Tutorial: Nonmonotonic Logic (Day 1)

Transcription

1 Tutorial: Nonmonotonic Logic (Day 1) Christian Straßer Institute for Philosophy II, Ruhr-University Bochum Center for Logic and Philosophy of Science, Ghent University September 2, 2015 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

2 Outline 1 Defeasible Reasoning Some Basic Concepts Some examples to warm up Some conceptual distinctions Nonmonotonic Logic in Context 2 The Dynamics of Defeasible Reasoning 3 Default Logic (in the tradition of Reiter) Warming up Defaults and Default Theories Inferring Alternatives and more examples Meta-Properties Introducing Priorities Summing up 4 Autoepistemic Logic 5 Bibliography Bibliography Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

3 Overview: the whole tutorial Day 1 1 motivation and basic concepts 2 default logic (Reiter, Poole, Horty) Reiter and variants many examples priorities (Horty) 3 autoepistemic logic (Moore, Konolige, Marek) basics Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

4 Overview: the whole tutorial Day 1 1 motivation and basic concepts 2 default logic (Reiter, Poole, Horty) Reiter and variants many examples priorities (Horty) 3 autoepistemic logic (Moore, Konolige, Marek) basics Day 2 1 Plausible Reasoning (Makinson, Rescher/Manor, Batens, Geffner/Pearl) default assumptions, adaptive logics reasoning with maximal consistent subsets bridging to preferential semantics meta-theory: interesting properties for non-monotonic logic 2 Preferential Semantics (Kraus/Lehmann/Magidor, Shoham, Batens) properties, limitations, variants enhancements (rational closure) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

5 Overview: the whole tutorial Day 1 1 motivation and basic concepts 2 default logic (Reiter, Poole, Horty) Reiter and variants many examples priorities (Horty) 3 autoepistemic logic (Moore, Konolige, Marek) basics Day 2 1 Plausible Reasoning (Makinson, Rescher/Manor, Batens, Geffner/Pearl) default assumptions, adaptive logics reasoning with maximal consistent subsets bridging to preferential semantics meta-theory: interesting properties for non-monotonic logic 2 Preferential Semantics (Kraus/Lehmann/Magidor, Shoham, Batens) properties, limitations, variants enhancements (rational closure) Yes,... I know: it s not realistic! Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

6 Golden Rule interrupt me ask complain comment Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

7 Slides can be found at Courses/Natal2015-NonMonLog/natal15-nonmonlog.html (already outdated... ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

8 Topic 1 Defeasible Reasoning Some Basic Concepts Some examples to warm up Some conceptual distinctions Nonmonotonic Logic in Context 2 The Dynamics of Defeasible Reasoning 3 Default Logic (in the tradition of Reiter) Warming up Defaults and Default Theories Inferring Alternatives and more examples Meta-Properties Introducing Priorities Summing up 4 Autoepistemic Logic 5 Bibliography Bibliography Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

9 What s defeasible reasoning: some examples Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

10 What s defeasible reasoning: some examples infer a (good!?) explanation α β and β thus α abductive inference Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

11 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

12 closed world assumption reasoning on the assumption that the given information is complete Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

13 Tweety is a bird. Thus,...? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

14 Tweety is a bird. Thus,...? stereotypical / default reasoning jump to a conclusion on the basis of what is usually/typically/normally/etc. the case Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

15 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

16 inductive generalisations Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

17 Domains of defeasible reasoning everyday reasoning expert reasoning (e.g. medical diagnosis) scientific reasoning Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

18 Commonalities tentative conclusions jumping to conclusions retraction possible if problems arise Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

19 Commonalities tentative conclusions jumping to conclusions retraction possible if problems arise Two tiers of defeasible reasoning 1 illative tier (support, concluding) 2 dialectic tier (retraction) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

20 Ampliative vs. Corrective approaches Corrective approaches In contrast to ampliative reasoning, each inference is in accordance with CL (or another deductive standard) and hence deductive. However, given an inconsistent theory, not all deductive inferences will be accepted. illative tier: strictly deductive (e.g., classical logic) dialectic tier: conflicting deductive inferences Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

21 Ampliative vs. Corrective approaches Corrective approaches In contrast to ampliative reasoning, each inference is in accordance with CL (or another deductive standard) and hence deductive. However, given an inconsistent theory, not all deductive inferences will be accepted. illative tier: strictly deductive (e.g., classical logic) dialectic tier: conflicting deductive inferences Examples nonmonotonic paraconsistent logics (Rescher/Manor, inconsistency-adaptive logics) deductive argumentation-based approaches Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

22 Example: Rescher and Manor s Free Consequences suppose we reason classically on the basis of a complex body of premises Γ (e.g., a mathematical or scientific theory, or code of law) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

23 Example: Rescher and Manor s Free Consequences suppose we reason classically on the basis of a complex body of premises Γ (e.g., a mathematical or scientific theory, or code of law) we do not know whether Γ is consistent Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

24 Example: Rescher and Manor s Free Consequences suppose we reason classically on the basis of a complex body of premises Γ (e.g., a mathematical or scientific theory, or code of law) we do not know whether Γ is consistent careful rationale for drawing inferences Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

25 Example: Rescher and Manor s Free Consequences suppose we reason classically on the basis of a complex body of premises Γ (e.g., a mathematical or scientific theory, or code of law) we do not know whether Γ is consistent careful rationale for drawing inferences call a formula ϕ in Γ free if it does not belong to a minimally inconsistent set Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

26 Example: Rescher and Manor s Free Consequences suppose we reason classically on the basis of a complex body of premises Γ (e.g., a mathematical or scientific theory, or code of law) we do not know whether Γ is consistent careful rationale for drawing inferences call a formula ϕ in Γ free if it does not belong to a minimally inconsistent set rationale: an inference in CL is retracted as soon as we find out that it relies on premises that are not free Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

27 Example: Rescher and Manor s Free Consequences suppose we reason classically on the basis of a complex body of premises Γ (e.g., a mathematical or scientific theory, or code of law) we do not know whether Γ is consistent careful rationale for drawing inferences call a formula ϕ in Γ free if it does not belong to a minimally inconsistent set rationale: an inference in CL is retracted as soon as we find out that it relies on premises that are not free Task: Free consequences What are the free consequences of Γ = {p q, p, r s}? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

28 Ampliative approaches While the truth of the premises does not warrant the truth of the conclusion as in deductive reasoning, the conclusion nevertheless holds in most/typical/etc. cases in which the premises hold. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

29 Ampliative approaches While the truth of the premises does not warrant the truth of the conclusion as in deductive reasoning, the conclusion nevertheless holds in most/typical/etc. cases in which the premises hold. illative tier: beyond truth-preservation typically: fixed minimal deductive (non-defeasible) standard of reasoning / core rules (e.g., classical logic) jump to more conclusions given additional warrants that allow for defeasible conclusions Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

30 Ampliative approaches While the truth of the premises does not warrant the truth of the conclusion as in deductive reasoning, the conclusion nevertheless holds in most/typical/etc. cases in which the premises hold. illative tier: beyond truth-preservation typically: fixed minimal deductive (non-defeasible) standard of reasoning / core rules (e.g., classical logic) jump to more conclusions given additional warrants that allow for defeasible conclusions dialectic tier: e.g., exceptional circumstances conflicting defeasible and deductive inferences/arguments Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

31 Ampliative approaches While the truth of the premises does not warrant the truth of the conclusion as in deductive reasoning, the conclusion nevertheless holds in most/typical/etc. cases in which the premises hold. illative tier: beyond truth-preservation typically: fixed minimal deductive (non-defeasible) standard of reasoning / core rules (e.g., classical logic) jump to more conclusions given additional warrants that allow for defeasible conclusions dialectic tier: e.g., exceptional circumstances conflicting defeasible and deductive inferences/arguments Examples inheritance networks default logic abductive logics inductive generalisation Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

32 Classical Logic only illative tier What s the inference ticket in classical logic (in short, CL)? β follows from α iff in all classical interpretations in which α is true, also β is true entailment: truth-preservation, deduction e.g. x(p(x) Q(x)) P(a) Q(a) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

33 Classical Logic only illative tier What s the inference ticket in classical logic (in short, CL)? β follows from α iff in all classical interpretations in which α is true, also β is true entailment: truth-preservation, deduction e.g. x(p(x) Q(x)) P(a) Q(a) Hence... CL seems not apt to characterize reasoning that is dynamic in the sense that reasoners are prepared to retract inferences. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

34 Summing up illative tier dialectic tier corrective approach deductive inferences conflicting ded. inferencces ampliative approach ded. + non-ded. inf. confl. non-ded. inferences Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

35 Summing up illative tier dialectic tier corrective approach deductive inferences conflicting ded. inferencces ampliative approach ded. + non-ded. inf. confl. non-ded. inferences In practice: distinction is not so clear-cut Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

36 Pessimism in the 60ies Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

37 Toulmin Scheme (Toulmin (1958)) Premises Conclusion Backing Warrant Defeat Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

38 A Toulmin Argument Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

39 Nonmonotonic Logic to the Rescue Figure: Artificial Intelligence, Volume 13, Issues 1 2, Pages 1-174,(April 1980), Special Issue on Non-Monotonic Logic Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

40 Nonmonotonic Logic to the Rescue Figure: Artificial Intelligence, Volume 13, Issues 1 2, Pages 1-174,(April 1980), Special Issue on Non-Monotonic Logic Aim capture defeasible reasoning in a mathematically precise way reproduce the success of CL in the domain of mathematical reasoning in the less sterile/idealized domain of defeasible reasoning where incompleteness and uncertainty play a central role Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

