Efficiently Estimating the Probability of Extensions in ... - CiteSeerX

Introduction Background Estimating Pr sem Experiments Conclusions and future work

Efficiently Estimating the Probability of Extensions in Abstract Argumentation Bettina Fazzinga, Sergio Flesca, Francesco Parisi DIMES Department University of Calabria Italy

SUM 2013 September 15-18, 2013 Washington DC Area, USA

Bettina Fazzinga, Sergio Flesca, Francesco Parisi

Efficiently Estimating the Probability of Extensions in AA

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Motivation Contribution

Argumentation in AI A general way for representing arguments and relationships (rebuttals) between them It allows representing dialogues, making decisions, and handling inconsistency and uncertainty Abstract Argumentation Framework (AAF) [Dung 1995]: arguments are abstract entities (no attention is paid to their internal structure) that may attack and/or be attacked by other arguments Example (a simple AAF) a= b= c=

Our friends will have great fun at our party on Saturday Saturday will rain (according to the weather forecasting service 1) Saturday will be sunny (according to the weather forecasting service 2) Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Argumentation in AI A general way for representing arguments and relationships (rebuttals) between them It allows representing dialogues, making decisions, and handling inconsistency and uncertainty Abstract Argumentation Framework (AAF) [Dung 1995]: arguments are abstract entities (no attention is paid to their internal structure) that may attack and/or be attacked by other arguments Example (a simple AAF) a= b= c=

Our friends will have great fun at our party on Saturday Saturday will rain (according to the weather forecasting service 1) Saturday will be sunny (according to the weather forecasting service 2) Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Probabilistic Abstract Argumentation Framework Arguments and attacks can be uncertain Example (modelling uncertainty in our simple AAF)

a

there is some uncertainty about the fact that our friends will have fun at the party about the truthfulness of the weather forecasting services about the fact that the bad weather forecast actually entails that the party will be disliked by our friends

90%

90%

b

70%

c

20%

In a Probabilistic Argumentation Framework (PrAF) [Li et Al. 2011] both arguments and defeats are associated with probabilities Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Semantics for Abstract Argumentations In the deterministic setting, several semantics (such as admissible, stable, complete, grounded, preferred, and ideal) have been proposed to identify “reasonable” sets of arguments Example (AAF)

a

For instance, {a, c} is admissible

b

c

These semantics do make sense in the probabilistic setting too: what is the probability that a set S of arguments is reasonable? (according to given semantics) Example (PrAF) the probability that {a, c} is admissible is 0.18 Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Semantics for Abstract Argumentations In the deterministic setting, several semantics (such as admissible, stable, complete, grounded, preferred, and ideal) have been proposed to identify “reasonable” sets of arguments Example (AAF)

a

For instance, {a, c} is admissible

b

c

These semantics do make sense in the probabilistic setting too: what is the probability that a set S of arguments is reasonable? (according to given semantics) Example (PrAF) the probability that {a, c} is admissible is 0.18 Bettina Fazzinga, Sergio Flesca, Francesco Parisi

90%

a

90%

70%

b


20%

c 4 / 24



Complexity of Probabilistic Abstract Argumentation P ROBsem (S) is the problem of computing the probability Pr sem (S) that a set S of arguments is reasonable according to semantics sem P ROBsem (S) is the probabilistic counterpart of the problem V ERsem (S) of verifying whether a set S is reasonable according to semantics sem admissible stable complete grounded preferred ideal

V ERsem (S) PTIME PTIME PTIME PTIME coNP-complete coNP-complete


P ROBsem (S) PTIME PTIME #P FP -complete FP #P -complete FP #P -complete FP #P -complete

both tractable

from tractability to intractability

both intractable


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Complexity of Probabilistic Abstract Argumentation P ROBsem (S) is the problem of computing the probability Pr sem (S) that a set S of arguments is reasonable according to semantics sem P ROBsem (S) is the probabilistic counterpart of the problem V ERsem (S) of verifying whether a set S is reasonable according to semantics sem admissible stable complete grounded preferred ideal

