VU Nichtmonotones Schließen

Transcription

1 VU Nichtmonotones Schließen WS 2010/11 Hans Tompits Institut für Informationssysteme Arbeitsbereich Wissensbasierte Systeme

2 1 Introduction 1.1 Phenomenology of Nonmonotonic Reasoning 1

3 Slogan Nothing is certain, but death and taxes. (Benjamin Franklin, 1789) Practically all inferences in daily life are uncertain. 2

4 Slogan (ctd.) Examples: We all believe that the sun will rise tomorrow... but we cannot prove it! I don t have an elder brother, because otherwise I would know it. However, it is nevertheless possible, unbeknownst to me, that I have an elder brother... The remaining lecture will not be taught in Japanese. Conclusions of this kind are ubiquitous!... But they are amazingly difficult to formalise. In this course we deal with formal methods to describe this kind of reasoning, generally referred to as common-sense reasoning. 3

5 Basic Ideas Traditional logics satisfy the monotonicity condition: If A is derivable from a theory T, then A is also derivable from T, for any T T. This principle is not adequate in the context of common-sense reasoning. Common-sense reasoning deals with incomplete information. We are constantly faced with situations for which not all relevant information is known. Yet, we are not paralysed by this obstacle and are able to draw plausible inferences. These inferences, however, are not ironclad they can be invalidated by new, more accurate information. They are defeasible, i.e., they may be retracted in the light of new information. 4

6 Example: Defeasible Conclusions Assume it is Saturday afternoon and James Bond must urgently meet M, his superior (and head of MI6). The following facts are known: M cannot be reached by phone. At Saturday afternoons, M usually visits his club. Given no information to the contrary, it is thus reasonable to assume that M is at his club and James Bond should therefore seek him there. On his way to the club (in his Aston Martin DB 5), Bond receives a phone call from Miss Moneypenny. She informs him: M had a car accident on Friday. Therefore, it is no longer reasonable to go to the club, rather to go to the hospital and meet M there. 5

7 General Definition Definition 1 (Minsky, 1975) 1. By nonmonotonic reasoning we understand the drawing of conclusions which may be invalidated in the light of new information. 2. A logical system is nonmonotonic iff its provability relation violates the property of monotonicity. 6

8 Monotonic vs. Nonmonotonic Logics Principal difference: Classical logic formalises truth and valid conclusions. Nonmonotonic logics formalise rationality and plausible conclusions. Note: Rationality is central for common-sense reasoning: although conclusions can be invalidated by new information, they are not chosen at random, rather our knowledge and experience is employed for accepting conclusions, demanding that they are rational. At least one rational justification is required to accept a conclusion in common-sense reasoning. 7

9 Rational Conclusions Rationality is a complex concept and a precise definition is nigh impossible. In fact, the different nonmonotonic reasoning formalisms introduced in the literature can be viewed as approaches for capturing different aspects of rationality. 8

10 Rational Conclusions (ctd.) Rationality has at least two specific properties distinguishing it from the concept of truth, namely: Rationality is agent-dependent different agents may have different opinions on what is rational in a certain situation; purpose-dependent the acceptance of a proposition as a rational conclusion depends on the purpose it is used for. Example: Given only cursory evidence, I may well assume that a person is honest and lend him 20 Euros. But I would be more cautious if I want to consider that person as a business partner, and would conduct a thorough investigation. 9

11 Beliefs To emphasise the subjective nature of common-sense conclusions, they are referred to as beliefs. Two fundamental questions: 1. What actually are common-sense conclusions (beliefs)? 2. What techniques do we use for reaching them? Concerning the first question: Definition 2 (Perlis, 1987) A proposition A is a belief of an agent G, i.e., G considers A as a rational conclusion, if G is prepared to use A as if it were true. 10

12 Beliefs (ctd.) Example (after Winograd (1980)): Assume I plan a trip by car. To begin with, I must decide where my car actually is. Given no better evidence, the following is plausible: (A) The car is there, where I parked it last. According to Def. 2, (A) is considered to be a belief if I act as if it were true. That is, if I ignore all circumstances in which (A) could be false, and I base my actions on the assumption that (A) is true, then I believe in (A), even if I cannot be sure that (A) is actually true. 11

13 Beliefs (ctd.) Example (ctd.): On the other hand: Even if I am almost certain that (A) holds, but at the same time I improve my chances by checking whether a bus will pass by, in case the car is missing, then (A) is not regarded as a belief, rather a very likely contingency. 12

14 Beliefs (ctd.) Now to the question how beliefs are reached. Consider the earlier example involving the famous British agent that (C) M is in his club since: (1) It is Saturday afternoon. (2) Usually, at Saturday afternoons, M visits his club. Obviously, in virtue of (2), (1) gives evidence supporting (C). It might seem that the belief (C) is inferred using the following rule: (R) If (1), conclude (C). But this cannot be the case! 13

15 Beliefs (Forts.) One cannot accept (R) and at the same time be prepared to reject (C) in the light of new information. The inference pattern must be more complex! The rule should be blocked whenever its use would be intuitively unacceptable. I.e., I am prepared to conclude (C) unless there is some evidence making this inference irrational. Instead of (R), the following ignorance-dependent rule should be used: (R ) From (1), in the absence of evidence to the contrary, infer (C). 14

16 Nonmonotonic Inference Rules To summarise: Definition 3 By a nonmonotonic inference pattern, or a nonmonotonic inference rule, we understand a rule of the following form: Given A, in the absence of evidence B, infer C. Remark: In practical applications, B is often identified with contradictory evidence, i.e., considering rules of the following form: Given A, in the absence of evidence to the contrary, infer C. 15

17 Nonmonotonic Inference Rules (ctd.) Def. 3 actually leaves the following fundamental questions open: How are nonmonotonic rules to be represented? Under which criteria is a piece of information to be considered as given (or known)? The answers to these questions are non-trivial! The formalisms we will deal with in this course are approaches to answer these questions! 16

18 1.2 Nonmonotonic Reasoning and Reasoning about Actions 17

19 Two Problems Two problems were fundamental in the development of nonmonotonic reasoning formalisms: frame problem; qualification problem. Both problems are relevant for reasoning about actions. 18

