L ogics for D ata and K nowledge R epresentation

1. Logics for Data and KnowledgeRepresentation Modeling First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

2. Outlines • Modeling • Logical Modeling • Domain • Language • Interpretation • Model • Theory • Expressiveness • Expressiveness VS. Efficiency • Efficiency VS. Complexity • Decidability • Algorithm First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

3. Modeling Language L Theory TH Data Knowledge World Model Domain D Model M Meaning Semantic Gap First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

4. Outlines • Modeling • Logical Modeling • Domain • Language • Interpretation • Model • Theory • Expressiveness • Expressiveness VS. Efficiency • Efficiency VS. Complexity • Decidability • Algorithm First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

5. Logical Modeling First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

6. Logical Modeling Elements The basic elements in the diagram are:1. Domain (objects, classes, relations, functions)2. Logical Language 3. Interpretation (of the language)4. Model5. Theory / Knowledge Base (data and knowledge)6. Truth-relation / logical entailment (|=) (deduction reasoning, inference) We illustrate each element in turn. First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

7. Domain • Domain (D) = the chosen objects from the world • Example (LDKR class): the members of the LDKR class define a domain D; • D is a finite set. • The “type” of the elements in D is: person. We will deal only with finite domains! Question: what are we leaving out? First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

8. Language • Language (L) = a logical language 1. L’s alphabet of symbolsΣ contains at least one of the logical symbols: ∧, ∨, ¬, →, ∀, ∃; 2. L has clear formation rules for formulas. • Example (cont’): any logical language with =, (, ), professor, student1, student2,... in Σ. • Note that English can be restricted to a propositional language, ...but it is not logical(informal syntax). First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

9. Logical Language (Syntax) • The first step in setting up a logical language (viz. a formal language) is to list the symbols, that is, the alphabet of (formal) symbols (Σ). • formal symbol = a character, or group of characters taken from some alphabet. • Symbols in Σ can be divided in ‘descriptive’ (non-logical) and ‘non-descriptive’ (logical). First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

10. Example (Monkey-Bananas Problem) • In the sentence “There is a monkey in a laboratory with some bananas”: • Descriptive symbols are:‘monkey’, ‘laboratory',‘bananas’, ... • Non-descriptive symbols are: ‘there is’, ‘with some’, ‘a’, ‘in’, ... First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

11. Formal Syntax • Formal Syntax: the set of “rules” saying how to construct the expressions of the language from the alphabet of symbols, (i.e., the syntax) is a grammar (i.e., formal). • Formal syntax is often called an abstractsyntax, in contrast to the concretesyntax used, e.g., in implementations. • Example: context-free grammars. First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

12. Interpretation • Interpretation (I) = a mapping of L into D. • I must be effective (i.e., computable) . • Exampe 1: D = {Fausto, Aundrey} • I(prof1) = Fausto, • I(studen2) = Audrey • Example: 2: D = <T,F> • I(HighBox) = T • I(canTakeExam) = F … First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

13. Model • Model (M) = the abstract (mathematical sense) representation of the intended truthsvia interpretation I of language L. M is called L-model of D. M |=A to be read as: M satisfies A A holds in M A is TRUE in M M yields A • Example (cont’): D= <T,F>, M such that • I(HighBox) = T then M |= Highbox • I(canTakeExam) = F then M |= ¬ canTakeExam ... First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

14. Theory • Theory T (also L-Theory) = set of facts of L. • A fact defines a piece of knowledge (about D), something true in the model. • A finite theory T is called a knowledge base (KB). • A database (DB) is the simplest kind of knowledge base. • Example (cont’): T = {¬ canTakeExam, HighBox}. We have: M |= T iff M |=A, for each Ai in T … A model M of a theory T is an interpretation function that satisfies all the facts in T First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

15. What is a Logic? Logic = <L, I, |= >where • L: Language = set of phrases/ formulas (alphabet + formation rules) • I: Interpretation Function (what phrases mean in a chosen domain D) I: L -> D • |= : how to compute the fact that a formula A holds in M (Ais TRUE in M), notationally M |= A with A |= B in M iff M|=A implies M |= B First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

16. How to use logics • Define a logic (most often by reseachers), once for all (not a trivial task!) • Given a problem the computer scientist must choose the right logic, most often one of the many available • The computer scientist writes a theory T • The computer scientist uses reasoning services to solve her programs First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

17. Reasoning Services Model Checking (EVAL): Is a sentence ψ true in model M? Check M |= ψ Satisfiability (SAT): Is there a model M where ψ is true? find M such that M |= ψ Validity(VAL): Is ψ true according to all possible models check whether for all M, M |= ψ Entailment(ENT): ψ1 true in M (all models) implies ψ2 is true in M (all models). check A |= B in M by checking M|=A implies M |= B First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

18. Outlines • Logical Modeling • Domain • Language • Interpretation • Model • Theory • Expressiveness • Expressiveness VS. Efficiency • Efficiency VS. Complexity • Decidability • Algorithm First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

19. 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 trading off expressiveness in the language for more tractable forms of reasoning services. First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

20. Examples • Propositional: • “I like skiing” : I-like-skiing • “Fausto likes skiing”: Fausto-like-skiing • Modal: • “I believe I like skiing”: B(I-like-skiing), BI(like-skiing) • First-order: • “Every person likes skiing”: ∀person.like-skiing(person) • “I like skiing” : like-skiing (I) • “Fausto likes skiing” : like-skiing(Fausto) • Description Logics • “Every person likes cars”: person < exists likes. Car • … First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

21. Efficiency VS. Complexity • Efficient (Webster). Performing in the best possible manner; satisfactory and economical to use. • In modeling it applies to reasoning; • In this case we use the more specific terminology computational complexity (time, space,...) in place of term efficiency. • Complexity: (or computational complexity) of reasoning is the difficulty to compute areasoning task expressed by using a logic. • with degrees of expressiveness, we may classify the logical languages according to some “degrees of complexity”. First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

22. Degrees of Complexity (NEW) • The basic “degreesofcomplexity” are: • Here comes the axis of complexity First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

23. Decidability First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

24. What is an Algorithm? • Decidability is strictly related to the notion of “algorithm.” There were two main approaches, and many equivalent definitions: • K. Gödel (1934): recursive functions.Alonzo Church (1936): lambda-calculus.S.C. Kleene (1936): functional equations. • Alan M. Turing (1936-37): Turing machines.Emil L. Post (1936): Post machines. Functional approaches Computational approaches First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang

25. Decision Procedures and Decidable Logics • A decisionprocedure is an algorithm that, given a decision problem, terminates with the correct yes/no answer. • A logic is decidable if there exists a decision procedure for that logic. • In this course we focus on logics that are expressive enough to model real problems but are still decidable. First version by Alessandro Agostini and Fausto Giunchiglia Second version by Fausto Giunchiglia and Rui Zhang