The Complexity of Reasoning for Fragments of Default Logic Heribert - PowerPoint PPT Presentation

The Complexity of Reasoning for Fragments of Default Logic Heribert Vollmer Joint work with O. Beyersdorff, A. Meier, M. Thomas Institut f¨ ur Theoretische Informatik Gottfried Wilhelm Leibniz Universit¨ at Hannover

Overview Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Default Logic Syntax and Semantics Generating Defaults Digression: Universal Algebra Clones Post’s Lattice Post’s Lattice and Computational Complexity The Complexity of Default Logic Extension Existence Credulous Reasoning Skeptical Reasoning Summary The Complexity of Reasoning for Fragments of Default Logic 2

What is Default Logic? Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e What is Default Logic? ◮ a non-monotone logic, introduced 1980 by Reiter ◮ models common-sense reasoning ◮ extends classical logic with default rules The Complexity of Reasoning for Fragments of Default Logic 3

What is Default Logic? Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e What is Default Logic? ◮ a non-monotone logic, introduced 1980 by Reiter ◮ models common-sense reasoning ◮ extends classical logic with default rules ◮ undecidable for first order logic (Reiter) ◮ here: propositional logic The Complexity of Reasoning for Fragments of Default Logic 3

Default Rules and Theories Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Definition (Reiter 80) A default rule is a triple α : β , where γ α is called the prerequisite, β is called the justification, and γ is called the consequent, for α, β, γ propositional formulae. Informally: infer a formula γ from a set of formulae W by a default rule α : β , if α ∈ W and ¬ β / ∈ W . γ The Complexity of Reasoning for Fragments of Default Logic 4

Default Theories Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Definition (Reiter 80) A default theory is a tuple � W , D � , where W is a set of formulae and D is a set of default rules. Example: Playing Football with Default Rules W = { football , rain , cold ∧ rain → snow } � football : ¬ snow � D = takesPlace ¬ snow is consistent with W . Hence we can infer takesPlace . The Complexity of Reasoning for Fragments of Default Logic 5

Default Theories Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Definition (Reiter 80) A default theory is a tuple � W , D � , where W is a set of formulae and D is a set of default rules. Example: Playing Football with Default Rules W = { football , rain , cold ∧ rain → snow , cold } � football : ¬ snow � D = takesPlace snow is consistent with W . Hence we cannot infer takesPlace . Default logics are non-monotone! The Complexity of Reasoning for Fragments of Default Logic 5

Semantics: Stable Extensions Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Definition (Reiter 80) For default theory � W , D � and set of formulae E , we define Γ( E ) as the smallest set, s.t. 1. W ⊆ Γ( E ), 2. Γ( E ) is closed under deduction, and 3. for all defaults α : β with α ∈ Γ( E ) and ¬ β / ∈ E , it holds that γ γ ∈ Γ( E ). A stable extension of � W , D � is a set E s.t. E = Γ( E ). The Complexity of Reasoning for Fragments of Default Logic 6

Semantics: Stable Extensions Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Definition (Reiter 80) For default theory � W , D � and set of formulae E , we define Γ( E ) as the smallest set, s.t. 1. W ⊆ Γ( E ), 2. Γ( E ) is closed under deduction, and 3. for all defaults α : β with α ∈ Γ( E ) and ¬ β / ∈ E , it holds that γ γ ∈ Γ( E ). A stable extension of � W , D � is a set E s.t. E = Γ( E ). Stable extensions correspond to possible views of an agent on the basis of � W , D � . The Complexity of Reasoning for Fragments of Default Logic 6

Stable Extensions Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Semantics: A Stage Construction (Reiter 80) Given: a default theory � W , D � and set of formulae E : E 0 := W E i +1 := Th ( E i ) ∪ { γ | α : β ∈ D , α ∈ E i and ¬ β / ∈ E } . γ Then: E a is stable extension of � W , D � iff E = � i ∈ N E i . The Complexity of Reasoning for Fragments of Default Logic 7

