axiom of choice, maximal independent sets, argumentation ... · the joy of sets: fundamentals of...

Axiom of Choice, Maximal Independent Sets,Argumentation and Dialogue Games

Christof Spanring

Department of Computer Science, University of Liverpool, UK

Institute of Information Systems, Vienna University of Technology, Austria

London, September 25, 2014

A minor example

Example (Games played on Argument Graphs)Can you defend an argument a beyond doubt, i.e. defeat anyattackers without running into conflict with your own argument base?

Who has a winning strategy, you as the proponent or your oponent?

a b c . . .d . . .

Christof Spanring, ICCSW 2014 Choice and Argumentation 2 / 15

The Why? of Infinities I

QuestionHow many prime numbers are there?

QuestionHow many rational numbers p

q are there?

QuestionHow many decimal numbers are there?

QuestionIs there a set of all sets?


The Why? of Infinities II

Example (|Q| = |N|)There are only as many rationalas natural numbers.

11

12

13

14

21

22

23

24

31

32

33

34

41

42

43

44

. . .

. . .

. . .

. . .

. . .

...

...

...

...

Example (|N| < |R|)There are more real than naturalnumbers.

i1 = 0.

i2 = 0.

i3 = 0.

i4 = 0.

i1,1

i2,1

i3,1

i4,1

i1,2

i2,2

i3,2

i4,2

i1,3

i2,3

i3,3

i4,3

i1,4

i2,4

i3,4

i4,4..

.

· · ·

· · ·

· · ·

· · ·

i1,1

i2,2

i3,3

i4,4


Set Theory

DefinitionZermelo-Fraenkel Set Theory (ZFC-Axioms)

1 Extensionality ∀x∀y (∀z (z ∈ x⇔ z ∈ y)⇒ x = y)2 Foundation ∀x (∃a (a ∈ x)⇒ ∃y (y ∈ x ∧ ¬∃z (z ∈ y ∧ z ∈ x)))3 Specification ∀z∀v1∀v2 · · · ∀vn∃y∀x (x ∈ y⇔ (x ∈ z ∧ ϕ))

4 Pairing ∀x∀y∃z (x ∈ z ∧ y ∈ z)5 Union ∀x∃z∀y∀v ((v ∈ y ∧ y ∈ x)⇒ v ∈ z)6 Replacement∀x∀v1∀v2 · · · ∀vn (∀y (y ∈ x⇒ ∃!zϕ)⇒ ∃w∀y (y ∈ x⇒ ∃!z(y ∈ w ∧ ϕ))

7 Infinity ∃x (∅ ∈ x ∧ ∀y (y ∈ x⇒ (y ∪ {y}) ∈ x))8 Power Set ∀x∃y∀z (z ⊆ x⇒ z ∈ y)9 Choice ∀x (∅ 6∈ x⇒ ∃f : x→

⋃x,∀a ∈ x (f (a) ∈ a))


Choice and Companions

Example (The Axiom of Choice)Every set of non-empty sets has a choice function, selecting exactly oneelement from each set.

Example (Basis Theorem for Vector Spaces)Every vector space has a basis.

Example (Well-ordering Theorem)Every set can be well-ordered.

Example (Zorn’s Lemma)If any chain of a non-empty partially ordered set has an upper boundthen there is at least one maximal element.


Determinacy

Example (A number game)

Some well-known set of sequences of natural numbers S ⊆ NN,defines the winning set.

Move i selects a number for position i, two players alternate,proponent starts with move 0.

Proponent wins if the played sequence is an element of S, otherwiseopponent wins.

Definition (Axiom of Determinacy)Every number game of the above form is predetermined, i.e. one of theplayers has a winning strategy.


Possibly infinite Games

Example (Some number game)Two players alternate stating moves.

Moves are decimal digits 0, 1, · · · 10.

Proponent wins if 0.i0i1i2i3 · · · ∈ Q.

Example (A slightly simpler number game)Two players alternate making moves i0, i1, i2, i3, . . .

Moves are binary digits 0 or 1.

The winning set (for proponent) consists of sequences where forsome n > 0 we have ij = ij+n for all j < n, i.e. the initial sequence isrepeated at least once.

For instance in 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, · · ·0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, · · · proponent wins.who wins?


Choice andvs. Determinacy

QuestionHow do the axioms of choice (AC) and determinacy (AD) relate to eachother?

Theorem (AD implies countable AC)(AD)⇒ (AC)fin

Theorem (AD implies Consistency of ZF Set Theory)(AD)⇒ Con(ZF)

Theorem (AC implies not AD)(AC)⇒ ¬(AD)


Abstract Argumentation I

Definition (Argumentation Frameworks)An argumentation framework (AF) is a pair F = (A,R).

A is an arbitrary set of arguments.

R ⊆ (A× A) is the attack relation.

For (a, b) ∈ R write a � b, and say a attacks b.

For a � b � c say a defends c against b.

Example

a b


Abstract Argumentation II

Definition (Argumentation Semantics)Some AF F = (A,R) and some set E ⊆ A.

E is conflict-free (cf) iff E 6� E.

E is admissible (adm) iff E ∈ cf (F) and for all a � E also E � a.

E is a preferred extension (pref) iff it is maximal admissible, i.e.E ∈ adm(F) and for any E′ ∈ adm(F) with E ⊆ E′ already E = E′.

Example

a b

cf (F) = {∅, {a} , {b}} adm(F) = {∅, {a}} prf (F) = {{a}}


(AC)⇒ prf (F) 6= ∅

Definition (Zorn’s Lemma)If any chain of a non-empty partially ordered set has an upper boundthen there is at least one maximal element.

Definition (Partial Order)A partial order (P,≤) is a set P with a binary relation ≤ that fulfills

reflexivity: a ≤ a,

antisymmetry: a ≤ b ∧ b ≤ a⇒ a = b,

transitivity: a ≤ b ∧ b ≤ c⇒ a ≤ c.

Definition (Axiom of Union)The union over the elements of a set is a set.

∀z∃y∀x∀u(x ∈ z ∧ u ∈ x)⇔ u ∈ y


(∀Fprf (F) 6= ∅)⇒(AC)

Definition (ZF-Axioms)Comprehension: we can construct formalizable subsets of sets.

Union: the union over the elements of a set is a set.

Replacement: definable functions deliver images of sets.

Power Set: we can construct the power set of any set.

Selecting Nodes/Elements: a choice function


References

Devlin, K. (1994).The Joy of Sets: Fundamentals of Contemporary Set Theory.Undergraduate Texts in Mathematics. Springer, Springer-Verlag 175 Fifth Avenue, New York, New York 10010, U.S.A., 2ndedition.

Dung, P. M. (1995).On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-persongames.Artif. Intell., 77(2):321–358.

Gödel, K. and Brown, G. W. (1940).The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory.Princeton University Press.

Kunen, K. (1983).Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics).North Holland.

Mycielski, J. (1964).On the axiom of determinacy.Fund. Math, 53:205–224II.

Walton, D. N. (1984).Logical Dialogue-Games.University Press of America, Lanham, Maryland.


axiom of choice, maximal independent sets, argumentation ... · the joy of sets: fundamentals of...

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