V ERsem (S) PTIME PTIME PTIME PTIME coNP-complete coNP-complete


P ROBsem (S) PTIME PTIME #P FP -complete FP #P -complete FP #P -complete FP #P -complete

both tractable

from tractability to intractability

both intractable


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Estimating the Probability of Extensions in Abstract Argumentation In [Li et Al. 2011] a Monte-Carlo-based simulation technique for estimating the probability P ROBsem (S), where sem is complete, grounded, preferred, is proposed. This method does not exploit the possibility of computing P ROB CF (S) and P ROB AD (S) in polynomial time. We propose a new method for estimating P ROBsem (S) which: 1 2 3

computes P ROB CF (S) (resp. P ROB AD (S)), sem|CF sem|AD computes an estimate of PrF (S) (resp., PrF (S)) sem|CF sem|AD CF returns PrF (S) × Pr (S) (resp., PrF (S) × Pr AD (S)) as an estimate of P ROBsem (S)

This method allows us to reduce the number of generated samples for obtaining the same level of accuracy compared to the one proposed in [Li et Al. 2011]. Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Abstract Argumentation Framework Probabilistic Argumentation Framework

Outline 1

Introduction Motivation Contribution

2

Background Abstract Argumentation Framework Probabilistic Argumentation Framework

3

Estimating Pr sem The state of the art approach Estimating PrFsem (S) by sampling AAFs wherein S is conflict-free Estimating PrFsem (S) by sampling AAFs wherein S is admissible

4

Experiments

5

Conclusions and future work Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Basic concepts of Abstract Argumentation An abstract argumentation framework consists of a set A of arguments, and a relation D ⊆ A × A, whose elements are defeats (or attacks) Example (AAF) A = {a, b, c} D = {hb, ai, hb, ci, hc, bi}

a

b

c

A set S ⊆ A of arguments is conflict-free if there are no a, b ∈ S such that a defeats b An argument a is acceptable w.r.t. S ⊆ A iff ∀b ∈ A such that b defeats a, there is c ∈ S such that c defeats b. Example (conflict-free and acceptable sets) {a}, {b}, {a, c} are conflict-free sets; a is acceptable w.r.t. {c} Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Basic concepts of Abstract Argumentation An abstract argumentation framework consists of a set A of arguments, and a relation D ⊆ A × A, whose elements are defeats (or attacks) Example (AAF) A = {a, b, c} D = {hb, ai, hb, ci, hc, bi}

a

b

c

A set S ⊆ A of arguments is conflict-free if there are no a, b ∈ S such that a defeats b An argument a is acceptable w.r.t. S ⊆ A iff ∀b ∈ A such that b defeats a, there is c ∈ S such that c defeats b. Example (conflict-free and acceptable sets) {a}, {b}, {a, c} are conflict-free sets; a is acceptable w.r.t. {c} Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Semantics for Abstract Argumentation Each semantics identifies “reasonable” sets of arguments semantics sem

A set S ⊆ A of arguments is reasonable according to sem iff

admissible stable complete

S is conflict-free and all its arguments are acceptable w.r.t. S S is conflict-free and S defeats each argument in A \ S S is admissible and S contains all the arguments that are acceptable w.r.t. S S is a minimal complete set of arguments S is a maximal admissible set of arguments

grounded preferred

Example (semantics for AAF) admissible sets: {a, c}, {b}, {c}, ∅ stable sets: {a, c}, {b} complete sets: {a, c}, {b}, ∅ grounded sets: ∅


a

b

c

preferred sets: {a, c}, {b}


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grounded preferred



a

b

c



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grounded preferred



a

b

c



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grounded preferred



a

b

c



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grounded preferred



a

b

c



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Basics of Probabilistic Argumentation A PrAF is a tuple hA, PA , D, PD i where hA, Di is an AAF, and PA and PD are functions assigning a probability value to each argument in A and defeat in D

PA (a) represents the probability that argument a actually occurs PD (ha, bi) represents the conditional probability that a defeats b given that both a and b occur Example (probabilities of arguments and defeats) PA (a) = .9 PA (b) = .7 PA (c) = .2

