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  1. An Introduction to Abstract Argumentation Dr. PierpaoloDondio, DIT – School of Computing

  2. Agenda • Introduction • What is argumentation theory? • Abstract Argumentation Frameworks (Dung 1995) • Stable, Grounded and Preferred Semantics • Instantiating Abstract Argumentation • Applications • Probabilistic and Uncertain Argumentation

  3. What is argumentation Theory • the interdisciplinary study of how conclusions can be reached through logical reasoning • Key Questions: • How arguments are built? • How humans negotiate, discuss, argue? • Who wins? i.e. how can we identify acceptable valid arguments and discard invalid? • We focus on AI developments • Computational Logic & Non-monotonic reasoning • Abstract Argumentation

  4. Nonmonotonic logic • Standard logic is monotonic: • If S |-  and S  S’ then S’ |-  • But commonsense reasoning is often nonmonotonic: • John is an adult, Adults are usually employed, so John is presumably employed • But suppose also that John is a student and students are usually not employed … • We often reason with rules that have exceptions • We apply the general rule if we have no evidence of exceptions • But must retract our conclusion if we learn evidence of an exception

  5. Source of non-monotonicity • Exceptions • Moral Rules • Legal Rules • (False) Generalizations • Limited knowledge • …

  6. Some nonmonotonic logics • Default logic (Ray Reiter) • Logic programming (Robert Kowalski) • … • Argumentation logics • Argumentation Logics, and in particular Abstract Argumentation Frameworks studied here, have the same expressive power as Default Logic

  7. Argumentations are nonmonotonic A Paul is a good at Maths Paul is not good at Maths B Paul got 90% in his final Maths test If you get high marks in a Maths test you are good at Maths Paul was never able to help me with my Maths homework Exam result is not a valid evidence Exam result is not a valid evidence What Mary said is not trustworthy D C E Exam was very easy this year Mary said Paul copied the exam Mary is a well-known layer

  8. Abstract Argumentation Frameworks A B E D C (Dung 1995) An argumentation framework is a pair where is a set of arguments, and is a binary relation on , i.e. . • For two arguments A,B, the meaning of is that A represents an attack against B.

  9. The Key Problem • I want to say something about arguments : • Which arguments are acceptable? Which are not? • When to abstain? • A argumentationsemantics sets the rules (postulates) used to answer the above questions • In the labelling approach, we label each argument • IN – Argument is accepted • OUT – Argument is rejected • UNDEC – Nothing can be said on argument status

  10. Starting Point • Given we define a labelling function as a total function over : • We also define the • Starting basic idea. We label all arguments according to these simple rules: • An argument in each labelling is either IN or OUT • An argument is ‘in’iff all arguments defeating it are ‘out’. • An argument is ‘out’iff it is defeated by an argument that is ‘in’.

  11. Example • A: Mark is good at Math, he got 90%! • B: John said Mark copied the test! • C: John is a well-known layer! • Our rule works fine. We expect • A in • B out • C in • This is called reinstatement; A is reinstated by C. A B C

  12. Example 2 • D: Sarah says that John is honest! • And now? The graph is cyclic! Our basic rule does not work anymore! • Multiple solutions: A B C D A B C D A B C D D A B C

  13. Complete Semantics • Grounded (Pollock, Dung) • Preferred (Dung) • Stable (Dung, Caminada) • Many more.. Semi-stable, CF2..

  14. Complete Semantics: Conflict-free A set S of arguments is said to be conflict-free if there are noarguments A,B inS such that A attacks B . B S A Arg (A  S & attack(A,B)) = > B S

  15. Complete Semantics - Admissibility An argument A is admissible with respect to a set Sif S can defend A with an argument B  S against all attacks C on A. We want to accept arguments for which there is an admissibility set A S C B (A Arg & attacks(C,A))

  16. Complete Semantics • It accepts all the conflict-tree and admissible arguments • In general multiplelabelings are valid • It can be proven that the following labelling rules exactly compute the complete semantics • if A is labelled in then all attackers of A are labelled out • if all attackers of A are labelled out then A is labelled in • if A is labelled out then A has a attacker that is labelled in, and • if A has a attacker that is labelled in then A is labelled out • A is labelled undeciff at least one attacker is undec and thre is no attacker labelled in

  17. Complete Semantics • Many sub-semantics have been defined over a complete labeling. Let Grounded • L is a complete labellings such as undec(L) is maximal w.r.t. to set inclusion Preferred • L is a complete labellings such as in(L) is maximalw.r.t. to set inclusion Stable • L is a complete labelling such that undec(L) = ∅

