Part 1: (Pills of) Knowledge Representation and Reasoning. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 1

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1 Part 1: (Pills of) Knowledge Representation and Reasoning Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 1

2 Knowledge base A knowledge base is a representation of the knowledge about the world (problem). intensional knowledge: general laws on the domain of interest extensional knowledge about a specific problem instance (situation) The knowledge base construction amounts to the assertion of the sentences representing both intensional and extensional knowledge (Tell). The language used to represent the knowledge is the knowledge representation language. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 2

3 Inference engine The implications (consequences) of the knowledge base are computed by an inference process (inference engine). Ask is used to characterize the answers that a kb can infer. A knowledge representation system defined in terms of Tell&Ask is said to be declarative: no need to know the details of the inference process to understand the answers of the knowledge base. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 3

4 Logic as a representation language Logic is declarative: Example: knowledge representation in propositional logic language: formulae semantics: = inference: deductive methods An inference algorithm is: sound if any formula that is derived, is logically entailed ( =). complete if any formula that is logically entailed, is derived. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 4

5 Plan of the lecture Logics for Knowledge Representation (in FOL) Rule-based Representations Taxonomic Representations Non monotonic reasoning Specific Logics: action, time, space, beliefs... Reasoning about action Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 5

6 Rule-based Representation Rules: originate from logical formulae (clauses). (Machine) Representation: Clausal Normal Form (Mechanized) Reasoning: Resolution Literals: predicate symbols (atoms) or negated atoms. A clause is a disjuntion of literals L 1 L 2 L n. usually written as: {L 1, L 2,..., L p }. A formula is in conjunctive normal form (CNF) or in clausal form or it is called a set of clauses if: it is C 1 C 2 C n where C i are clauses. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 6

7 Clauses (cont.) Since disjunction is commutative: A 1 A 2 A n B 1 B 2 B m where A i and B j are atoms. When n = m = 0 we have the empty clause, denoted {}. Every FOL formula can be translated in CNF. Here we focus on formulae in CNF Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 7

8 Horn Clauses A clause is Horn if n 1: example: P Q R corresponding to Q R P A CNF is Horn Horn if all its clauses are Horn If n = 1: A 1 B 1..., B m we have a definite (Horn) clause. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 8

9 Horn Data Bases Horn Data Bases (HDB): CNF without function symbols 1.F acts A(x) 2.Rules A 1 (x)... A n (x) B(x) 3.Goals A 1 (x)... A n (x) Data Bases emphasizes that: the language does not allow for function symbols; the domain is restricted to the known individuals. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 9

10 Reasoning in Horn Data Bases MP is a sound and complete rule for HDB: It is easy to build the deductive closure for a HDB: MP is applied in all possible ways, adding new conclusions HDB with MP is a formal system with deduction sound and complete wrt the semantics Complexity of the deduction in HDB: n d where n is the maximum number of premises in the rules and d is the dimension of the HDB HDB: restricting the language to lower the complexity of reasoninig Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 10

11 Logic programming Representation: Horn Clauses + Function Symbols Reasoning: Unification Resolution Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 11

12 Full first-order version: Resolution: brief sketch l 1 l k, m 1 m n (l 1 l i 1 l i+1 l k m 1 m j 1 m j+1 m n )θ where Unify(l i, m j ) = θ. Apply resolution steps to CN F (KB α); refutation-complete for FOL Select Linear Definite (SLD)-Resolution: one of the two clauses is the last one generated, while the other one comes from the initial set of clauses. Incomplete in general, but complete for definite clauses. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 12

13 Resolution: example Rich(x) U nhappy(x) Rich(Ken) U nhappy(ken) with θ = {x/ken} Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 13

14 Prolog systems Basis: backward chaining with Horn clauses + bells & whistles Program = set of clauses = head :- literal 1,... literal n. criminal(x) :- american(x), weapon(y), sells(x,y,z), hostile(z). Efficient unification Efficient retrieval of matching clauses Depth-first, left-to-right backward chaining Built-in predicates for arithmetic etc., e.g., X is Y*Z+3 negation as failure (see later) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 14

15 Prolog examples Depth-first search from a start state X: dfs(x) :- goal(x). dfs(x) :- successor(x,s),dfs(s). No loop over S: successor succeeds for each succ state Appending two lists to produce a third: append([],y,y). append([x L],Y,[X Z]) :- append(l,y,z). query: append(a,b,[1,2])? answers: A=[] B=[1,2] A=[1] B=[2] A=[1,2] B=[] Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 15

16 RULES KB has a representation of the form: IF A(x) A 1 (x)... A n (x) T HEN B(x) i.e.. a Rule System is a HDB. Why?? Facts and rules as in a HDB Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 16

