Outline. Golden Rule

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1 Outline Christian Straßer Institute for Philosophy II, Ruhr-University Bochum Center for Logic and Philosophy of Science, Ghent University 1 / 104 Overview: the whole tutorial 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 2 / 104 Golden Rule 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 Yes, 1 Defeasible... I know: Reasoning it s not realistic! 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 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) 3 / 104 Topic 5 Bibliography Bibliography 5 / 104 interrupt me ask complain comment 4 / 104 What s defeasible reasoning: some examples infer a (good!?) explanation α β and β thus α abductive inference 6 / 104 Tweety is a bird. Thus,...? closed world assumption reasoning on the assumption that the given information is complete stereotypical / default reasoning jump to a conclusion on the basis of what is usually/typically/normally/etc. the case 7 / / 104

2 Domains of defeasible reasoning everyday reasoning expert reasoning (e.g. medical diagnosis) scientific reasoning inductive generalisations 9 / 104 Commonalities 10 / 104 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. tentative conclusions jumping to conclusions retraction possible if problems arise illative tier: strictly deductive (e.g., classical logic) Two tiers of defeasible reasoning 1 illative tier (support, concluding) 2 dialectic tier (retraction) dialectic tier: conflicting deductive inferences Examples nonmonotonic paraconsistent logics (Rescher/Manor, inconsistency-adaptive logics) deductive argumentation-based approaches 11 / / 104 Ampliative approaches 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) 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 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 dialectic tier: e.g., exceptional circumstances conflicting defeasible and deductive inferences/arguments Examples Task: Free consequences inheritance networks What are the free consequences of Γ = {p q, p, r s? default logic abductive logics inductive generalisation 13 / 104 Classical Logic etc. 14 / 104 Summing up 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 corrective approach ampliative approach e.g. x(p(x) Q(x)) P(a) Q(a) illative tier deductive inferences ded. + non-ded. inf. dialectic tier conflicting ded. inferencces confl. non-ded. inferences In practice: distinction is not so clear-cut Hence... CL seems not apt to characterize reasoning that is dynamic in the sense that reasoners are prepared to retract inferences. 15 / / 104

3 Pessimism in the 60ies Toulmin Scheme (Toulmin (1958)) Premises Conclusion Backing Warrant Defeat 17 / 104 A Toulmin Argument 18 / 104 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 19 / 104 Nowadays cooperations between formal and informal logicians (Douglas Walton and formal argumentation) shift of normative standards in cognitive science (e.g., Stenning and Van Lambalgen (2008)) 21 / 104 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 20 / 104 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 22 / 104 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. 23 / / 104

4 Core property Be Cautious with Cut 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) (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? 25 / 104 What to replace Monotonicity with? 26 / 104 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. 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. 27 / 104 Schematic Inference Graphs (e.g., Inheritance Nets, Formal Argumentation, etc.) 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). 28 / 104 Resolution: Strict beats defeasible 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 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... So, we can read the diagram as follows: Penguins are birds (no exceptions); Birds usually fly; and Penguins usually don t fly. 29 / 104 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)) 31 / / 104 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 32 / 104

5 Some References to Classical Articles Short Reminder: 1st order logic 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) 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) 33 / 104 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) 35 / 104 Default Theory 34 / 104 What s a default conditional prerequisite α(x) justification : β 1 (x),..., β n (b) conclusion where x = x 1,..., x m, and α(x), β 1 (x),..., β n (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 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) the set of our beliefs is consistent with each β i (c) 36 / 104 Types of defaults Normal defaults set of facts, Φ set of defaults simple example { bird(x) : flies(x) flies(x) Φ = {bird(tweety), cat(sylvester) rather natural representation Semi-Normal defaults α(x) : α(x) : β(x) where β(x). E.g., 37 / 104 How to reason with default theories? α(x) : β(x) 38 / 104 Here s how it goes: Idea Apply iteratively modus ponens to defaults. This way build step-wise an extension (sets of beliefs that are obtained in this way) guess the extension Ξ init beliefs: Ξ = Φ ( ) take an c-ground instance of an default check whether: α(x) : β(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. 39 / / 104

6 Example: Tweety { 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. Question Are extensions always unique? 41 / 104 The Nixon Diamond 42 / 104 Another example { 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). Let, Φ be a default theory where { α(x) : β(x) : β(x), β(x) Φ = {α(c) Question Is there an extension? 43 / 104 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? 45 / 104 Going strictly procedural with Łukaszewicz (1988) 44 / 104 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) 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. Evaluation no initial guess guessing on-the-fly when we choose which triggered defaults to apply Christianonly Straßer in (RUB, the end UGENT) we see Tutorial: whether Nonmonotonic we guessed Logic (Dayluckily 1) 46 / 104 Some properties of the new procedure Let, Φ be a default theory. init: Φ = Φ ( ) take a c-instance of an arbitrary default check: α(x) : β(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. 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) : β(x), β(x) Φ = {α(c) 47 / / 104

