1. Logics for Data and Knowledge Representation Modeling Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. 2 Outline  Modeling  Logical Modeling (formal modeling)  Domain  Language  Interpretation  Theory  Model  How to use logical modeling  What is a logic?  Choosing the right logic and writing the theory  Reasoning services  Expressiveness  Expressiveness, Efficiency, Complexity  Decidability 2

3. 3 Modeling 3 World Language L Theory T Domain D Model M Data Knowledge Meaning Mental Model SEMANTIC GAP MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

4. 4 Modeling  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 :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

5. 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 :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

6. 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 :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

7. 7 Logical Modeling Elements  Domain: relevant objects  Logical Language: the set of formal words and rules we use to build complex sentences  Interpretation: the function that associates elements of the language to the elements in the domain  Model: the set of true facts in the language describing the mental model, in agreement with the theory  Theory / Knowledge Base (KB): data and knowledge  Truth-relation / logical entailment ( ⊨ ): deduction, reasoning, inference 7 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

8. 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 :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

9. 9 Domain  Domain (D): the chosen objects from the world  We will deal only with finite domains!  Question: what are we leaving out? 9 Example: the LDKR class  The members of the LDKR class define a domain D  D is a finite set  Two “kinds” of the elements in D: Professor Student Both are specializations of Person Fausto Mary PaulJane HugoSaraRui MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

10. 10 Language  Language (L) = a logical language  To each logical language we associate an alphabet of symbols Σ (sigma)  For instance, Σ may contain the logical symbols:  ∧ (and)  ∨ (or)  ¬ (not)  → (implication)  ∀ (for all, universal quantifier)  ∃ (exists, existential quantifier)  L has clear formation rules for formulas 10 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

11. 11 Logical Language (Syntax)  The first step in setting up a logical language is to list the symbols  the alphabet of (formal) symbols Σ  formal symbol = a character, or group of characters taken from some alphabet (it is formal because we specify the meaning)  Symbols in Σ can be divided in:  descriptive (non-logical)  non-descriptive (logical)  NOTE: English can be restricted to a propositional language,...but it is not logical (informal syntax) 11 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

12. 12 Logical language 12  The Language L for the LDKR class example with alphabet Σ  Logical symbols: ∧, ∨, ¬, →  Descriptive symbols: Person, Professor, Student Fausto Mary PaulJane HugoSaraRui MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

13. 13 Formal Syntax  The set of rules saying how to construct the sentences of the language from the alphabet of symbols (i.e. the syntax) is a grammar (i.e., is formal)  Example: ¬A, A ∧ B, A → B ¬Professor, Professor ∧ Student, Student → Person  Formal syntax is often called an abstract syntax, in contrast to the concrete syntax used, for instance in implementations.  Example: context-free grammars 13 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

14. 14 Interpretation  Interpretation (I) = a mapping of L into D  I must be effective (i.e., computable) 14 Intensional interpretation I(Professor) = T I(Student) = T I(Person) = T Extensional interpretation I(Professor) = {Fausto} I(Student) = {Mary, Paul, Jane, Rui, Sara, Hugo} I(Person) = {Fausto, Mary, Paul, Jane, Rui, Sara, Hugo} Fausto Mary PaulJane HugoSaraRui MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

15. 15 Theory  Theory T (also L-Theory) = set of facts of L  A fact defines a piece of knowledge (about D), namely something true  A theory is a way to put constraints on the intended models 15 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

16. 16 Theory  A finite theory T is called:  An ontology if it contains knowledge only (T-BOX)  A knowledge base (KB) if it contains knowledge (T-BOX) and data (A-BOX)  A database (DB) if it only contains data.  A DB + its schema is the simplest kind of knowledge base NOTE: Sometimes the terms Ontology and Knowledge Base are used interchangeably 16 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

17. 17 Theory 17 The set of (true) facts T over L Student → Person Professor → Person Student → ¬ Professor Professor → ¬ Student Fausto Mary PaulJane HugoSaraRui MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

18. 18 Model  Model (M) = the abstract (in mathematical sense) representation of the intended truths via interpretation I of the language L  M is called L-model of D:M ⊨ T M satisfies T T holds in M T is TRUE in M M yields T with T set of arbitrary complex formulas 18 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

19. 19 Model 19 Intensional interpretation I(Professor) = T I(Student) = T I(Person) = T The I is a model for the theories below: M ⊨ {Person} M ⊨ {Professor ∨ Student} M ⊨ {Person, Professor ∨ Student, Student → Person} … Fausto Mary PaulJane HugoSaraRui MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