17 In Production Rule Systems Production Rules - forward search (data driven) - decision on which rule is applicable: semi-unification, i.e. pattern matching on working memory - decision on which rule to apply: scheduling algorithm for rules - production rules allow for arbitrary modifications of the working memory Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 17

18 Disjunctive Data Bases Not every formula of FOL can be translated into Horn Clauses, e.g. rains (umbrella windjacket) Disjunctive Data Bases (DDB): A(x) A 1 (x)... A n (x) B 1 (x)... B n (x) The property of model intersection holds for HDBs: hence HDBs have a minimal model, but not for DDB Negation in the body: A 1 (x)... A 2 (x) B 1 (x) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 18

19 Taxonomic representations of knowledge 1. Semantic Networks 2. Frames 3. Description Logics and modern ontologies Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 19

20 Semantic networks Networks associative (psycologic model: at a different level, but they relate to neural networks) causal (bayesian networks) semantic (Quinlan 68) language: graphs with different bindings and annotations semantics: subset of FOL (the declarative part) otherwise informal/procedural inference: specialized methods for visiting the graph Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 20

21 nodes = objetcs or classes Inheritance networks arcs = relations (in particular is a, istance of) Cats Subset Mammals fufi Member Cats Cats Mammals Cat is a Mammal x Cat(x) Mammal(x) Cat(fufi) from the network one can infer that: fufi is a mammal Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 21

22 Semantic Networks Mammals SubsetOf HasMother Persons Legs 2 Female Persons SubsetOf SubsetOf Male Persons MemberOf MemberOf Mary SisterOf John Legs 1 Relationships between objects of different classes. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 22

23 Frames Minsky 75: Everything that characterizes an objects belongs to a frame. A frame is a prototype of the class elements, but since frames are connected to form networks, a set of frames is very similar to a semantic network, with nodes having a richer structure. language: graph/oo specification semantics: part in FOL (plus procedural aspects) inference: search in ad-hoc representations (graphs) Systems: KEE, KRSS,... Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 23

24 Example of frame definition Frame: Course in KB University Superclasses: Memberslot: ENROLLED ValueClass: Stud Cardinality.Min: 2 Cardinality.Max: 30 Memberslot: TAUGHTBY ValueClass: (UNION Grad Prof) Cardinality.Min: 1 Cardinality.Max: 1 Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 24

25 Example of instance definition rc Instance-of: AdvC in KB University Memberslot: TAUGHTBY ValueClass: nardi Memberslot: ENROLLS ValueClass: s1,s2 Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 25

26 Common between semantic networks and frames node properties (own) inherited properties (is a). class descriptions and instances (instance of) default values and defeasible inheritance* (delayed...) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 26

27 Additional features of frames Complex relationships between frames (slot values can be defined in terms of other frames) Logical operators in the slot definition Numerical restrictions Multiple inheritance Procedural Attachments (if needed, if added) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 27

28 Summarizing Taxonomic representations are based on the idea of organizing knowledge in classes of objects: objects are characterized by their properties inheritance is key to the representation complex relations hold between objects Formal KRR framework for semantic networks and frames: Description Logics Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 28

29 Taxonomic Knowledge Representation epistemological adequacy computational adequacy Thesis Taxonomic knowledge representations are efficient both epistemologically and computationally Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 29

30 Female Mother An example Person 6 Woman v/r Parent has-child (1,nil) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 30

31 Relations in a network ISA: Mother ISA Female(inheritance) Role restrictions: Parent A parent is a person having at least a child, and all his/her children are persons Deduction Discovery of implicit relations: let Woman represent the female persons, we have that Mother ISA Woman Simple inferences are simple to see in the network, but we need to characterize how inferences are computed. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 31

32 Logical reconstruction of networks classes or concepts are unary predicates relations between classes or roles are binary predicates Steps: Definition of a knowledge representation language to denote the network structures Definition of the meaning of the expressions of the language (interpretation structures) Definition of the inference problem as logical consequence Definition of inference algorithms Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 32

33 Syntax Given two disjoint sets of symbols (alphabets) for primitive (atomic) concepts, and primitive (atomic) roles. Terms are then defined using the alphabets by means of constructors/operators Ex. conjunction operator, C D, restricts the set to those individual objects belonging to both C and D. Note: no variables in the syntax, the expressions implicitly characterise a set of individual objects (in this case the intersection) DL-syntax ( PersonFemale) (abstract syntax with several concrete forms LISP-like, natural language, graphical interfaces) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 33

34 Examples of concept definitions AdvC Course ENROLLED.Grad ( 2 ENROLLED) ( 20 ENROLLED) IntC Course ENROLLED.UndGrad, IntC Course ENROLLED.Grad ENROLLED.UndGrad, Grad Stud DEGREE.Bachelor, Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 34