20 Frame Problem The frame problem deals with the following scenario: Suppose we perform an action. Which properties of the world change and which persist? Example: A cube is painted the same. = its colour changes, all other properties stay (1) t c(paint(t, c) color(t + 1, c)) (2) paint(0, red), weight(0, 500) We want to derive: weight(1, 500), color(2, red). Problem: This is not derivable from (1) and (2)! 19

21 Frame Problem (ctd.) We need additional frame axioms: (3) t c[(color(t, c) do nothing(t)) color(t + 1, c)]; do nothing(t)... no action at time point t. (4) t k(weight(t, k) weight(t + 1, k)). Together with fact do nothing(1). 20

22 Frame Problem (ctd.) Problem: In general, a large number of frame axioms is required = inefficient! Idea: Replace frame axioms by a simple nonmonotonic rule: All aspects of the world remain invariant, except those which explicitly change. 21

23 Qualification Problem The qualification problem describes the following scenario: Which preconditions ( qualifications ) must hold that an action succeeds? Example: Suppose a robot needs to move a block A on top of another block B. Requirements: B should have a clear top; A must not be too heavy; the robot s arm must not be broken; etc. In general, a large number of qualifications is necessary. 22

24 Qualification Problem (ctd.) Even worse: In real-world domains, the satisfiability of the qualifications is often unknown. For instance, suppose I plan a trip to the airport. I may know sufficiently enough to be convinced that my trip succeeds. BUT: I can always imagine situations making my trip impossible my car is broken, a meteor destroys the road to the airport, I will be abducted by aliens, etc. Idea: use a nonmonotonic rule In the absence of evidence to the contrary, assume that an action succeeds. 23

25 Ramification Problem Related problem: the ramification problem. Describes the implicit consequences from performed actions. Example: A robot moves a dusty block. = The movement of the dust particles should not be specified explicitly, rather it should be derivable from the fact that the cube has been moved. 24

26 1.3 Typology of Nonmonotonic Reasoning 25

27 Types of Nonmonotonic Inferences Different types of nonmonotonic inferences can be identified. The classification is according to the criteria under which nonmonotonic conclusions are accepted. Following Moore (1983), one should distinguish between genuine incomplete information; incomplete representation of principally complete information. 26

28 Types of Nonmonotonic Inferences (ctd.) Example: (1) In the absence of evidence to the contrary, assume that birds fly. (2) Unless your name is on a list of winners, assume that you are a loser (Gabbay & Sergot, 1986). Discussion: Rule (1) involves genuinely incomplete information. If all we know about Tweety is that it is a bird, we may conclude (at least tentatively) that it flies. This conclusion is plausible but not ironclad: if we later learn that Tweety is actually a penguin, then we have to retract our prior conclusion since penguins don t fly. 27

29 Types of Nonmonotonic Inferences (ctd.) Rule (2) describes a different scenario. Here, it is assumed that the list of winners is complete, so that it contains information about losers implicitly. That is, the rule refers not to incomplete information but to incomplete representation of virtually complete knowledge. The rule is logically valid: if it is true that a list of winners is complete, and it is true that your name is not on the list, then it must be true that you are a loser. This conclusion cannot be invalidated by providing better evidence. At the moment of announcing the list, our knowledge is complete and cannot be extended by a new piece of information. 28

30 Types of Nonmonotonic Inferences (ctd.) Rule (2) is nevertheless nonmonotonic since it is context-dependent: It refers to the absence of some evidence and thus depends on the context within it operates. Embedding the rule in a larger context may render previous conclusions underivable. Example: (3) The list of winners consists of James and Felix. Applying Rule (1), we obtain: ( ) Miss Moneypenny is a loser. Suppose (3) is replaced by (4) The list of winners consists of James, Felix, and Miss Moneypenny. = ( ) is no longer derivable. 29

31 Types of Nonmonotonic Inferences (ctd.) In summarising, we can distinguish between two kinds of nonmonotonic inferences: Default reasoning Refers to the drawing of rational conclusions from less than conclusive information, in the absence of evidence making these inferences implausible. Conclusions are defeasible. Autoepistemic reasoning Refers to the drawing of conclusions using incomplete representation of (theoretically) complete information. Principle: if a conclusion would be false, I would know it (by assuming that my knowledge is complete). Conclusions are not defeasible, yet nonmonotonic since context-dependent. 30

32 Default Reasoning: Further Classification Prototypical reasoning Principle: A describes a typical situation. Therefore, it is plausible that A holds. Conclusions are based on implicit statistical knowledge explicit probability values are not known and cannot be determined in general we simply know that they are sufficiently high. Example: The majority of birds fly. 31

33 Default Reasoning: Further Classification (ctd.) No-risk reasoning Principle: If A would be accepted, but turning out to be false, the consequences would be disastrous. Therefore, if one has to choose between A and A, select A. Example: In the absence of evidence to the contrary, assume that the accused is innocent. 32

34 Default Reasoning: Further Classification (ctd.) Best-guess reasoning Principle: No supporting evidence can be found, but a decision must be made. So, taking the currently available knowledge into account, choose A as the best conclusion. Example: It is Sunday and I want to go to a Sushi restaurant for lunch. There are two Sushi restaurants in my neighbourhood; one of them is closed on Sundays but I don t know which one. The best guess is to assume that the one nearest to my home is open. 33

35 Default Reasoning: Further Classification (ctd.) Probabilistic default reasoning Here, explicit numerical probability values are embedded in the inference process. Principle: If the probability for A is sufficiently high, infer A. In this lecture, we only deal with qualitative, logic-based methods for modelling nonmonotonic reasoning. Probabilistic methods are a separate field ( probabilistic reasoning, uncertain reasoning ) based on classical logic and probability theory (e.g., belief networks, Dempster-Shafer theory, ε-semantics due to Adams and Pearl, etc.). 34

36 Autoepistemic Reasoning: Further Classification Subjective autoepistemic reasoning Principle: In view of my own (subjective) knowledge, I would know if A is false. Since I do not know that A is false, A must be true. Example: I don t have an elder brother, otherwise I would know it (Moore, 1983). 35

37 Autoepistemic Reasoning: Further Classification (ctd.) Explicit autoepistemic reasoning Principle: According to an explicit convention, I would know whether A is false. Since I do not know that A is false, A must be true. Example: If you are not on a list of winners, assume that you are a loser (in view of the convention that lists of winners are always complete). 36

