VO Grundzüge der Artificial Intelligence SS Hans Tompits Institut für Informationssysteme Arbeitsbereich Wissensbasierte Systeme

Transcription

1 VO Grundzüge der Artificial Intelligence SS 2009 Hans Tompits Institut für Informationssysteme Arbeitsbereich Wissensbasierte Systeme

2 Knowledge Representation Folien adaptiert nach Vorlagen von Stuart Russell (U. Berkeley) Chapter 10 (incl. Material from Chapters 7 & 8)

3 Outline What is knowledge representation? Knowledge-based agents Logic in general models and entailment Review of propositional and first-order logic Aspects of knowledge and ontological engineering Chapter 10 (incl. Material from Chapters 7 & 8) 1

4 What is knowledge representation? The representation of knowledge and reasoning from knowledge are central for AI... after all, humans know things and do reasoning. Knowledge and reasoning play a crucial role in dealing with partially observable environments. A knowledge-based agent can combine general knowledge with current percepts to infer hidden aspects of the current state prior to selecting actions. E.g., a physician diagnoses a patient prior to choosing a treatment. = For diagnosing, the physician uses knowledge from textbooks and teachers as well as association patterns the physician cannot consciously describe. Chapter 10 (incl. Material from Chapters 7 & 8) 2

5 What is knowledge representation? (ctd.) Understanding natural language also involves inferring hidden states namely the intention of the speaker. E.g., when we hear John threw the stone against the mirror and broke it, we know that it refers to mirror and not to stone. In general, the goal of knowledge representation is the following: representing implicit knowledge about a certain area in such a way that it can be processed by computers original knowledge is encoded in suitable data structures and algorithms. Chapter 10 (incl. Material from Chapters 7 & 8) 3

6 What is knowledge representation? (Ctd.) Knowledge representation is a multidisciplinary field involving methods and techniques from: logic: provides the formal structures and rules for performing deductions ontology: defines the kinds of objects in the considered application area computer science: In short: supports the applications which distinguishes knowledge representation from pure philosophy. knowledge representation = application of logic and ontology for providing computational models. Chapter 10 (incl. Material from Chapters 7 & 8) 4

7 Declarative vs. procedural approaches Declarative knowledge representation techniques: knowledge is expressed as sentences in some suitable formal language which are accessed by the procedures using this knowledge = separation between the explicit representation of knowledge and the processing for answering queries. Advantages: increased versatility for performing complex tasks changes can be easily incorporated (modularity) Procedural techniques: knowledge is implicitly stored in a sequence of operations, manifested in the actual execution of the operations (i.e., directly as program code). Advantages: minimizing the role of explicit representation and reasoning can yield more efficient systems Chapter 10 (incl. Material from Chapters 7 & 8) 5

8 Declarative vs. procedural approaches (ctd.) In the 1970s and 1980s there were heated debates between advocates of the two approaches. Now it is understood that successful agents should combine both declarative and procedural elements in their designs. Chapter 10 (incl. Material from Chapters 7 & 8) 6

9 Knowledge-based agents Central components of a knowledge-based agent: a knowledge base a set of sentences in a formal language methods to add new sentences and methods to query what is known we use Tell and Ask as generic names for these tasks both tasks may involve inference i.e., deriving new sentences from old. In logical agents, answers to the Ask procedure is by means of logic! Schematic architecture: Inference engine domain independent algorithms Knowledge base domain specific content Chapter 10 (incl. Material from Chapters 7 & 8) 7

10 A simple knowledge-based agent The agent must be able to: represent states, actions, etc.; incorporate new percepts; update internal representations of the world; deduce hidden properties of the world; deduce appropriate actions. Each time the agent program is called, it does three things: 1. It Tells the knowledge base what it perceives; 2. it Asks the knowledge base what action it should perform; 3. it records its choice with Tell and executes the action. Chapter 10 (incl. Material from Chapters 7 & 8) 8