35 Semantics of concept expressions in DL Primitive concepts are subsets of the interpretation domain: A I I = { } I = I ( C) I = I \ C I (C D) I = C I D I (C D) I = C I D I ( R.C) I = {a I b. (a, b) R I b C I } ( R.C) I = {a I b. (a, b) R I b C I } ( n R) I = {a I {b I (a, b) R I } n} ( n R) I = {a I {b I (a, b) R I } n}. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 35

36 Semantics in FOL C can be phrased as φ C (x) (x free variable) such that for every interpretation I the set of elements I satisfying φ C (x) coincides with C I : φ C D (y) = φ C (y) φ D (y)... φ R.C (y) = x. R(y, x) φ C (x) φ R.C (y) = x. R(y, x) φ C (x) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 36

37 Knowledge Bases Two components: intensional (TBOX) extensional (ABOX) Σ = T, A T is a set of definitions: C D A is a set of assertions: C(a) P (a, b) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 37

38 Terminologies The TBOX is also called terminology and it includes concept definitions. Woman Person Female Definitions are interpreted as logical equivalences and thus represent necessary and sufficient conditions, for the defined concept. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 38

39 TBOX reasoning Classification is the basic reasoning task: It allows to position a new concept definition in the taxonomy, i.e., the classification of C determines: the most specific concepts subsuming C the most general concepts that are subsumed by C Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 39

40 A TBOX: family Woman Person Female Man Person Woman Mother Woman haschild.person Father Man haschild.person Parent Father Mother GrandMother Mother haschild.parent MotherWithManyChildren Mother ( 3 haschild) MotherWithoutDaughter Mother haschild. Woman Wife Woman hashusband.man Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 40

41 Reasoning techniques The main inference in DL is subsumption: C D D (the subsumer) is more general than (superset of) C (the subsumee). C is subsumed by D if C I D I for every interpretation I Other forms of reasoning reduced to subsumption: Unsatisfiability, Equivalence and Disjointness. E.g.: C is unsatisfiable iff C is subsumed by Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 41

42 Computing Subsumption As in early semantic networks and frames is reasoning based on the construction of graphs representing concepts and on the verification of their properties Structural subsumption: check through a structural matching that the subsumee graph can be mapped onto the subsumer graph. In simple languages Structural subsumption is sound, but not always complete: the answer yesis correct, but the answer no sometimes is not. In more expressive languages (sound and complete) reasoning is implemented by relying on more conventional approaches to reasoning in FOL. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 42

43 Complexity of reasoning Does the taxonomic representation allow for efficient reasoning? Brachman and Levesque s thesis There exists a fundamental tradeoff between the expressive power of a knowledge representation language and the complexity of reasoning in that language A language is defined by a set of allowed independent constructs. The complexity analysis of the basic reasoning tasks allowed to discover: Polynomial/NP-CONP/PSPACE/... / undecidable languages Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 43

44 The ABOX The ABox includes the extensional knowledge: Female Person(ANNA) haschild(anna, JACOPO) Female Person(JACOPO) haschild(anna, MICHELA) haschild(daniele, JACOPO) haschild(daniele, MICHELA) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 44

45 Semantics of the ABOX individual object intepretation a I I UNA: Unique Name Assumption a I b I An assertion C(a) is satisfiable in I if a I C I An interpretation I satisfies an ABOX if it satisfies all the assertions in the ABOX and it is called model of the ABOX. In order to apply the constraints in the TBOX we must expand the concepts in the ABOX. Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 45

46 ABOX reasoning instance checking, checks whether an individual is an instance of the concept (fundamental) A = C(a) if every model of A is also a model of C(a) knowledge base consistency, checks whether the KB admits a model realization, determines the most specific concept a given individual is an instance of retrieval, finds all the individuals that are instances of a given concept ABOX reasoning requires a significant extension of the techniques for TBOX reasoning Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 46

47 Differences wrt rule-based Indefinite information Rules vs Definitions Closed vs Open world Inheritance vs Classification Systems combining rules and taxonomic reasoning Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 47

48 Non monotonic reasoning Non monotonic vs common sense Closed-world defaults and exceptions Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 48

49 Monotonicity A formalism is monotonic when the addition of new knowledge can only increase the number of conclusions. If Γ and Γ A then A, or also Cn(Γ) Cn( ) In a non-monotonic formalism, the addition of new information can invalidate some of the previously derivable conclusions. Non-monotonicity as a property of the formalism, rather than of reasoning. Incomplete Representation (lack of information) Common Sense Reasoning Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 49

50 Negative Information flight(c 1, c 2 ) indicates a connection between two cities xyz.(flight(x, y) flight(y, z) flight(x, z)) In classical logic we cannot formally derive that two cities are not connected Assumption: when we cannot formally derive the existence of a connection, that connection does not exist For example, if we cannot derive f light(roma, Orte), we assume f light(roma, Orte) Non-monotonicity: if we add f light(roma, Orte) we cannot derive any longer that f light(roma, Orte) holds Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 50