41 Nowadays cooperations between formal and informal logicians e.g., Douglas Walton and formal argumentation (Gordon et al. (2007)) shift of normative standards in cognitive science, e.g. Stenning and Van Lambalgen (2008) Pfeifer (2014) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

42 Topic 1 Defeasible Reasoning Some Basic Concepts Some examples to warm up Some conceptual distinctions Nonmonotonic Logic in Context 2 The Dynamics of Defeasible Reasoning 3 Default Logic (in the tradition of Reiter) Warming up Defaults and Default Theories Inferring Alternatives and more examples Meta-Properties Introducing Priorities Summing up 4 Autoepistemic Logic 5 Bibliography Bibliography Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

43 Nonmonotonicity Nonmonotonic Logics Monotony: If Γ φ then Γ Γ φ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

44 Nonmonotonicity Nonmonotonic Logics Monotony: If Γ φ then Γ Γ φ. If we define a consequence function Cn( ) by Cn(Γ) = {φ Γ φ} we can equivalently express Monotony by: If φ Cn(Γ) then φ Cn(Γ Γ ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

45 Nonmonotonicity Nonmonotonic Logics Monotony: If Γ φ then Γ Γ φ. If we define a consequence function Cn( ) by Cn(Γ) = {φ Γ φ} we can equivalently express Monotony by: If φ Cn(Γ) then φ Cn(Γ Γ ). External and Internal dynamics (Pollock / Batens) External Dynamics retract conclusions under the influence of new information Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

46 Nonmonotonicity Nonmonotonic Logics Monotony: If Γ φ then Γ Γ φ. If we define a consequence function Cn( ) by Cn(Γ) = {φ Γ φ} we can equivalently express Monotony by: If φ Cn(Γ) then φ Cn(Γ Γ ). External and Internal dynamics (Pollock / Batens) External Dynamics retract conclusions under the influence of new information Internal Dynamics retract conclusions under the influence of a progressive analysis of the given information Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

47 What to replace Monotonicity with? Cautious Monotony If Γ ϕ and Γ ψ, then Γ, ϕ ψ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

48 What to replace Monotonicity with? Cautious Monotony If Γ ϕ and Γ ψ, then Γ, ϕ ψ. Cautious Monotony is the converse of Cut: (Cautious) Cut If Γ ϕ and Γ, ϕ ψ then Γ ψ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

49 What to replace Monotonicity with? Cautious Monotony If Γ ϕ and Γ ψ, then Γ, ϕ ψ. Cautious Monotony is the converse of Cut: (Cautious) Cut If Γ ϕ and Γ, ϕ ψ then Γ ψ. Cautious Monotonicity (Cut) states that adding a consequence ϕ back into the premise-set Γ does not lead to any decrease (increase) in inferential power. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

50 What to replace Monotonicity with? Cautious Monotony If Γ ϕ and Γ ψ, then Γ, ϕ ψ. Cautious Monotony is the converse of Cut: (Cautious) Cut If Γ ϕ and Γ, ϕ ψ then Γ ψ. Cautious Monotonicity (Cut) states that adding a consequence ϕ back into the premise-set Γ does not lead to any decrease (increase) in inferential power. Both together tell us that inference is a cumulative enterprise: we can keep drawing consequences that can in turn be used as additional premises, without affecting the set of conclusions. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

51 Core property central place in non-monotonic logic Gabbay Gabbay (1985) Kraus, Lehmann, Magidor ("KLM") Kraus et al. (1990) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

52 Core property central place in non-monotonic logic Gabbay Gabbay (1985) Kraus, Lehmann, Magidor ("KLM") Kraus et al. (1990) also empirically confirmed property of actual reasoning Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

53 Core property central place in non-monotonic logic Gabbay Gabbay (1985) Kraus, Lehmann, Magidor ("KLM") Kraus et al. (1990) also empirically confirmed property of actual reasoning however: misses e.g., from Reiter s Default Logic (see below) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

54 Be Cautious with Cut (Cautious) Cut If Γ ϕ and Γ, ϕ ψ then Γ ψ.... to be distinguished from: If Γ ϕ and Γ, ϕ ψ then Γ, Γ ψ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

55 Be Cautious with Cut (Cautious) Cut If Γ ϕ and Γ, ϕ ψ then Γ ψ.... to be distinguished from: If Γ ϕ and Γ, ϕ ψ then Γ, Γ ψ. Do you see why the second version of CUT is not suitable for nonmonotonic logic? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

56 What to replace Monotonicity with? Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

57 What to replace Monotonicity with? Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Adding formulas that are consistent with our current beliefs does not lead to a decrease in our set of conclusions. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

58 Stalnaker s problematic example for Rational Monotonicity Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

59 Stalnaker s problematic example for Rational Monotonicity Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Stalnaker s counter-example (Stalnaker (1994)) Stalnaker adopts the reading of α β as Given our initial set of beliefs Γ, if we learn α then (nonmonotonically) infer β. proposed in Makinson and Gärdenfors (1991). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

60 Stalnaker s problematic example for Rational Monotonicity Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Stalnaker s counter-example (Stalnaker (1994)) Stalnaker adopts the reading of α β as Given our initial set of beliefs Γ, if we learn α then (nonmonotonically) infer β. proposed in Makinson and Gärdenfors (1991). Consider the three composers: Verdi (v), Bizet (b), and Satie (s), and suppose that we initially accept (correctly but defeasibly) that Verdi is Italian (I(v)), while Bizet and Satie are French (F(b), F(s)). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

61 Stalnaker s problematic example for Rational Monotonicity Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Stalnaker s counter-example (Stalnaker (1994)) Stalnaker adopts the reading of α β as Given our initial set of beliefs Γ, if we learn α then (nonmonotonically) infer β. proposed in Makinson and Gärdenfors (1991). Consider the three composers: Verdi (v), Bizet (b), and Satie (s), and suppose that we initially accept (correctly but defeasibly) that Verdi is Italian (I(v)), while Bizet and Satie are French (F(b), F(s)). Suppose now that we learn that Verdi and Bizet are compatriots (C(v,b)). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

62 Stalnaker s problematic example for Rational Monotonicity Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Stalnaker s counter-example (Stalnaker (1994)) Stalnaker adopts the reading of α β as Given our initial set of beliefs Γ, if we learn α then (nonmonotonically) infer β. proposed in Makinson and Gärdenfors (1991). Consider the three composers: Verdi (v), Bizet (b), and Satie (s), and suppose that we initially accept (correctly but defeasibly) that Verdi is Italian (I(v)), while Bizet and Satie are French (F(b), F(s)). Suppose now that we learn that Verdi and Bizet are compatriots (C(v,b)). then: C(v, b) F (s) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

63 Stalnaker s problematic example for Rational Monotonicity Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Stalnaker s counter-example (Stalnaker (1994)) Stalnaker adopts the reading of α β as Given our initial set of beliefs Γ, if we learn α then (nonmonotonically) infer β. proposed in Makinson and Gärdenfors (1991). Consider the three composers: Verdi (v), Bizet (b), and Satie (s), and suppose that we initially accept (correctly but defeasibly) that Verdi is Italian (I(v)), while Bizet and Satie are French (F(b), F(s)). Suppose now that we learn that Verdi and Bizet are compatriots (C(v,b)). then: C(v, b) F (s) Now consider: C(v,s), then C(v, b) C(v, s). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

64 Stalnaker s problematic example for Rational Monotonicity Rational Monotony If it s not the case that Γ ϕ, and moreover Γ ψ, then Γ, ϕ ψ. Stalnaker s counter-example (Stalnaker (1994)) Stalnaker adopts the reading of α β as Given our initial set of beliefs Γ, if we learn α then (nonmonotonically) infer β. proposed in Makinson and Gärdenfors (1991). Consider the three composers: Verdi (v), Bizet (b), and Satie (s), and suppose that we initially accept (correctly but defeasibly) that Verdi is Italian (I(v)), while Bizet and Satie are French (F(b), F(s)). Suppose now that we learn that Verdi and Bizet are compatriots (C(v,b)). then: C(v, b) F (s) Now consider: C(v,s), then C(v, b) C(v, s). But if we add C(v,s) to our beliefs, then C(v, b), C(v, s) F (s). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

65 Schematic Inference Graphs (e.g., Inheritance Nets, Formal Argumentation, etc.) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

66 Schematic Inference Graphs (e.g., Inheritance Nets, Formal Argumentation, etc.) We use the following conventions: : signify deductive or strict (i.e., non-defeasible) inferences, : signify defeasible inferences, and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

67 Schematic Inference Graphs (e.g., Inheritance Nets, Formal Argumentation, etc.) We use the following conventions: : signify deductive or strict (i.e., non-defeasible) inferences, : signify defeasible inferences, and strikethrough (single resp. double) arrows signify that the negation of the pointed formula is (defeasibly resp. strictly) implied. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

68 Schematic Inference Graphs (e.g., Inheritance Nets, Formal Argumentation, etc.) We use the following conventions: : signify deductive or strict (i.e., non-defeasible) inferences, : signify defeasible inferences, and strikethrough (single resp. double) arrows signify that the negation of the pointed formula is (defeasibly resp. strictly) implied. Arguments: e.g., Penguin Bird flies Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

69 Schematic Inference Graphs (e.g., Inheritance Nets, Formal Argumentation, etc.) We use the following conventions: : signify deductive or strict (i.e., non-defeasible) inferences, : signify defeasible inferences, and strikethrough (single resp. double) arrows signify that the negation of the pointed formula is (defeasibly resp. strictly) implied. Arguments: e.g., Penguin Bird flies So, we can read the diagram as follows: Penguins are birds (no exceptions); Birds usually fly; and Penguins usually don t fly. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

