Principles of Knowledge Representation and Reasoning May 20 & 23, 2008 — Nonmonotonic Reasoning Introduction Motivation Different Forms of Reasoning Principles of Knowledge Representation and Reasoning Different Formalizations Nonmonotonic Reasoning Default Logic Basics Extensions Bernhard Nebel, Malte Helmert and Stefan W¨ olfl Properties of Extensions Normal Defaults Albert-Ludwigs-Universit¨ at Freiburg Default Proofs Decidability May 20 & 23, 2008 Propositional DL Complexity of Default Logic Complexity of DL Semi-Normal Defaults Open Defaults Outlook Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 1 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 2 / 39 Literature Introduction Motivation Introduction Motivation A Motivating Example: Defaults in Knowledge Bases A Motivating Example: Common Sense Reasoning 1. employee(anne) 1. Tweety is a bird like other birds. 2. employee(bert) 2. During the summer he stays in Northern Europe, in the winter he 3. employee(carla) stays in Africa. 4. employee(detlef) ◮ Would you expect Tweety to be able to fly? 5. employee(thomas) ◮ How does Tweety get from Northern Europe to Africa? 6. onUnpaidMPaternityLeave(thomas) 7. employee(X) ∧¬ onUnpaidMPaternityLeave(X) → gettingSalary(X) How would you formalize this in formal logic so that you get the expected answers? 8. typically: employee(X) → ¬ onUnpaidMPaternityLeave(X) Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 3 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 4 / 39
Introduction Motivation Introduction Different Forms of Reasoning A Formalization . . . Examples of Such Reasoning Patterns Closed world assumption: Data-base of ground atoms. All ground atoms 1. bird(tweety) not present are assumed to be false. 2. spend-summer(tweety,northern-europe) ∧ spend-winter(tweety,africa) Negation as failure: In PROLOG, NOT(P) means “P is not provable” 3. ∀ x (bird( x ) → can-fly( x )) instead of “P is provably false”. 4. far-away(northern-europe,africa) Non-strict inheritance: An attribute value is inherited only if there is no 5. ∀ xyz (can-fly( x ) ∧ far-away( y , z ) ∧ spend-summer( x , y ) ∧ more specialized information contradicting the attribute spend-winter( x , z ) → flies( x , y , z )) value. Reasoning about actions: When reasoning about actions, it is usually ◮ The implication (3) is just a reasonable assumption assumed that a property changes only if it has to change, ◮ What if Tweety is an emu? i.e., properties by default do not change. Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 5 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 6 / 39 Introduction Different Forms of Reasoning Introduction Different Formalizations Default, Defeasible, and Non-monotonic Reasoning Approaches to Non-Monotonic Reasoning Default Reasoning: Jump to a conclusion if there is no information that ◮ Consistency-based: Extend classical theory by rules that test whether contradicts the conclusion. an assumption is consistent with existing beliefs Defeasible Reasoning: Reasoning based on assumptions that can turn out ⇒ non-monotonic logics like DL (default logic), to be wrong, — i.e., conclusions are defeasible. In particular, NMLP (non-monotonic logic programming) default reasoning is defeasible. ◮ Entailment-based on normal models: Models are ordered by normality. Non-monotonic Reasoning: In classical logic, the set of consequence grows Entailment is determined by considering the most normal models only. monotonically with the set of premises. If reasoning is ⇒ Circumscription, Preferential and Cumulative Logics defeasible, then reasoning becomes non-monotonic. Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 7 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 8 / 39