11 A simple knowledge-based agent function KB-Agent( percept) returns an action static: KB, a knowledge base t, a counter, initially 0, indicating time Tell(KB, Make-Percept-Sentence( percept, t)) action Ask(KB, Make-Action-Query(t)) Tell(KB, Make-Action-Sentence(action, t)) t t + 1 return action Make-Percept-Sentence constructs a sentence asserting that the agent perceived the given percept at the given time. Make-Action-Query constructs a sentence that asks what action should be done at the current time. Make-Action-Sentence constructs a sentence asserting that the chosen action was executed. = Details of the inference mechanisms are hidden inside Tell and Ask! Chapter 10 (incl. Material from Chapters 7 & 8) 9

12 Logic in general Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language Semantics defines the meaning of sentences; i.e., defines truth of a sentence in a world E.g., the language of arithmetic x + 2 y is a sentence; x2 + y > is not a sentence x + 2 y is true iff the number x + 2 is no less than the number y x + 2 y is true in a world where x = 7, y = 1 x + 2 y is false in a world where x = 0, y = 6 Chapter 10 (incl. Material from Chapters 7 & 8) 10

13 Entailment Entailment means that one thing follows from another: knowledge base KB entails sentence α, symbolically KB = α, iff α is true in all worlds where KB is true. E.g., the KB containing Pooh laughs and Tigger laughs entails Either Pooh laughs or Tigger laughs. E.g., x + y = 4 entails 4 =x + y. Entailment is a relationship between sentences (i.e., syntax) that is based on semantics. Chapter 10 (incl. Material from Chapters 7 & 8) 11

14 Models Semantics is defined in terms of models, which are formally structured worlds with respect to which truth can be evaluated. We say m is a model of a sentence α if α is true in m. M(α) is the set of all models of α. Then KB = α if and only if M(KB) M(α). E.g. KB = Pooh laughs and Tigger laughs α = Tigger laughs M( ) x x x x x x x x x x x x x x x x x x x x x x x x x xx x xx x x x x x x x M(KB) x x x x x x x x x Chapter 10 (incl. Material from Chapters 7 & 8) 12

15 Important semantical notions Two sentences are logically equivalent iff true in the same models: α β if and only if α = β and β = α A sentence is valid if it is true in all models, A sentence is satisfiable if it is true in some model A sentence is unsatisfiable if it is true in no models Writing α for the negation of α (with the meaning that α is true precisely when α is not true), we can state: α is valid if and only if α is unsatisfiable; KB = α if and only if KB { α} is unsatisfiable i.e., prove α by reductio ad absurdum Chapter 10 (incl. Material from Chapters 7 & 8) 13

16 Inference KB i α sentence α can be derived from KB by procedure i Derivation from KB = a sequence of inference rule applications using axioms and elements of KB Intuitively: Consequences of KB are a haystack; α is a needle. Entailment = needle in haystack; inference = finding it Soundness: i is sound if whenever KB i α, it is also true that KB = α Completeness: i is complete if whenever KB = α, it is also true that KB i α Chapter 10 (incl. Material from Chapters 7 & 8) 14

17 Two fundamental logics There are many different logics, designed for different purposes. Two logics are pre-eminent: propositional logic first-order logic (FOL), a.k.a. predicate logic Propositional logic is simple, assuming that the world consists of facts which can be composed from atomic formulas using connectives S (negation), S 1 S 2 (conjunction), S 1 S 2 (disjunction), S 1 S 2 (implication), S 1 S 2 (biconditional). E.g. A (B C) states that if A is not the case, then one of B or C holds. This formula may represent, e.g., the following sentence: If the car is not proceeding, then it is broken or out of gas. Chapter 10 (incl. Material from Chapters 7 & 8) 15

18 Truth tables for connectives P Q P P Q P Q P Q P Q false false true false false true true false true true false true true false true false false false true false false true true false true true true true Chapter 10 (incl. Material from Chapters 7 & 8) 16