7 Here s how it works we start with Φ = Φ test the c-instance of the default α(x) : β(x) : Φ α(c), OK β(c) γ(c) is consistent with Φ, OK there are no previously used defaults, so 3 is OK hence, Φ = {α(c), γ(c). test the c-instance of the default : β(x) β(x) : Φ γ(c), OK β(c) is consistent with Φ, OK however β(c) γ(c) is not consistent with Φ { β(c). Ξ = Cn{α(c), γ(c) 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? 49 / 104 Difference: Grounding 50 / 104 Non-Procedural Fixed-Point Characterisations 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. 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 1 α(c) π Φ (Γ) (trigger) 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. 51 / 104 Zorn s Lemma Let (S, ) be a PO-set such that for every decreasing -chain there is a lower bound in S. There is at least one minimal element in S. Proof of the existence of π(γ). We first show that the supposition of Zorn s Lemma can be met. Let S be all sets that satisfy (1) (3) and is. (Note S since L S.) Let Γ 1, Γ 2,... be any members of S, e.g. a decreasing -chain. Let Γ = Γ i. We have to show (1) (3). 1 trivial 2 Suppose A Cn(Γ ). Hence, there is a finite Γ f Γ such that Γ f A. Hence, Γ f Γ i and Γ i A for all i. Since Cn(Γ i ) = Γ i, A Γ i for all i. Thus, A Γ i. 3 Suppose α(c) Γ and β i (c) / Γ for all i n. Hence, α(c) Γ i for all i and thus γ(c) Γ i for all i. Thus, γ(c) Γ. By Zorn s Lemma there is at least one minimal element in S. Assume there are two different Γ 1 and Γ 2 which are minimal. However, by the proof above Γ 1 Γ 2 S, a contradiction. 53 / 104 Cautious Cut for credulous version? 52 / 104 Cautious Cut What do you think? If, Φ A and, Φ {A B then, Φ B. 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). 54 / 104 The :noexport: special status of normal default theories Cautious cut (for skeptical version) follows immediately from the lemma. 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. 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. 55 / / 104

8 Floating conclusions Task 1 What are the two extensions of this default theory? 2 Is politically motivated(nixon) derivable? { Nixon, quaker, republican, dove, hawk, politically motivated { Nixon, quaker, republican, dove, hawk, politically motivated { Nixon, quaker, republican, dove, hawk, politically motivated 57 / 104 Specificity 58 / 104 Lukaszewicz s Fishing example { Sunday : I go fishing I wake up late Φ = {Sunday, Holidays. I go fishing, Holidays : I wake up late I wake up late Question Is flies(tweety) derivable? Nope There are two extensions: 1 one with flies(tweety) 2 one with flies(tweety) 59 / 104 Poole s Lottery Paradox 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? 60 / 104 Reflections about Extensions { 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. 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? 61 / 104 Problems with Disjunctions 62 / 104 "Naming defaults" { 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). 63 / 104 { : 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)) 64 / 104

9 A different modelling of the same example Semi-normal defaults { 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. { 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. 65 / 104 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. 66 / 104 Semi-normal defaults and the problem of inconsistent assumptions { 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. 67 / 104 Semi-normal vs. normal defaults 68 / 104 Semi-Normal defaults ala Brewka/Levesque Compare with has motive(x) : guilty(x) suspect(x) suspect(x) has motive(x) : guilty(x) suspect(x) guilty(x) suspect(x) 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. 69 / 104 The Finch-Example 70 / 104 Default logic and monotonicity Nonmonotonicity, both in the set of defaults in the set of facts Φ 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) 71 / 104 Not even cautious monotonic Here s an example that goes back to Makinson: { : p p, Φ 1 = Φ 2 = {p q Check, what happens! p q : p p 72 / 104

10 The problem with negative cycles Prioritized default theories (??) 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. for a way to deal with this problem see Antonelli (1999) 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). 73 / 104 Scenarios vs. extensions 74 / 104 Proper Scenarios 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. 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. 75 / 104 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) 76 / 104 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 77 / / 104 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) 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 79 / / 104

11 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 81 / 104 Horty s procedure for proper scenarios and extensions 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! 82 / 104 An example 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. 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. 83 / / 104 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 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). 85 / 104 Another problematic example: The Order Puzzle 86 / 104 The normative reading by Horty (p.392, 2007) 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. 87 / / 104