Introduction Different Formalizations Introduction Different Formalizations NM Logic – Consistency-Based NM Logic – Normal Models If ϕ typically implies ψ , then the models satisfying ϕ ∧ ψ should be more normal than those satisfying ϕ ∧ ¬ ψ . If ϕ typically implies ψ , ϕ is given, and it is consistent to assume ψ , then conclude ψ . Similarly, try to minimize the interpretation of “Abnormality” predicates. 1. Typically bird( x ) implies can-fly( x ) 1. ∀ x (bird( x ) ∧ ¬ Ab( x ) → can-fly( x )) 2. ∀ x (emu( x ) → bird( x )) 2. ∀ x (emu( x ) → bird( x )) 3. ∀ x (emu( x ) → ¬ can-fly( x )) 3. ∀ x (emu( x ) → ¬ can-fly( x )) 4. bird(tweety) 4. bird(tweety) ⇒ can-fly(tweety) Minimize interpretation of Ab. ⇒ can-fly(tweety) 5. . . . + emu(tweety) ⇒ ¬ can-fly(tweety) 5. . . . + emu(tweety) ⇒ Now in all models (incl. the normal ones): ¬ can-fly(tweety) Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 9 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 10 / 39 Default Logic Default Logic Basics Default Logic – Outline Motivation: Reiter’s Default Logic ◮ We want to express something like “typically birds fly”. Introduction ◮ Add non-logical inference rule Default Logic bird( x ) : can-fly( x ) Basics can-fly( x ) Extensions Properties of Extensions with the intended meaning: Normal Defaults If x is a bird and if it is consistent to assume that x can fly, then Default Proofs conclude that x can fly. Decidability ◮ Exceptions can be represented as formulae: Propositional DL ∀ x (penguin( x ) → ¬ can-fly( x )) Complexity of Default Logic ∀ x (emu( x ) → ¬ can-fly( x )) ∀ x (kiwi( x ) → ¬ can-fly( x )) Literature Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 11 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 12 / 39
Default Logic Basics Default Logic Extensions Formal Framework Extensions of Default Theories Default theories extend the theories given by W using the default rules D ◮ FOL with classical provability relation ⊢ and deductive closure: ( � extensions). There may be zero, one, or many extensions. Th(Φ) := { φ | Φ | = φ } Example ◮ Default rules: α : β γ α : Prerequisite: must have been derived before rule can be applied. W = { a , ¬ b ∨ ¬ c } β : Consistency condition: the negation may not be derivable. � a : b b , a : c � γ : Consequence: will be concluded. D = c ◮ A default rule is closed if it does not contain free variables. ◮ (Closed) default theory: A pair ( D , W ), where D is a countable set of One extension contains b , the other contains c . (closed) default rules and W is a countable set of FOL formulae. Intuitively: an extension is a set of beliefs resulting from W and D . Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 13 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 14 / 39 Default Logic Extensions Default Logic Extensions Decision Problems about Extensions in Default Logic Extensions – Informally Desirable properties of an extension E of ( D , W ): 1. Contains all facts W ⊆ E . 2. Is deductively closed: E = Th( E ). Existence of extensions: Does a default theory have an extension? 3. All applicable default rules have been applied: If Credulous reasoning: If ϕ is in at least one extension, ϕ is a credulous 3.1 ( α : β γ ) ∈ D , default conclusion. 3.2 α ∈ E , Skeptical Reasoning: If ϕ is in all extensions, ϕ is a skeptical default 3.3 ¬ β �∈ E conclusion. then γ ∈ E . ⇒ Requirement: Application of default rules must follow in sequence ( groundedness ). Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 15 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 16 / 39
Default Logic Extensions Default Logic Extensions Groundedness Extensions – Formally Definition Example Let ∆ = ( D , W ) be a closed default theory and let E be a set of closed formulae. Let W = ∅ E 0 = W � a : b b , b : a � � � � D = α : β � E i = Th( E i − 1 ) ∪ ∈ D , α ∈ E i − 1 , ¬ β �∈ E γ a � γ � Question : Should Th( { a , b } ) be an extension? Then E is an extension of ∆ iff Answer : No! a can only be derived if we already have derived b . ∞ � E = E i . b can only be derived if we already have derived a . i =0 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 17 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 18 / 39 Default Logic Extensions Default Logic Extensions How to Use This Definition? Examples � a : b b , b : a � D = W = { a ∨ b } a � a : b � D = W = ∅ ◮ The definition does not tell us how to construct an extension. ¬ b ◮ However, it tells us how to check whether a set is an extension. � a : b � D = W = { a } ◮ Guess a set E . ¬ b � : a � ◮ Then construct sets E i by starting with W . a , : b b , : c D = W = { b → ¬ a ∧ ¬ c } c ◮ If E = � ∞ i =0 E i , then E is an extension of ( D , W ). � : c ¬ d , : d ¬ e , : e � W = ∅ D = ¬ f � : c ¬ d , : d � W = ∅ D = ¬ c � a : b , a : d � D = W = { a , ¬ b ∨ ¬ d } c e Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 19 / 39 Nebel, Helmert, W¨ olfl (Uni Freiburg) KRR May 20 & 23, 2008 20 / 39
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