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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 γ , where 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 γ , if α ∈ W and ¬β / ∈ W .

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} D = football : ¬snow 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} D = football : ¬snow 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

γ with α ∈ Γ(E) and ¬β / ∈ E, it holds that 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

γ with α ∈ Γ(E) and ¬β / ∈ E, it holds that 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: E0 := W Ei+1 := Th (Ei) ∪ {γ | α : β γ ∈ D, α ∈ Ei and ¬β / ∈ E}. Then: E a is stable extension of W , D iff E =

i∈N Ei.

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: E0 := W Ei+1 := Th (Ei) ∪ {γ | α : β γ ∈ D, α ∈ Ei and ¬β / ∈ E}. Then: E a is stable extension of W , D iff E =

i∈N Ei.

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 G := α : β γ ∈ D

• α ∈ E and ¬β /

∈ E

• .

Then: E is stable a extension of W , D iff E = Th

• W ∪
• γ
• α : β

γ ∈ G

• .

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

G :=

• α:β

γ

∈ D

• α ∈ E and ¬β /

∈ E

• (generating defaults),

E = Th

• W ∪
• γ
• α:β

γ

∈ G

• (stable extension E).

Example 1

W = {B → ¬A ∧ ¬C}, D = ⊤: A A , ⊤: B B , ⊤: C C

• E1 = Th (W ∪ {A, C}) , E2 = 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

G :=

• α:β

γ

∈ D

• α ∈ E and ¬β /

∈ E

• (generating defaults),

E = Th

• W ∪
• γ
• α:β

γ

∈ G

• (stable extension E).

Example 2

W = ∅, D = ⊤: A ¬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 Σp

2-complete.

• 2. The Credulous Reasoning Problem is Σp

2-complete.

• 3. The Skeptical Reasoning Problem is Πp

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

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).

◮ [B] consists of those functions that can be defined by a

propositional formula with connectives from B.

The Complexity of Reasoning for Fragments of Default Logic 13

So What?

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Π – computational problem defined over Boolean circuits Π(B) – the restriction of Π to circuits with gates from B, e.g., CVP(B), the circuit value problem for circuits with gates from B

The Complexity of Reasoning for Fragments of Default Logic 14

So What?

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Π – computational problem defined over Boolean circuits Π(B) – the restriction of Π to circuits with gates from B, e.g., CVP(B), the circuit value problem for circuits with gates from B Then: If B ⊆ [B′] then Π(B) ≤p

m Π(B′),

e.g., CVP(B) ≤p

m CVP(B′).

The Complexity of Reasoning for Fragments of Default Logic 14

So What?

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Π – computational problem defined over Boolean circuits Π(B) – the restriction of Π to circuits with gates from B, e.g., CVP(B), the circuit value problem for circuits with gates from B Then: If B ⊆ [B′] then Π(B) ≤p

m Π(B′),

e.g., CVP(B) ≤p

m CVP(B′).

Upper bounds carry downwards. Lower bounds carry upwards.

The Complexity of Reasoning for Fragments of Default Logic 14

So What?

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Π – computational problem defined over Boolean circuits Π(B) – the restriction of Π to circuits with gates from B, e.g., CVP(B), the circuit value problem for circuits with gates from B Then: If B ⊆ [B′] then Π(B) ≤p

m Π(B′),

e.g., CVP(B) ≤p

m CVP(B′).

Upper bounds carry downwards. Lower bounds carry upwards. Complexity of Π(B) is determined by the clone [B].

The Complexity of Reasoning for Fragments of Default Logic 14

So What?

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Π – computational problem defined over propositional formulas Π(B) – the restriction of Π to circuits with connectives from B, e.g., SAT(B), the satisfiability problem for formulas with connectives from B Then: If B ⊆ [B′] then Π(B) ≤p

m Π(B′),

e.g., SAT(B) ≤p

m SAT(B′).

Upper bounds carry downwards. Lower bounds carry upwards. Complexity of Π(B) is determined by the clone [B].

The Complexity of Reasoning for Fragments of Default Logic 14

So What?

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Π – computational problem defined over propositional formulas Π(B) – the restriction of Π to circuits with connectives from B, e.g., SAT(B), the satisfiability problem for formulas with connectives from B Then: If B ⊆ [B′] then Π(B) ≤p

m Π(B′),

e.g., SAT(B) ≤p

m SAT(B′).

Upper bounds carry downwards. Lower bounds carry upwards. Complexity of Π(B) is determined by the clone [B]. Caveat: Explosion of formula size! (Usually does not happen . . .)

