DECOMPOSING ABSTRACT DIALECTICAL FRAMEWORKS

Artificial Intelligence, Computational Logic

DECOMPOSING ABSTRACT DIALECTICAL FRAMEWORKS

Sarah Gaggl and Hannes Strass

Pitlochry, 12th September 2014

Motivation

•

Computational complexityof semantics for ADFs is in general higher than for AFs [Strass and Wallner, 2014].

•

Algorithms based onSCC-recursive schemafor AF semantics show significant performance gain [Cerutti et.al. KR 2014].

•

We propose a similar approach based on a recursive decomposition along SCCs.

•

Allows to definecf2andstage2semantics for ADFs.

Motivation

•

Computational complexityof semantics for ADFs is in general higher than for AFs [Strass and Wallner, 2014].

•

Algorithms based onSCC-recursive schemafor AF semantics show significant performance gain [Cerutti et.al. KR 2014].

•

We propose a similar approach based on a recursive decomposition along SCCs.

•

Allows to definecf2andstage2semantics for ADFs.

Main Difference to AFs

1 Acceptance conditions of statements insub-frameworksmay still depend on statements not contained in sub-framework.

2 Elimination ofredundanciesfrom links and acceptance formulas.

3 Propagationof truth values to subsequent SCCs.

Agenda

1 Introduction and Background

– Abstract Dialectical Framework (ADFs)

2 Decomposing ADFs – Sub-Frameworks – Redundancies – Reduced Frameworks

– Decomposition-based Semantics

3 Conclusion and Future Work

ADFs - The Formal Framework

•

Like AFs, use graph to describe dependencies among nodes.

•

Unlike AFs, allow individual acceptance condition for each node.

•

Assignst(rue) orf(alse) depending on status of parents.

Definition

Anabstract dialectical framework(ADF) is a tupleD= (S,L,C)where

•

Sis a set ofstatements(positions, nodes),

•

L⊆S×Sis a set oflinks,

•

C={Cs}s∈Sis a set of total functionsCs:2^par(s)→ {t,f}, one for each statements.C_sis calledacceptance conditionofs.

Semantics

Definition

Letϕbe a propositional formula over vocabularySand for anM⊆Slet v:M→ {t,f,u}be athree-valued interpretation.

Thepartial valuationofϕbyvisϕ^v=ϕ[p/t:v(p) =t][p/f:v(p) =f].

Definition

LetD= (S,L,C)be an ADF. A three-valued interpretationvis

•

conflict-freeiff for alls∈Swe have:

– v(s) =timplies thatϕ^v_sis satisfiable, – v(s) =fimplies thatϕ^v_sis unsatisfiable;

•

naiveiff it is≤i-maximal with respect to being conflict-free;

Where≤_iis apartial orderover the truth values (resp. interpretations), i.e.

u<itandu<if.

Semantics ctd.

Definition

LetD= (S,L,C)be an ADF. Thepartial valuationofϕbyvis ϕ^v=ϕ[p/t:v(p) =t][p/f:v(p) =f].

A three-valued interpretationvis

•

conflict-freeiff for alls∈Swe have:

– v(s) =timplies thatϕ^v_sis satisfiable, – v(s) =fimplies thatϕ^v_sis unsatisfiable;

•

naiveiff it is≤_i-maximal with respect to being conflict-free;

Example

a

¬c

b

¬a

c

¬b

v={a7→f,b7→u,c7→t}is conflict-free, asϕ^v_a=¬tis unsatisfiable and ϕ^v_c=¬bis satisfiable.

Sub-Frameworks and Redundancies

a

¬c

b

¬a

c

¬b

d

c∨f

e

d∧f

f

e

•

independent setindD(∅) ={a,b,c}=M₀

•

independent moduloM₀:ind_D(M0) ={a,b,c,d,e,f}=S

•

Mindependent set:sub-frameworkD|_M = (M,L∩(M×M),{ϕs}s∈M)

Sub-Frameworks and Redundancies

a

¬c

b

¬a

c

¬b

d

c∨f

e

d∧f

f

e

•

independent setindD(∅) ={a,b,c}=M₀

•

independent moduloM₀:ind_D(M0) ={a,b,c,d,e,f}=S

•

Mindependent set:sub-frameworkD|_M = (M,L∩(M×M),{ϕs}s∈M)

•

Redundanciescan change dependencies between statements.