PD (hb, ai) = .9 PD (hb, ci) = 1 PD (hc, bi) = 1

The issue of how to assign probabilities to arguments/defeats has been investigated in [Hunter 2012, Hunter 2013] Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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90%

a

90%

70%

b

20%

c



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90%

a

90%

70%

b

20%

c



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Meaning of a probabilistic argumentation framework The meaning of a PrAF is given in terms of possible worlds A possible world represents a (deterministic) scenario consisting of some subset of the arguments and defeats of the PrAF given a PrAF F = hA, PA , D, PD i, a possible world w for F is an AAF hA0 , D 0 i such that A0 ⊆ A and D 0 ⊆ D ∩ (A0 × A0 ). Example (some possible worlds)



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Meaning of a probabilistic argumentation framework The meaning of a PrAF is given in terms of possible worlds A possible world represents a (deterministic) scenario consisting of some subset of the arguments and defeats of the PrAF given a PrAF F = hA, PA , D, PD i, a possible world w for F is an AAF hA0 , D 0 i such that A0 ⊆ A and D 0 ⊆ D ∩ (A0 × A0 ). Example (some possible worlds)

a

a

a

a

a

90%

90%

b c

b

b

b

b

b

70%

c

c

c

c

c

20%



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Probability of reasonable sets An interpretation I for a PrAF is a probability distribution over the set of possible worlds possible world w is assigned by I the probability I(w) equal to: Y Y Y Y PA (a) × (1 − PA (a)) × PD (δ) × (1 − PD (δ)) a∈Arg(w)

a∈A\Arg(w)

δ∈Def (w)

δ∈D(w)\Def (w)

where D(w) = D ∩ (Arg(w) × Arg(w)) is the set of defeats that may appear in w

The probability Pr sem (S) that a set S of arguments is reasonable according to a given semantics sem is defined as the sum of the probabilities of the possible worlds w for which S is reasonable according to sem



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a∈A\Arg(w)

δ∈Def (w)

δ∈D(w)\Def (w)





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a∈A\Arg(w)

δ∈Def (w)

δ∈D(w)\Def (w)





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The state of the art approach sem (S) by sampling AAFs wherein S is conflict-free Estimating PrF sem (S) by sampling AAFs wherein S is admissible Estimating PrF

Outline 1


2


3


4

Experiments

5



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Estimating Pr sem : The state of the art approach Algorithm (A1) sem State-of-the-art algorithm for approximating PrF (S) Input: F = hA, PA , D, PD i; S ⊆ A; sem; An error level ; A confidence level z1−α/2 sem sem sem sem b b b Output: Pr F (S) s.t. PrF (S) ∈ [Pr F (S)− , Pr F (S)+ ] with confidence z1−α/2 success = samples = maxsamples = 0; do Arg = Def = ∅

if S is an extension for hArg, Def i according to sem then success=success+1; samples=samples+1; update maxsamples while samples ≤ maxsamples success return samples



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Estimating Pr sem : The state of the art approach Algorithm (A1) sem State-of-the-art algorithm for approximating PrF (S) Input: F = hA, PA , D, PD i; S ⊆ A; sem; An error level ; A confidence level z1−α/2 sem sem sem sem b b b Output: Pr F (S) s.t. PrF (S) ∈ [Pr F (S)− , Pr F (S)+ ] with confidence z1−α/2 success = samples = maxsamples = 0; do Arg = Def = ∅ for each a ∈ A do With probability PA (a) do Arg = Arg ∪ {a}

if S is an extension for hArg, Def i according to sem then success=success+1; samples=samples+1; update maxsamples while samples ≤ maxsamples success return samples



