Default Reasoning and Theory Change

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1 Default Reasoning and Theory Change G. Antoniou A.Nayak A. Ghose MP2-1

2 Part I Introduction to Default Logic MP2-2

3 Nonmonotonic Reasoning Motivation How do you get to work? By bus! Usually, I go to work by bus. I walk to the bus stop and read: No buses today, we re on strike. Now I have to take back my previous conclusion: nonmonotonicity. MP2-3

4 Incomplete information Why not use the classical rule gotowork! strike " takebus? Can I list all potential obstacles? (icy streets, being in a hurry etc.). I may not have the time or the resources to establish the conditions of the left hand: Incomplete information. MP2-4

5 Plausible conjectures Intelligent systems need to make plausible conjectures, based on, say: Default rules Usually I go to work by bus. Introspection If the Rolling Stones were giving a concert in my city tonight, I would have heard of that. MP2-5

6 Reasons for being interested in NMR Reasoning with incomplete information. Maintaining "competing" information within the same knowledge base. Compact representation of information. Pieces of information that are stable under changes. MP2-6

7 Default Logics Overview The notion of a default Reiter s default logic - Syntax - Extensions - Properties Default logic variants - Constrained default logic - Priorities among defaults MP2-7

8 The notion of a default Defaults are rules of inference that can be applied if some information is given, and some assumptions can be made. Prototypical reasoning child(x) : hasparents(x) / hasparents(x) No-risk reasoning accused(x) : innocent(x) / innocent(x) MP2-8

9 Defaults Examples Defaults in law criminal(x)! foreigner(x) : expel(x) / expel(x) Exception: politicalrefugee(x) " expel(x) MP2-9

10 Defaults Examples Defaults in biology Typically molluscs are shell-bearers. Cephalopods are molluscs. Cephalopods are not shell-bearers. Expressed formally: mollusc(x) : shellbearer(x) / shellbearer(x) cephalopod(x) " mollusc(x)! shellbearer(x) MP2-10

11 Defaults Examples Closed World Assumption true : A / A for all ground facts A. MP2-11

12 Default logic Types of knowledge A default theory T consists of two kinds of knowledge: A set W of first order formulas called facts; certain information. A set D defaults; the assumptions that can be made. MP2-12

13 Defaults Definition A default d has the form A : B1,..., Bn / C with closed first order formulas A, Bi, C. A is called the prerequisite pre(d), B1,..., Bn the justifications just(d), and C the consequent cons(d) of d. MP2-13

14 Defaults with variables Defaults with free variables are read as schemes. bird(x) : flies(x) / flies(x) represents the set of defaults bird(tweety) : flies(tweety) / flies(tweety) bird(sam) : flies(sam) / flies(sam)... MP2-14

15 Extensions Informal idea Extensions are "world views" that are based on the given information (facts and defaults). Extensions are obtained from the application of some defaults in D. They include always the certain information W. MP2-15

16 Interpretation of defaults A : B1,..., Bn / C If A is currently known, and if it is consistent to assume B1,..., Bn, then conclude C. A : B1,..., Bn / C is applicable to a deductively closed set of formulas E iff A#E and B1$E,..., Bn$E. MP2-16

17 Extensions Desirable properties They should be deductively closed under classical reasoning. They should be maximal: no more defaults can be applied. Extensions represent maximal world views. In practice we may be interested in portions of extensions (query evaluation). MP2-17

18 In-set and Out-set P = (d0, d1,...) sequence of defaults from D without multiple occurrences. P[k] denotes the beginning part of P of length k. In(P) = Th(W % {cons(d) d occurs in P}). Out(P) = { B B # just(d) for some d in P}. MP2-18

19 In-set and Out-set Examples a a : b / b [d1] b : c / c [d2] Let P1 = (d1); In(P1) = Th({a, b}) and Out(P1) = {b}. Let P2 = (d2, d1); In(P2) = Th({a, c, b}), Out(P2) = { c, b}. MP2-19

20 Processes Enforce that defaults can indeed be applied in the given order. P is called a process iff the defaults can be applied in the given order. P is a process iff for every k such that P[k] is defined, dk is applicable to In(P[k]). MP2-20

21 Extensions Definition Given a process P we define the following: P is successful iff In(P) & Out(P) = ', otherwise it is failed. P is closed iff every default in D that is applicable to In(P) already occurs in P. E is an extension of a default theory T = (W,D) iff there is a closed and successful process P of T such that E = In(P). MP2-21

22 Processes & Extensions Example a a : b / d true : c / b [d1] [d2] P1 = (d1) is successful but not closed. P2 = (d1, d2) is closed but failed. P3 = (d2) is a closed and successful process. Thus E = Th({a,b}) is an extension, in fact the only extension of the default theory. MP2-22