51 Universal and General Assertions Violins have 4 cords Universal: Assertive properties that hold for all the instances General: Properties that generally hold Violins have 4 cords, but... when they have some trouble Non-monotonicity: If a violin looses a cord, it remains a violin with three cords Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 51

52 Exceptions Birds fly: x.bird(x) f lies(x) In this form the rule doesn t allow for exceptions, but kiwis do not fly x.bird(x) kiwi(x) f lies(x) In addition, to kiwis there are several other exceptions: x.bird(x) kiwi(x)... flies(x) In general, not all exceptions are known Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 52

53 Closed World Reasoning Basic Idea: There are many more false things than true things. If something is true and relevant, it has been put into the KB. So, if something is not present in the KB, it can reasonably be assumed to be false. Negation as finite failure (to prove) (P is a KB and A an atom) If from P we cannot derive A then from P we infer A Closed World Assumption Negation as Failure Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 53

54 Negation in the clause body A B C A B C Classical negation in the clause body introduces a disjunction, and we loose the uniqueness of the minimal model Negation as failure is weaker than negation not Q P is not considered equivalent to : P Q the model M = {P } is preferred Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 54

55 SLDNF = SLD-resolution + NF NF (Negation as Finite Failure): an atom A succeeds if the derivation tree corresponding to the goal A is finite and its leaves are all failure leaves. Consistent (with the various characterizations of negation), but incomplete, except under rather stringent conditions. Answer Set Programming extends SLDNF An extension of Logic Programming to disjunctive programs and computes the answer according the stable model semantics. Negation in Prolog is neither correct nor complete: it depends on the order of evaluation Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 55

56 Defeasible Inheritance Defaults and Exceptions Default Logic Circumscription Autoepistemic Logic Minimal Knowledge and Negation as Failure Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 56

57 The Situation Calculus First introduced by John McCarthy in Revisited by Raymond Reiter in the 90s A first order language with equality and sorts action object situation The language contains: do(a, s), whose result is a situation ( ), which applies to pairs of situations the constant S 0 of sort situation Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 57

58 The Situation Calculus contd Situation: a first order term denoting a possible world history (a sequence of actions) S 0 : the initial situation = null action sequence do(α, s): the successor situation to s resulting from performing the action α Actions may be parameterized: put(x, y) = put object x on object y. do(put(a, B), s) = that situation resulting from placing A on B when the current situation is s do(putdown(a), do(walk(l), do(pickup(a), S 0 ))) A situation denoting the sequence of actions pickup(a), walk(l), putdown(a) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 58

59 Fluents Fluent = relation or function whose values may vary A fluent differs from a (normal) predicate or function symbol as its value may change from situation to situation It is denoted by predicate/function symbols taking a situation term as one of its arguments closet o(x, y, s): x is close to y in situation s pos(x, s): x s position in situation s P (x 1,..., x n, s) f(x 1,..., x n, s) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 59

60 Examples (Relational): On(x, y, s), On(x, y, s ) Fluents ctnd Examples (Functional): Color(x, s) = red, Color(x, s ) = red P oss(a, s) is a fluent, meaning that action a is possible in situation s Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 60

61 Formalization of actions Preconditions: specify if an action can be executed defined by P oss(a, s) P oss(repair(w, x), s) hasglue(w, s) broken(x, s) Effects: characterizes the situation s, s = do(a, s) obtained by executing a in s f ragile(x, s) broken(x, do(drop(r, x), s)) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 61

62 Considerations about the Representation Qualification of the preconditions (qualification problem) Specification of the effects (frame problem) Static axioms and ramification of the effects (ramification problem) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 62

63 Theories of actions A Theory of Actions D consists of Precondition Axioms Effect (Successor) State Axioms Axioms for the Initial Situation Other Axioms for the Action theory Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 63

64 Deductive Planning A Deductive Planning Problem in the Situation Calculus is defined as D = s(executable(s) Goal(s)) A plan is a sequence of actions that brings the system from the initial situation to a situation that satisfies the condition Goal(s) The condition executable(s) guarantees that at each step of the plan the preconditions for action execution are satisfied Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 64

65 Other forms of reasoning Verification that a plan is executable D = executable(do([a 1,..., a n ], S 0 )) projection D = G(do([a 1,..., a n ], S 0 )) Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 65

66 Planning deductive planning (simple language to express plan) verification/monitoring of plans (expressive planning language) verification of state transitions search for a solution Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 66

67 What are the KRR assets? tools: Rule-based(Prolog, Answer Set Programming) Ontology-based (DL-languages, OWL and related reasoners) NMR:?? Specialized theories (and, sometimes, tools): Action Time Space... Needed: more qualitative representation and reasoning Lucia Winter School, Daniele Nardi, December 2013 Knowledge Representation and Reasoning 67