•

If(r,s)is redundant thenrhas no influence on the truth value ofϕs

whatsoever.

Example

Considerϕs=a∨(b∧c)and the interpretationv={a7→u,b7→f,c7→u}.

ϕ^v_s=a∨(f∧c)

Sub-Frameworks and Redundancies

a

¬c

b

¬a

c

¬b

d

c∨f

e

d∧f

f

e

•

independent setindD(∅) ={a,b,c}=M₀

•

independent moduloM₀:ind_D(M0) ={a,b,c,d,e,f}=S

•

Mindependent set:sub-frameworkD|_M = (M,L∩(M×M),{ϕs}s∈M)

•

Redundanciescan change dependencies between statements.

•

If(r,s)is redundant thenrhas no influence on the truth value ofϕs

whatsoever.

Example

Considerϕs=a∨(b∧c)and the interpretationv={a7→u,b7→f,c7→u}.

ϕ^v_s=a∨(f∧c)≡a chas no influence

Reduced ADF

Given an ADFD= (S,L,C), an independent setM⊆Sand an interpretation v:M→ {t,f,u}. The ADFDreduced withvonMis obtained by:

•

adapt the acceptance condition of statementsto – t(resp.f) ifv(s) =t(resp.v(s) =f) – ¬sifv(s) =u

– partial valuationϕ^v_sfor remaining statements and ifris redundant in ϕ^v_s, replacerwitht

•

remove redundant links

•

add links{(s,s)|v(s) =u}

Reduced ADF

Given an ADFD= (S,L,C), an independent setM⊆Sand an interpretation v:M→ {t,f,u}. The ADFDreduced withvonMis obtained by:

•

adapt the acceptance condition of statementsto – t(resp.f) ifv(s) =t(resp.v(s) =f) – ¬sifv(s) =u

– partial valuationϕ^v_sfor remaining statements and ifris redundant in ϕ^v_s, replacerwitht

•

remove redundant links

•

add links{(s,s)|v(s) =u}

Procedure

For a semanticsσand an ADFD, we obtain theσ2interpretations recursively by applyingσ₂(D) =σ₂(indD(∅),D)by:

1 Start with all statementsindependent modulo∅, i.e.M₀=ind_D(∅)

2 Compute allσ-interpretations of sub-frameworkD|_M

0 3 For eachσ-interpretationwofD|_M

0compute thereduced ADF

4 Call Step 1 with reduced ADF andM₁=ind_D(M0).

Example

a

¬c

b

¬a

c

¬b

d

c∨f

e

d∧f

f

e nai₂(D) =nai₂(indD(∅),D)

1 ind_D(∅) ={a,b,c}=M₀

Example

a

¬c

b

¬a

c

¬b

d

c∨f

e

d∧f

f

e nai₂(D) =nai₂(indD(∅),D)

1 ind_D(∅) ={a,b,c}=M₀

2 Then we obtainnai(D|_M

0) ={v₀,v1,v₂}:

v0={a7→u,b7→t,c7→f}, v₁={a7→f,b7→u,c7→t}, v₂={a7→t,b7→f,c7→u}.

Example

a

¬c

b

¬a

c

¬b

d

c∨f

e

d∧f

f

e nai₂(D) =nai₂(indD(∅),D)

1 ind_D(∅) ={a,b,c}=M₀

2 v₁={a7→f,b7→u,c7→t}

Example

a

f

b

¬b

c

t

d

t∨f

e

d∧f

f

e nai₂(D) =nai₂(indD(∅),D)

1 indD(∅) ={a,b,c}=M₀

2 v₁={a7→f,b7→u,c7→t}

3 Reduced ADF witht∨f ≡t, thus link(f,d)is redundant

Example

a

f

b

¬b

c

t

d

t∨t

e

d∧f

f

e nai₂(D) =nai₂(indD(∅),D)