19 Two fundamental logics (ctd.) Unlike natural language, propositional logic has, however, only very limited expressive power. E.g., the following argument (valid in natural language) cannot be adequately dealt with in propositional logic: All superheroes are brave. Superman is a superhero. Therefore: Superman is brave. In propositional logic, the three sentences would be formalized using atomic sentences A, B,C but A, B = C does not hold. = This is where FOL comes in! Chapter 10 (incl. Material from Chapters 7 & 8) 17

20 Two fundamental logics (ctd.) FOL assumes that the world contains Objects: people, houses, numbers, theories, Superman, Tigger, colors, centuries,... Relations: red, round, bogus, prime, multistoried..., brother of, bigger than, inside, part of, has color, occurred after, owns, comes between,... Functions: father of, best friend, addition, one more than, end of... Chapter 10 (incl. Material from Chapters 7 & 8) 18

21 Syntax of FOL: Basic elements Constants Superman, KingJohn, 2,... Predicates Friend, >,... Functions Sqrt, LeftLegOf,... Variables x, y, a, b,... Connectives Equality = Quantifiers (universal quantifier) (existential quantifier) Chapter 10 (incl. Material from Chapters 7 & 8) 19

22 Atomic sentences Atomic sentence = predicate(term 1,...,term n ) or term 1 = term 2 Term = function(term 1,...,term n ) or constant or variable E.g., Friend(Pooh, Tigger) > (Length(LeftLegOf (Hulk)), Length(LeftLegOf (Spider-Man))) Chapter 10 (incl. Material from Chapters 7 & 8) 20

23 Complex sentences Complex sentences are made from atomic sentences using connectives and quantifiers xs ( for all x, S ), xs ( for some x, S ) E.g. Archfiend(LexLuthor, Superman) Fights(LexLuthor, Superman) >(1, 2) (1, 2) >(1, 2) >(1, 2) x y(country(x) Capitol(y, x)) Chapter 10 (incl. Material from Chapters 7 & 8) 21

24 Truth in first-order logic Sentences are true with respect to a model and an interpretation Model contains 1 objects (domain elements) and relations among them Interpretation specifies referents for constant symbols objects predicate symbols relations function symbols functional relations An atomic sentence predicate(term 1,...,term n ) is true iff the objects referred to by term 1,...,term n are in the relation referred to by predicate. Chapter 10 (incl. Material from Chapters 7 & 8) 22

25 Truth example Consider the formula Brother(Richard,John) and the following interpretation: Richard Richard the Lionheart John the evil King John Brother the brotherhood relation Under this interpretation, Brother(Richard,John) is true just in case Richard the Lionheart and the evil King John are in the brotherhood relation in the model Chapter 10 (incl. Material from Chapters 7 & 8) 23

26 Translating natural language into logic Constructing a knowledge base means modeling knowledge in terms of logic. This process is also known as knowledge engineering. It usually involves translating pieces of knowledge expressed in natural language into symbols. We now discuss some aspects relevant for translating words into symbols. Chapter 10 (incl. Material from Chapters 7 & 8) 24

27 Translating natural language into logic (ctd.) First of all, translating natural language into logic is not automatic but requires experience and is often not easy. The passage to be translated must be understood clearly. Discourse in natural language often contain implicit assumptions which need to be made explicit. Ambiguities need to be resolved. Chapter 10 (incl. Material from Chapters 7 & 8) 25

28 Translating natural language into logic (ctd.) Statements often need to be paraphrased to exhibit their proper logical structure. This involves the following steps: 1. the direct translation of appropriate words into logical symbols; 2. rephrasing of component clauses to circumvent ambiguities in particular, to avoid the fallacy of equivocation (=misuse of a term with more than one meaning); 3. determining the intended grouping of paraphrased statements. Chapter 10 (incl. Material from Chapters 7 & 8) 26