The Complexity of Reasoning for Fragments of Default Logic 14

Properties of Boolean Functions

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Some Properties of Boolean Functions

◮ f is c-reproducing if f (c, . . . , c) = c, c ∈ {0, 1}. ◮ f is monotone if a1 ≤ b1, . . . , an ≤ bn implies

f (a1, . . . , an) ≤ f (b1, . . . , bn).

◮ f is c-separating if there exists an i ∈ {1, . . . , n} such that

f (a1, . . . , an) = c implies ai = c, c ∈ {0, 1}.

◮ f is self-dual if f ≡ dual(f ), where

dual(f )(x1, . . . , xn) = ¬f (¬x1, . . . , ¬xn).

◮ f is linear if f ≡ x1 ⊕ · · · ⊕ xn ⊕ c for a constant c ∈ {0, 1}

and variables x1, . . . , xn.

The Complexity of Reasoning for Fragments of Default Logic 15

Important Boolean Clones

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Name Definition Base BF All Boolean functions {∧, ¬} R0 {f : f is 0-reproducing} {∧, →} R1 {f : f is 1-reproducing} {∨, →} M {f : f is monotone} {∨, ∧, 0, 1} S0 {f : f is 0-separating} {→} S1 {f : f is 1-separating} {→} D {f : f is self-dual} {(x∧y) ∨ (x∧z) ∨ (y ∧z)} L {f : f is linear} {⊕, 1} V {f : f ≡ c0 ∨ n

i=1 cixi}

{∨, 0, 1} E {f : f ≡ c0 ∧ n

i=1 cixi}

{∧, 0, 1} N {f : f depends on only one variable} {¬, 0, 1} I {f : f is a projection or constant} {id, 0, 1}

The Complexity of Reasoning for Fragments of Default Logic 16

Post’s Lattice

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

◮ Refers to Emil Post,

1941.

The Complexity of Reasoning for Fragments of Default Logic 17

Post’s Lattice

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

◮ Refers to Emil Post,

1941.

◮ Each node is a finite

set of Boolean functions (basis).

The Complexity of Reasoning for Fragments of Default Logic 17

Post’s Lattice

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

◮ Refers to Emil Post,

1941.

◮ Each node is a finite

set of Boolean functions (basis).

◮ Many new decision

problems in default logic arise.

The Complexity of Reasoning for Fragments of Default Logic 17

Post’s Lattice

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

◮ Refers to Emil Post,

1941.

◮ Each node is a finite

set of Boolean functions (basis).

◮ Many new decision

problems in default logic arise.

◮ Hardness results carry

upwards.

The Complexity of Reasoning for Fragments of Default Logic 17

Post’s Lattice

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

◮ Refers to Emil Post,

1941.

◮ Each node is a finite

set of Boolean functions (basis).

◮ Many new decision

problems in default logic arise.

◮ Hardness results carry

upwards.

◮ Membership results

carry downwards.

The Complexity of Reasoning for Fragments of Default Logic 17

Post’s Lattice and Computational Complexity

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Motivation for a fine classification using Post’s Lattice

Given a hard problem involving Boolean functions, we aim to

◮ identify the sources of hardness of the general problem, ◮ design more efficient algorithms for tractable fragments.

The Complexity of Reasoning for Fragments of Default Logic 18

Post’s Lattice and Computational Complexity

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Motivation for a fine classification using Post’s Lattice

Given a hard problem involving Boolean functions, we aim to

◮ identify the sources of hardness of the general problem, ◮ design more efficient algorithms for tractable fragments.

Important problems have been classified via this approach:

◮ Lewis 79: classification of SAT

SAT(B) is NP-complete iff →∈ [B].

◮ Reith, Wagner 05: circuit value problem, QBF ◮ nonclassical logics: LTL, CTL, ...

The Complexity of Reasoning for Fragments of Default Logic 18

Post’s Lattice and Default Logic

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

The “new” decision problems

◮ We restrict the allowed Boolean functions for formulae in W

and D to functions from [B].

◮ EXT(B) is the extension existence problem for B-default

theories.

◮ CRED(B) and SKEP(B) are defined analogously.

The Complexity of Reasoning for Fragments of Default Logic 19

Extension Existence Problem

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

EXT(B):

Instance: a B-default theory W , D Question: Does W , D have a stable extension?

The Complexity of Reasoning for Fragments of Default Logic 20

Extension Existence Problem

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Theorem

Let B be a finite set of Boolean functions. Then EXT(B) is

• 1. Σp

2-complete if S1 ⊆ [B] ⊆ BF or D ⊆ [B] ⊆ BF,

• 2. NP-complete if [B] ∈ {N, N2, L, L0, L3},
• 3. trivial in all other cases (i. e., if [B] ⊆ R1 or [B] ⊆ M).

