1. LDK R Logics for Data and Knowledge Representation PL of Classes

2. Outline  Introduction  Syntax  Semantics  Reasoning  Properties 2

3. Overview The 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. 3

4. Outline  Introduction  Syntax  Semantics  Reasoning  Properties 4

5. 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... 5

6. 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) 6

7. 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- Σ 0 + FRs above define a propositional language of classes (“classPL”). 7

8. Outline  Introduction  Syntax  Semantics  Reasoning  Properties 8

9. Semantics  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 of the language (viz. its symbols, formulas, etc.). 9

10. 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. 10

11. 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. 11

12. 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. 12

13. Extensions / Intensions  Compare:  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 intension of a proposition P consists of all the properties the proposition implies 13

14. 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’) = { Boeing747-300 1, Boeing747-300 n, piper 1, piper k,... } = … 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. 14

15. 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 15

16. Class-valuation σ  σ ( ⊥ ) = ∅  σ ( ⊤ ) = U (Universal Class, or Universe)  σ (P) subset of U, as defined by σ  σ (¬P) = {a  U | a ∉ σ (P)} = comp( σ (P))  σ (P ⊓ Q) = σ (P) ∩ σ (Q)  σ (P ⊔ Q) = σ (P) ∪ σ (Q) 16

17. 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 ( ⊤ ). 17

18. 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. 18

19. 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). 19

20. 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. 20

21. Venn Diagram of ¬P, ⊤  σ (¬P) :  σ ( ⊤ ) : 21

22. Venn Diagram of P ⊓ Q, P ⊔ Q  σ (P ⊓ Q) :  σ (P ⊔ Q) : ` 22

23. 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 |= Γ iff for all θ ∈ Γ, σ |= θ. 23

24. 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 ). 24

25. 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. 25

26. Outline  Introduction  Syntax  Semantics  Reasoning  Properties 26

27. 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 27

28. 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 ) ? 28

29. 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 29

30. Example NEED A FEW EXAMPLES OF REASONING, ONE PER KIND 30