The Computational Complexity of Ideal Semantics I Abstract Argumentation Frameworks

1. Paul E. Dunne Dept. Of Computer Science Univ. Of Liverpool ped@csc.liv.ac.uk The Computational Complexityof Ideal Semantics IAbstract Argumentation Frameworks COMMA08 – Toulouse, May 2008

2. COMMA08 – Toulouse, May 2008 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.

3. COMMA08 – Toulouse, May 2008 Abstract Argument Frameworks • H(X,A) – X finite set of arguments; A set of ordered pairs of arguments (AX×X) called the set of attacks. • <x,y>A read as “xattacksy”. • “Collection of justifiable arguments” = “Subset, S of X which is internally consistent” AND (some property P)

4. COMMA08 – Toulouse, May 2008 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 rangeS  (arguments S attacks)

5. COMMA08 – Toulouse, May 2008 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 oneE-extension of <X,A>. • x in X is sceptically accepted w.r.t. E if in everyE-extension of <X,A>.

6. COMMA08 – Toulouse, May 2008 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.

7. COMMA08 – Toulouse, May 2008 Computational Problems in AFs • Given an argumentation semantics, E: • Does SX satisfy E’s constraints? • Is xX credulously accepted w.r.t. E? • Is xX sceptically accepted w.r.t. E? • Does <X,A> have any E-extension? • Does <X,A> have any non-emptyE-extension?

8. COMMA08 – Toulouse, May 2008 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.

9. COMMA08 – Toulouse, May 2008 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)

10. COMMA08 – Toulouse, May 2008 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.

11. COMMA08 – Toulouse, May 2008 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).

12. COMMA08 – Toulouse, May 2008 Adaptive and non-adaptive oracles • PNPallows 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)

13. COMMA08 – Toulouse, May 2008 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)

14. COMMA08 – Toulouse, May 2008 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|),

15. COMMA08 – Toulouse, May 2008 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.

16. COMMA08 – Toulouse, May 2008 Features • The Valiant-Vazirani reduction has a low success probability - 1/(4n) BUT • CAwrt 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).

17. COMMA08 – Toulouse, May 2008 Upper Bound Proofs • The coNP bound for verifying S is an ideal set uses a characterisation of these as “admissible sets of which noattacker isCAwrtPE”. • The PNP[||] bounds follow from an algorithm to construct the ideal extension: its complexity being FPNP[||] the function class arising from PNP[||]

18. COMMA08 – Toulouse, May 2008 Finding the Ideal Extension of H(X,A) • Use |X| queries (in parallel) to decide which arguments of X are not CA wrt PE. • Partition X into 3 sets – XOUTarguments that are notCA wrt PE; XPSAthe arguments attacking and attacked by those in XOUT(but not themselves in XOUT); XCAother args. • Find the maximal admissible subset of XPSAin the bipartite graph (XPSA; XOUT). • This forms the Ideal extension of H(X,A).

19. COMMA08 – Toulouse, May 2008 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.

20. COMMA08 – Toulouse, May 2008 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)