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Estimating Pr sem : The state of the art approach Algorithm (A1) sem State-of-the-art algorithm for approximating PrF (S) Input: F = hA, PA , D, PD i; S ⊆ A; sem; An error level ; A confidence level z1−α/2 sem sem sem sem b b b Output: Pr F (S) s.t. PrF (S) ∈ [Pr F (S)− , Pr F (S)+ ] with confidence z1−α/2 success = samples = maxsamples = 0; do Arg = Def = ∅ for each a ∈ A do With probability PA (a) do Arg = Arg ∪ {a}

for each ha, bi ∈ D s.t. a, b ∈ Arg do With probability PD (ha, bi) do Def = Def ∪ {ha, bi} if S is an extension for hArg, Def i according to sem then success=success+1; samples=samples+1; update maxsamples while samples ≤ maxsamples success return samples



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Estimating PrFsem (S) by sampling AAFs wherein S is conflict-free Algorithm (A2) cf Compute PrF (S) success = samples = maxsamples = 0; do Arg = S; Def = ∅;

if S is an extension for hArg, Def i according to sem then success=success+1; samples=samples+1; update maxsamples while samples ≤ maxsamples return

success samples ·

cf PrF (S)



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Estimating PrFsem (S) by sampling AAFs wherein S is conflict-free Algorithm (A2) cf Compute PrF (S) success = samples = maxsamples = 0; do Arg = S; Def = ∅; for each a ∈ A \ S do With probability Pr (a|CF) do Arg = Arg ∪ {a}


success samples ·

cf PrF (S)



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Estimating PrFsem (S) by sampling AAFs wherein S is conflict-free Algorithm (A2) cf Compute PrF (S) success = samples = maxsamples = 0; do Arg = S; Def = ∅; for each a ∈ A \ S do With probability Pr (a|CF) do Arg = Arg ∪ {a}

for each ha, bi ∈ D such that a, b ∈ Arg do if a ∈ / S∨b ∈ / S then With probability Pr (ha, bi |CF) do Def = Def ∪ {ha, bi} if S is an extension for hArg, Def i according to sem then success=success+1; samples=samples+1; update maxsamples while samples ≤ maxsamples return

success samples ·

cf PrF (S)



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Estimating PrFsem (S) by sampling AAFs wherein S is admissible Algorithm (A3) success = samples = maxsamples = 0; ad Compute PrF (S) do Arg = S; Def = ∅; defeatS = ∅;


success samples ·

ad PrF (S)



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Estimating PrFsem (S) by sampling AAFs wherein S is admissible Algorithm (A3) success = samples = maxsamples = 0; ad Compute PrF (S) do Arg = S; Def = ∅; defeatS = ∅; for each a ∈ A \ S do With probability Pr (a|AD) do Arg = Arg ∪ {a} With probability Pr (a → S|AD ∧ a) do Def = Def ∪ generateAtLeastOneDefeatAndDefend(F , hArg, Def i, S, a) defeatS = defeatS ∪ {a}


success samples ·

ad PrF (S)



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Estimating PrFsem (S) by sampling AAFs wherein S is admissible Algorithm (A3) success = samples = maxsamples = 0; ad Compute PrF (S) do Arg = S; Def = ∅; defeatS = ∅; for each a ∈ A \ S do With probability Pr (a|AD) do Arg = Arg ∪ {a} With probability Pr (a → S|AD ∧ a) do Def = Def ∪ generateAtLeastOneDefeatAndDefend(F , hArg, Def i, S, a) defeatS = defeatS ∪ {a} for each ha, bi ∈ D s.t. (a, b ∈ Arg \ S) ∨ (a ∈ S ∧ b ∈ Arg \ S ∧ b 6∈ defeatS) do With probabilty Pr (ha, bi|AD ∧ b 6→ S) do Def = Def ∪ {ha, bi} if S is an extension for hArg, Def i according to sem then success=success+1; samples=samples+1; update maxsamples while samples ≤ maxsamples

return

success samples ·

ad PrF (S)