23 Process tree Nodes: (In, Out) Edges: (In, Out) " (In, Out ) iff there is a default d = A : B1,..., Bn / C such that - d is applicable to In - In = Th(In % {C}) - Out = Out % { B1,..., Bn} MP2-23

24 Process tree Root: (Th(W), ') Paths starting at the root correspond to processes. MP2-24

25 A procedure for determining extensions Given a default theory T. 1. Build the process tree of T. 2. Traverse the tree, and collect all closed & successful nodes (corresponding to closed & successful processes of T). MP2-25

26 Extensions Example true : a / a MP2-26

27 Extensions Example true : p / q true : q / r MP2-27

28 Extensions Example green : likescars / likescars aaamember : likescars / likescars MP2-28

29 Reiter s definition of extensions Let E and F be deductively closed sets of formulas, and D a set of defaults. A default A : B1,..., Bn / C is applicable to F with respect to belief set E iff A # F and B1 $ E,..., Bn $ E. F is closed under D w.r.t. belief set E iff, for every default d#d that is applicable to F w.r.t. E, cons(d)#f. MP2-29

30 Reiter s definition of extensions For T = (W,D) let ( T (E) be the smallest set of formulas that includes W, is deductively closed, and is closed under D w.r.t. E. THEOREM [Antoniou 1996] E is an extension of T iff E = ( T (E). Note: Tradeoff between guessing and search. MP2-30

31 Default logic Properties Existence of extensions Joint consistency of justifications Cumulativity MP2-31

32 Existence of extensions We saw that it is not guaranteed. Is this undesirable? One may say no: user is responsible, as in programming The opposite view expects a logic to be more "fault tolerant". Different information sources (distributed systems, information superhighway). MP2-32

33 Ensuring existence of extensions Maintain the notion of extensions, but restrict the classes of default theories considered. Modify the concept of an extension. MP2-33

34 Normal default theories If we go the first way we may restrict attention to normal defaults: A : B / B [Reiter 1980] shows that normal default theories have always extensions, essentially because all processes are successful. MP2-34

35 Normal defaults Limitations They may be unable to express interactions among defaults. dropout(bill) dropout(x) : adult(x) / adult(x) adult(x) : employed(x) / employed(x) Two extensions, but we would prefer one (which?) In general, normal default theories are strictly less expressive than general default theories. MP2-35

36 Semi-normal defaults One possible solution: adult(x) : employed(x)! dropout(x) / employed(x) Semi-normal defaults: A : B! C / C [Etherington 1987] gives a sufficient condition for the existence of extensions. MP2-36

37 Preference among extensions Another way to ensure extensions is to use normal defaults, but add a priority relation which can model interactions among defaults. See Priorities among defaults later on. MP2-37

38 Modifications of the extension concept To ensure the existence of extensions. Justified Default Logic [Lukaszewicz 1988] Constrained Default Logic [Schaub 1992] MP2-38

39 Joint consistency of justifications Given the defaults true : p / q true : p / r there is a single extension, Th({q, r}). Jumping to conclusions versus consistent set of beliefs. MP2-39

40 Approaches based on joint consistency Constrained Default Logic [Schaub 1992] Rational Default Logic [Mikitiuk & Truszczynski 1993] MP2-40

41 Joint consistency Discussion Is joint consistency a desirable property? Which is the right DL approach? It depends very much on the problem at hand! Reiter s approach may lead to counterintuitive results: true : usable(l)! broken(l) / usable(l) true : usable(r)! broken(r) / usable(r) broken(l) ) broken(r) MP2-41

42 Joint consistency Discussion But joint consistency may also lead to counterintuitive results. travel : goodweather / takeswimsuit travel : badweather / takeraincoat travel goodweather ) badweather Here a cautious traveler would wish to apply both defaults, even though the assumptions contradict one another. MP2-42

43 Inference relations In classical logic: M A. In default logic: Skeptical reasoning W ~ D A iff A is included in all extensions of (W,D). Credulous reasoning W ~ D A iff A is included in at least one extension of (W,D). MP2-43

44 Cumulativity Ensures the safe use of lemmas (in the skeptical approach): If W ~ D A, then for all formulas B, W ~ D B * W%{A} ~ D B. MP2-44

45 Default logic violates cumulativity Consider the default theory T = (W,D) [Makinson 1989]: true : p / p p ) q : p / p The only extension is Th({p}), so ' ~ D p ) q. But if we add p ) q to T, then we get two extensions Th({p}) Th({ p, q}) p is not included in both extensions. MP2-45

