LLogics for DData and KKnowledge RRepresentation

PL of Classes

Outline

IntroductionSyntaxSemanticsReasoningProperties

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OverviewThe logic of classes is a propositional logic with

a different alphabet and semantics, i.e.:

Logical constants: (“operators”): ⊓ (“and, intersection”), ⊔ (“or, disjunction”), (“not”)

Defined and meta-logical symbols: ⊥; ⊤

Propositions are interpreted using an extensional interpretation.

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Outline IntroductionSyntaxSemanticsReasoningProperties

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Language (Syntax)

Propositional constants: A, B, “LDKR”,... Remark: not only single characters but also words

are propositional constants.Logical constants (“Boolean operators”):

⊓ (“and”), ⊔ (“or”), (“not”)Parentheses (auxiliary symbols): (, )Propositional variables: P, Q,... ψ,θ,... α,β...

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Logical constants

Similarly to PL, we have logical constants ⊥ (falsehood symbol), ⊤(truth symbol)

Remark: - ⊥ =df P∧¬P is a defined symbol- ⊥ ≡ P ⊓ ¬P is a logical fact (a theorem)

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Formation Rules (FRs)Atomic Formulas (atomic propositions):

1. A, B,..., P, Q,...; ⊥, ⊤. Class-Propositional Formulas (propositions, or

also class-propositions, in short wff’s): 2. All the atomic formulas.3.P, P ⊓ Q, P ⊔ Q for all wff’s P, Q.

The alphabet Class-Σ0 + FRs above define a propositional language of classes (“classPL”).

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Outline IntroductionSyntaxSemanticsReasoningProperties

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SemanticsSo far the elements of our propositional

language are simply strings of symbols

without formal meaningThe meanings which are intended to be

attached to the symbols and propositions form the intended interpretation of the language (viz. its symbols, formulas, etc.).

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Extensional Semantics-Extensions

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 (domain elements) to which the proposition applies.

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Examples

Take the proposition ‘Rome’: its extension is simply the singleton set whose element is the city of Rome.

Take the proposition ‘red apple’: its extension is the class containing all the red apples.

Take the proposition ‘lion’: its extension includes not only living lions, but also all the lions of the past, and those of the future.

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Extensions - Remarks

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.

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Extensions / IntensionsCompare:

Observe:

- The central notion of PL is “property”.- The central notion of ClassPL is “class”.

Exercise: build the formal model, in the two cases above of a real world situation with three lions

The extension of a proposition P is the class of things or objects to which P applies

The extension of a proposition P is the class of things or objects to which P applies

The intension of a proposition P consists of all the properties the proposition implies

The intension of a proposition P consists of all the properties the proposition implies

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Extensional Interpretation

Given a Domain (or Universe) of Intepretation U, the extensional interpretation I of a proposition P, denoted by I(P), is I(P) subset of U.

Example: Take P = ‘airplane’. Then:I(‘airplane’) = {Boeing747-3001, Boeing747-300n,

piper1, piperk, ...} = … all airplanes occuring inthe part of world being modeled

Note: By assuming one world, i.e., on domain, the extension of a proposition is unique.

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Class-valuation σ 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 namely

σ: L U

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Class-valuation σ

σ(⊥) = ∅σ(⊤) = U (Universal Class, or Universe) σ(P) subset of U, as defined by σσ(¬P) = {a U | a ∉ σ(P)} = comp(σ(P))σ(P ⊓ Q) = σ(P)∩σ(Q)σ(P ⊔ Q) = σ(P)∪σ(Q)

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Example

Suppose Person and Female are atomic concepts;

Then Person ⊓ Female and Person ⊓Female are the concepts describing, intuitively, those persons that are female, and those that are not female.

And Person ⊔Person is the concept describing the all the world (⊤).

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Venn Diagrams and Class-Values 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’value in polynomial time.

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Venn Diagrams 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 a proposition P’s extension by the device of shading in a region corresponding to the proposition, as to indicate that P has a meaning (i.e., P’s extension is not empty).

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Venn Diagram of P, ⊥Now we illustrate the Venn diagram for each

proposition of our language in Universe (4).σ(P) :

σ(⊥) :

NB: Venn diagrams are build starting from a “main box” which is used to represent the Universal Class.

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Venn Diagram of ¬P,⊤

σ(¬P) :

σ(⊤) :

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Venn Diagram of P⊓Q, P⊔Qσ(P ⊓ Q) :

σ(P ⊔ Q) :

`

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Truth Relation(Satisfaction Relation)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 |= Γiff for all θ ∈ Γ, σ |= θ.

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Model, Satisfiable

Let σ be a class valuation on language L. σ is a model of a proposition P, in (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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Truth and Validity

Let σ be a class valuation: (1) P is true under σ if σ |= P.(2) P is valid (and P is called a tautology), if σ |= P for all σ (notation: |= P).

Note 1: 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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Outline IntroductionSyntaxSemanticsReasoningProperties

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Language

Add two logical symbols

⊑ (subsumption), with σ |= C⊑D iff σ(C)⊆ σ(D) ≡ (equivalence), with σ |= C≡D iff σ(C)= σ(D)

Where A ⊑ B reads A is subsumed (more specific than / less general than) by B

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Reasoning on Class-Propositions

Given two class-propositions P, Q, we want to reason about the following relations between P and Q:

Satisfibability: Is P satisfiable (with respect to σ) ? Equivalence: Is P equivalent to Q (with respect to σ)? Subsumption: Is P less general than Q (with respect

to σ)? Supsumption: Is P more general than Q (with respect

to σ)? Disjointness: Are P and Q disjoint (with respect to σ)

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Reasoning on Class-Propositions

Satisfiability: σ |= P?Subsumption: σ |= P ⊑ Q?Equivalence: σ |= P ⊑ Q and σ |= Q ⊑ P?Disjointness: σ |= P ⊓ Q ⊑ ⊥?

Note: Two versions of subsumption, equivalence, disjointness Single model: model checking, satisfiability All models: validity, unsatisfiability

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Example

NEED A FEW EXAMPLES OF REASONING, ONE PER KIND

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