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Outline 1


2


3


4

Experiments

5



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Theoretical analysis of the efficiency of A2 and A3 Theorem Let z1−α/2 be a confidence level, be an error level and let n1 , n2 and n3 be the number of Monte-Carlo iterations of A1, A2, and A3, respectively. Let i1 , i2 , i3 , i4 and i5 be the following inequalities: (i1 ) Pr sem (S) ≥ k · , (i2 ) Pr sem|CF (S) ≥ k 0 · , (i3 ) Pr cf (S) ≤ 1 − k20 , (i4 ) Pr sem|AD (S) ≥ k 00 · , (i5 ) Pr ad (S) ≤ 1 − k200 . If there exist k and k 0 greater than 1 such that i1 , i2 and i3 hold, then k ·(k 0 +1) cf 2 n2 ≤ n1 · (k −1)·k 0 · PrF (S), with confidence level z1−α/2 . If there exist k and k 00 greater than 1 such that i1 , i4 and i5 hold, then k ·(k 00 +1) ad 2 n3 ≤ n1 · (k −1)·k 00 · PrF (S), with confidence level z1−α/2 .



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Experimental validation: data sets

We performed experiments on 75 PrAFs each obtained considering a set A of arguments whose size ranges from 12 to 40. For each |A|, we considered 5 PrAFs having different sets of defeats. For each of the so obtained PrAFs, we considered 5 sets S of arguments, whose size was chosen in the interval [20%, 40%] · |A|, and such that PrFcf (S) and PrFad (S) ranged in the interval [.5, .8] and [.4, .7], respectively.



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Experimental validation: performace measures

samples(A3) ImpS(A2) = samples(A2) samples(A1) and ImpS(A3) = samples(A1) , for measuring the improvement of A2 and A3 w.r.t. A1, in terms of number of generated samples; time(A2) ImpT(A2) = time(A1) and ImpT(A3) = time(A3) time(A1) , for measuring the improvement of A2 and A3 w.r.t. A1, in terms of execution time.

where samples(Ak ) and time(Ak ), with k ∈ {1, 2, 3}, are the average number of samples and the average execution time of the runs of algorithm Ak , respectively.



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Experimental validation: results On average, ImpT(A2) is equal to 70%, and ImpS(A2) is equal to 65%. On average, ImpT(A3) is equal to 60%, and ImpS(A3) is equal to 55%.

Improvements of A2 and A3 vs A1 for (a) complete, (b) grounded, (c) preferred semantics. Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Outline 1


2


3


4

Experiments

5



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Conclusions and future work In this paper, we focused on estimating the probability PrFsem (S) that a set S of arguments is an extension for a F according to a semantics sem, where sem is the complete, the grounded, or the preferred semantics. In particular, we proposed two algorithms for estimating PrFsem (S), which outperform the state-of-the-art algorithm proposed in [Li et Al. 2011], both in terms of number of generated samples and evaluation time. Future work will be devoted to: experimentally characterizing, on larger data sets, when A2 is preferable to A3, applying the proposed algorithms to other semantics (e.g. the ideal set semantics) for which computing Pr sem is hard.



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Thank you! ... any question?



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Appendix

References PrAF

Selected References Phan Minh Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell., 77(2):321–358, 1995. Paul E. Dunne and Michael Wooldridge. Complexity of abstract argumentation. In Argumentation in Artificial Intelligence, 85–104, 2009. Paul E. Dunne. The computational complexity of ideal semantics. Artif. Intell., 173(18):1559–1591, 2009. Bettina Fazzinga, Sergio Flesca, and Francesco Parisi. On the Complexity of Probabilistic Abstract Argumentation. In IJCAI, 2013. Anthony Hunter. Some foundations for probabilistic abstract argumentation. In COMMA, 117–128, 2012. Anthony Hunter. A probabilistic approach to modelling uncertain logical arguments. Int. J. Approx. Reasoning, 54(1):47–81, 2013. Hengfei Li, Nir Oren, and Timothy J. Norman. Probabilistic argumentation frameworks. In TAFA, 1–16, 2011.



