Logics for Data and Knowledge Representation Exercises: ClassL Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

SYNTAX

Symbols in ClassL Which of the following symbols are used in ClassL? ⊓  ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨ Which of the following symbols are in well formed formulas? 3

Symbols in ClassL (solution) Which of the following symbols are used in ClassL? ⊓  ⊤ ∨ ≡ ⊔ ⊑ → ↔ ⊥ ∧ ⊨ Which of the following symbols are in well formed formulas? 4

Extended formation rules The basic BNF grammar: <Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤ <wff> ::= <Atomic Formula> | ¬<wff> | <wff> ⊓ <wff> | <wff> ⊔ <wff> TBox: <definition> ::= <Atomic Formula> ≡ <wff> <specialization> ::= <Atomic Formula> ⊑ <wff> ABox: <individual> ::= a | b | ... | <assertion> ::= <Atomic Formula> (<individual>) 5

Formation rules Which of the following is not a wff in ClassL?  MonkeyLow ⊔ BananaHigh   MonkeyLow ⊓ BananaHigh ⊑  GetBanana MonkeyLow  ⊓ BananaHigh MonkeyLow   GetBanana NUM 2, 3, 4 ! 6

MODELING

Formalization of simple sentences Propositional DL (ClassL) has pretty poor expressiveness. For instance, we cannot represent attributes and relations effectively. The set of games which are not legal Game ⊓ Legal Lakes are locations Lake ⊑ Location Lakes are locations made of water Lake ⊑ Location ⊓ MadeofWater Persons can be distinguished into male and female Male ⊑ Person Female ⊑ Person Male and Female are disjoint Male ⊑  Female Persons have a birthplace Person ⊑ hasBirthDate The set of documents about “programming in Java” are a subset of the documents about “programming languages” and “computer science” JavaProgramming ⊑ ProgrammingLanguage ⊓ ComputerScience 8

Formalization of a problem in ClassL Unicorns are mythical horses having a horn. Pegasus is a unicorn while Mike is not. Unicorn ⊑ mythical ⊓ horse ⊓ hasHorn Unicorn(Pegasus), Unicorn(Mike) There are two kinds of students: master students and PhD students. All PhD students do research. Ronald is a master student that does research. MasterStudent ⊑ Student PhDStudent ⊑ Student ⊓ doResearch MasterStudent(Ronald) doResearch(Ronald) 9

Formalization of a semantic network This can be only partially expressed in ClassL: T = {BodyOfWater ⊑ Location, PopulatedPlace ⊑ Location, Lake ⊑ BodyOfWater, City ⊑ PopulatedPlace, Country ⊑ PopulatedPlace} A = {Person(GiorgioNapolitano), Lake(GardaLake), City(Trento), Country(Italy),} 10

Defining the TBox and ABox: the LDKR Class Define a TBox and ABox for the following database: ABox = {Italian(Fausto), Italian(Enzo), Chinese(Rui), Indian(Bisu), BlackHair(Enzo), BlackHair(Rui), BlackHair(Bisu), WhiteHair(Fausto)} TBox = {Italian ⊑ LDKR, Indian ⊑ LDKR, Chinese ⊑ LDKR, BlackHair ⊑ LDKR, WhiteHair ⊑ LDKR} LDKR Name Nationality Hair Fausto Italian White Enzo Black Rui Chinese Bisu Indian NOTE: ClassL is not expressive enough to represent database constrains such as keys involving two fields. 11

SEMANTICS

Semantics of ⊓ Suppose that A and B are satisfiable. Is A ⊓ B always satisfiable in ClassL? We can easily observe that the fact that A and B are satisfiable does not imply that A ⊓ B is also satisfiable. Think to the case in which their extensions are disjoint. This might not be the case even when they are satisfiable by the same model. A A B B A B 13

TBOX REASONING

Satisfiability with respect to a TBox T RECALL: Satisfiability in one model A concept P is satisfiable w.r.t. a terminology T, if there exists an interpretation I with I ⊨ θ for all θ ∈ T, and such that I ⊨ P, namely I(P) is not empty Satisfiability in all models (validity) A concept P is satisfiable w.r.t. a terminology T, if for all interpretations I with I ⊨ θ for all θ ∈ T, and such that I ⊨ P, namely I(P) is not empty ADD EXAMPLE 15

Satisfiability with respect to a TBox (I) Given the TBox T={A⊑B, B⊑A}, is (A⊓B) satisfiable in ClassL? This corresponds to the problem: T ⊨ (A⊓B) To prove satisfiability in one model it is enough to find one model; we can use Venn Diagrams. A B To prove satisfiability in all models we need to prove validity; this can be proved analytically, or with DPLL as follows: DPLL( (RewriteInPL(A⊑B)  RewriteInPL(B⊑A)  RewriteInPL((A⊓B) ))) DPLL( ((A B)  (B A))  (A  B)) 16

Satisfiability with respect to a TBox (II) Given the TBox T={A⊑B, B⊑A}, is (A⊓B) satisfiable in ClassL? This corresponds to the problem: T ⊨ (A⊓B) To prove that it is not satisfiable in all models it is instead enough to find a counter example (i.e. one model of T which is not a model for the proposition). A B 17