20. 20 Theory and Model  We have: M ⊨ T iff M ⊨ A, for each formula A in T  A model M of a theory T is any interpretation function that satisfies all the facts in T  There can be many models satisfying the theory T. They are a subset of all possible interpretation functions over L.  In case there are no models for T, we say that the theory T is unsatisfiable. 20 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

21. 21 Theory and Model 21 T Student → Person Professor → Person Student → ¬ Professor Professor → ¬ Student Fausto Mary PaulJane HugoSaraRui M I(Professor) = {Fausto} I(Student) = {Mary, Paul, Jane, Rui, Sara, Hugo} I(Person) = {Fausto, Mary, Paul, Jane, Rui, Sara, Hugo} M is a model for T (M ⊨ T ) M’ I(Professor) = {Fausto, Mary} I(Student) = {Mary, Paul, Jane, Rui, Sara, Hugo} I(Person) = {Fausto, Mary, Paul, Jane, Rui, Sara, Hugo} M’ is not, because Mary is both a student and a professor MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

22. 22 Same theory, different models (MODEL#1) L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , ,  } T = {  (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  (  MonkeyLow   BananaHigh  MonkeyGetBanana)} Informal Semantics: “If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “ Formal Semantics: I(MonkeyLow) = T I(BananaHigh) = T I(MonkeyClimbBox) = F I(MonkeyGetBanana) = F MODEL#1 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

23. 23 Same theory, different models (MODEL#2) L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , ,  } T = {  (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  (  MonkeyLow   BananaHigh  MonkeyGetBanana)} Informal Semantics: “If the monkey climbs onto the box, he becomes high in position (not low anymore) and can get the banana.” Formal Semantics: I(MonkeyLow) = F I(BananaHigh) = T I(MonkeyClimbBox) = T I(MonkeyGetBanana) = T MODEL#2 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

24. 24 Same theory, different models (MODEL#3) L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , ,  } T = {  (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  (  MonkeyLow   BananaHigh  MonkeyGetBanana)} Informal Semantics: “If the monkey is low and the banana is also low, then the monkey can get the banana. “ Formal Semantics: I(MonkeyLow) = T I(BananaHigh) = F I(MonkeyClimbBox) = F I(MonkeyGetBanana) = T MODEL#3 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

25. 25 How to use logical modeling 25 1. Define a logic  most often by reseachers  once for all (not a trivial task!) 2. Choose the right logic for the problem  Given a problem the computer scientist must choose the right logic, most often one of the many available 3. Write the theory  The computer scientist writes a theory T 4. Use reasoning services  The computer scientist uses reasoning services to solve her programs MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: EXPRESSIVENESSMODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

26. 26 What is a Logic?  Logic = where L Language Set of phrases/sentences/formulas (alphabet + formation rules) I Interpretation function What phrases mean in a chosen domain D with I: L -> D ⊨ Satisfiability relation (over M) How to compute the fact that a formula A is TRUE in M, notationally M ⊨ A A ⊨ B in M iff M ⊨ A implies M ⊨ B 26 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

27. 27 Choose the right logic for the problem 27 Problem: the LDKR class Specification: “In the LDKR class Fausto is the professor. There are 6 students. They are Mary, Paul, Jane, Rui, Hugo and Sara”. Formalization:  We want to represent both classes of objects (Professor, Student) and individuals (Mary, Paul…)  Choose a logic that allows for them, e.g. ClassL with Individuals Fausto Mary PaulJane HugoSaraRui L = {Professor, Student, ∧, ∨, ¬, → } D = {Fausto, Mary, Paul, Jane, Rui, Sara, Hugo} MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

28. 28 Define the theory for the problem 28 L = {Professor, Student, ∧, ∨, ¬, → } D = {Fausto, Mary, Paul, Jane, Rui, Sara, Hugo} In the theory we want to model the fact that the set of professors is always disjoint from the set of students… Fausto Mary PaulJane HugoSaraRui T = {Student → ¬ Professor; Professor → ¬ Student} M I(Professor) = {Fausto} I(Student) = {Mary, Paul, Jane, Rui, Sara, Hugo} M ⊨ T Then we can use reasoning services to handle our problem MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

29. 29 Reasoning Services: EVAL Model Checking (EVAL) Is a sentence ψ true in model M? Check M ⊨ ψ EVAL ψ, M Yes, M ⊨ ψ No, M ⊭ ψ MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

30. 30 EVAL L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , ,  } T = {  (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  (  MonkeyLow   BananaHigh  MonkeyGetBanana)} I(MonkeyLow) = T I(BananaHigh) = T I(MonkeyClimbBox) = F I(MonkeyGetBanana) = F Evaluate ψ 1 and ψ 2 in M ψ 1 =  MonkeyClimbBox is true in M (YES) ψ 2 = MonkeyGetBanana is false in M (NO) M MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