38 1.4 Formal Approaches to Nonmonotonic Reasoning 37

39 Sandewall s Formalism First attempt to formalise nonmonotonic inferences (1972). Extends classical first-order logic by introducing a unary operator UNLESS. Intuitive meaning: for a theory T and a proposition A: UNLESS(A) is derivable from T (in Sandewall s formalism) A is not derivable from T by means of first-order logic. Symbolically: T UNLESS(A) T A. 38

40 Sandewall s Formalism (ctd.) Sandewall s formalism possesses undesired properties: There are cases where both A as well as UNLESS(A) is derivable. Example: T = {C, (C UNLESS(B)) A}. = T A and T UNLESS(A). The formalism is too strong to deal with conflicting rules in an adequate manner: Example: Assume T consists of Quaker(Nixon) Republican(Nixon); ( (Quaker(x) ) ) x UNLESS( Pacifist(x)) Pacifist(x) ; ( (Republican(x) ) ) x UNLESS(Pacifist(x)) Pacifist(x) ; = T is inconsistent: It holds that T UNLESS(Pacifist(Nixon)) and T UNLESS( Pacifist(Nixon)), so T Pacifist(Nixon) and T Pacifist(Nixon) hold as well. This lead to the development of more refined methods! 39

41 Modal Nonmonotonic Logics Refers to a family of formalisms employing the language of modal logic for representing the notions of consistency and belief. Employ a unary operator M for representing the notion of consistency directly in the object language: MA: A can be consistently assumed. Dual operator L = M : LA: A is nonmonotonically derivable (A is a belief). Example: Birds usually fly: x ( (Bird(x) MFlies(x)) Flies(x) ) Interpretation: For all objects x, if x is a bird and it is consistent to assume that x flies, then infer that x flies. 40

42 Modal Nonmonotonic Logics (ctd.) Important formalism: autoepistemic logic (Moore, 1983) models autoepistemic reasoning describes inference patterns of an ideally rational agent reasoning about his own knowledge is a reconstruction of earlier (problematic) modal nonmonotonic logics due to McDermott & Doyle (1980) and McDermott (1982). 41

43 Default Logic Formalises default reasoning. Proposed by Reiter in Uses special inference rules, called defaults, for allowing nonmonotonic inferences. Example: Birds usually fly is represented by Bird(x) : Flies(x) Flies(x). Interpretation: For all objects x, if x is a bird and it is consistent to assume that x flies (i.e., Flies(x) is not derivable), then infer that x flies. 42

44 Circumscription Circumscription was proposed by McCarthy in 1980 and is a form of minimal-model reasoning: Circumscription of a predicate P is tantamount to minimising the extension of P (i.e., the set of objects satisfying P ). Principle: Only those objects satisfy a property E for which it can be shown that they satisfy E. This is realised by means of a schema of second-order logic. Example: Birds usually fly is represented here by means of x ( (Bird(x) Abnormal(x)) Flies(x) ), and by minimising the interpretation of abnormal birds (i.e., minimising the extension of Abnormal(x)). 43

45 Circumscription (ctd.) Model theoretically, circumscription is reasoning under minimal models: For checking whether A follows from T, not all models of T are taken into consideration, but only those which are minimal in a certain sense. To this end, a partial pre-order,, over the models is defined a model is minimal iff it is a minimal element with respect to. A similar model-theoretic method has been introduced by Shoham (1988). 44

46 Closed-World Assumption The Closed-world assumption (CWA) was postulated by Reiter in 1978 in the context of database theory. Idea: Only positive information is stored in a database. All facts which are not derivable from the database are assumed to be false. Example: train schedule all train connections which are not explicitly given in the train schedule are assumed to be non-existent. 45

47 1.5 Literature (Selection of Standard Texts) 46

48 1. Witold Lukaszewicz. Non-monotononic Reasoning. Ellis Horwood, Viktor Marek und Miroslaw Truszczyński. Nonmonotonic Logic. Springer, Gerd Brewka. Nonmonotonic Reasoning. Cambridge University Press, Philippe Besnard. An Introduction to Default Logic. Springer, Dov M. Gabbay, Christopher J. Hogger und J.A. Robinson. Handbook of Logic in Artificial Intelligence and Logic Programming. Volume 3, Oxford University Press,

49 2 Prerequisites from Classical Logic 48

50 Language Two important forms of classical logic: Classical propositional logic Classical first-order logic (or predicate logic) 49

51 Language of Propositional Logic Formulas of propositional logic are built from a set of atomic formulas using the primitive logical connectives and and the logical constant (verum) in the usual manner. The remaining operators,,, and (falsum) are defined from the primitive ones as usual, i.e., by setting (A B) = ( A B), (A B) = ( A B), (A B) = ((A B) (B A)), =. 50

52 Language of First-Order Logic Formulas of first-order logic are built from a set of atomic formulas using the operators from propositional logic and the universal quantification symbol,, in the usual manner. Here, we assume a first-order language with function symbols. We denote (object) variables by x, y, z,..., and (object) constants by a, b, c,... or by proper names. Terms are built from the variables and the function symbols. The existential quantification is defined by setting xa = x A. 51

53 Some Concepts and Notation Let Q {, }. For any formula QxA, A is the scope of Qx. Each occurrence of a variable x in A as well as the occurrence of x directly preceded by Q is in QxA bound. Each occurrence of a variable x in A is free iff it is not bound. A formula in which all variable occurrences are bound is closed, otherwise open. A theory is a set of formulas, in case of propositional logic, and a set of closed formulas, in case of first-order logic. 52

54 Some Concepts and Notation (ctd.) A term containing no variables is called ground. Let x 1,..., x n be the free variables in a formula A. The universal closure of A is A = x 1... x n A. Let A(x 1,..., x n ) be a formula containing the free variables x 1,..., x n and let α 1,..., α n be terms. A[x 1 /α 1,..., x n /α n ] denotes the formula resulting from A(x 1,..., x n ) by uniformly replacing all free occurrences of x i by α i, for 1 i n. E.g., for A(x, y) = (p(x) x q(x, y)) r(x, y), we have that A[x/a, y/f(b)] = (p(a) x q(x, f(b))) r(a, f(b)). 53

55 Semantics vs. Proof Theory Each proper logic is assigned with two basic ingredients: semantics; proof theory. The semantics associates a meaning to the language elements i.e., it defines the truth values to formulas. The proof theory takes care for defining the notion of a formal proof or a formal derivation it is defined in terms of a calculus, comprising axioms and inference rules. 54

