20081comma08 – toulouse, may 2008 the computational complexity of ideal semantics i abstract...

2008 1COMMA08 – Toulouse, May 2008

The Computational Complexityof Ideal Semantics I

Abstract Argumentation Frameworks

Paul E. DunneDept. Of Computer Science

Univ. Of [email protected]


Overview

• Argumentation Frameworks (brief review).• Collections of “justified arguments” –

extension based semantics.• Ideal sets and extensions.• Established complexity properties in

extension-based argumentation semantics.• Decision and construction problems for Ideal

semantics and their complexity.• Conclusions and Open Issues.


Abstract Argument Frameworks

• H(X,A) – X finite set of arguments; A set of ordered pairs of arguments (AX×X) called the set of attacks.

• <x,y>A read as “x attacks y”.

• “Collection of justifiable arguments” = “Subset, S of X which is internally consistent” AND (some property P)


Property P = Extension semantics

• “Internally consistent” = “conflict-free” – no argument in S attacks any other in S.

• Additional (choices for property P)• Admissible – S attacks all its attackers.• Preferred – S is maximal admissible set.• Stable – S attacks X-S.• Semi-stable – S is admissible and has

maximal range S (arguments S attacks)


Credulous vs. Sceptical

• Let E be one of preferred, stable, semi-stable.

• x in X is credulously accepted w.r.t. E if in at least one E-extension of <X,A>.

• x in X is sceptically accepted w.r.t. E if in every E-extension of <X,A>.


Ideal Sets and Extensions

• S is an ideal set if it is both admissible and a subset of every preferred extension of <X,A>.

• S is an ideal extension if it is a maximal such set.

• Every AF, <X,A>, has at least one ideal set and a unique ideal extension.


Computational Problems in AFs

• Given an argumentation semantics, E:

• Does SX satisfy E’s constraints?

• Is xX credulously accepted w.r.t. E?

• Is xX sceptically accepted w.r.t. E?

• Does <X,A> have any E-extension?

• Does <X,A> have any non-empty E-extension?


Previous work on Computational Complexity in AFs

• Properties of admissible sets, preferred and stable extensions have been studied in work of Dung (1995); Dimopoulos & Torres (1996); Dunne & Bench-Capon (2002) for AFs.

• Dimopoulos, Nebel, and Toni (2002) presents detailed analyses of these for Assumption-based Argumentation Frameworks (ABFs).

• Recent work of Dunne & Caminada (2008) addresses semi-stable semantics.


Computational Complexity

• Verification: P (adm, stable); coNP-complete (pref, semi-stable).

• Credulous acceptance: NP-complete (pref, stable).

• Sceptical acceptance: 2 –complete (pref); coNP-complete/Dp –complete (stable).

• Existence: NP-complete (stable); trivial (pref, adm, semi-stable);

• Non-empty: NP-complete (adm,pref,stable, semi-stable)


Computational Complexity of Ideal Semantics

• Verification (is S an ideal set?) – coNP-complete (preferred & semi-stable).

• Verification (is S the ideal extension?); non-emptiness; credulous acceptance –

• Upper Bound: PNP[||]

• Lower bound: PNP[||] –hard (“probably”)

• Credulous=Sceptical in ideal semantics.


Meaning?• PNP : suppose we can obtain answers

about instances of some NP problem by asking an “oracle”, e.g. we can construct a propositional formula and ask if it is satisfiable.

• PNP is the class of problems we can solve in polynomial time using such an oracle (each call taking a single step).


Adaptive and non-adaptive oracles

• PNP allows the form of successive queries to depend on earlier answers, e.g. we could construct different formulae at the second call for each of the answers to the first. (Adaptive)

• PNP[||] requires the form of all queries to be fixed in advance. (non-adaptive)

• Non-adaptive queries can be made in a single parallel step (involving all the different call instances)


Relationship to other classes

• Standard assumptions/conjectures:• “adaptive queries” are more powerful than

non-adaptive, i.e. PNP[||] PNP

• Both are more powerful than NP, coNP

• Both are less powerful than 2 2 .

• In other words: CA (w.r.t Ideal) is (“probably”) harder than CA (w.r.t. Pref) but “definitely” easier than SA (w.r.t Pref)


Why “probably”?

• “standard” hardness proofs for F map instances of a (known) difficult problem to instances of F. Such mappings are deterministic and always succeed.

• The hardness proof for CA w.r.t Ideal semantics uses a randomized reduction: an instance of SAT, F, is mapped to a random <H,x>

• F unsatisfiable: x is never in the ideal extension; • F satisfiable: <H,x> has x in the ideal extension with

probability >1-exp(-|X|),


CA w.r.t. Ideal Semantics

• The randomized element of the proof is built into the Valiant-Vazirani transformation from CNF-SAT to unique satisfiability (USAT) (Given F does it have exactly one satisfying instantiation?).

• We then use a (standard, deterministic) reduction from USAT to CA wrt Ideal which gives an NP-hardness (via randomized reductions) lower bound.


Features

• The Valiant-Vazirani reduction has a low success probability - 1/(4n)

BUT

• CA wrt Ideal has a number of structural properties which are used for the PNP[||] hardness proof and allow the success probability of the (composite) reduction to be amplified from 1/(4n2) up to 1-exp(-n).


Upper Bound Proofs

• The coNP bound for verifying S is an ideal set uses a characterisation of these as “admissible sets of which no attacker is CA wrt PE”.

• The PNP[||] bounds follow from an algorithm to construct the ideal extension: its complexity being FPNP[||] the function class arising from PNP[||]


Finding the Ideal Extension of H(X,A)

1. Use |X| queries (in parallel) to decide which arguments of X are not CA wrt PE.

2. Partition X into 3 sets – XOUT arguments that are not CA wrt PE; XPSA the arguments attacking and attacked by those in XOUT (but not themselves in XOUT); XCA other args.

3. Find the maximal admissible subset of XPSA

in the bipartite graph (XPSA ; XOUT).4. This forms the Ideal extension of H(X,A).


Summary

• Constructing Ideal Extensions and verifying that S is an ideal set are easier than testing if an argument is sceptically accepted wrt PE.

• This is despite sceptical acceptance being a precondition for S to be ideal.

• The upper bound arguments rely on the fact that it is not necessary explicitly to test sceptical acceptance in order to verify S is an ideal set or to construct the ideal extension.


Open Problems

• Complexity of Ideal semantics in ABFs.• Direct (i.e. non-randomized) reductions for

CA wrt Ideal? NB it is “highly unlikely” that CA wrt Ideal has equivalent complexity to USAT.

• Conditions on AFs under which Ideal semantics becomes “more tractable”: known cases – bipartite, bounded treewidth (P); no change (planar, bounded attacks; tripartite)

20081comma08 – toulouse, may 2008 the computational complexity of ideal semantics i abstract...

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