1. 20081COMMA08 – Toulouse, May 2008 The Computational Complexity of Ideal Semantics I Abstract Argumentation Frameworks Paul E. Dunne Dept. Of Computer Science Univ. Of Liverpool ped@csc.liv.ac.uk

2. 20082COMMA08 – 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. 20083COMMA08 – 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.  A read as “x attacks y”. “Collection of justifiable arguments” = “Subset, S of X which is internally consistent” AND (some property P)

4. 20084COMMA08 – 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 range S  (arguments S attacks)

5. 20085COMMA08 – 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 one E -extension of. x in X is sceptically accepted w.r.t. E if in every E -extension of.

6. 20086COMMA08 – 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. S is an ideal extension if it is a maximal such set. Every AF,, has at least one ideal set and a unique ideal extension.

7. 20087COMMA08 – 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 have any E -extension? Does have any non-empty E - extension?

8. 20088COMMA08 – 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. 20089COMMA08 – 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/D p –complete (stable). Existence: NP-complete (stable); trivial (pref, adm, semi-stable); Non-empty: NP-complete (adm,pref,stable, semi-stable)

10. 200810COMMA08 – 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: P NP[||] Lower bound: P NP[||] –hard (“probably”) Credulous=Sceptical in ideal semantics.

11. 200811COMMA08 – Toulouse, May 2008 Meaning? P NP : 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. P NP is the class of problems we can solve in polynomial time using such an oracle (each call taking a single step).

12. 200812COMMA08 – Toulouse, May 2008 Adaptive and non-adaptive oracles P NP 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) P NP[||] 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. 200813COMMA08 – Toulouse, May 2008 Relationship to other classes Standard assumptions/conjectures: “adaptive queries” are more powerful than non-adaptive, i.e. P NP[||]  P NP 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. 200814COMMA08 – 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 F unsatisfiable: x is never in the ideal extension; F satisfiable: has x in the ideal extension with probability >1-exp(-|X|),

15. 200815COMMA08 – 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. 200816COMMA08 – Toulouse, May 2008 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 P NP[||] hardness proof and allow the success probability of the (composite) reduction to be amplified from 1/(4n 2 ) up to 1-exp(-n).

17. 200817COMMA08 – 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 no attacker is CA wrt PE”. The P NP[||] bounds follow from an algorithm to construct the ideal extension: its complexity being FP NP[||] the function class arising from P NP[||]

18. 200818COMMA08 – Toulouse, May 2008 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 – X OUT arguments that are not CA wrt PE; X PSA the arguments attacking and attacked by those in X OUT (but not themselves in X OUT ); X CA other args. 3.Find the maximal admissible subset of X PSA in the bipartite graph (X PSA ; X OUT ). 4.This forms the Ideal extension of H(X,A).

19. 200819COMMA08 – 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. 200820COMMA08 – 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)