56 Semantics The semantics is defined in terms of interpretations and the semantic consequence operator =: In propositional logic, an interpretation is a function assigning each atom a truth value. In first-order logic, an interpretation is a bit more complicated, involving a domain for the values of variables and objects, assignments of the predicate letters and function symbols. 55

57 Semantics (ctd.) Interpretations induce truth values for composite formulas in an inductive fashion. A model of a formula A is an interpretation making A true. Similarly, a model of a theory T is an interpretation which is a model of all elements of T. The relation T = A holds if all models of T are models of A. In particular: for T =, T = A means that A is valid, i.e., A is true under every interpretation. We write = A instead of = A. 56

58 Proof Theory The proof theory involves the following concepts: An axiom system, consisting of a set of formulas, called axioms, a set of relations between formulas, called inference rules. If R is an inference rule and if X 1,..., X n, Y R, then Y is a direct consequence of X 1,..., X n under R. A proof is a finite sequence A 1,..., A n of formulas s.t. for each A i (1 i n) A i is an axiom, or A i is a direct consequence of earlier elements in the sequence under some inference rule. A formula A is provable, symbolically A, iff there is a proof A 1,..., A n s.t. A n = A holds. 57

59 Proof Theory (ctd.) A derivation from a theory T is a finite sequence A 1,..., A n of formulas s.t. for each A i (1 i n) A i T, or A i is an axiom, or A i is a direct consequence of earlier elements in the sequence under some inference rule. A (closed) formula A is derivable from a theory T, symbolically T A, iff there is a derivation A 1,..., A n of T s.t. A n = A holds. 58

60 Proof Theory (ctd.) Central properties: Soundness of an axiom system: if T A, then T = A. Completeness of an axiom system: if T = A, then T A. Concrete sound and complete axiom systems for propositional logic and first-order logic can be readily found in the literature, like Hilbert-type systems, sequent-type calculi, tableau calculi, resolution calculi, natural deduction systems, etc. = We present a sequent-type calculus for propositional logic later on! 59

61 Deductive Closure The deductive closure of a theory T is given by Th(T ) = {A T A}, in case of propositional logic, and Th(T ) = {A T A and A is closed}, in case of first-order logic. Properties: T Th(T ); ( inflationaryness ) Th(T ) = Th(Th(T )) ( idempotency ); T T implies Th(T ) Th(T ) ( monotonicity ). 60

62 Prenex Normal Form A formula A is in prenex normal form if A is of the form where Q l {, }, l = 1,..., k, Q 1 x 1... Q k x k ( n i=1 ( m(i) j=1 B ij ) ), x 1,..., x k are pairwise distinct variables of ( n i=1 ( m(i) j=1 B ij)), B ij is an atom or the negation of an atom. For each formula A there is a formula B in prenex normal form s.t. = (A B). 61

63 Skolemisation The skolemisation of a closed formula A in prenex normal form is constructed as follows: Consider a subformula of A of the form y B(y, x 1,..., x n ) x 1,..., x n are those free variables of y B(y, x 1,..., x n ) which are bound by -quantifiers in A. Replace y B(y, x 1,..., x n ) by B(f(x 1,..., x n ), x 1,..., x n ) f is a new n-ary function symbol ( Skolem function ). Repeat this for any subformula of A of the form y B(y). A skolemised form of a closed formula A is a closed formula resulting from a prenex form of A by skolemisation. 62

64 3 Default Logic 3.1 Preliminaries 63

65 Default Rules In default logic, nonmonotonic conclusions are represented in terms of special inference rules. Let A( x), B 1 ( x),..., B n ( x), C( x) be first-order formulas, whose free variables are among x = x 1,..., x k. A default, d, is an expression of form Intuitive meaning: A( x) : B 1 ( x),..., B n ( x) C( x) for all objects x = x 1,..., x k, if A( x) is known and B 1 ( x),..., B n ( x) can be consistently assumed, then infer C( x). Notation: A( x) ist the prerequisite, {B 1 ( x),..., B n ( x)} is the justification, and C( x) is the consequent of d. 64

66 Examples Prototypical reasoning: Bird(x) : Flies(x) Flies(x) Typically, birds fly. No-risk reasoning: Accused(x) : Innocent(x) Innocent(x) The accused is innocent unless proven otherwise. 65

67 Examples (ctd.) Best-guess reasoning: Solution found so far(x) : Best Solution(x) Best Solution(x) In the absence of evidence to the contrary, assume that the best solution found so far is the best one (McCarthy, 1984). Subjective autoepistemic reasoning: : Have elder brother Have elder brother In the absence of evidence to the contrary, assume that you have no elder brother (Moore, 1983). 66

68 Notation Let d be a default of form A : B 1,..., B n C If A =, then d is called prerequisite-free. In this case, d is also written as : B 1,..., B n C If n = 0, then d is justification-free. 67

69 Default Theories In default logic, knowledge is represented in terms of default theories. A default theory is an ordered pair T = W, D, where W is a set of closed first-order formulas; D is a set of defaults. Elements of W are the premisses of T, representing certain yet in general incomplete information about the world. D represents plausible although not necessarily true conclusions (i.e., conclusions which hold typically rules of thumb ). 68

70 Default Theories (ctd.) Problem: What totality of knowledge is induced by a default theory? Answer: notion of an extension. Basic idea of obtaining extensions: Apply the defaults in D to the premisses in W to extend certain knowledge by plausible conclusions; apply the defaults as long as possible, until no new knowledge is generated the result is an extension of T. 69

71 Fundamental Questions Recall: intuitive meaning of A : B 1,..., B n C If A is known and B 1,..., B n can be consistently assumed, then infer C. Problem: What means A is known? When can B 1,..., B n be consistently assumed? Can default conclusions be obtained from applications of the premisses in W alone? 70

72 Example Consider the following default theory T = W, D : W = {Friend(Tom, Bob), Friend(Bob, Sally), Friend(Sally, Tina)} D = { } Friend(x, y) Friend(y, z) : Friend(x, z) Friend(x, z) On the basis of the premisses (by using the defaults), Friend(Tom, Sally), Friend(Bob, Tina) can be inferred. But it cannot be further derived that Friend(Tom, Tina) holds, since Friend(Tom, Sally) and Friend(Bob, Tina) can only be derived by default, i.e., they do not belong to the premisses. The default inference mechanism must be more involved! 71