70 Resolution: Strict beats defeasible Two types of conflicts 1 conflicts between defeasible conclusions and "hard facts,": A B vs. A B 2 conflicts between one potential defeasible conclusion and another: A B vs. A B Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

71 Resolution: Strict beats defeasible Two types of conflicts 1 conflicts between defeasible conclusions and "hard facts,": A B vs. A B 2 conflicts between one potential defeasible conclusion and another: A B vs. A B Resolution Concerning 1: hard facts are prioritized. Concerning 2: this is more complicated... Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

72 Specificity / Preemption According to the Specificity Principle an inference with a more specific antecedent overrides a conflicting defeasible inference with a less specific antecedent. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

73 Specificity / Preemption According to the Specificity Principle an inference with a more specific antecedent overrides a conflicting defeasible inference with a less specific antecedent. lot of work in inheritance nets is devoted to this problem (see Horty (1994)) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

74 Topic 1 Defeasible Reasoning Some Basic Concepts Some examples to warm up Some conceptual distinctions Nonmonotonic Logic in Context 2 The Dynamics of Defeasible Reasoning 3 Default Logic (in the tradition of Reiter) Warming up Defaults and Default Theories Inferring Alternatives and more examples Meta-Properties Introducing Priorities Summing up 4 Autoepistemic Logic 5 Bibliography Bibliography Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

75 Some References to Classical Articles A logic for default reasoning. Artificial Intelligence, 1 2(13). Reiter (1980) A logical framework for default reasoning. Artificial intelligence, 36(1), Poole (1988) The effect of knowledge on belief: conditioning, specificity and the lottery paradox in default reasoning. Artificial Intelligence, 49(1-3), Poole (1991) Considerations on default logic: an alternative approach. Computational intelligence, 4(1), Łukaszewicz (1988) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

76 Short Reminder: 1st order logic Logical symbols quantifiers, logical connectives,,, brackets variables non-logical symbols predicate / relation symbols with specific arity function symbols with specific arity constants (0-ary functions) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

77 Short Reminder: 1st order logic, special terminology terms: variables, f (t 1,..., t n ) where t i are terms atomic formula: P(t 1,..., t n ) formulas:,,,,, closure of atomic formulas free / bound variables sentence: formula without free variables instance of a formula ϕ: substitution of some free variables for terms ground term: term without variables ground instance: instance that is a sentence (obtained by substituting all free variables by ground terms) Example bird(tweety) flies(tweety) is a ground instance of bird(x) flies(x) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

78 What s a default conditional prerequisite α(x) : β 1 (x),..., β n (b) γ(x) where x = x 1,..., x m, and α(x), β 1 (x),..., β n (x), γ(x) are formulas whose free variables are among x 1,..., x m. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

79 What s a default conditional prerequisite α(x) justification : β 1 (x),..., β n (b) γ(x) where x = x 1,..., x m, and α(x), β 1 (x),..., β n (x), γ(x) are formulas whose free variables are among x 1,..., x m. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

80 What s a default conditional prerequisite α(x) justification : β 1 (x),..., β n (b) γ(x) conclusion where x = x 1,..., x m, and α(x), β 1 (x),..., β n (x), γ(x) are formulas whose free variables are among x 1,..., x m. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

81 What s a default conditional prerequisite α(x) justification : β 1 (x),..., β n (b) γ(x) conclusion where x = x 1,..., x m, and α(x), β 1 (x),..., β n (x), γ(x) are formulas whose free variables are among x 1,..., x m. Application of a default The default is applied in order to derive the c-ground instance of γ in case trigger: α(c) belongs to our set of depending on the perspective we have beliefs/(defeasible) knowledge/plausible assumptions/etc. (henceforth I will speak only about beliefs) justification: the set of our beliefs is consistent with each β i (c) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

82 Default Theory, Φ set of defaults Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

83 Default Theory set of facts, Φ set of defaults Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

84 Default Theory set of facts, Φ set of defaults simple example { } bird(x) : flies(x) = flies(x) Φ = {bird(tweety), cat(sylvester)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

85 Types of defaults Normal defaults rather natural representation α(x) : γ(x) γ(x) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

86 α(x) : γ(x) β(x) γ(x) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104 Types of defaults Normal defaults rather natural representation Semi-Normal defaults α(x) : γ(x) γ(x) α(x) : β(x) γ(x) where β(x) γ(x). E.g.,

87 How to reason with default theories? Idea Apply iteratively modus ponens to defaults. This way build step-wise an extension (sets of beliefs that are obtained in this way) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

88 Here s how it goes: guess the extension Ξ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

89 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

90 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

91 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: 1 trigger?: Ξ α(c) α(x) : β(x) γ(x) and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

92 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

93 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

94 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

95 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: try another triggered default in (goto ( )) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

96 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: try another triggered default in (goto ( )) if there isn t: terminate. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

97 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: try another triggered default in (goto ( )) if there isn t: terminate. if Ξ = Cn(Ξ ): extension found. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

98 Here s how it goes: guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: try another triggered default in (goto ( )) if there isn t: terminate. if Ξ = Cn(Ξ ): extension found. Problem (?) We have to guess and use our guess when adding new defaults. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

99 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

100 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) and Φ = {bird(tweety), cat(sylvester)}. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

101 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) and Φ = {bird(tweety), cat(sylvester)}. we have only the two constants Tweety and Sylvester in the language Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

102 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) and Φ = {bird(tweety), cat(sylvester)}. we have only the two constants Tweety and Sylvester in the language Building up the extensions: guess: Ξ = Cn({flies(Tweety)} Φ}) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

103 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) and Φ = {bird(tweety), cat(sylvester)}. we have only the two constants Tweety and Sylvester in the language Building up the extensions: guess: Ξ = Cn({flies(Tweety)} Φ}) our initial knowledge is Φ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

104 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) and Φ = {bird(tweety), cat(sylvester)}. we have only the two constants Tweety and Sylvester in the language Building up the extensions: guess: Ξ = Cn({flies(Tweety)} Φ}) our initial knowledge is Φ note that the Sylvester-instance of our default is not applicable to Φ since Φ bird(sylvester) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

105 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) and Φ = {bird(tweety), cat(sylvester)}. we have only the two constants Tweety and Sylvester in the language Building up the extensions: guess: Ξ = Cn({flies(Tweety)} Φ}) our initial knowledge is Φ note that the Sylvester-instance of our default is not applicable to Φ since Φ bird(sylvester) however, we have bird(tweety) and fly(tweety) is consistent with Ξ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

106 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) and Φ = {bird(tweety), cat(sylvester)}. we have only the two constants Tweety and Sylvester in the language Building up the extensions: guess: Ξ = Cn({flies(Tweety)} Φ}) our initial knowledge is Φ note that the Sylvester-instance of our default is not applicable to Φ since Φ bird(sylvester) however, we have bird(tweety) and fly(tweety) is consistent with Ξ. fixed point reached Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

107 Example: Tweety Let T =, Φ where { } bird(x) : flies(x) = flies(x) and Φ = {bird(tweety), cat(sylvester)}. we have only the two constants Tweety and Sylvester in the language Building up the extensions: guess: Ξ = Cn({flies(Tweety)} Φ}) our initial knowledge is Φ note that the Sylvester-instance of our default is not applicable to Φ since Φ bird(sylvester) however, we have bird(tweety) and fly(tweety) is consistent with Ξ. fixed point reached the only extension is Ξ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

108 Question Are extensions always unique? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

109 The Nixon Diamond Let T =, Φ where = { quaker(x) : pacifist(x) pacifist(x), } republican(x) : pacifist(x) pacifist(x) Φ = {Quaker(Nixon), republican(nixon)}. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

110 The Nixon Diamond Let T =, Φ where = { quaker(x) : pacifist(x) pacifist(x), } republican(x) : pacifist(x) pacifist(x) Φ = {Quaker(Nixon), republican(nixon)}. There are two extensions: 1 one that contains pacifist(nixon), 2 and one that contains pacifist(nixon). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

111 Another example Let, Φ be a default theory where { } α(x) : β(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Question Is there an extension? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

112 Another example Let, Φ be a default theory where { } α(x) : β(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Question Is there an extension? Guess: Cn({α(c), γ(c)}) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

113 Another example Let, Φ be a default theory where { } α(x) : β(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Question Is there an extension? Guess: Cn({α(c), γ(c)}) first run: Φ = Φ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

114 Another example Let, Φ be a default theory where { } α(x) : β(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Question Is there an extension? Guess: Cn({α(c), γ(c)}) first run: Φ = Φ α(x) : β(x) γ(x) is triggered and justified: apply Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

115 Another example Let, Φ be a default theory where { } α(x) : β(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Question Is there an extension? Guess: Cn({α(c), γ(c)}) first run: Φ = Φ α(x) : β(x) γ(x) is triggered and justified: apply Φ = Φ {γ(c)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

116 Another example Let, Φ be a default theory where { } α(x) : β(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Question Is there an extension? Guess: Cn({α(c), γ(c)}) first run: Φ = Φ α(x) : β(x) γ(x) is triggered and justified: apply Φ = Φ {γ(c)} second run: γ(x) : β(x) β(x) justified is triggered and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

117 Another example Let, Φ be a default theory where { } α(x) : β(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Question Is there an extension? Guess: Cn({α(c), γ(c)}) first run: Φ = Φ α(x) : β(x) γ(x) is triggered and justified: apply Φ = Φ {γ(c)} second run: γ(x) : β(x) β(x) justified is triggered and Φ = Φ {γ(c), β(c)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