  18. Hierarchies of Semantics (Caminada)

  19. Example • Grounded: all undecided • Stable: IN={b,d} ; OUT={a,c,e} • Preferred: • IN={b,d} ; OUT={a,c,e} • IN={a} ; OUT={b} ; UNDEC={c,d,e}

  20. The example of floating assignment • Grounded: all undecided • Preferred: • IN={b} ; OUT={c,a} • IN={c} ; OUT={b,a} • Stable: same as preferred A C B

  21. The Nixon Diamonds & 3-Cycle • Grounded: undec • Preferred: • IN={a}, OUT={b} • IN={b}, OUT={a} • Stable: same as preferred • Grounded: undec • Preferred: undec • Stable: none C B A C B

  22. Credulous vs ScepticalAcceptability • After each labelling, we are left with three set of arguments In general, there are multiple labelings (one for grounded, maybe many for preferred or stable) How can I accept arguments? • Credulous acceptance. • If there is at least one labelling where argument A is labeled IN, accept it • Sceptical acceptance • An argument must have the same labels in all the labelings • Grounded acceptance implies sceptical preferred or stable acceptance

  23. Instanciating an AA • Nothing is said about argument internal arguments structure • Arguments as modus-ponens rules • How can these rule be attacked? • Rebuttals: attack • Undercutting: attack • Undermining : attack • The red light example • If an object looks red, it is red • What if the object is illuminated by red light?

  24. Arguments as logical consequences • propositional language whose atoms is a finite set and its connectives are . • The symbol means logical consequence. • for and a formula in , is an argumentiff and it does not exits so that • Given argument , we call the support of and the claim of . • Given two arguments and , we define rebuttal and undercut attacks in the following way: • rebuts if • undercuts if there is such that

  25. Some Applications • Legal Reasoning (Prakken) • Computational Trust • Dondio (2007 – 2013, Phd) • Multi-Agents Conflict Resolution • Decision Support Systems • Healtchare (Longo 2012) • … many more

  26. Trust as a form of Argumentation • if agent has high level of past performance, then trust it • if agent is similar to Carol, then trust him • if agent has low reputation, then distrust him • if task context is new to the agent, then invalidate argument PP • if past performance are high an reputation is low, then prefer past performance • if past-performance are out of date, then invalidate argument PP PP+ + - + Sim+ R- = = A2 = A1 A3

  27. Problems with Abstract Argumentation • Nothing is said about arguments. There is the danger to model impossible situations or derive useless conclusions • Too coarse! • Many times, you are left with no arguments or multiple labelings and nothing to choose about • Arguments are perceived with difference strength, importance, maybe based on their likelihood or certainty level or subjective preferences.. • How can we build a numerical argumentation?

  28. Towards Argumentation with Strengths • First attempt: Pollock • Arguments are modus ponens rules • Two premises: fact and assumption • Strength are numbers in [0,1] • Strength of a conclusion – Weakest Link: • If C attacks B, B strength is • Multiple attacks. No accrual, chose the max only • What is this strength? Ad-hoc? • Rejection of probability, is this justified? A D C B

  29. Weighted Argumentation • Abstract Argumentation with weights • Inconsistency Tolerance (Dunne 2009) • Baroni / Toni 2013 proposal (ordinal functions for attack and support), Pollock-like • Same criticism: what’s the meaning of the numbers used to quantify argument importance? • Social Argumentation • Weights (importance) attached to each arguments come form a voting systems (online forums etc..)

  30. Probabilistic Argumentation Li (2011), Hunter (2012), Dondio (2012) • Real world arguments are clearly affected by uncertainty. • Probabilistic uncertainty is well studied (probability calculus) and clear understood (maybe….) • Allow arguments to be probabilistic in nature. Source of probability: • Randomness, stochastic processes • Statistical information • Subjective Beliefs • Example: If you have fever, 80% is flu

  31. Probabilistic Argumentation • A PAF is a tuple (AF,P) where • AF = (Ar,R) is an abstract argumentation framework and • is a joint probability over arguments. If statistical independence holds, is a scalar function: 0.7 0.4 0.5

  32. How to compute a PAF? • We need to find the probability that an argument is labelled IN, OUT, UNDEC • Probabilistic arguments implies multiple scenarios () each obtained by assuming that each argument claim hold or not. • Each scenario has its own probability (computed using P) • Each scenario corresponds to a sub-graph of the argumentation framework

  33. Notation for group of sub-graphs • = 2 sub-graphs, all the sub-graphs containing and not • = 5 sub-graphs not containing or all the sub-graphs containing togheter • = 1 sub-graph,

  34. Computing PAF • We can label arguments in each sub-graph, assigning the OUT label if the argument is not present in the sub-graph (defeated by its own)! • We then group all the sub-graphs where a generic argument is labelled • We call these sets • We compute the probabilities of these sets summing up the probabilities of all the sub-graphs in each set.