73 Extensions An extension E of a default theory T = W, D should satisfy the following conditions: 1. E should be a set of first-order formulas. 2. E should contain the premisses, i.e., W E. 3. E should be deductively closed, i.e., Th(E) = E. 4. E should contain a maximal number of formulas which are derivable by defaults. 5. E should contain no unfounded formulas. Conditions 4 and 5 are informal and need to be recast into formal terms! 72

74 Example: Tweety Consider the following default theory T = W, D : W = {Bird(Tweety))}; D = { Bird(x) : Flies(x) d = Flies(x) }. An extension E of T should contain Bird(Tweety) and Flies(Tweety), and all formulas which can be classically derived from them. = It should hold that E = Th({Bird(Tweety), Flies(Tweety)}). 73

75 Example: Tweety (ctd.) Consider now T = W, D, where W = W {Penguin(Tweety), x(penguin(x) Flies(x))}. Extension E of T should contain W, but not Flies(Tweety), since the application of d is blocked : with Penguin(Tweety) E and x(penguin(x) Flies(x)) E, it follows that Flies(Tweety) E (since E is closed under classical logic). Flies(Tweety) can no longer be consistently assumed! It should hold that E = Th(W ). 74

76 Example: Nixon Diamond Consider the following default theory T = W, D : W = {Republican(Nixon), Quaker(Nixon))}; { Republican(x) : Pacifist(x) D = d 1 =, Pacifist(x) } Quaker(x) : Pacifist(x) d 2 =. Pacifist(x) Defaults d 1 and d 2 are mutually conflicting! (The application of d 1 blocks d 2 and vice versa.) There are two alternatives for extensions: E 1 = Th({Republican(Nixon), Quaker(Nixon), Pacifist(Nixon)}); E 2 = Th({Republican(Nixon), Quaker(Nixon), Pacifist(Nixon)}). Formal definition of an extension is given later on! 75

77 3.2 Open Defaults 76

78 Open Defaults A default d = (A : B 1,..., B n /C) is open iff one of the formulas A, B 1,..., B n, C is open; otherwise d is closed. A set D of defaults is open iff at least one default d D is open; otherwise D is closed. A default theory T = W, D is open iff D is open; otherwise T is closed. 77

79 Open Defaults (ctd.) Let d be a default of form A( x) : B 1 ( x),..., B n ( x) C( x), where x = x 1,..., x k comprises the free variables in A( x), B 1 ( x),..., B n ( x), C( x). Let furthermore α 1,..., α k be ground terms. Then, the closed default A[x 1 /α 1,..., x k /α k ] : B 1 [x 1 /α 1,..., x k /α k ],..., B n [x 1 /α 1,..., x k /α k ] C[x 1 /α 1,..., x k /α k ] is called an instance of d. 78

80 Open Defaults (ctd.) Open defaults represent general inference schemata which can be applied to arbitrary objects. Open defaults can be identified with the set of all its instances. We will replace open defaults by their closed instances. It therefore suffices to consider only closed default theories in the following. To this end, we define the closure of an open default theory. 79

81 Closure of a Default Theory: Motivation An instance of a default is obtained by replacing the free variables by ground terms. Ground terms can intuitively be viewed as representing individuals. Problem: which totality of individuals is determined by a default theory? 80

82 Examples Let T = W, D be the following default theory: W = {Musician(Lemmy), Musician(Phil), Philosopher(Immanuel))}; D = { } Musician(x) : Deaf (x) d =. Deaf (x) Lemmy, Phil, and Immanuel are the only individuals mentioned by T. The closure of T should therefore be comprised of W, together with the following instances of d: Musician(Lemmy) : Deaf (Lemmy) ; Deaf (Lemmy) Musician(Phil) : Deaf (Phil) ; Deaf (Phil) Musician(Immanuel) : Deaf (Immanuel). Deaf (Immanuel) 81

83 Examples (ctd.) Suppose T = W, D is the following default theory: W = {Kryptonian(Kal-El), x(kryptonian(x) Kryptonian(father(x)))}; { } Kryptonian(x) : Superpowers(x) D = d =. Superpowers(x) Clearly, here, the set of individuals induced by T should be given by {Kal-El, father(kal-el), father(father(kal-el)),...}. The closure of T should be given by W, together with the following infinitely many instances of d: Kryptonian(Kal-El) : Superpowers(Kal-El) ; Superpowers(Kal-El) Kryptonian(father(Kal-El)) : Superpowers(father(Kal-El)) ; Superpowers(father(Kal-El)) Kryptonian(father(father(Kal-El))) : Superpowers(father(father(Kal-El))) ;... Superpowers(father(father(Kal-El))) 82

84 Examples (ctd.) Let T = W, D now be the following default theory: W = { x Kryptonian(x)}; D = { Kryptonian(x) : Superpowers(x) d = Superpowers(x) }. T makes here only implicit reference to an object, namely one which is determined by the existential quantifier. We intuitively expect that this object possesses superpowers, i.e., x (Kryptonian(x) Superpowers(x)) should be contained in an extension of T. 83

85 Examples (ctd.) To achieve this, we introduce a new constant, a, and replace the premiss x Kryptonian(x) by Kryptonian(a). The closure of T is then given as follows: Kryptonian(a), Kryptonian(a) : Superpowers(a) Superpowers(a) This default theory has one extension, namely E = Th({Kryptonian(a), Superpowers(a)}) and it holds that x (Kryptonian(x) Superpowers(x)) E. 84

86 Examples (ctd.) Remark: In the last example, we have the problem that E is defined over another language than T. In particular: E contains formulas which have no direct connection to T (namely formulas containing a). One identifies extensions of T as those sets, which are extensions of the closure of T, but restricted to the original language of T. 85

87 Examples (ctd.) As final example, we consider T = W, D given as follows: W = {Superhero(Batman)}; { : x Archfiend(x, Batman) D = d 1 = x Archfiend(x, Batman), d 2 = Archfiend(x, y) : Fights(x, y) Fights(x, y) T explicitly mentions the person Batman and refers implicitly to an archfiend of Batman. We expect that x (Archfiend(x, Batman) Fights(x, Batman)) should be contained in an extension of T. To this end, we introduce a new constant, b, and replace d 1 by its skolemised form : x Archfiend(x, Batman). Archfiend(b, Batman) }. 86