Satisfiability with respect to a TBox (III) Given the TBox T={C⊑A, C⊑B} is (A⊓B) satisfiable? (in one model) A B C 18

Satisfiability with respect to a TBox (IV) Suppose we model the Monkey-Banana problem as follows: “If the monkey is low in position then it cannot get the banana. If the monkey gets the banana it survives”. TBox T MonkeyLow ⊑  GetBanana GetBanana ⊑ Survive Is T satisfiable? YES! Look at the Venn diagram Survive GetBanana MonkeyLow 19

Satisfiability with respect to a TBox (V) Suppose we model the Monkey-Banana problem as follows: TBox T MonkeyLow ⊑  GetBanana GetBanana ⊑ Survive Is it possible for a monkey to survive even if it does not get the banana? We can restate the problem as follow: does T ⊨  GetBanana ⊓ Survive ? YES! Look at the Venn diagram Survive GetBanana MonkeyLow 20

Subsumption Suppose we describe the students/attendees in a course: Are all assistants also undergraduates? T ⊨ Assistant ⊑ Undergraduate Assistant ≡ PhD ⊓ Teach ≡ Master ⊓ Research ⊓ Teach ≡ Student ⊓  Undergraduate ⊓ Research ⊓ Teach Assistants are actually students who are not undergraduate. Undergraduate ⊑  Teach Bachelor ≡ Student ⊓ Undergraduate Master ≡ Student ⊓  Undergraduate PhD ≡ Master ⊓ Research Assistant ≡ PhD ⊓ Teach TBox T

Disjointness Suppose we describe the students/attendees in a course: Are Bachelor and master disjoint? T ⊨ Bachelor ⊓ Master ⊑ ⊥ (Student ⊓ Undergraduate) ⊓ (Student ⊓  Undergraduate) Student ⊓ (Undergraduate ⊓  Undergraduate) ≡ ⊥ Undergraduate ⊑  Teach Bachelor ≡ Student ⊓ Undergraduate Master ≡ Student ⊓  Undergraduate PhD ≡ Master ⊓ Research Assistant ≡ PhD ⊓ Teach TBox T

Normalization of a TBox Normalize the TBox below: MonkeyLow ⊑  GetBanana GetBanana ≡ Survive Possible solution: MonkeyLow ≡  GetBanana ⊓  ClimbBox 23

Expansion of a TBox Expand the TBox below: MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive T’, expansion of T (The Venn diagram gives a possible model): MonkeyLow ≡  Survive ⊓  ClimbBox Survive GetBanana MonkeyLow ClimbBox Notice that the fact that a monkey climbs the box does not necessarily mean that it survives. 24

ABOX REASONING

ABox: Consistency Check the consistency of A w.r.t. T via expansion. T MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive A MonkeyLow(Cita) Survive(Cita) Expansion of A is consistent: MonkeyLow(Cita)  GetBanana(Cita) ClimbBox(Cita)  Survive(Cita) 26

ABox: Instance checking Given T and A below Is Cita an instance of MonkeyLow? YES Is Cita an instance of ClimbBox? NO Is Cita an instance of GetBanana? T MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive A MonkeyLow(Cita) Survive(Cita) Expansion of A MonkeyLow(Cita)  GetBanana(Cita) ClimbBox(Cita)  Survive(Cita) GetBanana(Cita) 27

Instance Retrieval. Consider the following expansion… Undergraduate ⊑  Teach Bachelor ≡ Student ⊓ Undergraduate Master ≡ Student ⊓  Undergraduate PhD ≡ Master ⊓ Research Assistant ≡ PhD ⊓ Teach A Master(Chen) PhD(Enzo) Assistant(Rui) The expansion of A Master(Chen) Student(Chen) Undergraduate(Chen) PhD(Enzo) Master(Enzo) Research(Enzo) Student(Enzo) Undergraduate(Enzo) Assistant(Rui) PhD(Rui) Teach(Rui) Master(Rui) Research(Rui) Student(Rui) Undergraduate(Rui) 28

Instance Retrieval. … find the instances of Master Undergraduate ⊑  Teach Bachelor ≡ Student ⊓ Undergraduate Master ≡ Student ⊓  Undergraduate PhD ≡ Master ⊓ Research Assistant ≡ PhD ⊓ Teach A Master(Chen) PhD(Enzo) Assistant(Rui) The expansion of A Master(Chen) Student(Chen) Undergraduate(Chen) PhD(Enzo) Master(Enzo) Research(Enzo) Student(Enzo) Undergraduate(Enzo) Assistant(Rui) PhD(Rui) Teach(Rui) Master(Rui) Research(Rui) Student(Rui) Undergraduate(Rui) 29

ABox: Concept realization Find the most specific concept C such that A ⊨ C(Cita) Notice that MonkeyLow directly uses GetBanana and ClimbBox, and it uses Survive. The most specific concept is therefore MonkeyLow. T MonkeyLow ≡  GetBanana ⊓  ClimbBox GetBanana ≡ Survive A MonkeyLow(Cita) Survive(Cita) 30