118 Another example Let, Φ be a default theory where { } α(x) : β(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Question Is there an extension? Guess: Cn({α(c), γ(c)}) first run: Φ = Φ α(x) : β(x) γ(x) is triggered and justified: apply Φ = Φ {γ(c)} second run: γ(x) : β(x) β(x) justified is triggered and Φ = Φ {γ(c), β(c)} our guess is wrong (similar problems with other guesses) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

119 How to define the consequences of a default theory? Two approaches: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

120 How to define the consequences of a default theory? Two approaches: Skeptical approach, Φ skp A iff A Extensions(, Φ ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

121 How to define the consequences of a default theory? Two approaches: Skeptical approach, Φ skp A iff A Extensions(, Φ ) Credulous approach, Φ crd A iff A Extensions(, Φ ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

122 How to define the consequences of a default theory? Two approaches: Skeptical approach Credulous approach, Φ skp A iff A Extensions(, Φ ), Φ crd A iff A Extensions(, Φ ) Question: When is which approach useful? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

123 Alternative: Consistency Check in the end guess the extension Ξ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

124 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

125 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(x) γ(x) and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

126 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: 1 trigger?: Ξ α(c) α(x) : β(x) γ(x) and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

127 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default α(x) : β(x) γ(x) and check whether: 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

128 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default α(x) : β(x) γ(x) and check whether: 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

129 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default α(x) : β(x) γ(x) and check whether: 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

130 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default α(x) : β(x) γ(x) and check whether: 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: try another triggered default in (goto ( )) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

131 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default α(x) : β(x) γ(x) and check whether: 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: try another triggered default in (goto ( )) if there isn t: terminate Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

132 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default α(x) : β(x) γ(x) and check whether: 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: try another triggered default in (goto ( )) if there isn t: terminate if Ξ is consistent with all the justifications of the applied defaults and if Ξ = Cn(Ξ ) we found an extension, otherwise try again from the start. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

133 Alternative: Consistency Check in the end guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default α(x) : β(x) γ(x) and check whether: 1 trigger?: Ξ α(c) 2 conflicted?: each β i (c) (1 i n) is consistent with Ξ (!!) if yes: update beliefs: Ξ := Ξ {γ(c)} if no: try another triggered default in (goto ( )) if there isn t: terminate if Ξ is consistent with all the justifications of the applied defaults and if Ξ = Cn(Ξ ) we found an extension, otherwise try again from the start. no initial guess guessing on-the-fly when we choose which triggered defaults to apply only in the end we see whether we guessed luckily Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

134 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

135 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ ( ) take a c-instance of an arbitrary default check: α(x) : β(x) γ(x) and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

136 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ ( ) take a c-instance of an arbitrary default check: 1 Φ α(c) (trigger) α(x) : β(x) γ(x) and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

137 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ α(x) : β(x) ( ) take a c-instance of an arbitrary default γ(x) check: 1 Φ α(c) (trigger) 2 each β i (c) is consistent with Φ (justification 1) and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

138 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ ( ) take a c-instance of an arbitrary default check: α(x) : β(x) γ(x) and 1 Φ α(c) (trigger) 2 each β i (c) is consistent with Φ (justification 1) 3 each justification of previously applied defaults is consistent with Φ {γ(c)} (justification 2) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

139 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ ( ) take a c-instance of an arbitrary default check: α(x) : β(x) γ(x) and 1 Φ α(c) (trigger) 2 each β i (c) is consistent with Φ (justification 1) 3 each justification of previously applied defaults is consistent with Φ {γ(c)} (justification 2) if yes: Φ = Φ {γ(c)} and goto ( ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

140 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ ( ) take a c-instance of an arbitrary default check: α(x) : β(x) γ(x) and 1 Φ α(c) (trigger) 2 each β i (c) is consistent with Φ (justification 1) 3 each justification of previously applied defaults is consistent with Φ {γ(c)} (justification 2) if yes: Φ = Φ {γ(c)} and goto ( ) if no: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

141 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ ( ) take a c-instance of an arbitrary default check: α(x) : β(x) γ(x) and 1 Φ α(c) (trigger) 2 each β i (c) is consistent with Φ (justification 1) 3 each justification of previously applied defaults is consistent with Φ {γ(c)} (justification 2) if yes: Φ = Φ {γ(c)} and goto ( ) if no: if there is another instance of a default in that wasn t tasted, goto ( ) and test it Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

142 Going strictly procedural with Łukaszewicz (1988) Let, Φ be a default theory. init: Φ = Φ ( ) take a c-instance of an arbitrary default check: α(x) : β(x) γ(x) and 1 Φ α(c) (trigger) 2 each β i (c) is consistent with Φ (justification 1) 3 each justification of previously applied defaults is consistent with Φ {γ(c)} (justification 2) if yes: Φ = Φ {γ(c)} and goto ( ) if no: if there is another instance of a default in that wasn t tasted, goto ( ) and test it otherwise: if Ξ = Cn(Φ ), we found an extension. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

143 Some properties of the new procedure No guess needed. real procedural character guarantees existence of an extension hence: yields sometimes different results from Reiter s account Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

144 Some properties of the new procedure No guess needed. real procedural character guarantees existence of an extension hence: yields sometimes different results from Reiter s account Question What happens in the new approach when plugging in the default theory, Φ where { } α(x) : β(x) γ(x) γ(x) : β(x) =, γ(x) β(x) Φ = {α(c)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

145 Recall the three conditions 1 Φ α(c) (trigger) 2 each β i (c) is consistent with Φ (justification 1) 3 each justification of previously applied defaults is consistent with Φ {γ(c)} (justification 2) = { α(x) : β(x) γ(x) γ(x), Φ = {α(c)} } γ(x) : β(x) β(x)

146 Non-Procedural Fixed-Point Characterisations What about the following definition? Definition: Extension Ξ is an extension of a default theory, Φ iff it is a minimal set that satisfies the following conditions: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

147 Non-Procedural Fixed-Point Characterisations What about the following definition? Definition: Extension Ξ is an extension of a default theory, Φ iff it is a minimal set that satisfies the following conditions: 1 Φ Ξ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

148 Non-Procedural Fixed-Point Characterisations What about the following definition? Definition: Extension Ξ is an extension of a default theory, Φ iff it is a minimal set that satisfies the following conditions: 1 Φ Ξ 2 Cn(Ξ) = Ξ (fixed-point) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

149 Non-Procedural Fixed-Point Characterisations What about the following definition? Definition: Extension Ξ is an extension of a default theory, Φ iff it is a minimal set that satisfies the following conditions: 1 Φ Ξ 2 Cn(Ξ) = Ξ (fixed-point) 3 if α(c) : β(c) γ(c) is a c-instance of some default in and then γ(c) Ξ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

150 Non-Procedural Fixed-Point Characterisations What about the following definition? Definition: Extension Ξ is an extension of a default theory, Φ iff it is a minimal set that satisfies the following conditions: 1 Φ Ξ 2 Cn(Ξ) = Ξ (fixed-point) 3 if α(c) : β(c) γ(c) 1 α(c) Ξ (trigger) then γ(c) Ξ is a c-instance of some default in and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

151 Non-Procedural Fixed-Point Characterisations What about the following definition? Definition: Extension Ξ is an extension of a default theory, Φ iff it is a minimal set that satisfies the following conditions: 1 Φ Ξ 2 Cn(Ξ) = Ξ (fixed-point) 3 if α(c) : β(c) γ(c) 1 α(c) Ξ (trigger) is a c-instance of some default in and 2 β i (c) is consistent with Ξ for all 1 i n (justification) then γ(c) Ξ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

152 Non-Procedural Fixed-Point Characterisations What about the following definition? Definition: Extension Ξ is an extension of a default theory, Φ iff it is a minimal set that satisfies the following conditions: 1 Φ Ξ 2 Cn(Ξ) = Ξ (fixed-point) 3 if α(c) : β(c) γ(c) 1 α(c) Ξ (trigger) is a c-instance of some default in and 2 β i (c) is consistent with Ξ for all 1 i n (justification) then γ(c) Ξ Question Is this equivalent to the procedural approach? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

153 Difference: Grounding Take { :p p }, }. Note that Cn({ p}) is a minimal set satisfying the previous conditions. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

154 Difference: Grounding Take { :p p }, }. Note that Cn({ p}) is a minimal set satisfying the previous conditions. However, the only extension is Cn({p}). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

155 Difference: Grounding Take { :p p }, }. Note that Cn({ p}) is a minimal set satisfying the previous conditions. However, the only extension is Cn({p}). We face the Problem of grounding We expect that all members of the extension can be generated iteratively by chaining and detaching defaults. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

156 Non-Procedural Fixed-Point Characterisations Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

157 Non-Procedural Fixed-Point Characterisations Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

158 Non-Procedural Fixed-Point Characterisations Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

159 Non-Procedural Fixed-Point Characterisations Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) is a c-instance of some default in and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

160 Non-Procedural Fixed-Point Characterisations Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) 1 α(c) π Φ (Γ) (trigger) is a c-instance of some default in and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

161 Non-Procedural Fixed-Point Characterisations Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) 1 α(c) π Φ (Γ) (trigger) is a c-instance of some default in and 2 β i (c) / Γ for all 1 i n then γ(c) π Φ (Γ) (justification). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

162 Non-Procedural Fixed-Point Characterisations Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) 1 α(c) π Φ (Γ) (trigger) is a c-instance of some default in and 2 β i (c) / Γ for all 1 i n then γ(c) π Φ (Γ) (justification). Definition: Extension A set of formulas Γ is an extension of, Φ iff π(γ) = Γ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

