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Mathematical Logic

Mathematical Logic. Data and Knowledge Representation. Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese. Outline. Modeling and logical modeling Domain Language Theory Model Languages Logic: formal languages

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Mathematical Logic

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  1. Mathematical Logic Data and Knowledge Representation Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

  2. Outline • Modeling and logical modeling • Domain • Language • Theory • Model • Languages • Logic: formal languages • Using Logic • Specification • Automation • Expressiveness • Efficiency VS. Complexity • Decidability

  3. Modeling the world MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC A model is an abstraction of a part of the world INDIVIDUAL SET RELATION (e.g. Near, Sister) DOMAIN (e.g. the girls in foreground)

  4. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Modeling Language L Theory T Data Knowledge World Mental Model Domain D Model M Meaning SEMANTIC GAP

  5. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Modeling: explained • 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 • Theory: the set of sentences (constraints) about the world expressed in the language that limit the possible models • Model: the formalization of the mental model, i.e. the set of true facts in the language, in agreement with the theory • Semantic gap: the impossibility to capture the entire reality NOTE: this does not necessarily need to be in formal semantics

  6. Mental models MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC There is a monkey in a laboratory with some bananas hanging out of reach from the ceiling. A box is available that will enable the monkey to reach the bananas if he climbs on it. The monkey and box have height Low, … … but if the monkey climbs onto the box he will have height High, the same as the bananas. [...]” NOTE: each picture is a different model

  7. Example of informal Modeling MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Mental Model Language L Domain D Theory T Model M World L: natural language D: {monkey, banana, tree} T: “If the monkeyclimbs on the tree, he can get the banana” M: “The monkey actually climbs on the tree and gets the banana” SEMANTIC GAP

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

  9. Logical Modeling: explained MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC • 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 • Theory / Knowledge Base (KB): the set of facts which are always true (in data and knowledge) • Model: the set of true facts in the language describing the mental model (the part of the world observed), in agreement with the theory • Truth-relation / logical entailment (⊨): deduction, reasoning, inference

  10. Intensional vs Extensional semantics MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC “Bananas are yellow” “Bananas have a curve shape” “All men are mortal” man  mortal GirlsPlaying = True • Extensional: I provides the objects of the domain corresponding to the proposition “The author of Romeo and Juliet is Shakespeare” author(R&G, Shakespeare) GirlsPlaying = {Mary, Sara, Julie} • Intensional: I can only express the fact that a given proposition is true or false

  11. Example of formal (intentional) modeling MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Mental Model Language L Domain D Theory T Model M World L = {ThereIsMonkey, MonkeyClimbsTree, GetBanana, , , } D= {T, F} T = { (ThereIsMonkey MonkeyClimbsTree) GetBanana} A possible model M: I(ThereIsMonkey) = T I(MonkeyClimbsTree) = T I(GetBanana) = T SEMANTIC GAP

  12. Example of formal (extensional) modeling MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Mental Model Language L Domain D Theory T Model M World L = {Monkey, Climbs, GetBanana, , , } D= {Cita, Banana#1, Tree#1} T = { (Monkey Climbs) GetBanana} A possible model M: I(Monkey) = {Cita} I(Climbs) = {(Cita, Tree#1)} I(GetBanana) = {(Cita, Banana#1)} SEMANTIC GAP

  13. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Language • A (usually finite) set of words (the alphabet) and rules to compose them to build “correct sentences” • e.g. Monkey and GetBanana are words • e.g. Monkey  GetBanana is a sentence (rule: A  B) • A tool for codifying our (mental) model (what we have in mind): • Sentences (syntax) with an intended meaning (semantics) • e.g. with the word Monkey we mean

  14. Language and correctness Yes, correct! PARSER s, L No MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC • We need an algorithm for checking correctness of the sentences in a language • We say that a language is decidable if it is possible to create such a tool that in finite time can take the decision

  15. Syntax and Semantics MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC • Syntax: the way a language is written: • Syntax is determined by a set of rules saying how to construct the expressions of the language from the set of atomic tokens (i.e., terms, characters, symbols). • The set of atomic tokens is called alphabet of symbols, or simply the alphabet). • Semantics: the way a language is interpreted: • It determines the meaning of the syntactic constructs (expressions), that is, the relationship between syntactic constructs and the elements of some universe of meanings (the intended model). • Such relationship is called interpretation.

  16. Example of Syntax and Semantics MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC • Suppose we want to represent the fact that Mary and Sara are near each other.

