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Knowledge Representation École Supérieure Polytechnique Dakar, Senegal Winter 2011 by Fabian M. Suchanek This document is available under a Creative Commons Attribution Non-Commercial License

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Organisation 1h class on Knowledge Representation Slides are available at the Web-site http://suchanek.name/work/teaching Language of the Slides is English, lecturer will do his best to talk in French, questions are welcome throughout the session in French or English 2

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Check 3 To help assess your background knowledge, please answer the following questions on a sheet of paper: 1.say “All men are mortal” in first order logic. 2.reduce ~ x: y: (loves(x,y) & married(x,y)) 3.write a rule in PROLOG. 4.what is the minimum of f(x)=x 2 ? Do not worry if you cannot answer. Hand in your answers anonymously. This check just serves to adjust the level of the lecture. You will not be graded. E A

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Definition of KR 4 Knowledge representation (KR) is a research field of artificial intelligence that aims to formalize information so that it can be used for automated reasoning.

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Motivation 5 Is Elvis Presley still alive? alive(Elvis) ? sings(Elvis) bornIn(Elvis, Tupelo) locatedIn(Tupelo, USA) sings(X) => alive(X) … Elvis is alive / not alive formalization reasoning KR is required to find formal answers to questions:

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Big Picture 6 Is Elvis Presley still alive? Knowledge Base (KB) facts / assertions query rules / formulas implied / deduced facts / statements inference alive(Elvis) sings(Elvis) bornIn(Elvis, Tupelo) locatedIn(Tupelo, USA) sings(X) => alive(X) …

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Query Answering 7 Is Elvis Presley still alive? Knowledge Base (KB) query alive(Elvis) Answering a query : Finding out whether the query or its contrary can be deduced from the KB sings(Elvis) bornIn(Elvis, Tupelo) locatedIn(Tupelo, USA) sings(X) => alive(X) …

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KR Formalism 8 Knowledge Base (KB) KR formalism : The language of the KB sings(Elvis) bornIn(Elvis, Tupelo) locatedIn(Tupelo, USA) sings(X) => alive(X) …

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KR Formalisms, 1/2 9 There are different ways to represent knowledge: PROLOG-like: alive(Elvis), is(Elvis, alive) graphically in natural language “Elvis is alive” is alive in propositional logic elvis_alive.

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KR Formalisms, 2/2 10 in a programming language elvis.alive=true in first order logic x: rocksinger(x) => alive(x) A in a mathematical notation elvis Alive in completely different formalism ф elvis → ☺☺☺

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Canonicity 11 A KR formalism is canonic if one piece of knowledge can be represented in only one way alive(Elvis) is(Elvis, alive) alive(Elvis, true) vivant(Elvis) not very canonic

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Canonicity is Desirable 12 Canonicity is in general desirable because it facilitates co-operation by different people alive(Elvis) ? is(Elvis, alive) is(Elvis, alive) ! no

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Achieving Canonicity 13 A formalism can be made more canonic by restricting it e.g., by allowing only unary predicates: alive(Elvis) is(Elvis, alive) providing best practice guidelines e.g., by prescribing certain conventions: alive(Elvis) Alive(elvis) providing standard vocabularies e.g., by listing predicates that should be used in a certain domain: {alive, dead, young, old, happy, sad}

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Expressiveness 14 A KR formalism is more expressive than another one if we can say things in the first formalism that we cannot say in the second. First order logic x: rocksinger(x) => alive(x) A Propositional logic ?

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Expressiveness is Desirable 15 In general, a higher expressiveness is desirable from a modeling point of view x: alive(x) A … but it comes at a cost…

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Decidability 16 A KR formalism is decidable, if there is an algorithm (computer program) that can answer any query on a knowledge base in that formalism. Some formalisms are so expressive that they are undecidable: Is sentence (*) true? (Technically: A logical system is decidable iff there is an effective method for determining whether arbitrary formulas are theorems of the logical system.) (*) This sentence is false. Natural language is undecidable

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Decidability of FOL 17 First order logic is also undecidable First order logic is so powerful that it can express sentences of which it is impossible to determine whether they are true or not. This means that there are some first-order-logic sentences, which we can never prove or disprove. Worse, we cannot even determine whether a given sentence is of this type.

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Achieving Decidability 18 In general, decidability is desirable. But the more expressive a formalism is, the more likely it is to be undecidable. Some formalisms can be made decidable by restricting them Propositional logic is decidable First order logic is decidable if all formulae are of the following form: x, y,…. z, q,… : p(x,y) … => … A E existential universal arbitrary formula quantifiers quantifiers without quantifiers and function symbols

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Closed World Assumption 19 A KR formalism follows the closed world assumption (CWA), if any statement that cannot be proven is assumed to be false. PROLOG, e.g., follows the CWA: ?- assert(bornIn(Elvis, Tupelo)). yes ?- alive(Elvis). no

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Open World Assumption 20 In many contexts, the open world assumption (OWA) is more appropriate. Under the OWA, a statement can be provably false provable true unknown ?- alive(Elvis). I have no clue

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Reification 21 A KR formalism allows reification, if it can treat statements like entities. thinks(Fabian, alive(Elvis)). => alive(Elvis) reification, a statement appears as argument

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Modal Logic 22 Modal logic allows talking about necessity and possibility “It is possible that Elvis is alive” alive(Elvis) Thus, modal logic implements a (restricted) type of reification.