163 Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

164 Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) 1 α(c) π Φ (Γ) (trigger) is a c-instance of some default in and 2 β i (c) / Γ for all 1 i n then γ(c) π Φ (Γ) (justification). Proof of the existence of π(γ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

165 Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) 1 α(c) π Φ (Γ) (trigger) is a c-instance of some default in and 2 β i (c) / Γ for all 1 i n then γ(c) π Φ (Γ) (justification). Proof of the existence of π(γ). Let S be all sets that satisfy (1) (3). (Note S since L S.) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

166 Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) 1 α(c) π Φ (Γ) (trigger) is a c-instance of some default in and 2 β i (c) / Γ for all 1 i n then γ(c) π Φ (Γ) (justification). Proof of the existence of π(γ). Let S be all sets that satisfy (1) (3). (Note S since L S.) Let Γ = S. We have to show (1) (3). 1 trivial Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

167 Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) 1 α(c) π Φ (Γ) (trigger) is a c-instance of some default in and 2 β i (c) / Γ for all 1 i n then γ(c) π Φ (Γ) (justification). Proof of the existence of π(γ). Let S be all sets that satisfy (1) (3). (Note S since L S.) Let Γ = S. We have to show (1) (3). 1 trivial 2 Suppose A Cn(Γ ). Hence (by monotonicity), Γ A for all Γ S. Since Cn(Γ ) = Γ, A Γ. Thus, A S. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

168 Let, Φ be a default theory. Define the operator π Φ such that for any set of formulas Γ, π Φ (Γ) the smallest set satisfying: 1 Φ π Φ (Γ) 2 π Φ (Γ) = Cn(π Φ (Γ)) (fixed point) 3 if α(c) : β(c) γ(c) 1 α(c) π Φ (Γ) (trigger) is a c-instance of some default in and 2 β i (c) / Γ for all 1 i n then γ(c) π Φ (Γ) (justification). Proof of the existence of π(γ). Let S be all sets that satisfy (1) (3). (Note S since L S.) Let Γ = S. We have to show (1) (3). 1 trivial 2 Suppose A Cn(Γ ). Hence (by monotonicity), Γ A for all Γ S. Since Cn(Γ ) = Γ, A Γ. Thus, A S. 3 Suppose α(c) Γ and β i (c) / Γ for all i n. Hence, α(c) Γ for all Γ S and thus γ(c) Γ. Thus, γ(c) Γ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

169 Cautious Cut If, Φ A and, Φ {A} B then, Φ B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

170 Cautious Cut What do you think? If, Φ A and, Φ {A} B then, Φ B. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

171 Cautious Cut If, Φ A and, Φ {A} B then, Φ B. What do you think? Lemma (from this Cut follows immediately for skeptical consequence) If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

172 Cautious Cut If, Φ A and, Φ {A} B then, Φ B. What do you think? Lemma (from this Cut follows immediately for skeptical consequence) If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). Proof Suppose, Φ A and let Ξ Ext(, Φ ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

173 Cautious Cut If, Φ A and, Φ {A} B then, Φ B. What do you think? Lemma (from this Cut follows immediately for skeptical consequence) If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). Proof Suppose, Φ A and let Ξ Ext(, Φ ). We know that Ξ = π Φ (Ξ) and have to show that Ξ = π Φ {A} (Ξ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

174 Cautious Cut If, Φ A and, Φ {A} B then, Φ B. What do you think? Lemma (from this Cut follows immediately for skeptical consequence) If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). Proof Suppose, Φ A and let Ξ Ext(, Φ ). We know that Ξ = π Φ (Ξ) and have to show that Ξ = π Φ {A} (Ξ). Clearly, since A Ξ, Ξ satisfies (1) (3) (relative to Φ {A}). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

175 Cautious Cut If, Φ A and, Φ {A} B then, Φ B. What do you think? Lemma (from this Cut follows immediately for skeptical consequence) If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). Proof Suppose, Φ A and let Ξ Ext(, Φ ). We know that Ξ = π Φ (Ξ) and have to show that Ξ = π Φ {A} (Ξ). Clearly, since A Ξ, Ξ satisfies (1) (3) (relative to Φ {A}). Assume π Φ {A} (Ξ) Ξ. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

176 Cautious Cut If, Φ A and, Φ {A} B then, Φ B. What do you think? Lemma (from this Cut follows immediately for skeptical consequence) If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). Proof Suppose, Φ A and let Ξ Ext(, Φ ). We know that Ξ = π Φ (Ξ) and have to show that Ξ = π Φ {A} (Ξ). Clearly, since A Ξ, Ξ satisfies (1) (3) (relative to Φ {A}). Assume π Φ {A} (Ξ) Ξ. But then π Φ {A} (Ξ) also satisfies (1) (3) relative to Φ which contradicts Ξ = π Φ (Ξ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

177 Cautious Cut If, Φ A and, Φ {A} B then, Φ B. What do you think? Lemma (from this Cut follows immediately for skeptical consequence) If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). Proof Suppose, Φ A and let Ξ Ext(, Φ ). We know that Ξ = π Φ (Ξ) and have to show that Ξ = π Φ {A} (Ξ). Clearly, since A Ξ, Ξ satisfies (1) (3) (relative to Φ {A}). Assume π Φ {A} (Ξ) Ξ. But then π Φ {A} (Ξ) also satisfies (1) (3) relative to Φ which contradicts Ξ = π Φ (Ξ). Hence, Ξ = π Φ {A} (Ξ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

178 Cautious Cut for credulous version? Recall: Lemma If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

179 Cautious Cut for credulous version? Recall: Lemma If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). What do you think, does this help? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

180 Cautious Cut for credulous version? Recall: Lemma If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). What do you think, does this help? Counter-example Take, Φ where = { :p p }, p q: p p. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

181 Cautious Cut for credulous version? Recall: Lemma If, Φ A, Ext(, Φ ) Ext(, Φ {A} ). What do you think, does this help? Counter-example Take, Φ where =, Φ cred p q. { :p p, Φ {p q} cred p. But,, Φ cred p. }, p q: p p. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

182 The special status of normal default theories A normal default theory is a default theory that only consists of normal defaults. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

183 The special status of normal default theories A normal default theory is a default theory that only consists of normal defaults. A normal default theory always has an extension both in Reiter s and in Lukaszewicz s approach. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

184 The special status of normal default theories A normal default theory is a default theory that only consists of normal defaults. A normal default theory always has an extension both in Reiter s and in Lukaszewicz s approach. For normal theories the set of Reiter extensions and the set of Lukaszewicz extensions coincides. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

185 Floating conclusions Task 1 What are the two extensions of this default theory? 2 Is politically motivated(nixon) derivable? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

186 { Nixon, quaker, republican, dove, hawk, politically motivated } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

187 { Nixon, quaker, republican, dove, hawk, politically motivated } { Nixon, quaker, republican, dove, hawk, politically motivated } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

188 { Nixon, quaker, republican, dove, hawk, politically motivated } { Nixon, quaker, republican, dove, hawk, politically motivated } { Nixon, quaker, republican, dove, hawk, politically motivated } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

189 Specificity Question Is flies(tweety) derivable? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

190 Specificity Question Is flies(tweety) derivable? Nope There are two extensions: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

191 Specificity Question Is flies(tweety) derivable? Nope There are two extensions: 1 one with flies(tweety) 2 one with flies(tweety) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

192 Lukaszewicz s Fishing example Let T =, Φ where = { Sunday : I go fishing I wake up late Φ = {Sunday, Holidays}. I go fishing, } Holidays : I wake up late I wake up late Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

193 Lukaszewicz s Fishing example Let T =, Φ where = { Sunday : I go fishing I wake up late Φ = {Sunday, Holidays}. I go fishing, } Holidays : I wake up late I wake up late Reiter there is only the extension containing Sunday, Holidays, I wake up late (by first applying the second and then the first default) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

194 Lukaszewicz s Fishing example Let T =, Φ where = { Sunday : I go fishing I wake up late Φ = {Sunday, Holidays}. I go fishing, } Holidays : I wake up late I wake up late Reiter there is only the extension containing Sunday, Holidays, I wake up late (by first applying the second and then the first default) Lukaszewicz we also(!) have the extension that is the result of first applying the first default Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

195 Lukaszewicz s Fishing example Let T =, Φ where = { Sunday : I go fishing I wake up late Φ = {Sunday, Holidays}. I go fishing, } Holidays : I wake up late I wake up late Reiter there is only the extension containing Sunday, Holidays, I wake up late (by first applying the second and then the first default) Lukaszewicz we also(!) have the extension that is the result of first applying the first default Question What do you make of it? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

196 Poole s Lottery Paradox Let T =, Φ where { bird(x) : flies(x) penguin(x) =, flies(x) penguin(x) } bird(x) : treenest(x) sandpiper(x),... treenest(x) sandpiper(x) Φ = {bird(tweety)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

197 Poole s Lottery Paradox Let T =, Φ where { bird(x) : flies(x) penguin(x) =, flies(x) penguin(x) } bird(x) : treenest(x) sandpiper(x),... treenest(x) sandpiper(x) Φ = {bird(tweety)} Problem However, then we conclude penguin(x) sandpiper(x)... for all bird-species. But then Tweety does not belong to any species of birds. Typical birds (in an ideal sense) do not exist. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

198 Reflections about Extensions should there always be extensions? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

199 Reflections about Extensions should there always be extensions? what do extensions represent? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

200 Reflections about Extensions should there always be extensions? what do extensions represent? equilibrium states of a rational reasoner? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

201 Reflections about Extensions should there always be extensions? what do extensions represent? equilibrium states of a rational reasoner? different, possibly conflicting conclusion sets as rational outcomes based on initial information (Horty, 2005) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