Stable Extensions Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Semantics: A Stage Construction (Reiter 80) Given: a default theory � W , D � and set of formulae E : E 0 := W E i +1 := Th ( E i ) ∪ { γ | α : β ∈ D , α ∈ E i and ¬ β / ∈ E } . γ Then: E a is stable extension of � W , D � iff E = � i ∈ N E i . The Complexity of Reasoning for Fragments of Default Logic 7

Generating Defaults Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Semantics: Generating Defaults (Reiter 80) Given: a default theory � W , D � and set of formulae E : Define the set of generating defaults as � α : β � � G := ∈ D � α ∈ E and ¬ β / ∈ E . � γ Then: E is stable a extension of � W , D � iff � � α : β �� � � E = Th W ∪ γ ∈ G . � γ The Complexity of Reasoning for Fragments of Default Logic 8

Example: More than one stable extension! Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Recap � � � α : β ∈ D � α ∈ E and ¬ β / ∈ E G := (generating defaults), � γ � � � �� � α : β E = Th W ∪ γ ∈ G (stable extension E ). � γ Example 1 � ⊤ : A , ⊤ : B , ⊤ : C � W = { B → ¬ A ∧ ¬ C } , D = A B C E 1 = Th ( W ∪ { A , C } ) , E 2 = Th ( W ∪ { B } ) The Complexity of Reasoning for Fragments of Default Logic 9

Example: No stable extension! Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Recap � � � α : β ∈ D � α ∈ E and ¬ β / ∈ E G := (generating defaults), � γ � � � �� � α : β E = Th W ∪ γ ∈ G (stable extension E ). � γ Example 2 � ⊤ : A � W = ∅ , D = ¬ A � W , D � has no stable extension. The Complexity of Reasoning for Fragments of Default Logic 9

Three Important Decision Problems Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Extension Existence Problem Instance: a default theory � W , D � Question: Does � W , D � have a stable extension? Credulous Reasoning Problem Instance: a formula ϕ and a default theory � W , D � Question: Is there a stable extension of � W , D � that includes ϕ ? Skeptical Reasoning Problem Instance: a formula ϕ and a default theory � W , D � Question: Does every stable extension of � W , D � include ϕ ? The Complexity of Reasoning for Fragments of Default Logic 10

Known Complexity Results Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Theorem (Gottlob 92) 1. The Extension Existence Problem is Σ p 2 -complete. 2. The Credulous Reasoning Problem is Σ p 2 -complete. 3. The Skeptical Reasoning Problem is Π p 2 -complete. The Complexity of Reasoning for Fragments of Default Logic 11

Motivation Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e What about the complexity if ... ◮ ... we allow only the Boolean functions ∧ and ∨ in W ? ◮ ... we allow only monotone functions in the default rules? The Complexity of Reasoning for Fragments of Default Logic 12

Motivation Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e What about the complexity if ... ◮ ... we allow only the Boolean functions ∧ and ∨ in W ? ◮ ... we allow only monotone functions in the default rules? � Parameterization of the three decision problems by a set B of Boolean functions The Complexity of Reasoning for Fragments of Default Logic 12

Motivation Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e What about the complexity if ... ◮ ... we allow only the Boolean functions ∧ and ∨ in W ? ◮ ... we allow only monotone functions in the default rules? � Parameterization of the three decision problems by a set B of Boolean functions We need a suitable characterisation for sets of Boolean functions. The Complexity of Reasoning for Fragments of Default Logic 12

A Little Bit of Universal Algebra Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Definition ◮ A clone is a set B of Boolean functions that contains all projections and is closed under composition. ◮ For a set B of Boolean functions, we denote by [ B ] the smallest clone containing B . ◮ B is called a base for [ B ]. The Complexity of Reasoning for Fragments of Default Logic 13

A Little Bit of Universal Algebra Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e Definition ◮ A clone is a set B of Boolean functions that contains all projections and is closed under composition. ◮ For a set B of Boolean functions, we denote by [ B ] the smallest clone containing B . ◮ B is called a base for [ B ]. Thus: ◮ [ B ] consists of those functions that can be computed by a Boolean circuit with basis B (gates from B ). The Complexity of Reasoning for Fragments of Default Logic 13