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Unique Name Assumption 23 A KR formalism follows the unique name assumption (UNA), if different names always refer to different objects. Elvis The King

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Unique Name Assumption 24 The UNA is not useful if different people want to use different identifiers: alive(TheKing) TheKing=Elvis => alive(Elvis) Elvis The King

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Schema 25 A KR formalism is schema-bound, if one has to decide upfront which entities can have which properties. In schema-bound formalisms, one has to decide a priori for classes of things and their properties: person: alive/dead birthday profession camera: resolution shutter speed weight gender

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Advantages of Schemas 26 ?- assert(alive(Elvis)). yes ?- assert(hasResolution(Elvis,five_megapixel)) yes PROLOG is schema-free, any entity can have any property: A schema-bound formalism puts more modeling constraints, but can exclude non-sensible statements.

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Databases 27 Databases are a particular schema-bound KR formalism. A database can be seen as a set of tables. NameProfessionBirth ElvisSinger1935 ObamaPresident1961 ……… each table corresponds to one class of things Person each column is one property each row is one thing

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SQL 28 A database can be queried in the Structured Query Language ( SQL ). SELECT name, birth FROM person WHERE profession=‘Singer’ AND birth>1930 Elvis, 1935 JohnLennon, 1940 … NameProfessionBirth ElvisSinger1935 ObamaPresident1961 ………

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Uses of Databases 29 Databases are used in practically every major enterprise, with the main database systems being Oracle Microsoft SQL Server IMB’s DB2 Postgres MySQL Headquarters of Oracle in Redwood Shores, CA, USA

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Inheritance 30 A KR formalism supports inheritance, if properties specified for one class of things can be automatically transferred to a more specific class. person: name birthday gender singer: name birthday gender instrument

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Inheritance 31 person: name birthday gender singer: inherits from person instrument more general class, fewer properties more specific class, more properties additional properties

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Restrictions 32 person: name birthday gender profession singer: inherits from person profession=singer instrument restriction The more specific class can restrict properties inherited from the more general class.

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Object Oriented Languages 33 Object-oriented programming languages (such as e.g., Java) support inheritance. public class Person { String name; String profession; } public class Singer extends Person { String profession=“Singer”; String instrument; } Singer elvis=new Singer(); elvis.name=“Elvis”; elvis.instrument=“guitar”; System.out.println(elvis.profession) Singer super-class * declares two properties sub-class * overwrites a property * adds a property Elvis is a singer * but he inherited properties * and has predefined values

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Object Oriented Languages 34 Most programming languages support object-orientation today, with the most important ones being Java C++ C# Visual Basic Python Ruby

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Monotonicity 35 A KR formalism is monotonous, if adding new knowledge does not undo deduced facts. elvis_is_person elvis_alive elvis_dead => ~elvis_alive + Monotonicity can be very counter-intuitive. It requires everything to be known upfront. First order logic and propositional logic are monotonous: => elvis_alive elvis_is_person elvis_alive elvis_dead=> ~elvis_alive elvis_is_dead => elvis_alive => m_jackson_alive => elvis_is_dead

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Default Logic 36 Default logic is not monotonous: elvis_is_person: elvis_is_alive elvis_is_alive => elvis_is_alive elvis_is_dead ~elvis_is_alive if Elvis is a person and nothing says he’s not alive then he is alive if Elvis is dead then he is not alive elvis_is_person elvis_is_dead + prerequisite conclusion justification

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Fuzziness 37 A KR formalism is fuzzy, if certain statements can hold to a certain degree. fantastic(Bush ) 0.1 fantastic(Madonna) 0.8 fantastic(Elvis) 1.0 The opposite of fuzzy is crisp. First order logic, PROLOG and propositional logic are all crisp.

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Fuzzy Logic 38 Fuzzy logic is a fuzzy KR formalism. rainy => bad_weather rainy (0.8) bad_weather (0.8)

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Fuzzy Logic Operators 39 rainy \/ windy => bad_weather rainy (0.8) bad_weather (??) windy(1.0) Fuzzy logic defines how to compute fuzzy values for complex combinations of fuzzy predicates, e.g. OR is computed as maximum AND is computed as minimum NOT is computed as 1-x 1.0

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Contradictions 40 A KR formalism is tolerant to contradictions if it allows contradictions in the knowledge base. Elvis is alive Elvis is not alive Knowledge base Elvis is a person Madonna is alive

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FOL & Contradictions 41 alive(Elvis). ~alive(Elvis). => life_is_beautiful First order logic and propositional logic are not tolerant to contradictions: ex falso quod libet… Some domains require handling of contradictions (e.g. information extraction from the Web) Elvisfans.com: Elvis is alive Wikipedia: Elvis is dead.