202 Reflections about Extensions should there always be extensions? what do extensions represent? equilibrium states of a rational reasoner? different, possibly conflicting conclusion sets as rational outcomes based on initial information (Horty, 2005) good reasons approach: if A is in an extension then there are good reasons to suppose A (see Nixon) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

203 Reflections about Extensions should there always be extensions? what do extensions represent? equilibrium states of a rational reasoner? different, possibly conflicting conclusion sets as rational outcomes based on initial information (Horty, 2005) good reasons approach: if A is in an extension then there are good reasons to suppose A (see Nixon) are some extensions preferable to others? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

204 Reflections about Extensions should there always be extensions? what do extensions represent? equilibrium states of a rational reasoner? different, possibly conflicting conclusion sets as rational outcomes based on initial information (Horty, 2005) good reasons approach: if A is in an extension then there are good reasons to suppose A (see Nixon) are some extensions preferable to others? should some extensions be filtered out? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

205 Reflections about Extensions should there always be extensions? what do extensions represent? equilibrium states of a rational reasoner? different, possibly conflicting conclusion sets as rational outcomes based on initial information (Horty, 2005) good reasons approach: if A is in an extension then there are good reasons to suppose A (see Nixon) are some extensions preferable to others? should some extensions be filtered out? how to build extensions (naturally) in order to explicate actual default reasoning? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

206 Reflections about Extensions should there always be extensions? what do extensions represent? equilibrium states of a rational reasoner? different, possibly conflicting conclusion sets as rational outcomes based on initial information (Horty, 2005) good reasons approach: if A is in an extension then there are good reasons to suppose A (see Nixon) are some extensions preferable to others? should some extensions be filtered out? how to build extensions (naturally) in order to explicate actual default reasoning? should floating conclusions be accepted? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

207 Problems with Disjunctions Let T =, Φ where { Quaker(x) : dove(x) =, dove(x) } republican(x) : hawk(x), hawk(x) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

208 Problems with Disjunctions Let T =, Φ where { } Quaker(x) : dove(x) republican(x) : hawk(x) =,, dove(x) hawk(x) Φ = {Quaker(Peter) republican(peter), Quaker(Anne) Quaker(George)}. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

209 Problems with Disjunctions Let T =, Φ where { } Quaker(x) : dove(x) republican(x) : hawk(x) =,, dove(x) hawk(x) Φ = {Quaker(Peter) republican(peter), Quaker(Anne) Quaker(George)}. Problem we don t get hawk(peter) dove(peter), Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

210 Problems with Disjunctions Let T =, Φ where { } Quaker(x) : dove(x) republican(x) : hawk(x) =,, dove(x) hawk(x) Φ = {Quaker(Peter) republican(peter), Quaker(Anne) Quaker(George)}. Problem we don t get hawk(peter) dove(peter), dove(anne) dove(george). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

211 "Naming defaults" Let T =, Φ where = { : birdsfly(x) birdsfly(x) } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

212 "Naming defaults" Let T =, Φ where = { : birdsfly(x) birdsfly(x) Φ consists of } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

213 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

214 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) x(bird(x) baby(x) birdsfly(x)) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

215 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) x(bird(x) baby(x) birdsfly(x)) bird(tweety), bird(polly) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

216 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) x(bird(x) baby(x) birdsfly(x)) bird(tweety), bird(polly) bird(anne) bird(george) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

217 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) x(bird(x) baby(x) birdsfly(x)) bird(tweety), bird(polly) bird(anne) bird(george) baby(polly), baby(keith) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

218 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) x(bird(x) baby(x) birdsfly(x)) bird(tweety), bird(polly) bird(anne) bird(george) baby(polly), baby(keith) flies(fred)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

219 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) x(bird(x) baby(x) birdsfly(x)) bird(tweety), bird(polly) bird(anne) bird(george) baby(polly), baby(keith) flies(fred)} The good: flies(anne) flies(george) flies(tweety) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

220 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) x(bird(x) baby(x) birdsfly(x)) bird(tweety), bird(polly) bird(anne) bird(george) baby(polly), baby(keith) flies(fred)} The good: flies(anne) flies(george) flies(tweety) The bad: But, in some respect this proposal is too radical: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

221 "Naming defaults" Let T =, Φ where { } = : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) bird(x) flies(x)) x(bird(x) baby(x) birdsfly(x)) bird(tweety), bird(polly) bird(anne) bird(george) baby(polly), baby(keith) flies(fred)} The good: flies(anne) flies(george) flies(tweety) The bad: But, in some respect this proposal is too radical: bird(keith) bird(fred) for any ground term t Polly: birdsfly(t) bird(t) (flies(t) baby(t)) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

222 A different modelling of the same example Let T =, Φ where Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

223 A different modelling of the same example Let T =, Φ where { = bird(x) : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) flies(x)) x(baby(x) birdsfly(x)) bird(tweety), bird(polly) bird(anne) bird(george) baby(polly), baby(keith) flies(fred)} } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

224 A different modelling of the same example Let T =, Φ where { = bird(x) : birdsfly(x) birdsfly(x) Φ consists of x(birdsfly(x) flies(x)) x(baby(x) birdsfly(x)) bird(tweety), bird(polly) bird(anne) bird(george) baby(polly), baby(keith) flies(fred)} } Task Try to see why some of the negative too strong consequences of the previous slide are avoided in this proposal. Try to see why flies(anne) flies(george) is not anymore in any extension. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

225 Semi-normal defaults Let T =, Φ where = { bird(x) : flies(x) baby(x) flies(x) } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

226 Semi-normal defaults Let T =, Φ where = { bird(x) : flies(x) baby(x) flies(x) Φ consists of } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

227 Semi-normal defaults Let T =, Φ where { = bird(x) : } flies(x) baby(x) flies(x) Φ consists of bird(tweety), bird(pete), bird(mary) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

228 Semi-normal defaults Let T =, Φ where { = bird(x) : } flies(x) baby(x) flies(x) Φ consists of bird(tweety), bird(pete), bird(mary) baby(polly) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

229 Semi-normal defaults Let T =, Φ where { = bird(x) : } flies(x) baby(x) flies(x) Φ consists of bird(tweety), bird(pete), bird(mary) baby(polly) flies(fred) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

230 Semi-normal defaults Let T =, Φ where { = bird(x) : } flies(x) baby(x) flies(x) Φ consists of bird(tweety), bird(pete), bird(mary) baby(polly) flies(fred) bird(oscar) bird(sylvester) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

231 Semi-normal defaults Let T =, Φ where { } = bird(x) : flies(x) baby(x) flies(x) Φ consists of bird(tweety), bird(pete), bird(mary) baby(polly) flies(fred) bird(oscar) bird(sylvester) baby(pete) baby(mary)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

232 Semi-normal defaults Let T =, Φ where { } = bird(x) : flies(x) baby(x) flies(x) Φ consists of bird(tweety), bird(pete), bird(mary) baby(polly) flies(fred) bird(oscar) bird(sylvester) baby(pete) baby(mary)} This prevents the conclusion baby(tweety). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

233 Semi-normal defaults Let T =, Φ where { } = bird(x) : flies(x) baby(x) flies(x) Φ consists of bird(tweety), bird(pete), bird(mary) baby(polly) flies(fred) bird(oscar) bird(sylvester) baby(pete) baby(mary)} This prevents the conclusion baby(tweety). Task Try to see what problems we have with disjunctions for this example. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

234 The expressive power of semi-normal defaults Lukasziewicz writes: Assume, for instance, that on Sundays I usually go fishing, and suppose that you should remain agnostic about my fishing in rainy Sundays. It seems that the only appropriate representation of this situation is to use the following non-normal default: Sunday : I go fishing rain I go fishing Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

235 The expressive power of semi-normal defaults Lukasziewicz writes: Assume, for instance, that on Sundays I usually go fishing, and suppose that you should remain agnostic about my fishing in rainy Sundays. It seems that the only appropriate representation of this situation is to use the following non-normal default: Critically evaluated this claim. Sunday : I go fishing rain I go fishing 1 Why is a normal representation of this default suboptimal? 2 Do you agree with L. s assessment that the proposed non-normal representation is adequate? Suppose your only knowledge is that it is Sunday. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

236 Semi-normal defaults and the problem of inconsistent assumptions Let T =, Φ where { bird(x) : flies(x) dead(x) = flies(x) } of ancient species(x) : fossilised(x) dead(x) fossilised(x) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

237 Semi-normal defaults and the problem of inconsistent assumptions Let T =, Φ where { bird(x) : flies(x) dead(x) = flies(x) } of ancient species(x) : fossilised(x) dead(x) fossilised(x) Φ = {bird(tweety), of ancient species(tweety)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

238 Semi-normal defaults and the problem of inconsistent assumptions Let T =, Φ where { bird(x) : flies(x) dead(x) = flies(x) } of ancient species(x) : fossilised(x) dead(x) fossilised(x) Φ = {bird(tweety), of ancient species(tweety)} Task Try to see what s the problem here. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

239 Semi-normal vs. normal defaults Compare with has motive(x) : guilty(x) suspect(x) suspect(x) has motive(x) : guilty(x) suspect(x) guilty(x) suspect(x) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

240 Semi-Normal defaults ala Brewka/Levesque Let T =, Φ where = { : flies(x) bird(x) flies(x) } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

241 Semi-Normal defaults ala Brewka/Levesque Let T =, Φ where = { : flies(x) bird(x) flies(x) Φ consists of } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

242 Semi-Normal defaults ala Brewka/Levesque Let T =, Φ where { } = : flies(x) bird(x) flies(x) Φ consists of bird(tweety), bird(polly) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

