Venn Diagram of P ⊓ Q and P ⊔ Q
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σ(P ⊓ Q)
The intersection of P and Q P Q
σ(P ⊔ Q)
The union of P and Q P Q 20

21. How to use Venn diagrams: exercise 1
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Prove by Venn diagrams that σ(P) = σ(¬¬P)
Case σ(P) = ∅
σ(P) ⊥
σ(¬P)
σ(¬¬P) ⊥ 21

22. How to use Venn diagrams: exercise 1
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Prove by Venn diagrams that σ(P) = σ(¬¬P)
Case σ(P) = U
σ(P)
σ(¬P) ⊥
σ(¬¬P) 22

23. How to use Venn diagrams: exercise 1
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Prove by Venn diagrams that σ(P) = σ(¬¬P)
Case σ(P) not empty and different from U
σ(P) P
σ(¬P) P
σ(¬¬P) P 23

24. How to use Venn diagrams: exercise 2
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Prove by Venn diagrams that σ(¬(A ⊔ B)) = σ(¬ A ⊓ ¬ B)
Case σ(A) and σ(B) not empty (other cases as homework)
σ(A ⊔ B)
σ(¬(A ⊔ B)) A B A B
σ(¬ A)
σ(¬ A ⊓ ¬ B) A B A B
σ(¬ B) A B 24

25. Truth Relation (Satisfaction Relation)
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Let σ be a class-valuation on language L, we define the truth- relation (or class-satisfaction relation) ⊨ and write
σ ⊨ P
(read: σ satisfies P) iff σ(P) ≠ ∅
Given a set of propositions Γ, we define
σ ⊨ Γ
iff σ ⊨ θ for all formulas θ ∈ Γ

26. Model and Satisfiability
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Let σ be a class valuation on language L. σ is a model of a proposition P (set of propositions Γ) iff σ satisfies P (Γ).
P (Γ) is class-satisfiable if there is a class valuation σ such that σ ⊨ P (σ ⊨ Γ).

27. Satisfiability, an example
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Is the formula P = ¬(A ⊓ B) satisfiable?
In other words, there exist a σ that satisfies P? YES!
In order to prove it we use Venn diagrams and it is enough to find one.
σ is a model for P A B 27

28. Truth, satisfiability and validity
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Let σ be a class valuation on language L.
P is true under σ if P is satisfiable (σ ⊨ P)
P is valid if σ ⊨ P for all σ (notation: ⊨ P)
In this case, P is called a tautology (always true)
NOTE: the notions of ‘true’ and ‘false’ are relative to some truth valuation.
NOTE: A proposition is true iff it is satisfiable

29. Validity, an example Is the formula P = A ⊔ ¬A valid?
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Is the formula P = A ⊔ ¬A valid?
In other words, is P true for all σ? YES!
In order to prove it we use Venn diagrams, but we need to discuss all cases.
Case σ(A) empty:
if σ(A) is empty, then σ(¬A) is the universe U ⊥
Case σ(A) not empty:
if σ(A) is not empty, σ(¬A) covers all the other elements of U A 29

30. Reasoning on Class-Propositions
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Given a class-propositions P we want to reason about the following:
Model checking Does σ satisfy P? (σ ⊨ P?)
Satisfiability Is there any σ such that σ ⊨ P?
Unsatisfiability Is it true that there are no σ satisfying P?
Validity Is P a tautology? (true for all σ) 30

31. Class-Values and Truth-Values
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Intensional Interpretation: the intentional interpretation ν of a proposition P determines a truth-value ν(P)
“P holds”
Extensional Interpretation: the extensional interpretation of σ of P determines a class of objects σ(P)
“x belongs to P” or “x in P” or “x is an instance of P”
What is the relation between ν(P) and σ(P)? (see next slides) 31

32. PL and ClassL: table of the symbols
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ClassL
Syntax ∧ ⊓ ∨ ⊔  ⊤ ⊥
P, Q...
Semantics
∆={true, false}
∆={o, …} (compare models)
RECALL: A proposition P is true (in a model) iff P is satisfiable
NOTE: There is no logical implication (yet) 32

33. PL and ClassL are notational variants
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Theorem: P is satisfiable w.r.t. an intensional interpretation ν if and only if P is satifisfiable w.r.t. an extensional interpretation σ
ν(P) implies σ(P):
Build σ(P) from ν(P) by substituting true with U and false with empty set.
σ(P) implies ν(P):
Less trivial. Build first a σ’(P) which is equivalent to σ(P) and which uses only U and empty set. 33

34. ClassL reasoning using DPLL
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Given the theorem and the correspondences above, ClassL reasoning can be implemented using DPLL.
The first step consists in translating P into P’ expressed in PL
Model checking Does σ satisfy P? (σ ⊨ P?)
Find the corresponding model ν and check that v(P’) = true
Satisfiability Is there any σ such that σ ⊨ P?
Check that DPLL(P’) succeeds and returns a ν
Unsatisfiability Is it true that there are no σ satisfying P?
Check that DPLL(P’) fails
Validity Is P a tautology? (true for all σ)
Check that DPLL(P’) fails 34

35. Entailment in ClassL
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Logical consequence (entailment) is not preserved in ClassL
Intersection
σ ⊨ P and σ ⊨ Q may not imply that σ ⊨ P ⊓ Q
implies σ ⊨  (P ⊓ Q) (sometimes)
Satisfiability in an extensional interpretation is “richer” than in an intensional interpretation
NOTE about union
σ ⊨ P and σ ⊨ Q, implies σ ⊨ P ⊔ Q (always)
NOTE about complement
If σ ⊨ P implies σ ⊨ P (sometimes)
MODIFIED 35

36. Entailment in ClassL Suppose that Male and Female are satisfiable.
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Suppose that Male and Female are satisfiable.
Is Male ∧ Female satisfiable in PL?
And Male ⊓ Female in ClassL?
It is clear that if we assume that they are disjoint:
1. It cannot be ν ⊨ Male and ν ⊨ Female for the same ν
2. σ ⊨ Male and σ ⊨ Female do not imply that σ ⊨ Male ⊓ Female
3. We have that σ ⊨ Male ⊔ Female
4. We have that σ ⊨ Male and σ ⊨ Male
IMPORTANT NOTE:
In PL, ν ⊨ A and ν ⊨ B implies that ν ⊨ A ∧ B (same ν!)
In ClassL, σ ⊨ A and σ ⊨ B may not imply that σ ⊨ A ⊓ B (same σ!)
Think to the case Male and Male above. 36