88 As closure of T, we obtain: Superhero(Batman), Examples (ctd.) : x Archfiend(x, Batman), Archfiend(b, Batman) Archfiend(Batman, Batman) : Fights(Batman, Batman), Fights(Batman, Batman) Archfiend(Batman, b) : Fights(Batman, b), Fights(Batman, b) Archfiend(b, Batman) : Fights(b, Batman), Fights(b, Batman) Archfiend(b, b) : Fights(b, b). Fights(b, b) Note that the extension of the closure of T contains Archfiend(b, Batman) Fights(b, Batman), and thus x (Archfiend(x, Batman) Fights(x, Batman)) as desired. 87

89 Closure of a Default Theory: Definition A skolemised form of a default d = (A : B 1,..., B n /C) results from d as follows: 1. Let C 1 be a skolemised form of C such that no Skolem function already occurs in A, B 1,..., B n. 2. Eliminate all quantifiers in C 1. The result is C Replace C in d by C 2. This is the desired skolemised form of d. A skolemised form of a default theory T = W, D results from T by replacing each premiss in W and each default in D by their skolemised forms, respectively. All introduced Skolem functions must be pairwise distinct and may not already occur in T. 88

90 Closure of a Default Theory: Definition (ctd.) Let T = W, D be a default theory. The set TERMS(T ) consists of all ground terms which can be constructed from the function symbols occurring in T. The closure, T, of T is defined as follows: If T is closed, then T = T. Otherwise: Let T 1 be a skolemised form of T. Replace all open defaults in T 1 by their instances over TERMS(T 1 ). The result is T. From now on, we only deal with closed default theories. 89

91 3.3 Extensions 90

92 Properties of Extensions: Reminder An extension E of a default theory T = W, D should satisfy the following conditions: W E. Th(E) = E. E should contain a maximal number of formulas which can be derived by means of defaults. E should contain no unfounded formulas. 91

93 Definition of Extensions We define extensions of a closed default theory T = W, D by means of a two-step procedure: Step 1: We define a closure operator, Γ T, mapping sets of closed formulas to sets of closed formulas. Intuitive meaning: For constructing Γ T (E), E serves as a context for testing the consistency conditions of the defaults in D. Γ T (E) itself collects all formulas derivable from W by means of classical logic and those defaults in D satisfying the consistency condition relative to E. Step 2: E is an extension of T iff Γ T (E) = E, i.e., if E is a fixed point of Γ T. Intuitively, extensions of a default theory are those contexts which reproduce themselves under the closure operator Γ T. 92

94 Definition of Extensions (ctd.) Extensions of an open default theory T are defined in terms of extensions of the closure T of T. Formally: Let T = W, D be an open default theory. E is an extension of T iff E = F L T, where F is an extension of the closure T of T, and L T is the language generated by T, i.e., L T is the set of all formulas which can be constructed from the function and predicate symbols occurring in T. 93

95 The Operator Γ T We now formally define Γ T, for each closed default theory T. Let S be a set of closed formulas, and let D be a set of closed defaults. The classical reduct of D relative to S is the following set of classical inference rules: { A D S = C A : B 1,..., B n C } D, B 1 / S,..., B n / S. Let D S be the derivability relation resulting from the classical derivability relation by adding the inference rules in D S. For every theory W, we define Th D S (W ) = {A W D S A, A is closed}. Then, for T = W, D, Γ T (S) is defined as follows: Γ T (S) = Th D S (W ). 94

96 The Operator Γ T (ctd.) Remarks: The set Th D S ( ) can be expressed in terms of Th( ): Th D S (W ) = Th(W i 0 E i), where E 0 = {C A/C D S and W A}; E i = {C A/C D S and W E i 1 A}. Consequently: Each extension E of a closed default theory T = W, D is necessarily of the form E = Th(W G), where G {C (A : B 1,..., B n /C) D}. This property allows to select suitable candidates of sets for being an extension of a default theory! 95

97 Example: Nixon Diamond We consider the following propositional form of the Nixon diamond: W = {q, r}; D = {(q : p/p), (r : p/ p)}. Intuitively: Nixon is both a Quaker and a Republican. A Quaker is normally a pacifist. A Republican is normally not a pacifist. 96

98 Example: Nixon Diamond (ctd.) D = {(q : p/p), (r : p/ p)} has two defaults = four candidates: E 1 = Th({q, r}); E 2 = Th({q, r, p}); E 3 = Th({q, r, p}); E 4 = Th({q, r, p, p}). Construction of Γ T (E i ) (= Th D E i (W )): D E1 = {q/p, r/ p}; Γ T (E 1 ) = Th({q, r, p, p}) = E 4 ; D E2 = {q/p}; Γ T (E 2 ) = Th({q, r, p}) = E 2 ; D E3 = {r/ p}; Γ T (E 3 ) = Th({q, r, p}) = E 3 ; D E4 = ; Γ T (E 4 ) = Th({q, r}) = E 1. = E 2 and E 3 are extensions of T (and they are the only extensions of T ). 97

99 Example: Nixon Diamond (ctd.) Assume we extend T to T = W, D, where W = W {s, s p} = {q, r, s, s p} (s: supporter of the military industry). We have again four candidates: F 1 = Th(W ); F 2 = Th(W {p}); F 3 = Th(W { p}); F 4 = Th(W {p, p}). Note: it holds that F 1 = F 3 and F 2 = F 4. Construction of Γ T (F i ) (= Th D F i (W )): D F1 = {r/ p}; Γ T (F 1 ) = Th(W ) = F 1 ; D F2 = ; Γ T (F 2 ) = Th(W ) = F 1. = F 1 is the only extension of T. 98

100 Example: Nixon Diamond (ctd.) Remark: For T = W, D and T = W, D it holds that W W, but the beliefs have not increased monotonically (as p E 2 but p / F 1, i.e., E 2 F 1 ). 99

101 Example: Non-Existence of Extensions Consider T = W, D, where: W = ; D = {(: p/ p)}. We have two candidates: E 1 = Th( ); E 2 = Th({ p}). Construction of Γ T (E i ) (= Th D E i (W )): D E1 = { / p}; Γ T (E 1 ) = Th({ p}) = E 2 ; D E2 = ; Γ T (E 2 ) = Th( ) = E 1. = T has no extension (as E 1 E 2 )! 100