243 Semi-Normal defaults ala Brewka/Levesque Let T =, Φ where { } = : flies(x) bird(x) flies(x) Φ consists of bird(tweety), bird(polly) baby(polly), flies(fred) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

244 Semi-Normal defaults ala Brewka/Levesque Let T =, Φ where { } = : flies(x) bird(x) flies(x) Φ consists of bird(tweety), bird(polly) baby(polly), flies(fred) bird(oscar) bird(sylvester) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

245 Semi-Normal defaults ala Brewka/Levesque Let T =, Φ where { } = : flies(x) bird(x) flies(x) Φ consists of bird(tweety), bird(polly) baby(polly), flies(fred) bird(oscar) bird(sylvester) flies(pete) flies(mary)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

246 Semi-Normal defaults ala Brewka/Levesque Let T =, Φ where Task { } = : flies(x) bird(x) flies(x) Φ consists of bird(tweety), bird(polly) baby(polly), flies(fred) bird(oscar) bird(sylvester) flies(pete) flies(mary)} We cannot conclude bird(fred). See why. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

247 Semi-Normal defaults ala Brewka/Levesque Let T =, Φ where Task { } = : flies(x) bird(x) flies(x) Φ consists of bird(tweety), bird(polly) baby(polly), flies(fred) bird(oscar) bird(sylvester) flies(pete) flies(mary)} We cannot conclude bird(fred). See why. Observe what happens to Pete and Mary. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

248 The Finch-Example Let T =, { Φ where = ruffed finch(x) : green island(x) green island(x), least ruffed finch(x) : green island(x) sand island(x) green island(x) sand island(x) } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

249 The Finch-Example Let T =, { Φ where = ruffed finch(x) : green island(x) green island(x), least ruffed finch(x) : green island(x) sand island(x) green island(x) sand island(x) Φ consists of } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

250 The Finch-Example Let T =, { Φ where = ruffed finch(x) : green island(x) green island(x), least ruffed finch(x) : green island(x) sand island(x) green island(x) sand island(x) Φ consists of least ruffed finch(frank) } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

251 The Finch-Example Let T =, { Φ where = ruffed finch(x) : green island(x) green island(x), least ruffed finch(x) : green island(x) sand island(x) green island(x) sand island(x) Φ consists of least ruffed finch(frank) x(least ruffed finch(x) ruffed finch(x))} } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

252 The Finch-Example Let T =, { Φ where = ruffed finch(x) : green island(x) green island(x), least ruffed finch(x) : green island(x) sand island(x) green island(x) sand island(x) Φ consists of least ruffed finch(frank) x(least ruffed finch(x) ruffed finch(x))} } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

253 The Finch-Example Let T =, { Φ where = ruffed finch(x) : green island(x) green island(x), least ruffed finch(x) : green island(x) sand island(x) green island(x) sand island(x) Φ consists of least ruffed finch(frank) x(least ruffed finch(x) ruffed finch(x))} Problem } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

254 The Finch-Example Let T =, { Φ where = ruffed finch(x) : green island(x) green island(x), least ruffed finch(x) : green island(x) sand island(x) green island(x) sand island(x) Φ consists of least ruffed finch(frank) x(least ruffed finch(x) ruffed finch(x))} Problem the unique extension includes both green island(frank) and green island(frank) sand island(frank) (since both defaults are triggered) } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

255 Default logic and monotonicity Nonmonotonicity, both in the set of defaults in the set of facts Φ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

256 Default logic and monotonicity Nonmonotonicity, both in the set of defaults in the set of facts Φ Not even cautious monotonic Here s an example that goes back to Makinson: = Φ 1 = { : p p, Φ 2 = {p q} Check, what happens! } p q : p p Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

257 The problem with negative cycles Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

258 The problem with negative cycles for a way to deal with this problem see Antonelli (1999) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

259 Prioritized default theories (Horty (2007, 2012)) Φ,, where we have a strict partial order on the defaults:. This represents a priority/preference relation. Depending on the application this may indicate the rank of an authority from which the information stems, the reliability of the source, specificity relations, etc. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

260 Prioritized default theories (Horty (2007, 2012)) Φ,, where we have a strict partial order on the defaults:. This represents a priority/preference relation. Depending on the application this may indicate the rank of an authority from which the information stems, the reliability of the source, specificity relations, etc. A strict partial order is (i) irreflexive (A A), (ii) asymmetric (If A < B then B A), and (iii) transitive (A < B and B < C implies A < C). Graphically they are represented by directed acyclic graphs (the transitive closure is usually not represented). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

261 Scenarios vs. extensions Given an ordered default theory Φ,, <, Horty distinguishes between scenarios, that is sets of defaults in, and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

262 Scenarios vs. extensions Given an ordered default theory Φ,, <, Horty distinguishes between scenarios, that is sets of defaults in, and the sets of beliefs generated by scenarios given the facts Φ, i.e., Cn(Φ Conclusion(S)) where S is a scenario. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

263 Proper Scenarios like in the non-prioritised case, not just any superset of Φ constitutes an extension, we are also now interested in scenarios that in some sense represent the set of defaults a rational agent would select/use given Φ and <. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

264 Proper Scenarios like in the non-prioritised case, not just any superset of Φ constitutes an extension, we are also now interested in scenarios that in some sense represent the set of defaults a rational agent would select/use given Φ and <. E.g., a scenario should not generate conflicting beliefs. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

265 Proper Scenarios like in the non-prioritised case, not just any superset of Φ constitutes an extension, we are also now interested in scenarios that in some sense represent the set of defaults a rational agent would select/use given Φ and <. E.g., a scenario should not generate conflicting beliefs. Moreover, priorities should be taken into account. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

266 Proper Scenarios like in the non-prioritised case, not just any superset of Φ constitutes an extension, we are also now interested in scenarios that in some sense represent the set of defaults a rational agent would select/use given Φ and <. E.g., a scenario should not generate conflicting beliefs. Moreover, priorities should be taken into account. Hence, we are interested in what Horty calls proper scenarios and the belief sets generated by them. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

267 The idea is again to build up scenarios stepwise similar as in the procedural approaches to build extensions. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

268 The idea is again to build up scenarios stepwise similar as in the procedural approaches to build extensions. we start with our facts Φ Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

269 The idea is again to build up scenarios stepwise similar as in the procedural approaches to build extensions. we start with our facts Φ What are interesting defaults to take into account? Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

270 The idea is again to build up scenarios stepwise similar as in the procedural approaches to build extensions. we start with our facts Φ What are interesting defaults to take into account? we have the choice between triggered defaults: Triggered(S) = {δ Φ Conclusion(S) Premise(δ)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

271 The idea is again to build up scenarios stepwise similar as in the procedural approaches to build extensions. we start with our facts Φ What are interesting defaults to take into account? we have the choice between triggered defaults: Triggered(S) = {δ Φ Conclusion(S) Premise(δ)} However, some triggered defaults may bad candidates since they (i.e., their conclusions) conflict with our belief set. Hence, we want to neglect: Conflicted(S) = {δ Φ Conclusion(S) Conclusion(δ)} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

272 What about the priorities? Also, we have to take into account our priorities. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

273 What about the priorities? Also, we have to take into account our priorities. A first idea would be to pick one of the highest ranked triggered defaults. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

274 What about the priorities? Also, we have to take into account our priorities. A first idea would be to pick one of the highest ranked triggered defaults. this is not Horty s approach (see later: the order puzzle) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

275 What about the priorities? Also, we have to take into account our priorities. A first idea would be to pick one of the highest ranked triggered defaults. this is not Horty s approach (see later: the order puzzle) We also have to take into account that sometimes sets of defaults defeat other defaults Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

276 What about the priorities? Also, we have to take into account our priorities. A first idea would be to pick one of the highest ranked triggered defaults. this is not Horty s approach (see later: the order puzzle) We also have to take into account that sometimes sets of defaults defeat other defaults We write S < S where S, S and δ < δ for all δ S and all δ S Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

277 What about the priorities? Also, we have to take into account our priorities. A first idea would be to pick one of the highest ranked triggered defaults. this is not Horty s approach (see later: the order puzzle) We also have to take into account that sometimes sets of defaults defeat other defaults We write S < S where S, S and δ < δ for all δ S and all δ S Example E.g., where δ 1 = a b, δ 2 = b c and δ 3 = a c and δ 3 < δ 1, δ 2, we have {δ 3 } < {δ 1, δ 2 } Note that {a} Conclusion({δ 1, δ 2 }) Conclusion(δ 3 ) in this sense {δ 1, δ 2 } defeats δ 3 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

278 Horty makes the idea above precise relative to a given scenario S. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

279 Horty makes the idea above precise relative to a given scenario S. The set Defeated(S) is the set of all δ such that there is a d Triggered(S) (a defeating set) for which Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

280 Horty makes the idea above precise relative to a given scenario S. The set Defeated(S) is the set of all δ such that there is a d Triggered(S) (a defeating set) for which {δ} < d and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

281 Horty makes the idea above precise relative to a given scenario S. The set Defeated(S) is the set of all δ such that there is a d Triggered(S) (a defeating set) for which {δ} < d and there is a S a S (an accommodation set) for which Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

282 Horty makes the idea above precise relative to a given scenario S. The set Defeated(S) is the set of all δ such that there is a d Triggered(S) (a defeating set) for which {δ} < d and there is a S a S (an accommodation set) for which 1 S a < d Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

283 Horty makes the idea above precise relative to a given scenario S. The set Defeated(S) is the set of all δ such that there is a d Triggered(S) (a defeating set) for which {δ} < d and there is a S a S (an accommodation set) for which 1 S a < d 2 Φ Conclusion((S \ S a ) d ) is consistent Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

