The Computational Complexity of Ideal Semantics I Abstract Argumentation Frameworks

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The Computational Complexityof Ideal Semantics
IAbstract Argumentation Frameworks

• Paul E. Dunne
• Dept. Of Computer Science
• Univ. Of Liverpool
• ped_at_csc.liv.ac.uk

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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.

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Abstract Argument Frameworks

• H(X,A) X finite set of arguments A set of
  ordered pairs of arguments (A?XX) called the set
  of attacks.
• ltx,ygt?A read as x attacks y.
• Collection of justifiable arguments Subset,
  S of X which is internally consistent AND (some
  property P)

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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)

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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 ltX,Agt.
• x in X is sceptically accepted w.r.t. E if in
  every E-extension of ltX,Agt.

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Ideal Sets and Extensions

• S is an ideal set if it is both admissible and a
  subset of every preferred extension of ltX,Agt.
• S is an ideal extension if it is a maximal such
  set.
• Every AF, ltX,Agt, has at least one ideal set and a
  unique ideal extension.

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Computational Problems in AFs

• Given an argumentation semantics, E
• Does S?X satisfy Es constraints?
• Is x?X credulously accepted w.r.t. E?
• Is x?X sceptically accepted w.r.t. E?
• Does ltX,Agt have any E-extension?
• Does ltX,Agt have any non-empty E-extension?

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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.

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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)

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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)
• CredulousSceptical in ideal semantics.

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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).

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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)

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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)

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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 ltH,xgt
• F unsatisfiable x is never in the ideal
  extension
• F satisfiable ltH,xgt has x in the ideal extension
  with probability gt1-exp(-X),

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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.

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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).

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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

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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).

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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.

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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)