102 The Operator Γ T Revisited Theorem 3.1 Let T = W, D be a closed default theory. Furthermore, let I T (S) be the family of all sets K satisfying the following conditions: (1) K = Th(K); (2) W K; (3) If (A : B 1,..., B n /C) D, A K, B 1 / S,..., B n / S, then C K. Then, Γ T (S) = K I T (S) K. Furthermore, Γ T (S) is the smallest set satisfying Conditions (1)-(3). 101

103 3.4 Properties of Extensions 102

104 Closure Property Let E be a set of closed formulas and D a set of closed defaults. E is closed relative to D iff every (A : B 1,..., B n /C) D satisfies: if A E and B 1 / E,..., B n / E, then C E. Theorem 3.2 Each extension of a closed default theory T = W, D is closed relative to D. Corollary 3.1 If a set E is not closed under D, then E cannot be an extension of T = W, D. 103

105 Semi-Recursive Characterisation of Extensions Theorem 3.3 Let E be a set of closed formulas and let T = W, D be a closed default theory. Define a sequence (E i ) i 0 of sets of formulas as follows: E 0 = W ; E i+1 = Th(E i ) {C (A : B 1,..., B n /C) D, E i A and B 1,..., B n / E}. Then, E is an extension of T iff E = E i. i 0 104

106 Semi-Recursive Characterisation of Extensions (ctd.) Theorem 3.4 Let T = W, D be a closed default theory without justification-free defaults. Furthermore, let E be a set of closed formulas such that E = i 0 E i, where (E i ) i 0 is defined as in Theorem 3.3. Then: E is consistent W is consistent. 105

107 Proof of Theorem 3.4 Proof: ( ) Trivial, since W E = i 0 E i. ( ) Suppose E is inconsistent. Consider H i = {C (A : B 1,..., B n /C) D, E i A, B 1,..., B n / E}. = E i+1 = Th(E i ) H i, for all i 0. Since it is easy to see that E = Th(E), and E is assumed to be inconsistent, it holds that F E, for every closed formula F. Since D contains no justification-free defaults, it follows that H i =, for all i 0. = i 0 E i = Th(E 0 ) = Th(W ). = E = Th(W ), since E = i 0 E i. = W is inconsistent since E is inconsistent. 106

108 Semi-Recursive Characterisation of Extensions (ctd.) Corollary 3.2 Let T = W, D be a closed default theory without justification-free defaults. T has an inconsistent extension W is inconsistent. Proof: ( ) Let W be inconsistent. Then, it is easy to verify that Th(W ) is an extension of T. Moreover, Th(W ) is clearly inconsistent. ( ) Let E be an inconsistent extension of T. In view of Theorem 3.3, we have that E = i 0 E i (where E i is defined as in Theorem 3.3). From Theorem 3.4 it follows that W is inconsistent. 107

109 Semi-Recursive Characterisation of Extensions (ctd.) Corollary 3.3 Let T = W, D be a closed default theory without justification-free defaults. If T has an inconsistent extension, then it is the only extension of T. Proof: Let E be an inconsistent extension of T. According to Corollary 3.2, W is inconsistent. Let F be an arbitrary extension of T. Since W F, we obtain that F is also inconsistent. Moreover, it holds that Th(E) = E and Th(F ) = F. = It follows that E = F, since both Th(E) and Th(F ) coincide with the set of all formulas. 108

110 Semi-Recursive Characterisation of Extensions (ctd.) Theorem 3.5 ( Maximality of Extensions ) Let T = W, D be a closed default theory and let E, F be extensions of T. If E F, then E = F. Proof: Consider sequences (E i ) i 0 and (F i ) i 0 s.t. E = i 0 E i and F = i 0 F i as in Theorem 3.3, and assume that E F. By induction, it is easy to show that F i E i, for all i 0. = i 0 F i i 0 E i, i.e. F E. Since E F by hypothesis, E = F follows. 109

111 Generating Defaults Let E be a set of closed formulas. A default (A : B 1,..., B n /C) is active in E iff E A and B 1 / E,..., B n / E. The set of generating defaults of E with respect to T = W, D is defined as GD(E, T ) = {d D d is active in E}. For a set D of defaults, define CONS(D) = {C (A : B 1,..., B n /C) D}. Prior results imply: Let T = W, D be a closed default theory and let E be an extension of T. Then: E = Th(W CONS(GD(E, T ))). 110

112 Properties of Active/Non-Active Defaults Theorem 3.6 Let d = (A : B 1,..., B n /C) be a default and E a set of closed formulas. 1. If d is not active in E, then: E is an extension of W, D E is an extension of W, D {d}. 2. If d is active in E, then: E is an extension of W, D {d} = E is an extension of W {C}, D. 3. If W A and B 1 / E,..., B n / E, then: E is an extension of W {C}, D = E is an extension of W, D {d}. 111

113 3.5 Normal Default Theories 112

114 Definition A default d is normal iff it is of the form A : B B. A default theory T = W, D is normal iff all defaults in D are normal. 113

115 Properties of Normal Default Theories Theorem 3.7 Each closed default theory which is normal has an extension. Proof: Let T = W, D be a closed normal default theory. If W is inconsistent, then T has an extension, namely the inconsistent set Th(W ) (cf. Corollary 3.2). Assume now that W is consistent. We define a sequence (E i ) i 0 as follows: E 0 = W ; E i+1 = Th(E i ) T i, where T i is a maximal set of closed formulas satisfying the following conditions: 1. E i T i is consistent; 2. if B T i, then there is a default (A : B/B) D s.t. E i A. 114

116 Proof of Theorem 3.7 Let E = i 0 E i. Claim: E is an extension of T. Proof of claim: Let H i = {B (A : B/B) D, E i A, B / E}. We show that T i = H i, for all i. From this, the claim follows. 1. We first show that T i H i. Let B T i. Then, by definition of T i, there is some default (A : B/B) D s.t. E i A. Suppose B E. Since E = i 0 E i, there is some k 0 s.t. B E k. Since B T i and T i E i+1, we have that B E i+1. Let m = max(i + 1, k). Since E l E l+1, for all l 0, it follows that B E m and B E m. = E m is inconsistent. Contradiction. Therefore, B / E must hold. We obtain B H i. = This proves that T i H i. 115