46 Default Logics Overview The notion of a default Reiter s default logic - Syntax - Extensions - Properties Default logic variants - Constrained default logic - Priorities among defaults MP2-46

47 Constrained default logic JDL does not guarantee joint consistency of justifications. true : p / q true : p / r has the single modified extension Th({q, r}). Constrained Default Logic [Schaub 1992] guarantees joint consistency. It does so by maintaining a consistent set of supporting beliefs. MP2-47

48 Default applicability in CDL CDL adopts the idea of JDL not to run blindly into failure: "look ahead". A default A : B1,..., Bn / C is applicable to deductively closed sets of formulas E (current knowledge) and Con (set of supporting beliefs) iff A#E, and {B1,..., Bn, C} % Con is consistent. MP2-48

49 The supporting belief set Let T = (W,D) be a default theory. Let P be a sequence of defaults (d0, d1,...) of defaults without multiple occurrences. Con(P) = Th(W % cons(p) % just(p)). Example true : p / q true : p / r [d1] [d2] For the sequence P = (d1) we have Con(P) = Th({p, q}). MP2-49

50 Processes and extensions in CDL P is a constrained process iff dk is applicable to In(P[k]) and Con([k]), for all k such that P[k] is defined. P is a closed constrained process iff every default in D that is applicable to In(P) and Con(P) already occurs in P. A pair (E, C) of deductively closed sets of formulas is a constrained extension iff there is a closed constrained process P such that E = In(P) and C = Con(P). MP2-50

51 Constrained extensions Example p p : r / q p : r / r MP2-51

52 Constrained extensions Example true : usable(l)! broken(l) / usable(l) true : usable(r)! broken(r) / usable(r) MP2-52

53 Properties of CDL THEOREM Every default theory has at least one constrained extension. THEOREM Let E = In(P) for a closed and successful process P of T. If E % Out(P) is consistent then (E, Th(E % Out(P))) is a constrained extension of T. The converse is not true: true : p / p. MP2-53

54 Properties of CDL THEOREM If (E,C) is a constrained extension of T then there is a modified extension F of T such that E + F. MP2-54

55 Comparison of default logics THEOREM Let T be a normal default theory. The following statements are equivalent: (1) E is an extension of T. (2) E is a modified extension of T. (3) There is a set C such that (E,C) is a constrained extension of T. MP2-55

56 Priorities among defaults Default logics give overview of all possible extensions. But what if we wish to choose the most "important" (probable) possibilities? - Medical diagnosis. - Law. Technically achieved by introducing priorities among defaults. MP2-56

57 Default theories with priorities penguin : flies / flies bird : flies / flies Give the first default higher priority than the second. Total ordering among defaults: apply the default with the highest priority that is applicable. In the example above we get only one extension, Th({ flies}). MP2-57

58 Partial priority orders Sometimes it is not reasonable to expect that the preference order be total. conservative : taxcut! spengingscut / taxcut! spengingscut conservative! radical : taxcut! spengingscut / taxcut! spengingscut socialdemocrat : taxcut! spengingscut / taxcut! spengingscut MP2-58

59 Partial priority orders politician politician : respected / respected [d1] politician : wellpaid / wellpaid [d2] respected : wellpaid / wellpaid [d3] d2 < d3 MP2-59

60 Consider linearizations of partial orders According to all three total orderings which include <, d1 << d2 << d3 d2 << d1 << d3 d2 << d3 << d1 we conclude wellpaid. MP2-60

61 References G. Antoniou (1997). Nonmonotonic Reasoning. MIT Press (in press) G. Brewka (1994). Reasoning about Priorities in Default Logic. In Proc. 12th National Conference on Artificial Intelligence, AAAI/MIT Press, D. Etherington (1987). Formalizing Nonmonotonic Reasoning Systems. Artificial Intelligence 31: W. Lukaszewicz (1988). Considerations on Default Logic. Computational Intelligence 4: 1-16 D. Makinson (1989). General theory of cumulative inference. In Proc. 2nd International Workshop on Non-Monotonic Reasoning, Springer LNAI 346 MP2-61

62 References (continued) W. Marek & M. Truszczynski (1993). Nonmonotonic Logic. Springer R. Reiter (1980). A Logic for Default Reasoning. Artificial Intelligence 13: T. Schaub (1992). On Constrained Default Theories. In Proc. 10th European Conference on Artificial Intelligence MP2-62

Section 3 Default Logic. Subsection 3.1 Introducing defaults and default logics

Section 3 Default Logic. Subsection 3.1 Introducing defaults and default logics Section 3 Default Logic Subsection 3.1 Introducing defaults and default logics TU Dresden, WS 2017/18 Introduction to Nonmonotonic Reasoning Slide 34 Introducing defaults and default logics: an example