S1: base {→} D: f ≡ ¬f (¬x1, . . . , ¬xn) N: base {¬, 0, 1} L: base {⊕, 1} M: monotone functions R1: f (1, . . . , 1) = 1

The Complexity of Reasoning for Fragments of Default Logic 20

Complexity Overview: Extension Existence

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

trivial NP-complete Σp

2-complete

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

The Complexity of Reasoning for Fragments of Default Logic 21

• 1. Σp

2-complete if S1 ⊆ [B] ⊆ BF or D ⊆ [B] ⊆ BF

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Proof.

◮ The results from Gottlob (1992) can be generalized to BF. ◮ For each finite set B of Boolean functions,

EXT(B) ≡ EXT(B ∪ {1}). S1: base {→} D: f ≡ ¬f (¬x1, . . . , ¬xn)

The Complexity of Reasoning for Fragments of Default Logic 22

• 2. NP-complete if [B] ∈ {N, N2, L, L0, L3}

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Proof (Membership in NP for [B] ⊆ L).

Let W , D be a B-default theory.

◮ Guess a set of generating defaults G ⊆ D.

Let G ′ = W ∪ {γ | α:β

γ

∈ G}.

◮ Inductively compute generators Gi for each Ei until Ei = Ei+1. ◮ G0 = W ◮ For each α:β γ

∈ D check: If Gi | = α and G ′ | = ¬β, then put γ into Gi+1 (this test is possible in polynomial time).

◮ If Gi = Gi+1, then check whether Gi = G ′.

N: base {¬} L: base {⊕}

The Complexity of Reasoning for Fragments of Default Logic 23

• 2. NP-complete if [B] ∈ {N, N2, L, L0, L3}

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Proof (NP-hardness for N ⊆ [B]).

Let W , D be a B-default theory. Reduction from 3SAT:

◮ Let ϕ = n i=1(li1 ∨ li2 ∨ li3) in variables x1, . . . , xm. ◮ ϕ → ∅, Dϕ, where

Dϕ := 1 : xi xi | 1 ≤ i ≤ m

• ∪

1 : ¬xi ¬xi | 1 ≤ i ≤ m

• ∪

¬liπ(1) : ¬liπ(2) liπ(3)     1 ≤ i ≤ n, π is a permutation of {1, 2, 3}

• Then σ: {x1, . . . , xm} → {0, 1} corresponds to

E = Th ({xi | σ(xi) = 1} ∪ {¬xi | σ(xi) = 0})

The Complexity of Reasoning for Fragments of Default Logic 23

• 3. Trivial Complexity ([B] ⊆ R1 or [B] ⊆ M)

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Proof (easy cases for [B] ⊆ R1 or [B] ⊆ M).

Let B be a finite set of Boolean functions s.t. [B] ⊆ R1 or [B] ⊆ M, and W , D be a B-default theory with finite D. Then W , D has a unique stable extension. R1: 1-reproducing functions (f (1, . . . , 1) = 1) M: monotone functions

The Complexity of Reasoning for Fragments of Default Logic 24

Complexity Overview: Extension Existence

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

trivial NP-complete Σp

2-complete

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

The Complexity of Reasoning for Fragments of Default Logic 25

Credulous Reasoning

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Credulous Reasoning Problem, CRED(B)

Instance: a B-formula ϕ and a B-default theory W , D Question: Is there a stable extension of W , D that includes ϕ?

The Complexity of Reasoning for Fragments of Default Logic 26

Credulous Reasoning

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Theorem

Let B be a finite set of Boolean functions. Then CRED(B) is

• 1. Σp

2-complete if S1 ⊆ [B] ⊆ BF or D ⊆ [B] ⊆ BF,

• 2. coNP-complete if X ⊆ [B] ⊆ Y , where X ∈ {S00, S10, D2}

and Y ∈ {R1, M},

• 3. NP-complete if [B] ∈ {N, N2, L, L0, L3},
• 4. P-complete if V2 ⊆ [B] ⊆ V, E2 ⊆ [B] ⊆ E or [B] ∈ {L1, L2},
• 5. NL-complete if I2 ⊆ [B] ⊆ I.