117 Proof of Theorem 3.7 (ctd.) 2. Suppose that T i H i holds. Then, there is some B H i s.t. B / T i. By the maximality of T i, we have that E i T i {B} is inconsistent. Since E i T i E i+1 E holds, E {B} is also inconsistent. It follows that E B. Furthermore, it is easy to see that E = Th(E). Hence, B E. But B H i, and so B / E must hold, by definition of H i. Contradiction. 116

118 Properties of Normal Default Theories (ctd.) Theorem 3.8 ( Orthogonality of Extensions ) If a closed normal default theory T = W, D has two distinct extensions E, F, then E F is inconsistent. Intuitive meaning: An agent cannot consistently accept both extensions simultaneously. Theorem 3.9 ( Semi-Monotonicity ) Let D, D be sets of closed normal defaults s.t. D D, and let E be an extension of T = W, D. Then, T = W, D has an extension E s.t. E E. 117

119 Semi-Normal Defaults A default is semi-normal iff it is of the form A : B C C. A default theory T = W, D is semi-normal iff all defaults in D are semi-normal. 118

120 Semi-Normal Defaults (ctd.) Semi-normal defaults possess in general not the properties of normal default theories. In particular: for semi-normal default theories, it is not guaranteed that they always have extensions. Example: Consider T = W, D, where W =, { : p q D =, q T has no extension! : q r r, } : r p. p 119

121 3.6 Proof Theory for Default Logic 120

122 Basic Reasoning Tasks Due to the multiplicity of extensions, two canonical inference relations can be defined in the context of default logic: Brave reasoning: a formula A is a brave consequence of a default theory T iff there is an extension of T containing A. Skeptical reasoning: a formula A is a skeptical consequence of a default theory T iff A is contained in all extensions of T. In what follows, we describe a sequent calculus for brave default reasoning (in the propositional case) due to Bonatti (1993). 121

123 A Sequent Calculus for Default Logic The sequent calculus consists of three parts: a sequent calculus, LK, for classical propositional logic; a complementary sequent calculus, LK c, formalising invalid propositions; and special default inference rules. Remarks: The complementary calculus is needed for formalising the consistency condition of default rules. LK c is due to Bonatti (1993), although a similar system was independently developed by Goranko (1994). 122

124 The Sequent Calculus LK Sequent-type calculi are axiom systems optimised for proof search. = Important for automated deduction! The sequent method was introduced by Gerhard Gentzen in his Ph.D. thesis in Actually, the system LK was introduced by him (but our version of LK is slightly different from Gentzen s original one). Formal objects of LK are sequents, which are ordered pairs of form Γ Σ, where Γ, Σ are finite sets of propositional formulas. Notation: Γ, A Σ for Γ {A} Σ ; Γ, Σ for Γ Σ ; Σ for Σ ; etc. 123

125 The Sequent Calculus LK (ctd.) Semantical meaning: a sequent S = A 1,..., A n B 1,..., B m has the same semantical meaning as the formula n m ι(s) = ( A i ) ( B i ) i=1 i=1 empty conjunctions are identified with, and empty disjunctions with. In particular, the following is defined: S is valid ι(s) is valid. 124

126 The Sequent Calculus LK (ctd.) Axioms of LK: ; A A, for each formula A. Inference rules of LK: Γ Σ, A, B Π Γ,, (A B) Σ, Π l Γ, A Σ, B Γ Σ, (A B) r Γ Σ, A Γ, A Σ l Γ, A Σ Γ Σ, A r Γ Σ Γ, A Σ wl Σ Γ Σ Γ, A wr 125

127 The Sequent Calculus LK (ctd.) From the inference rules for the primitive connectives and, we obtain derived rules for the defined connectives,, and, like: Γ, A Σ Γ, (A B) Σ l 1 Γ, B, Σ Γ, (A B) Σ l 2 Γ Σ, A Λ Π, B Γ, Λ Σ, Π, (A B) r Γ, A Σ Π, B Σ Γ, Π, (A B) Σ l Γ Σ, A Γ Σ, (A B) r 1 Γ Σ, B Γ Σ, (A B) r 2 126

128 The Sequent Calculus LK (ctd.) Proofs in LK are written in the form of trees. The root of such a tree is the sequent to be proven; top-most sequents are axioms; transitions between adjacent nodes are sanctioned by means of the inference rules. Adequacy of LK: Γ Σ is provable in LK Γ Σ is valid. 127

129 The Complementary Sequent Calculus LK C LK c formalises sequents which are not provable in LK. Formal objects of LK c are anti-sequents: An anti-sequent is an ordered pair of form Γ Σ, where Γ, Σ are finite sets of propositional formulas. An anti-sequent Γ Σ is true iff the sequent Γ Σ is not valid. 128

130 The Complementary Sequent Calculus LK C (ctd.) Axioms of LK c : anti-sequents of form Φ Ψ, where Φ, Ψ are finite sets of atomic formulas s.t. Φ Ψ =, and { } Ψ =. Inference rules of LK c : Γ Θ, A Γ, (A B) Θ lc 1 Γ, B Θ Γ, (A B) Θ lc 2 Γ, A Θ, B Γ Θ, (A B) rc Γ Θ, A Γ, A Θ Γ, A Θ lc Γ, Θ, A rc 129

131 The Complementary Sequent Calculus LK C (ctd.) For and we obtain the following derived rules: Γ, A, B Θ Γ, (A B) Θ lc Γ Θ, A Γ Θ, (A B) rc 1 Γ Θ, B Γ Θ, (A B) rc 2 Γ, A Θ Γ, (A B) Θ lc 1 Γ, B Θ Γ, (A B) Θ lc 2 Γ Θ, A, B Γ Θ, (A B) rc 130

132 The Complementary Sequent Calculus LK C (ctd.) Central property: Γ Θ is provable in LK c Γ Θ is true. Immediate consequence: Γ Θ is provable in LK c Γ Θ is not provable in LK. 131

133 Default Sequents A default sequent is an ordered quadruple of form Γ; Σ; Θ, where Γ, Σ, Θ are finite sets of propositional formulas, and is a finite set of propositional defaults (i.e., defaults containing only propositional formulas). A default sequent Γ; Σ; Θ is true iff there is an extension E of the default theory Γ, s.t. Σ E and Θ E = (E is called witness of Γ; Σ; Θ). 132