1. Logics for Data and Knowledge Representation Exercises: Modeling Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. Outline  Modeling  Logical Modeling  Exercises with intensional models  Forest  Exercises with extensional models  Classroom  Family  My friends  Databases 2

3. Modeling: from the world to its representation 3 World Language L Theory T Domain D Model M Data Knowledge Meaning Mental Model SEMANTIC GAP MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

4. What and How  World: the phenomenon we want to describe  Domain: the abstract relevant elements in the real world  Mental Model: what we have in mind. It is the first abstraction of the world (subject to the semantic gap)  Language: the set of words and rules we use to build sentences used to express our mental model  Model: the formalization of the mental model, i.e. the set of true facts in the language, in agreement with the theory  Theory: the set of sentences (constraints) about the world expressed in the language that limit the possible models  NOTE: this does not necessarily need to be in formal semantics 4 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

5. Example of informal Modeling 5 World Mental Model SEMANTIC GAP Model M L: Informal description in NL D: {monkey, banana, tree} T: If the monkey climbs on the tree, he can get the banana M: The monkey actually climbs on the tree and gets the banana Theory T NOTE: a database can be seen as an informal model Language L Domain D MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

6. Logical Modeling 6 Modeling Realization World Language L Theory T Domain D Model M Data Knowledge Meaning Mental Model SEMANTIC GAP Interpretation I Entailment ⊨ NOTE: the key point is that in logical modeling we have formal semantics MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

7. What and How  World: the phenomenon that we are observing and want to model  Domain (D) = the abstract relevant objects from the world  Language (L) = a logical language with formal syntax and semantics:  The formal syntax is given by the set of rules to construct complex sentences (the grammar)  The formal semantics is given by the interpretation function I: L → D  Model (M) = the abstract (mathematical sense) representation of the intended truths via the interpretation I of the language L.  M is called L-model of D  M ⊨ P, indicates that M satisfies P  Theory (T) = the set of facts/constraints expressed in the language L.  A fact defines a piece of knowledge (about D), something true in the model. 7 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

8. Example of formal (intentional) Modeling 8 World SEMANTIC GAP L = {Monkey, Climbs, GetBanana, , ,  } D= {T, F} T = {  (Monkey  Climbs)  GetBanana} A possible model M: I(Monkey) = T I(Climbs) = T I(GetBanana) = T Mental Model M Theory T Language L Domain D MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

9. Example of formal (extensional) Modeling 9 World SEMANTIC GAP L = {Monkey, Climbs, GetBanana, , ,  } D= {Cita, ThatBanana} T = {  Climbs  GetBanana} A possible model M: I(Monkey) = Cita I(Climbs) = Cita I(GetBanana) = ThatBanana Mental Model M Theory T Language L Domain D MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

10. Modeling Exercise: Forest  Description: There are two lions, Kimba and Simba, in the forest. They are in competition for the food. There is a nice antelope they want to hunt. If they want to survive they have to catch it.  Problem: Model the problem by identify relevant objects, defining the domain, the language, the theory and providing a possible intentional model. 10 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

11. Solution: Forest (I)  Description: There are two lions, Kimba and Simba, in the forest. They are in competition for the food. There is a nice antelope they want to hunt. If they want to survive they have to catch it. Relevant objects are in red D = {T, F} L = {Lion, Antelope, Survive, Catch} 11 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

12. Solution: Forest (II)  A possible model: I (Lion) = TI (Antelope) = T I (Catch) = TI (Survive) = T  The theory T: Antelope  (Catch   Survive) Antelope   Catch  I above is a model for T  I below is NOT a model for T I (Lion) = TI (Antelope) = F I (Catch) = FI (Survive) = T 12 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

13. Modeling Exercise: Classroom  Description: In a class there are several persons. Usually there is one professor who teaches to some students. Students can be Master students or PhD students. Among PhD students there might be some Assistants of the professor.  Problem: Model the problem by identify relevant objects, defining the domain and the language, and providing a possible extensional model for it. 13 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

14. Solution: Classroom (I)  Description: In a class there are several persons. Usually there is one professor who teaches to some students. Students can be Master students or PhD students. Among PhD students there might be some Assistants of the professor. Relevant objects are in red L = {Person, Professor, Student, Master, PhD, Assistant} D = {Fausto, Mary, Paul, Jane} 14 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

15. Solution: Classroom (II)  The corresponding Venn diagram 15 Person Student Professor PhD Master Assistan t MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS U

16. Solution: Classroom (III)  A possible model: I (Person) = {Fausto, Mary, Paul, Jane} I (Professor) = {Fausto} I (Student) = {Mary, Paul, Jane} I (Master) = {Mary} I (PhD) = {Paul, Jane} I (Assistant) = {Paul} 16 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

17. Modeling Exercise: Family  Description: My family consists of several members. There is a grandparent and my parents. Then there are some children, i.e. two sisters, one brother and me  Problem: Model the problem by identify relevant objects, defining the domain and the language, and providing a possible extensional model for it. 17 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

18. Solution: Family (I)  Description: My family consists of several members. There is a grandparent and my parents. Then there are some children, i.e. two sisters, one brother and me Relevant objects are in red L = {member, grandparent, parent, child, brother, sister, me} D = {Bob, Fausto, Mary, Paul, Jane, Hugo, Robert} 18 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

19. Solution: Family (II) 19 Member Parent Grandparent Brother Sister Child  The corresponding Venn diagram Me MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS U

20. Solution: Family (III)  A possible model: I (Member) = {Bob, Fausto, Mary, Paul, Jane, Hugo, Robert} I (Grandparent) = {Bob} I (Parent) = {Fausto, Mary, Bob} I (Brother) = {Robert, Paul} I (Sister) = {Jane} I (Me) = {Hugo} 20 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

21. Modeling Exercise: My friends  Description: I have a lot of friends. I met some of them on the forum of my website. However, only a few of them are close to me. In particular, I use to play chess with Paul.  Problem: Model the problem by identify relevant objects, defining the domain and the language, and providing a possible extensional model for it. 21 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

22. Solution: My friends (I)  Description: I have a lot of friends. I met some of them on the forum of my website. However, only a few of them are close to me. In particular, I use to play chess with Paul. Relevant objects are in red L = {Friend, Forum, Close, PlayingChess} D = {Bob, Fausto, Mary, Paul, Jane, Hugo, Robert} 22 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

23. Solution: My friends (II) 23 Friend Forum  The corresponding Venn diagram Close PlayingChess MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS U

24. Solution: My friends (III)  A possible model: I (Friend) = {Bob, Paul, Jane, Robert, Richard, Samuel} I (Forum) = {Bob, Paul, Jane} I (Close) = {Bob, Paul, Samuel} I (PlayingChess) = {Paul} 24 MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS

25.  Closed world assumption (CWA): The assumption that what is not currently known to be true, is false. I (Italian) = {Fausto, Enzo} I (BlackHair) = {Enzo, Rui} …  Open world assumption (OWA): anything which is not explicitly asserted is unknown. Is Rui Italian? This is not asserted in the DB, therefore it is unknown. A Database 25 IDNameNationalityHair ColorAffiliation 1FaustoItalianWhiteProfessor 2EnzoItalianBlackPhD 3RuiChineseBlackAssistant 4… 5… …… Class Italian BlackHair PhD MODELING :: LOGICAL MODELING :: INTENTIONAL MODELS :: EXTENSIONAL MODELS