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Possible Worlds 42 Given propositional predicates, a possible world is an assignment of truth values to the predicates. w1: elvis_alive elvis_dead w2: elvis_alive elvis_dead w4: elvis_alive elvis_dead w3: elvis_alive elvis_dead Given the predicates elvis_alive, elvis_dead there are 4 possible worlds:

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Markov Logic 43 f1: elvis_alive [0.8] f2: elvis_dead => ~elvis_alive [0.5] f3: elvis_dead [0.6] A Markov Logic Network is a set of propositional logic formulas with weights: A Markov Logic Network assigns a probability to each possible world: P(w) ~ Π e weight(f) satisfied formula f w1: elvis_alive elvis_dead f1 satisfied, f2 satisfied, f3 not satisfied P(w1) ~ e 0.8. e 0.5 = 3.6

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Markov Logic Example 44 f1: elvis_alive [0.8] f2: elvis_dead => ~elvis_alive [0.5] f3: elvis_dead [0.6] w1: elvis_alive elvis_dead w2: elvis_alive elvis_dead w4: elvis_alive elvis_dead w3: elvis_alive elvis_dead f1,f2 f2 f2,f3 f1,f3 Satisfied formulas: Markov Logic Network: Possible worlds: 3.6 1.6 3.0 4.1

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Markov Logic Summary 45 A Markov Logic Network is a KR formalism that allows contradictions. It assigns a probability to all possible worlds: larger probabilities for worlds that satisfy more constraints smaller values for worlds that violate more constraints

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Explicitness 46 A KR formalism is explicit, if it can explain why a certain statement was deduced. rich(BillGates) rich(X) => happy(X) rich(BillGates) => happy(BillGates) happy(BillGates) Instantiation Modus Ponens Logic-based formalisms are usually explicit:

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Explicitness 47 Explicitness may not always be useful, consider pattern recognition: a a a a a hard to describe explicitly a... or image recognition: hard to describe explicitly Elvis Presley

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Neural Networks 48 A neural network is a a non-explicit KR formalism. It tries to model the human brain. connections neuron (nerve cell)

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Perceptrons 49 A perceptron is a simplified model of a nerve cell (neuron). It is defined by a vector of n real-valued weights, w 1,...,w n. It can be visualized as follows: nerve cell x 1 x 2... x n input values perceptron y real-valued output value w 1 w 2... w n weights

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Perceptron Function 50 A perceptron computes a function R n {0,1} from the n input values to the set {0,1} as follows: 1 if w i · x i > 0 0 else x 1 x 2... x n y = w 1 w 2... w n

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Perceptron Example 1 51 x 1 x 2 x 3 0.3 0.8 0.2 w i · x i = 0.3· 1 + 0.8· 2 + 0.2· 1 = 2.1 > 0 ~~~~> Output y=1 1 if w i · x i > 0 0 else y = input: 1 2 1

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Perceptron Example 2 52 x 1 x 2 x 3 0.3 0.8 0.2 w i · x i = 0.3· 0 + 0.8· -2 + 0.2· 0 = -0.8 < 0 ~~~~> Output y=0 1 if w i · x i > 0 0 else y = input: 0 -2 0

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Combining Perceptrons 53 final output of the neural network 1 Perceptrons can be combined to a neural network. input of the neural network output of one perceptron is the input for another one

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Imitating the Retina 54 imitation of our eye retina 1

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Network Training 55 a 1 net will also recognize this one a Given examples: a a a a a...adjust the weights so that the output is 1 for all examples (= training the net ). Then the net will also recognize similar inputs:

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Neural Networks 56 a 1 A Neural Network is a non-explicit KR model. Neural Networks are used, e.g, to recognize postal codes on letters to optimize search engine results This network “knows” what the letter “a” is. But this knowledge is not explicit.

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Distributedness 57 A KR formalism is distributed, if it encourages use and co-operation by different people and systems across different places and organizations. Every rock singer is alive! The King is a rock singer. I am Elvis, the King. Elvis is alive

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The Semantic Web 58 The Semantic Web is a distributed KR formalism. 1935 Grammy Award sameAs born won

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KR Design: Summary 59 In general, a KR formalism serves to represent knowledge infer facts from that knowledge answer queries on the knowledge

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KR Design: Summary 60 There are many KR formalisms with different properties: Canonicity (does the formalism allow only one form to represent a statement?) Expressiveness (how powerful is the formalism) Decidability (can every query be answered?) Closed World (does the formalism assume that everything unknown is false?) Unique Name (do two names always mean two different things?)

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KR Design: Summary 61 Schema-bound (do we have to decide upfront on properties of things?) Databases/SQL is a schema-bound KR formalism Inheritance (can properties be transferred from one class to another?) Object-oriented programming languages support inheritance Monotonicity (will new knowledge never undo existing knowledge?) Default logic allows non-monotonicity

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KR Design: Summary 62 Fuzziness (can properties be fufilled to a certain degree?) Fuzzy logic is a fuzzy KR formalism Tolerance to contradictions (is it OK to have a contradiction in the KB?) Markov Logic Networks can deal with contradictions Explicitness (is knowledge always explainable?) Neural Networks store knowledge only implicitly Distributedness (can the knowledge be distributed over sources?) The Semantic Web is a distributed formalism

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