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

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

Introduction: ClassL, the logic of classes INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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.

Intensional vs Extensional interpretation INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL Intentional interpretation: D = {T, F} Lion Monkey Tree . T F . Lion1 Lion2 The World The Mental Model The Formal Model 4

Intensional vs Extensional interpretation INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL Extensional interpretation: D = {Cita, Kimba, Simba} Lion Monkey Tree Kimba . Cita . . Simba Lion1 Lion2 The World The Mental Model The Formal Model 5

they can be substituted by any proposition or formula Language (Syntax) INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Additional Symbols Auxiliary symbols Parentheses: ( ) INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Formation Rules (FR): well formed formulas INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Semantics means providing an interpretation INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Extensional Semantics: Extensions INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Extensions - Remarks INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Examples Take the proposition lion: INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Extensional Interpretation INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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) = {Boeing747-3001, Boeing747-300n, piper1, piperk, ...} = … all airplanes occurring in the part of the world being modeled

Class-valuation σ INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Class-valuation σ σ(⊥) = ∅ σ(⊤) = U (Universal Class, or Universe) INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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)

Example INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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 (⊤)

Venn Diagrams and Class-Values INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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).

Venn Diagram of P, ⊥ σ(P) σ(⊥) INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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. ⊥

Venn Diagram of ¬P,⊤ σ(¬P) σ(⊤) INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Venn Diagram of P ⊓ Q and P ⊔ Q INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL σ(P ⊓ Q) The intersection of P and Q P Q σ(P ⊔ Q) The union of P and Q P Q 20

How to use Venn diagrams: exercise 1 INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL Prove by Venn diagrams that σ(P) = σ(¬¬P) Case σ(P) = ∅ σ(P) ⊥ σ(¬P) σ(¬¬P) ⊥ 21

How to use Venn diagrams: exercise 1 INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL Prove by Venn diagrams that σ(P) = σ(¬¬P) Case σ(P) = U σ(P) σ(¬P) ⊥ σ(¬¬P) 22

How to use Venn diagrams: exercise 1 INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

How to use Venn diagrams: exercise 2 INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Truth Relation (Satisfaction Relation) INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Model and Satisfiability INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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).

Satisfiability, an example INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Truth, satisfiability and validity INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Validity, an example Is the formula P = A ⊔ ¬A valid? INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Reasoning on Class-Propositions INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Class-Values and Truth-Values INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

PL and ClassL: table of the symbols INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL PL 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

PL and ClassL are notational variants INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

ClassL reasoning using DPLL INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Entailment in ClassL INTRODUCTION :: SYNTAX :: SEMANTICS :: REASONING :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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

Entailment in ClassL Suppose that Male and Female are satisfiable. INTRODUCTION :: SYNTAX :: SEMANTICS :: PL AND CLASSL :: CLASSL REASONING USING DPLL 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