31. 31 Reasoning Services: SAT Satisfiability (SAT) Is there a model M where ψ is true? find M such that M ⊨ ψ SAT ψ M No MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

32. 32 SAT L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , ,  } T = {  (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  (  MonkeyLow   BananaHigh  MonkeyGetBanana)} I(MonkeyLow) = F I(BananaHigh) = T I(MonkeyClimbBox) = T I(MonkeyGetBanana) = T ψ = MonkeyGetBanana Is there a model M where ψ is true? (YES, the model on the left) SAT is a search problem (find a model M) MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

33. 33 Reasoning Services: VAL Validity (VAL) Is ψ true according to all possible models? Check whether for all M, M ⊨ ψ VAL ψ Yes No MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

34. 34 VAL L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , ,  } T = {  (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  (  MonkeyLow   BananaHigh  MonkeyGetBanana)} I(MonkeyLow) = F I(BananaHigh) = T I(MonkeyClimbBox) = T I(MonkeyGetBanana) = F ψ = MonkeyLow  MonkeyGetBanana Is ψ true according to all possible model M? (No, the monkey can be high without getting the banana) VAL is a search problem (check ψ in all M) NOTE: It is enough to find a counterexample to conclude for non validity MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

35. 35 Reasoning Services: ENT Entailment (ENT) ψ 1 true in M (all models) implies ψ 2 true in M (all models) check A ⊨ B in M by checking M ⊨ A implies M ⊨ B ENT ψ 1, ψ 2, M Yes, ψ 1 ⊨ ψ 2 No MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

36. 36 ENT L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , ,  } T = {  (MonkeyLow  BananaHigh  MonkeyGetBanana)  (MonkeyLow  MonkeyClimbBox)  (  MonkeyLow   BananaHigh  MonkeyGetBanana)} ψ1 = MonkeyClimbBox ψ2 =  MonkeyLow ψ1 true in M (all models) implies ψ2 is true in M (all models). (Yes) NOTE: ψ 1 entails ψ 2 in all models MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

37. 37 An Important Trade-Off  There is a trade-off between:  expressive power (expressiveness) and  computational efficiency provided by a (logical) language  This trade-off is a measure of the tension between specification and automation  To use logic for modeling, the modeler must find the right trade off between expressiveness in the language for more tractable forms of reasoning services. 37 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

38. 38 Examples of Expressiveness 38 LANGUAGENL SENTENCEFORMULA Propositional logic Fausto likes skiing I like skiing Fausto-likes-skiing I-like-skiing Modal logic I believe I like skiingB(I-like-skiing) First-order logic Every person likes skiing I like skiing Fausto likes skiing ∀ person.like-skiing(person) like-skiing(I) like-skiing(Fausto) Description Logic Every person likes cars person ⊑ ∃ likes.Car … MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

39. 39 Efficiency VS. Complexity  Efficiency  Performing in the best possible manner; satisfactory and economical to use [Webster]  In modeling it applies to reasoning  We use the more specific term computational complexity (time, space,...)  Complexity (or computational complexity) of reasoning  It is the difficulty to compute a reasoning task expressed by using a logic  With degrees of expressiveness, we may classify the logical languages according to some “degrees of complexity” 39 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

40. 40 Degrees of Complexity 40  The more you specify, the more cost grows COST & PRECISION OF THE SPECIFICATION Natural Language Diagram Logics FORMALITY MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

41. 41 When is the use of logics appropriate?  When logic is used we always pay a performance price  We therefore use it when it is cost-effective  To prove correctness (offline use)  To draw conclusions (online use) MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

42. 42 Examples of offline use (specification)  For data and knowledge representation  In safety-critical applications  Trains  Planes  …  In security critical applications  bank transactions  … MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

43. 43 Examples of online use (reasoning)  To let programs interoperate Web 1.0 Web 2.0 LET PEOPLE INTEROPERATE (informal semantics) Web 3.0 LET PROGRAMS INTEROPERATE (formal semantics) MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS

44. 44 Decidability  The existence of an effective method to determine the validity of formulas in a logical language  A logic is decidable if there is an effective method to determine whether arbitrary formulas are included in a theory  A decision procedure is an algorithm that, given a decision problem, terminates with the correct yes/no answer.  In this course we focus on logics that are expressive enough to model real problems but are still decidable 44 MODELING :: LOGICAL MODELING :: HOW TO USE LOGICAL MODELING :: REASONING SERVICES :: EXPRESSIVENESS