1 indD(∅) ={a,b,c}=M₀

2 v₁={a7→f,b7→u,c7→t}

3 Reduced ADF

Example

a

f

b

¬b

c

t

d

t∨t

e

d∧f

f

e nai₂(D) =nai₂(indD(∅),D)

1 indD(∅) ={a,b,c}=M₀

2 v₁={a7→f,b7→u,c7→t}

3 Reduced ADF

4 Call Step 1 with reduced ADFD₁andM₁=indD₁(M₀).

Example

a

f

b

¬b

c

t

d

t∨t

e

d∧f

f

e nai₂(M1,D₁)

1 M₁=indD₁(M₀) ={a,b,c,d}

Example

a

f

b

¬b

c

t

d

t∨t

e

d∧f

f

e nai₂(M1,D₁)

1 M₁=indD₁(M₀) ={a,b,c,d}

2 nai(D|_M

1) =v₃=v₁∪ {d7→t}

Example

a

f

b

¬b

c

t

d

t

e

t∧f

f

e nai₂(M1,D₁)

1 M₁=indD₁(M₀) ={a,b,c,d}

2 nai(D|_M

1) =v₃=v₁∪ {d7→t}

3 Reduced ADF

Example

a

f

b

¬b

c

t

d

t

e

t∧f

f

e nai₂(M1,D₁)

1 M₁=indD₁(M₀) ={a,b,c,d}

2 nai(D|_M

1) =v₃=v₁∪ {d7→t}

3 Reduced ADF

4 Call Step 1 with reduced ADFD₂andM₂=ind_D2(M1)

Example

a

f

b

¬b

c

t

d

t

e

t∧f

f

e nai₂(M2,D₂)

1 M₂=indD₂(M₁) ={a,b,c,d,e,f}=S

Example

a

f

b

¬b

c

t

d

t

e

t∧f

f

e nai₂(M2,D₂)

1 M₂=indD₂(M₁) ={a,b,c,d,e,f}=S

2 nai(D2) ={v₄,v6}:

v₄=v₃∪ {e7→t,f 7→t}, v₆=v₃∪ {e7→f,f7→f}.

Main Theorem

Theorem

1. Letσ∈ {cfi,adm,pre,com,mod}. Thenσ≤σ₂. 2. Letσ∈ {nai,stg}. Thenσ6≤σ2. 3. Letσ∈ {cfi,nai,adm,pre,com,mod}. Thenσ2≤σ.

4. Letσ∈ {stg}. Thenσ26≤σ.

Conclusion

•

We proposed a decomposition schema for semantics for ADFs.

•

We introducednai₂andstg₂for ADFs.

•

Due to the relation of ADFs to logic programms we also getnai₂andstg₂ semantics for LPs.

•

Also in the paper:composingADFs.

Future Work

•

Analysis of the complexity of the approach.

•

Implementation.

•

Studysplittingsof ADFs andequivalences.

References

Leila Amgoud and Claudette Cayrol and Marie-Christine Lagasquie and Pierre Livet, On Bipolarity in Argumentation Frameworks

International Journal of Intelligent Systems 23(10): 1062–1093 (2008)

P. Baroni, F. Cerutti, M. Giacomin and G. Guida.

AFRA: Argumentation Framework with Recursive Attacks.

Int. J. Approx. Reasoning 52(1): 19–37 (2011).

Baroni, P., Giacomin, M., and Guida, G. (2005).

Scc-recursiveness: A general schema for argumentation semantics.

Artif. Intell., 168(1-2):162–210.

G. Brewka and S. Woltran.

Abstract Dialectical Frameworks.

KR 2010, pp. 102–111, AAAI Press, 2010.

Dung, P. M. (1995).

On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games.

Artif. Intell., 77(2):321–358.

S. Modgil.

Reasoning about Preferences in Argumentation Frameworks.

Artif. Intell. 173(9-10): 910–934 (2009).

H. Strass and J. Wallner.

Analyzing the Computational Complexity of Abstract Dialectical Frameworks via Approximation Fixpoint Theory.

KR 2014, AAAI Press, 2014.