  35. Brute Force Brute force approach for Given Chose argument in foreach sub-graph of assign a label to in using the chosen semantics if add to if add to if add to are computed using and they are the probabilities we were looking for

  36. 2-Layer Approach • is not • is the probability that argument a claim holds in isolation • is the probability that the argument claims holds after the argumentation process

  37. Example / Grounded Case • Not a very clever approach (w.r.t. computation) • We can skip many scenarios and reduce the problem space

  38. Notation for Grounded labelling • , , ,is the set of all the sub-graphs of • Under grounded semantic there is one unique labelling for each sub-graph called . Sets of interests: (2 sub-graphs) (5 sub-graphs) (1 subgraph)

  39. Preferred Case • There are multiple labelings for each sub-graph • An argument can be IN and OUT in the same scenario (but in different labelings) • Solution 1: give a probability for credulous acceptance (possibility) and 1 for sceptical acceptance (necessity) • Solution 2: use principle of indifference, splitting the probability of a single sub-graph among all the valid preferred labeling

  40. Preferred Sets • = set of labelings for a scenario s • Preferred credulous sets: • While the skeptical sets are:

  41. Preferred Case • (note that also in ) • (6 scenarios) • Necessity is = 0.25 • Possibility is = 0.375 • Indifferent probability = 0.3125

  42. Brute force is not efficient • w.r.t. computational time • w.r.t. the length of the expression of • 56 sub-graphs in • It can be reduced to 3 clauses only • Wasted of computational time. Some cases do not need to be comptued / recomputed • Idea: assign sets of sub-graphs at each computational step • Recursive and decision-tree-like argument (Dondio 2013)

  43. A Recursive Algorithm (grounded) • It decomposes the grounded semantics computation. It applies the rules of a complete labelling + maximise undec. • The algorithm. Visit the transpose graph from root and imposes the following two rules: • Terminal conditions • if is required to be in then • if node is required to be out then • Cycles • If a cycle is detected, end the recursive step and return • Some optimizations

  44. Recursive Algorithm / Example

  45. Some Optimizations • Generate non-overlapping solutions can be rewritten as disjoint sets in the form , condition 2 is rewritten as: • Optimizing condition 1: returning empty set 3. Exploiting Rebuttals it is instead of when rebutts 4. Re-using computations if

  46. Recursive Algorithm Example /2

  47. ADT – Arguments Decision Tree • Decision Tree-like Algorithm. Select an argument, split, analyse the two spit sub-graphs • Which is the criteria for selecting the splitting argument? • Dialectical Strength. the dialectical strength of an argument w.r.t. , called , is defined as follows: • If is initial, is the number of arguments that are defeated by plus the arguments that result disconnected from once the arguments defeated by are removed from . • Note that, if directly attacks , then . • If x is not initial, is the number of arguments that are disconnected from after is removed.

  48. ADT /2 - Example

  49. Applications / Legal Reasoning Paul and John are under trial for the assassination of Sam. Evidence collected: • John was alone in the room between 1 to 3; the medical test says that Sam died between 1 and 3 [0.6]→ John shoot Sam • Paul was alone in the room between. 3 and 5; the medical test says that Sam died between 3 and 5 [0.4]→ Paul shoot Sam • The medical test is void [0.1] → nothing can be said on Sam’s time of death • We also , since Sam either died btw. 1 and 3 or btw. 3 and 5. • The fingerprints are Paul’s [0.7]→ Paul shot Sam btw 3 and 5 • The weapon was tampered and the test is void [0.5] → fingerprints are not a valid evidence • A witness heard a shot at 2pm [0.8], John was in the room at 2 → John shot Sam and Sam died between 1 and 3 • The number in square brackets is the probability of each premise (assume 1 if no number is specified)

  50. Argumentation Graph for the Legal Case • John is guilty when argument is in or is in. Therefore: How is changing if changes? Good points of PAF • Solid axioms • Much richer computation • Maybe useful for prob. reasoning?