12 A similar problem Let RC = being a resident of Cuba ; RN = being a resident of North-America; CC = being a citizen of Cuba ; CU = being a citizen of the USA ; and VU = having voting rights in the USA. Let T = Φ,, < where {δ 1, δ 2, δ 3 and δ 1 = RN CU δ 2 = RC CC δ 3 = CU VU Φ = {RC, RC RN, (CC CU), (CC VU) δ 1 < δ 2 < δ 3 There are two proper scenarios: S 1 = {δ 2 and S 2 = {δ 1, δ 3. (this is problematic) 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 No extension/proper scenario! 89 / 104 Multiple occurrences of the same defaults Let T = Φ,, < where {δ 1, δ 2, δ 3, δ 4 and δ 1 = A δ 2 = A δ 3 = A δ 4 = A Φ = δ 1 < δ 2 and δ 3 < δ 4 We have two proper scenarios: S 1 = {δ 2, δ 4 S 2 = {δ 1, δ 3 (problematic!) 90 / 104 Some philosophical Questions in the end What should we derive in default logic? What is our standard of adequacy, what is our ideal rational agent? How descriptive of actual reasoning should default logic be, and how normative should it be? What are the core properties of default reasoning? Solution... redefine < for sets of defaults? 91 / 104 Looking back... : what did we learn? What is a default theory? We learned about two different notions for extensions. There may be multiple extensions, for Reiter s logic there may be none. We have highlighted various problems for default logic: specificity disjunctive facts disjunctive exceptions consistency of assumptions etc. There are various ways of phrasing defaults (normal vs. semi-normal vs. naming defaults, etc.) 93 / 104 Idea (Moore (1984, 1985)) 92 / 104 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 94 / 104 Stable Autoepistemic Theories Logic to model the reasoning of an (ideal) agent reflecting about her beliefs Autoepistemic Theory features the beliefs A of an epistemic agent including her introspective beliefs about her beliefs BA (and lack thereof) closed under rationality constraints (next slide!) Stalnaker (1993), stability criteria on a autoepistemic theory Γ 1 A Γ implies BA Γ 2 A / Γ implies BA Γ. 3 classical closure: Cn(Γ) = Γ. If Γ is consistent, this implies... 1 A Γ iff BA Γ 2 A / Γ iff BA. You see why? 95 / / 104

13 Stability is not enough... Groundedness Take Ξ = { Bbrother brother. Two consistent stable autoepistemic supersets of Ξ: do you see which? Ξ 1 containing brother Can you see what else is contained? by (1), B i brother Ξ 1 by cons., brother / Ξ 1 by (2), Bbrother Ξ 1 by (1), B i Bbrother Ξ 1 Do you see a problem? Ξ 2 containing brother The belief in brother in Ξ 2 is not grounded given Ξ! Can you see what else is contained? by (1), B i brother Ξ 2 by cons., brother / Ξ 2 by (2), B brother Ξ 2 Grounding Autoepistemic Theories Here s how it s done: Autoepistemic Extension of Γ where Ξ = {A A / Γ BΞ = {BA A Ξ Ξ = Cn({Γ BΞ BΞ) Do you see where there is no autoepistemic extension of Γ = { Bbrother brother that contains brother? Suppose there were one Ξ! But then Γ BΞ BΞ brother but this is impossible! 97 / 104 Extension do not always exist 98 / 104 Are autoepistemic extensions of some Γ always consistent? Recall: Ξ = Cn({Γ BΞ BΞ) Recall: Ξ = Cn({Γ BΞ BΞ) Take Γ = { BA A. Let Ξ be any classically closed superset of Γ. suppose A Ξ: but how to ground A? in view of Γ the only way is to have also BA by we have BA Ξ hence, Ξ is trivial (by classical closure) but then BΞ =, a contradiction, since now there is no way to derive BA suppose A / Ξ: then BA but then also A by modus ponens, contradiction If Γ is inconsistent: nope! What if Γ is consistent? Take Γ = { Bp if p is in we get Bp together with Γ we can derive anything 99 / 104 Another Example Take Γ = { Bp q, Bq p. What are the autoepistemic extensions of Γ? 1 what about one with p and q? but how to ground p and q? We would need Bp to ground q and Bq to ground p 2 what about one with p and/or q? again: how to ground p and/or q? only possibility is if Γ were inconsistent: but it isn t 3 what about one with p and without q? then Bq (neg. introspection) thus, p is grounded 4 similarly, one with q and without p 101 / 104 Bootstrapping problem continued Recall: Ξ = Cn({Γ BΞ BΞ) Take Γ = { Bp q, Bp p Two (minimal) autoepistemic extensions: 1 including Bq, q and not including p 2 including Bp, p and not including q (and thus including Bq) (see Konolige (1988)) 103 / / 104 Bootstrapping problem (Konolige (1988)) Recall: The problem Ξ = Cn({Γ BΞ BΞ) Take Γ = {BA A. There is an autoepistemic extension that contains A (and BA and A B and B(A B) and B A, etc.). Note that A is grounded in view of Γ once we adopt BA. Solution Take the minimal autoepistemic extensions! Note there is another autoepistemic extension Ξ of Γ that only contains (introspective) beliefs in classical tautologies and for all non-tautologies C it contains B i BC (where i 0). If we compare the subsets of Ξ and Ξ without occurrences of B, written Ξ 0 and Ξ 0, then Ξ 0 Ξ / 104 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 104 / 104

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