29 Translating sentential connectives The following tables list (not exhaustively) expressions given on the right which can be translated by the symbols on the left. (N.B. The subsequent discussion largely follows Kleene (1967).) A B If A, then B. B if A. A only if B. When A, then B. B when A. A only when B. In case A, B. B in case A. A only in case B. B provided that A. A is a sufficient condition for B. B is a necessary condition for A. A (materially) implies B. A B A if and only if B. (Abbreviation: A iff B.) A if B, and B if A. If A then B, and conversely. A exactly if B. A exactly when B. A just in case B. A is a necessary and sufficient condition for B. A is (materially) equivalent to B. Chapter 10 (incl. Material from Chapters 7 & 8) 27

30 Translating sentential connectives (ctd.) A B A and B. Both A and B. A but B. Not only A but B. A although B. A despite B. A yet B. A while B. A B A or B. A and/or B [in legal documents]. A or B or both. A unless B. A except when B. (A B) (A B) A or B but not both. (is equivalent to A or else B. (A B), (A B), A or B [sometimes]. ( A B)) Either A or B. A unless B [sometimes]. A except when B [sometimes]. Chapter 10 (incl. Material from Chapters 7 & 8) 28

31 Translating sentential connectives (ctd.) (A B) Neither A nor B. (equivalent to A B) A Not A (or the result of transforming A to put not just after the verb or an auxiliary verb). A doesn t hold. A isn t so. It is not the case that A. Chapter 10 (incl. Material from Chapters 7 & 8) 29

32 Translating sentential connectives (ctd.) Note: Some nuances of meaning can get lost during the translation. Examples: In propositional logic A B is equivalent to B A, but the following sentences will be interpreted differently: Mary-Jane had a baby and got married. Mary-Jane got married and had a baby. = The sentences suggest a certain temporal order which is lost in the translation. Similarly, a young man would react differently if his girlfriend tells him one of the following sentences: I love you and I love your brother almost as well. I love you but I love your brother almost as well. Chapter 10 (incl. Material from Chapters 7 & 8) 30

33 Translating sentential connectives (ctd.) The fallacy of equivocation can be illustrated thus: Consider the following two statements: (a) He went to Albany and I went along. (b) He went to Schenectady but I did not go along. A direct (naive) translation of the two sentences in propositional logic would be of the form A W and S W. = The conjunction of both formulas is inconsistent. Chapter 10 (incl. Material from Chapters 7 & 8) 31

34 Translating sentential connectives (ctd.) The problem here is that I went along in (a) must be distinguished from the I went along whose negation appears in (b): I went along means in (a) I went along to Albany, but in (b) I went along to Schenectady. Hence, the correct translations of (a) and (b) in propositional logic are: A W A and S W S... or better yet in FOL: went(he, Albany) went(i, Albany) and went(he, Schenectady) went(i, Schenectady). Chapter 10 (incl. Material from Chapters 7 & 8) 32

35 Translating quantifiers xa(x) For all x, A(x). For every x, A(x). For each x, A(x). For arbitrary x, A(x). Whatever x is, A(x). A(x) always holds. Everyone is A. Everybody is A. Everything is A. Each one is A. Each person is A. Each thing is A. xa(x) For some x, A(x). For suitable x, A(x). There exists an x such that A(x). There is an x such that A(x). There is some x such that A(x). Someone is A. Somebody is A. Something is A. For at least one x, A(x). At least one is A. Chapter 10 (incl. Material from Chapters 7 & 8) 33

36 Common mistakes to avoid Typically, is the main connective with as in: all S are P: x (S(x) P(x)) Common mistake: using as the main connective with : x At(x,Berkeley) Smart(x) means Everyone is at Berkeley and everyone is smart Typically, is the main connective with as in: some S are P: x (S(x) P(x)) Common mistake: using as the main connective with : x (At(x,Stanford) Smart(x)) is true if there is anyone who is not at Stanford! Chapter 10 (incl. Material from Chapters 7 & 8) 34

37 Some ambiguities In natural language, all S are P would normally not be asserted if it is already known that S does not hold. Indeed, people would not consider all S are P true if S is false. = all S are P would in this sense be translated as rather than as x(s(x) P(x)). xs(x) x(s(x) P(x)) Chapter 10 (incl. Material from Chapters 7 & 8) 35

