L ogics for D ata and K nowledge R epresentation

1. Logics for Data and KnowledgeRepresentation Exercise 1: Model and Language

2. Outline • Modeling • Logical Modeling • What and How • Exercises • Languages • BNF • Exercises

3. Modeling (Abstraction)Modeling Representation Model Language + Theory The World Realization Interpretation Monkey Banana Box …

4. Logical Modeling

5. What? • Domain (D) = the chosen objects from the world who can figure out the domain of the LDKR course? • From the person point of view: • students, professor; • Italian, Chinese, … • white-haired, black-eyed,… • From the material point of view: • courseware, homework, exam,… • logics, modeling, … • From … ?

6. What else? • Language (L) = a logical language • Syntax • L’s alphabet of symbolsΣ contains at least one of the logical symbols: ∧, ∨, ¬, →, ∀, ∃; • L has clear formation rules for formulas. • 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). • Semantics • Interpretation (I) = a mapping of L into D. • Formal Semantics: the relationship between syntactic constructs and the elements of an universe of meanings is a function in mathematical sense.

7. How? • Model (M) = the abstract (mathematical sense) representation of the intended truths via interpretation I of language L. M is called L-model of D. • M |=A • reads? • satisfies, yields, holds, is true. • 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).

8. Modeling Exercises • Select from the following domain to model (5 minutes preparation) • Classroom Student, Master & Doctor, Professor, Assistant,… • Family Parent, Grandparent, Male, Female, Sibling,… • Friend Close, Hiking, Chess, Forum, …

9. Possible Solutions 1 • Classroom Person Student PhD Professor Master

10. Possible Solutions 2 • Family Male Female Grandparent Parent Brother Sibling Sister

11. Possible Solutions 3 • Friend Friend Chess Hiking Close Forum

12. A Database • Let’s look at this sheet in a DB: • What’s it like? LDKR Italian Black Hair Master

13. Closed world vs. Open world • DB follows CWA, which assumes negative when no record found. • Closed word assumption (CWA) is the presumption that what is not currently known to be true, is false. • In contrast, ClassL assumes OWA, which allows ‘new’ knowledge emerges. • Open world assumption (OWA) is the assumption that the truth-value of a statement is independent of whether or not it is known by any single observer or agent to be true. • NOTE: In general no single agent or observer has complete knowledge, and therefore cannot make the closed world assumption.

14. Example • Recall the DB table in previous slide: • A theory of this world in ClassL: T={}, A={Italian(Fausto), Italian(Enzo), Chinese(Rui), White-Hair(Fausto), Black-Hair(Enzo), Black-Hair(Rui), Professor(Fausto), PhD(Enzo), TA(Rui), …}

15. Outline • Modeling • Logical Modeling • What and How • Exercises • Languages • BNF • Exercises

16. Backus–Naur Form (BNF) • In computer science, Backus–Naur Form (BNF) is a syntax used to express context-free grammars: that is, a formal way to describe formal languages. • Optional items enclosed in square brackets []. • Items repeating 0 or more times are enclosed in curly brackets or suffixed with an asterisk. {} or * • Items repeating 1 or more times are followed by a '+' • Terminals may appear in bold and NonTerminals in plain text rather than using italics and angle brackets <>. • Alternative choices in a production are separated by the ‘|’ symbol. • Where items need to be grouped they are enclosed in simple parentheses ().

17. Example of BNF • Who can give examples of the above syntaxes? • An example of mathematical expression • <expression> ::=<value> [<operator> <expression>] • <value  > ::= [<sign>] <unsigned> [ . <unsigned>] • <unsigned> ::=<digit> {<digit>}* • <digit> ::=0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 • <sign> ::=+ | - • <operator> ::=+ | - | * | /

18. Exercises of BNF • Is the following a well-formed formula of expression? • 00123 • 199+299 • +20*200 • 345/(123+456-789) • Recall the BNF of PL, and differentiate WFFs from others below. • A⊔B⊒A • A⊓B⊑B • A∧¬B→A • A∧B⊨A∨B • ¬A∨B⊢A→B