The Complexity of Reasoning for Fragments of Default Logic 26

Complexity Overview: Credulous Reasoning

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Complexity of CRED(B): NL-complete P-complete NP-complete coNP-complete Σp

2-complete

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

The Complexity of Reasoning for Fragments of Default Logic 27

• 2. Proof (coNP-Membership for [B] ⊆ R1 or [B] ⊆ M)

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Require: W , D, ϕ Gnew ← W repeat Gold ← Gnew for all α:β

γ

∈ D do if Gold | = α then // coNP-oracle call Gnew ← Gnew ∪ {γ} end if end for until Gnew = Gold return true iff Gnew | = ϕ // coNP-oracle call

The Complexity of Reasoning for Fragments of Default Logic 28

coNP-hardness for X ⊆ [B], X ∈ {S00, S10, D2}

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

◮ The implication problem IMP(B) is already coNP-complete. ◮ (ϕ, ψ) → ({ϕ}, ∅ , ψ) reduces IMP(B) ≤ CRED(B)

BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10

The Complexity of Reasoning for Fragments of Default Logic 29

Proof Ideas for 3. and 4.

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

• 3. NP-complete if [B] ∈ {N, N2, L, L0, L3}

◮ Hardness: Reduce 3SAT to CRED(B). ◮ Membership: Use the NP-algorithm for EXT({⊕}).

The Complexity of Reasoning for Fragments of Default Logic 30

Proof Ideas for 3. and 4.

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

• 3. NP-complete if [B] ∈ {N, N2, L, L0, L3}

◮ Hardness: Reduce 3SAT to CRED(B). ◮ Membership: Use the NP-algorithm for EXT({⊕}).

• 4. P-complete if V2 ⊆ [B] ⊆ V, E2 ⊆ [B] ⊆ E or [B] ∈ {L1, L2}

◮ Hardness: Reduce the hypergraph accessibility problem to

CRED({∧}).

◮ Membership: use the previous coNP-Algorithm.

As implications can now be tested in poly-time, we get a poly-time algorithm.

The Complexity of Reasoning for Fragments of Default Logic 30

• 5. NL-complete if I2 ⊆ [B] ⊆ I

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

◮ Hardness: GAP ≤ CRED(∅) via

(G, s, t) →

• {ps},

pu : pu pv | (u, v) ∈ E

• , pt
• The Complexity of Reasoning for Fragments of Default Logic

31

• 5. NL-complete if I2 ⊆ [B] ⊆ I

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

◮ Hardness: GAP ≤ CRED(∅) via

(G, s, t) →

• {ps},

pu : pu pv | (u, v) ∈ E

• , pt
• ◮ Membership: CRED({0, 1}) ≤ GAP

The Complexity of Reasoning for Fragments of Default Logic 31

Skeptical Reasoning

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Skeptical Reasoning Problem, SKEP(B)

Instance: a B-formula ϕ and a B-default theory W , D Question: Does every stable extension of W , D include ϕ?

The Complexity of Reasoning for Fragments of Default Logic 32

Skeptical Reasoning

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Theorem

Let B be a finite set of Boolean functions. Then SKEP(B) is

• 1. Πp

2-complete if S1 ⊆ [B] ⊆ BF or D ⊆ [B] ⊆ BF,

• 2. coNP-complete if X ⊆ [B] ⊆ Y , where X ∈ {S00, S10, N2, L0}

and Y ∈ {R1, M, L},

• 3. P-complete if V2 ⊆ [B] ⊆ V, E2 ⊆ [B] ⊆ E or [B] ∈ {L1, L2},

and

• 4. NL-complete if I2 ⊆ [B] ⊆ I.

The Complexity of Reasoning for Fragments of Default Logic 32

Complexity Overview: Credulous/Skeptical Reasoning

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Complexity of CRED(B): NL-complete P-complete NP-complete coNP-complete Σp

2-complete BF R1 R0 R2 M M1 M0 M2 S2 S2

02

S2

01

S3 S2

00

S3

02

S3

01

S3

00

S0 S02 S01 S00 D D1 D2 V V1 V0 V2 L L1 L3 L0 L2 N N2 I I1 I0 I2 S2

1

S2

12

S2

11

S3

1

S2

10

S3

12

S3

11

S3

10

S1 S12 S11 S10 E E0 E1 E2

Complexity of SKEP(B): NL-complete P-complete coNP-complete coNP-complete Πp

2-complete

The Complexity of Reasoning for Fragments of Default Logic 33

Summary

Defaults Stages Generators Post Existence Credulous Skeptical R´ esum´ e

Complexity for ...

◮ EXT(B) is a trichotomy

(Σp

2-, NP-complete, and trivial cases) ◮ CRED(B) is a pentachotomy

(Σp

2-, coNP-, NP-, P- and, NL-complete cases) ◮ SKEP(B) is a tetrachotomy

(Πp

2-, coNP-, P- and, NL-complete cases)

The complexity is determined by two parameters:

◮ whether there exist unique extensions and ◮ how hard it is to test for formula implication.

The Complexity of Reasoning for Fragments of Default Logic 34