  17. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Formal vs. Informal languages • Language = Syntax(what we write)+ Semantics(what we mean) • Formal syntax • Infinite/finite (always recognizable) alphabet • Finite set of formal constructors and building rules for phrase construction • Algorithm for correctness checking (a phrase in a language) • Formal Semantics • The relationship between syntactic constructs in a language L and the elements of an universe of meanings D is a (mathematical) function I: L  pow(D) • Informal syntax/semantics • the opposite of formal, namely the absence of the elements above.

  18. Levels of Formalization MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Both Syntax and Semantics can be formal or informal. ProgrammingLanguages Diagrams NL Logics Level1 Leveln PLFOLDL... EnglishItalianRussianHindi... SQL... ERUML... FORMALITY semi-formal informal formal 18

  19. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Informal languages: natural language • Let us try to recognize relevant entities, relations and properties in the NL text below • The Monkey-Bananas (MB) problem by McCarthy, 1969 “There is a monkey in a laboratory with some bananas hanging out of reach from the ceiling. A box is available that will enable the monkey to reach the bananas if he climbs on it. The monkey and box have height Low, but if the monkey climbs onto the box he will have height High, the same as the bananas. [...]” • Question: How shall the monkey reach the bananas?

  20. 0..1 0..n Climb Monkey Box Height Banana MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Languages with formal syntax • In the Entity-Relationship (ER) Model [Chen 1976] the alphabet is a set of graphical objects, that are used to construct schemas (the sentences). Relation Attribute Entity • Examples of ER sentences:

  21. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Languages with formal syntax and semantics • Logics has two fundamental components: • L is a formal language (in syntax and semantics) • I is an interpretation function which maps sentences into a formal model M (over all possible ones) with a domain D • Domain D = {T, F} or D’ = {o1, …, on} • Language L= {A,,,} • Interpretation I: L  D is intensional or I: L  pow(D’) is extensional • Theory: a set of sentences which are always true in the language (facts) • Model: the set of true facts in the language describing the mental model (the part of the world observed), in agreement with the theory

  22. Theory MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC A theory T is set of facts: • A fact written in the language L defines a piece of knowledge (something true) about the domain D • A theory can be seen as a set of constraints on possible models to filter out all undesired ones

  23. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Data and Knowledge • Knowledge is factual information about what is true in general (axioms and theorems) e.g. singers sing songs • Data is factual information about specific individuals (observed knowledge) e.g. Michael Jackson is a singer A finite theory T is called: • An ontology if it contains knowledge only • A knowledge base (KB) if it contains knowledge and data • A database (DB) if it only contains data. • A database + its schema is the simplest kind of knowledge base NOTE: Sometimes the terms Ontology and KB are used interchangeably

  24. Model MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC A model M is an abstract (in mathematical sense) representation of the intended truths via interpretation I of the language L e.g. Madonna is a singer, Sting is a singer, … M ⊨ T (M entails 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 (M satisfies 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.

  25. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Usages of Representation Languages The two purposes in modeling • Specification: • Representation of the problem at the proper level of abstraction • Allow informal/formal syntaxandinformal/formal semantics • Automation (Automated Reasoning) • Computing consequences or properties of the chosen specifications • It requires formal semantics. • Logics have formal syntax and formal semantics!

  26. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Why Natural Languages?

  27. Why Diagrams? MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC 27

  28. Why Logic? MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC 28

  29. Logics for Automation: reasoning MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC 29 • Logics provides a notion of deduction • axioms, deductive machinery, theorem • Deduction can be used to implement reasoners • Reasoners allow inferring conclusions from known facts (i.e., a set of “premises”, premises can be axioms or theorems). • From implicit knowledge to explicit knowledge • Reasoning services: • Model Checking (EVAL) Is a sentence ψ true in model M? • Satisfiability (SAT) Is there a model M where ψ is true? • Validity (VAL) Is ψ true according to all possible models? • Entailment (ENT) ψ1true implies ψ2 true (in all models)

  30. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC How to use logical modeling in practice • Define a logic • most often by reseachers • once for all (not a trivial task!) • 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 • Write the theory • The computer scientist writes a theory T • Use reasoning services • Computer scientists use reasoning services to solve their programs 30

  31. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC 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.

  32. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Examples of Expressiveness

  33. MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Efficiency VS. Complexity • Efficiency • Performing in the best possible manner; satisfactory and economical to use [Webster] • In modeling it applies to reasoning • In time, space, consumption of resources • Complexity(or computational complexity) of reasoning • 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” NOTE: When logic is used we always pay a performance price and therefore we use it when it is cost-effective

  34. 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 decisionprocedure 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 MODELING :: LOGICAL MODELING :: LANGUAGES :: LOGIC :: USING LOGIC Decidability

  35. Appendix: Greek letters 35

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