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Appendix

References PrAF

How to assign probabilities

probability theory is recognized as a fundamental tool to model uncertainty The issue of how to assign probabilities to arguments and defeats in abstract argumentation, with particular reference to the PrAF proposed in [Li et Al. 2011], has been investigated in [Hunter 2012, Hunter 2013], where a connection among argumentation theory, classical logic, and probability theory was investigated In this paper, we do not address this issue, but, assuming that the probabilities of arguments and defeats are given, we tackle the probabilistic counterpart of the problem V ERsem (S)



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Appendix

References PrAF

Other approaches to model uncertainty Besides the approaches that model uncertainty in AAFs by relying on probability theory, many proposals have been made where uncertainty is represented by exploiting weights or preferences on arguments and/or defeats, or by relying on the possibility theory Although the approaches based on weights, preferences, possibilities, or probabilities to model uncertainty have been proved to be effective in different contexts, there is no common agreement on what kind of approach should be used in general we believe that the probability-based approaches may take advantage from relying on a well-established and well-founded theory our complexity characterization, along with that of other approaches, may help in deciding what approach is better from a computational point of view Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Appendix

References PrAF

Estimating PrFsem (S) by sampling AAFs wherein S is conflict-free Lemma Given a PrAF F = hA, PA , D, PD i and a set S ⊆ A of arguments, then ∀a ∈ S, Pr (a|CF)=1;

∀a ∈ A \ S, Pr (a|CF)=PA (a);

∀ha, bi ∈ D such that a, b ∈ S, Pr (ha, bi|CF) = 0; ∀ha, bi ∈ D \ {ha, bi ∈ D s.t. a, b ∈ S}, Pr (ha, bi|CF) = PD (ha, bi).

Theorem sem

c (S) Let be an error level, and z1−α/2 a confidence level. The estimate Pr F sem sem sem c c returned by Algorithm 2 is such that PrF (S) ∈ [Pr F (S)−, Pr F (S)+] with confidence level z1−α/2 . Bettina Fazzinga, Sergio Flesca, Francesco Parisi


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Appendix

References PrAF

Estimating PrFsem (S) by sampling AAFs wherein S is admissible Fact (Prad F (S)) PrFad (S) = PrFcf (S)·

Q

d∈A\S

P1 (S, d)+ P2 (S, d)+ P3 (S, d) , where:

P1 (S, d) = 1−PA (d), i.e., the probability that d is false. Q P2 (S, d) = PA (d) · hd, bi ∈ D 1−PD (hd, bi) , ∧b ∈ S

i.e., the probability that d is true but do not attack any argument in S. Q Q P3 (S, d) = PA (d)· 1− hd, bi ∈ D 1−PD (hd, bi) · 1− ha, di ∈ D 1−PD (ha, di) , ∧b ∈ S

∧a ∈ S

i.e., the probability that d is true, d attack an argument in S but it is counterattacked by an argument in S.



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Appendix

References PrAF

Estimating PrFsem (S) by sampling AAFs wherein S is admissible Lemma Given a PrAF F = hA, PA , D, PD i and a set S ⊆ A of arguments, then ∀a ∈ S, Pr (a|AD)=1; P2 (S,a)+P3 (S,a) 1 (S,a)+P2 (S,a)+P3 (S,a)

∀a ∈ A \ S, Pr (a|AD)= P

;

∀ha, bi ∈ D s.t. a, b ∈ S, Pr (ha, bi|AD) = 0; ∀ha, bi ∈ D s.t. a, b ∈ A \ S, Pr (ha, bi|AD ∧ b 6→ S) = PD (ha, bi); ∀a ∈ A \ S, Pr (a → S|AD ∧ a) =

P3 (S,a) ; P2 (S,a)+P3 (S,a)

∀ha, bi ∈ D s.t. a ∈ S ∧ b ∈ A \ S, Pr (ha, bi|AD ∧ b 6→ S) = PD (ha, bi).

where P1 (S, a), P2 (S, a), and P3 (S, a) are defined as in Fact (Prad F (S)).



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Appendix

References PrAF

Estimating PrFsem (S) by sampling AAFs wherein S is admissible

Theorem sem

c (S) Let be an error level, and z1−α/2 a confidence level. The estimate Pr F sem sem c (S)−, Pr c (S)+] returned by Algorithm 3 is such that PrFsem (S) ∈ [Pr F F with confidence level z1−α/2 .



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