Logics for Data and Knowledge
Representation
ClassL (part 1): syntax and semantics
Outline
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Introduction
Syntax
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Semantics
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Alphabet
Formation rules
Class-valuation
Venn diagrams
Satisfiability
Validity
Reasoning
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Comparing PL and ClassL
ClassL reasoning using DPLL
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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.
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Intensional vs Extensional interpretation
Intentional interpretation: D = {T, F}
BeingLion
Monkey
Tree
.T
Lion1
The World
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The Mental Model
F.
Lion2
The Formal Model
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Intensional vs Extensional interpretation
Extensional interpretation: D = {Cita, Kimba, Simba}
BeingLion
Monkey
Tree
Kimba .
. Simba
Lion1
The World
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The Mental Model
Cita .
Lion2
The Formal Model
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Language (Syntax)
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The syntax of ClassL is similar to PL
Alphabet of symbols Σ0
Σ0
Logical
⊓, ⊔, 
Descriptive
Constants
Variables
one proposition only
they can be substituted by any
proposition or formula
A, B, C …
P, Q, ψ …
NOTE: not only characters but also words (composed by several
characters) like “monkey” are descriptive symbols
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Additional Symbols
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Auxiliary symbols
 Parentheses: ( )
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Additional logical constants:
 Logical constants are, for all propositions P:
⊥ (falsehood symbol, false, bottom)
T (truth symbol, true, top)
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⊥  P ⊓ ¬P
T¬⊥
Note that differently from PL, in ClassL they are not defined
symbols but they are logical facts, i.e. theorems
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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>
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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
ψ, ClassL
Yes, ψ is correct!
PARSER
No
Σ0 + FR define a propositional language
(*) BNF = Backus–Naur form (formal grammar)
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Semantics means providing an interpretation
 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
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Extensional Semantics: Extensions
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The semantics of a propositional language of classes L are
extensional (semantics)
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The extensional semantics of L is based on the notion of
“extension” of a formula (proposition) in L
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The extension of a proposition is the totality, or class, or set of
all objects D (domain elements) to which the proposition
applies
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Examples
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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
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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)
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Take the proposition red ⊓ apple:
its extension can the class containing all the red apples
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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.
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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.
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In ClassL, a proposition is also called a concept
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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
Take P = ‘airplane’.
I(airplane) = {Boeing747-3001, Boeing747-300n, piper1, piperk, ...}
= … all airplanes occurring in the part of the world being modeled
This is fundamental to make the language formal.
NOTE: By assuming one world, i.e. one domain, the extension
of a proposition is unique.
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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
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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
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Class-valuation σ
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σ(⊥) = ∅
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σ(⊤) = U (Universal Class, or Universe)
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σ(P)  U, as defined by σ
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σ(¬P) = {a  U | a ∉ σ(P)} = comp(σ(P))
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σ(P ⊓ Q) = σ(P) ∩ σ(Q) (Intersection)
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σ(P ⊔ Q) = σ(P) ∪ σ(Q) (Union)
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(Complement)
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Example
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Suppose Person and Female are atomic formulas (also called
concepts)
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Person ⊓ Female
denotes those persons that are female
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Person ⊓ Female
denotes those that are not female
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Person ⊔ Person
is the concept describing the whole world (⊤)
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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).
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Venn Diagram of P, ⊥
σ(P)
P
σ(⊥)
⊥
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Venn diagrams are
built starting from a
“main box” which is
used to represent
the Universe U.
The falsehood
symbol corresponds
to the empty set.
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Venn Diagram of ¬P,⊤
σ(¬P)
P
σ(⊤)
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¬P corresponds to
the complement of P
w.r.t. the universe U.
The truth symbol
corresponds to the
universe U.
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Venn Diagram of P ⊓ Q and P ⊔ Q
σ(P ⊓ Q)
The intersection of P
and Q
P
Q
σ(P ⊔ Q)
The union of P and
Q
P
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Q
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How to use Venn diagrams: exercise 1
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Prove by Venn diagrams that σ(P) = σ(¬¬P)
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Case P = ∅
⊥
σ(P)
σ(¬P)
⊥
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σ(¬¬P)
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How to use Venn diagrams: exercise 1
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Prove by Venn diagrams that σ(P) = σ(¬¬P)
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Case P = U
σ(P)
⊥
σ(¬P)
σ(¬¬P)
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How to use Venn diagrams: exercise 1
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Prove by Venn diagrams that σ(P) = σ(¬¬P)
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Case P not empty and different from U
P
P
P
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σ(P)
σ(¬P)
σ(¬¬P)
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How to use Venn diagrams: exercise 2
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Prove by Venn diagrams that σ(¬(A ⊔ B)) = σ(¬ A ⊓ ¬ B)
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Case A and B not empty (try the other cases as homework)
σ(A ⊔ B)
A
B
σ(¬(A ⊔ B))
A
B
σ(¬ A)
A
B
A
σ(¬ B)
A
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B
σ(¬ A ⊓ ¬ B)
B
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Truth Relation (Satisfaction Relation)
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Let σ be a class-valuation on language L, we define the truthrelation (or class-satisfaction relation) ⊨ and write
σ⊨P
(read: σ satisfies P) iff σ(P) ≠ ∅
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Given a set of propositions Γ, we define
σ⊨Γ
iff σ ⊨ θ for all formulas θ ∈ Γ
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Model and Satisfiability
 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).
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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.
A
B
σ is a model for P
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Truth, satisfiability and validity
 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
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Validity, an example
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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.
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⊥
Case A empty:
if A is empty, then ¬A is the universe U
A
Case A not empty:
if A is not empty, ¬A covers all the other
elements of U
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Reasoning on Class-Propositions
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Given a class-propositions P we want to reason about the following:
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Model checking
Does σ satisfy P? (σ ⊨ P?)
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Satisfiability
Is there any σ such that σ ⊨ P?
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Unsatisfiability
Is it true that there are no σ satisfying P?
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Validity
Is P a tautology? (true for all σ)
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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”
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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”
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What is the relation between ν(P) and σ(P)? (see next slides)
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PL and ClassL: table of the symbols
Syntax
Semantics
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PL
∧
ClassL
∨
⊔


⊤
⊤
⊥
⊥
P, Q...
P, Q...
∆={true, false}
∆={o, …} (compare models)
⊓
RECALL: A proposition P is true (in a model) iff it is satisfiable
NOTE: There is no logical implication (yet)
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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.
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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
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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
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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 negation
If σ ⊨ P implies σ ⊨ P (sometimes)
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Entailment in ClassL
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Suppose that Male and Female are satisfiable.
Is Male ∧ Female satisfiable in PL?
And Male ⊓ Female in ClassL?
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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.
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