284 Horty makes the idea above precise relative to a given scenario S. The set Defeated(S) is the set of all δ such that there is a d Triggered(S) (a defeating set) for which {δ} < d and there is a S a S (an accommodation set) for which 1 S a < d 2 Φ Conclusion((S \ S a ) d ) is consistent 3 Φ Conclusion((S \ S a ) d ) Conclusion(δ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

285 Some examples Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Suppose S = {B F }. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

286 Some examples Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Suppose S = {B F }. P F Triggered(S) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

287 Some examples Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Suppose S = {B F }. P F Triggered(S) {B F } is an accommodation set for the defeating set {P F } w.r.t. B F Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

288 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

289 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

290 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

291 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

292 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

293 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

294 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Let S 1 = {a c} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

295 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Let S 1 = {a c} a b is triggered by S Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

296 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Let S 1 = {a c} a b is triggered by S no defeating takes place Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

297 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Let S 1 = {a c} a b is triggered by S no defeating takes place Let S 2 = {a c, a b} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

298 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Let S 1 = {a c} a b is triggered by S no defeating takes place Let S 2 = {a c, a b} b c is triggered Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

299 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Let S 1 = {a c} a b is triggered by S no defeating takes place Let S 2 = {a c, a b} b c is triggered b c is conflicted by S Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

300 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Let S 1 = {a c} a b is triggered by S no defeating takes place Let S 2 = {a c, a b} b c is triggered b c is conflicted by S {a c} is defeated by S Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

301 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Let S 1 = {a c} a b is triggered by S no defeating takes place Let S 2 = {a c, a b} b c is triggered b c is conflicted by S {a c} is defeated by S note that {a c} is an accommodation set for the defeating set {b c} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

302 An intuitive problem? Take T = Φ,, < where = {δ 1, δ 2, δ 3 } with Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

303 An intuitive problem? Take T = Φ,, < where = {δ 1, δ 2, δ 3 } with δ 1 = a b Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

304 An intuitive problem? Take T = Φ,, < where = {δ 1, δ 2, δ 3 } with δ 1 = a b δ 2 = b c Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

305 An intuitive problem? Take T = Φ,, < where = {δ 1, δ 2, δ 3 } with δ 1 = a b δ 2 = b c δ 3 = c b Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

306 An intuitive problem? Take T = Φ,, < where = {δ 1, δ 2, δ 3 } with δ 1 = a b δ 2 = b c δ 3 = c b Φ = {a} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

307 An intuitive problem? Take T = Φ,, < where = {δ 1, δ 2, δ 3 } with δ 1 = a b δ 2 = b c δ 3 = c b Φ = {a} δ 1 < δ 3 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

308 An intuitive problem? Take T = Φ,, < where = {δ 1, δ 2, δ 3 } with δ 1 = a b δ 2 = b c δ 3 = c b Φ = {a} δ 1 < δ 3 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

309 An intuitive problem? Take T = Φ,, < where = {δ 1, δ 2, δ 3 } with δ 1 = a b δ 2 = b c δ 3 = c b Φ = {a} δ 1 < δ 3 Take S = {a b, b c} Note that a b Defeated(S) {a b} is an accommodation set for the defeating set {c b} Note that Φ Conclusion(b c, c b) b However, neither b c nor c b is triggered w.r.t. Φ! Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

310 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

311 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

312 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = 3 do the following until a fixed point is reached Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

313 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = 3 do the following until a fixed point is reached 1 S i+1 : add all δ to S i that satisfy the following conditions Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

314 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = 3 do the following until a fixed point is reached 1 S i+1 : add all δ to S i that satisfy the following conditions 1 δ Triggered(S i ) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

315 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = 3 do the following until a fixed point is reached 1 S i+1 : add all δ to S i that satisfy the following conditions 1 δ Triggered(S i ) 2 δ / Conflicted(S) (here you need to make use of your guess!) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

316 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = 3 do the following until a fixed point is reached 1 S i+1 : add all δ to S i that satisfy the following conditions 1 δ Triggered(S i ) 2 δ / Conflicted(S) (here you need to make use of your guess!) 3 δ / Defeated(S) (also here) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

317 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = 3 do the following until a fixed point is reached 1 S i+1 : add all δ to S i that satisfy the following conditions 1 δ Triggered(S i ) 2 δ / Conflicted(S) (here you need to make use of your guess!) 3 δ / Defeated(S) (also here) 4 let your fixed point be S. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

318 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = 3 do the following until a fixed point is reached 1 S i+1 : add all δ to S i that satisfy the following conditions 1 δ Triggered(S i ) 2 δ / Conflicted(S) (here you need to make use of your guess!) 3 δ / Defeated(S) (also here) 4 let your fixed point be S. 1 If S = S you re done. Then S is a proper scenario and Ξ = Cn(Φ Conclusion(S)) is an extension. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

319 Horty s procedure for proper scenarios and extensions Given an ordered default theory Φ,, < we construct proper scenarios as follows. 1 guess S 2 the initial scenario is S 0 = 3 do the following until a fixed point is reached 1 S i+1 : add all δ to S i that satisfy the following conditions 1 δ Triggered(S i ) 2 δ / Conflicted(S) (here you need to make use of your guess!) 3 δ / Defeated(S) (also here) 4 let your fixed point be S. 1 If S = S you re done. Then S is a proper scenario and Ξ = Cn(Φ Conclusion(S)) is an extension. 2 Otherwise, start anew with another guess. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

320 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

321 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

322 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

323 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

324 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

325 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

326 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

327 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Guess: S = {P F } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

328 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Guess: S = {P F } First round: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

329 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Guess: S = {P F } First round: S 0 = Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

330 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Guess: S = {P F } First round: S 0 = B F, P F Triggered(S 0) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

331 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Guess: S = {P F } First round: S 0 = B F, P F Triggered(S 0) B F Conflicted(S) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

332 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Guess: S = {P F } First round: S 0 = B F, P F Triggered(S 0) B F Conflicted(S) P F / Defeated(S) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

333 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Guess: S = {P F } First round: S 0 = B F, P F Triggered(S 0) B F Conflicted(S) P F / Defeated(S) hence: S 1 = {P F } Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

334 An example Take T = {P, P B}, {δ 1, δ 2 }, {(δ 1, δ 2 )} where δ 1 = B F and δ 2 = P F. Guess S = {B F }. First round: S 0 = B F, P F Triggered(S 0) P F Conflicted(S) but: B F Defeated(S) nothing is added! Guess: S = {P F } First round: S 0 = B F, P F Triggered(S 0) B F Conflicted(S) P F / Defeated(S) hence: S 1 = {P F } second round: fixed point reached. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

335 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

336 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

337 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

338 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

339 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

340 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

341 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Guess: S = {a c} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

342 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Guess: S = {a c} Round 1: S 0 = Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

343 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Guess: S = {a c} Round 1: S 0 = Triggered: a b and a c Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

344 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Guess: S = {a c} Round 1: S 0 = Triggered: a b and a c a b / Defeated(S) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

345 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Guess: S = {a c} Round 1: S 0 = Triggered: a b and a c a b / Defeated(S) a b / Conflicted(S) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

346 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Guess: S = {a c} Round 1: S 0 = Triggered: a b and a c a b / Defeated(S) a b / Conflicted(S) hence, a b has to be added! Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

347 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Guess: S = {a c} Round 1: S 0 = Triggered: a b and a c a b / Defeated(S) a b / Conflicted(S) hence, a b has to be added! Guess: S = {a b, b c} Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

348 Take T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = a b δ 2 = b c δ 3 = a c Φ = {a} δ 3 < δ 1 and δ 3 < δ 2 Guess: S = {a c} Round 1: S 0 = Triggered: a b and a c a b / Defeated(S) a b / Conflicted(S) hence, a b has to be added! Guess: S = {a b, b c} This works: check out why Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

349 Theorem Where Φ,, < is an ordered default theory and < =, then the associated extensions are exactly the same as the Reiter-extensions of Φ, (where is the translation of into normal defaults). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

350 Theorem Where Φ,, < is an ordered default theory and < =, then the associated extensions are exactly the same as the Reiter-extensions of Φ, (where is the translation of into normal defaults). Theorem Where T < = Φ,, < is an ordered default theory and Ξ is an extension of T <, then Ξ is also a Reiter-extension of Φ, (where is the translation of into normal defaults). Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

351 Another problematic example: The Order Puzzle Let T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = A δ 2 = B δ 3 = A B Φ = δ 1 < δ 2 < δ 3 Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

352 Another problematic example: The Order Puzzle Let T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = A δ 2 = B δ 3 = A B Φ = δ 1 < δ 2 < δ 3 We have one proper scenarios: Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

353 Another problematic example: The Order Puzzle Let T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = A δ 2 = B δ 3 = A B Φ = δ 1 < δ 2 < δ 3 We have one proper scenarios: S 1 = {δ 1, δ 2 }? Nope! Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

354 Another problematic example: The Order Puzzle Let T = Φ,, < where = {δ 1, δ 2, δ 3 } and δ 1 = A δ 2 = B δ 3 = A B Φ = δ 1 < δ 2 < δ 3 We have one proper scenarios: S 1 = {δ 1, δ 2 }? Nope! S 2 = {δ 1, δ 3 } What do you think? Interpret the premises in terms of conditional commands, and in terms of various pieces of information with different degrees of reliability. Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104

355 The normative reading by Horty (p.392, 2007) Christian Straßer (RUB, UGENT) Tutorial: Nonmonotonic Logic (Day 1) September 2, / 104