19. Something challenging • Can we build the BNF of ER diagram? • What about the BNF of natural language of English?

20. Recall: ER Diagram • In software engineering, an Entity-Relationship Model (ERM) is an abstract and conceptual representation of data. • The basic components of ER in Lecture 2: • Entity • Relation • Cardinality of Relation • Cardinality of Attribute • Attribute • Primary Key 0..1 Climb 0..n Monkey Box Height Banana ID

21. BNF of ER Diagram • Build the Backus–Naur Form (BNF) of ER diagram system. <Entity>::= <Relation>::= <Attribute>::= <Connector>::= <Diagram>::={<Entity>}| <Entity>+[<connector ><Relation><connector>]<Entity>+| Entity+[<connector>< Attribute>] Entity Relation Attribute <Unsigned>”.. “ n | m

22. BNF of Yahoo Directories • The Yahoo! Directory is an online guide to the World Wide Web. It is a catalog of sites created by a staff of editors who visit and evaluate web sites, and then organize them into subject-based categories and sub-categories. • Yahoo! editors distinguish between a number of factors when organizing web sites, including commercial vs. non-commercial, regional vs. global, and so forth. All of the site listings in the Directory are contained in an appropriate place within the 14 main categories seen on the front page of the Yahoo! Directory.

23. So take a look!

24. Preliminaries: Open/Close Word Class • In linguistics, an open class (or open word class) is a word class that accepts the addition of new items, through such processes as compounding, derivation, coining, borrowing, etc. Typical open word classes are nouns, verbs and adjectives. • A closed class (or closed word class) is a word class to which no new items can normally be added, and that usually contains a relatively small number of items. Typical closed classes found in many languages are adpositions (prepositions and postpositions), determiners, conjunctions, and pronouns.

25. Parts of Speech Open class WORD CLASSEXAMPLE JJ//Adjective blue green soft NN//Noun, singular or mass apple sugar NNS//Noun, plural apples NNP//Proper noun, singular Rui RB//Adverb slowly VB//Verb, base form go VBD//Verb, past tense went VBZ//Verb, 3rd person singular present goes

26. Parts of Speech (2) Closed class WORD CLASS EXAMPLE CC//Coordinating conjunction and or CD//Cardinal number DT//Determiner the an a IN//Preposition or subordinating conjunction in for but POS//Possessive ending TO//to

27. BNF for Yahoo Directory (1) ForwardPhrase::= [VB] [IN] DisPhrase {Conn } DisPhrase (2) DisPhrase::= Phrase [“(”ProperDis | NounDis“)”] [“(”Period“)”][“:” Phrase] (3) Phrase::=[DT] Adjectives [Nouns] | [Proper] Nouns (4) Adjectives::= Adjective|CD {[CC] Adjective} (5) Nouns::= Noun {Noun} (6) Conn::= ConjunctionConn | PrepositionConn (7) Noun::= NN [POS] | NNS [POS]

28. BNF for Yahoo Directory (2) (8) Adjective::= JJ (9) ConjunctionConn::= CC | “,” (10) PrepositionConn::= IN | TO (11) Proper::= NNP {NNP|POS} (12) NounDis::= Period|Nouns|Adjectives [Nouns] (13) ProperDis::= ProperSeq [CC ProperSeq] (14) Period::= [NN] CD [“-”] [CD] [NN] (15) ProperSeq::= Proper [“,” Proper]

29. Example: Provinces and Districts ForwardPhrase Conn ForwardPhrase DisPhrase ConjunctionConn DisPhrase Phrase CC Phrase Nouns Nouns Noun Noun NNS NNS

30. Exercises • Directory > Science > Computer Science > Artificial Intelligence > Natural Language Processing > Web Directories • Computer Science NN NN • Artificial Intelligence JJ NN • Natural Language Processing JJ NN VBG • Web Directories NN NNS