38 Some ambiguities (ctd.) Sometimes all S are not-p is understood as not all S are P. Examples: All that glisters is not gold (Shakespeare, Merchant of Venice). All women are not gold diggers. = Translations would be of form x(s(x) P(x)) but not of form x(a(x) P(x)). The indefinite article a or an has sometimes different meaning: A child needs affection. = x(c(x) A(x)) A man climbed the Mount Everest. = x(m(x) E(x)) Chapter 10 (incl. Material from Chapters 7 & 8) 36

39 Some ambiguities (ctd.) Also, the meaning of the expression any depends on the context: When an any-expression stands by itself, it has the same force as all. But when an any-expression D is put into contexts D or D E, the meaning of any normally changes from all to some. Examples: I would do that for anyone. = xa(x) I wouldn t do that for anyone. = xa(x) If any man is godfearing, he is just. = x(g(x) J(x)) If any man is just, Aristides is just. = ( xj(x)) J(a) If Superman is a villain, then any man is a villain. = V (s) xv (x) Chapter 10 (incl. Material from Chapters 7 & 8) 37

40 The problem of grouping For determining the intended grouping of complex sentences, it is advisable to analyze the outermost structure first and then to paraphrase inward, step by step. Example: Consider the following argument (after Quine, 1950): Premisses: The guard searched all who entered the building except those who were accompanied by members of the firm. Some of Fiorecchio s men entered the building unaccompanied by anyone else. The guard searched none of Fiorecchio s men. Conclusion: Some of Fiorecchio s men were members of the firm. Chapter 10 (incl. Material from Chapters 7 & 8) 38

41 The problem of grouping (ctd.) The first premiss can be paraphrased as follows: For every person x it holds that, if x entered the building, then x was searched by the guard except x was accompanied by some member of the firm. The outermost structure of this sentence is thus: x (x entered the building (x was searched by the guard except x was accompanied by some member of the firm)). We next consider the consequent of the implication: x (x entered the building (x was searched by the guard x was accompanied by some member of the firm)). Chapter 10 (incl. Material from Chapters 7 & 8) 39

42 The problem of grouping (ctd.) For the second clause of the consequent we obtain: x (x entered the building (x was searched by the guard y ( x was accompanied by y y was member of the firm)). Symbolically: x[b(x) (S(x) y(a(x,y) M(y)))] The second premiss Some of Fiorecchio s men entered the building unaccompanied by anyone else can be translated as follows (using F(x) for x was one of Fiorecchio s men ): x(f(x) B(x) x was unaccompanied by anyone else) Chapter 10 (incl. Material from Chapters 7 & 8) 40

43 The problem of grouping (ctd.) The sentence x was unaccompanied by anyone else can be interpreted as anyone accompanying x was one of Fioreccio s men = y(a(x, y) F(y)). For the last premiss ( the guard searched none of Fiorecchio s men ) and the conclusion ( some of Fiorecchio s men were members of the firm ) we get: x(f(x) S(x)) and x(f(x) M(x)), respectively. One can show: x[b(x) (S(x) y(a(x,y) M(y)))], x(f(x) B(x) y(a(x, y) F(y)), x(f(x) S(x)) = x(f(x) M(x)). Chapter 10 (incl. Material from Chapters 7 & 8) 41

44 The story so far Logical agents apply inference to a knowledge base to derive new information and make decisions Basic concepts of logic: syntax: formal structure of sentences semantics: truth of sentences wrt models entailment: necessary truth of one sentence given another inference: deriving sentences from other sentences soundness: derivations produce only entailed sentences completeness: derivations can produce all entailed sentences First-order logic: objects and relations are semantic primitives syntax: constants, functions, predicates, equality, quantifiers = Increased expressive power: sufficient to capture many aspects of natural language Chapter 10 (incl. Material from Chapters 7 & 8) 42