Games, Queries, and Argumentation Frameworks: Time for a Family Reunion!

1
Games, Queries, and Argumentation Frameworks:
Time for a Family Reunion!
Bertram Ludäscher1, Shawn Bowers2, Yilin Xia1
1 School of Information Sciences, University of Illinois, Urbana-Champaign, IL, USA
2 Department of Computer Science, Gonzaga University, WA, USA
{ludaesch,yilinx2}@illinois.edu
bowers@gonzaga.edu
7th Workshop on Advances in Argumentation in Artificial Intelligence (AI3)
AIxIA 2023: 22nd International Conference of the Italian Association for Artificial Intelligence

Games, Queries, Argumentation
Outline
1. What’s this? (a query puzzle)
2. Identical Twins (triplets)
3. A Correspondence (a defeatist’s game)
4. Harvesting Time (translational research)
5. Time for a Family Reunion!
7th Workshop on Advances in Argumentation in Artificial Intelligence (AI3) 2

What’s this? (an easy query puzzle ..)
• q(X,Y) :- e(X,A), e(A,B), e(B,Y).
• Input: digraph with edges e(From,To)
• Output: binary answer relation q(X,Y)
• We can interpret e/2 differently => output q/2 is a different relation
• e/2 ≅ parent/2 => q/2 ≅ great_grandparent/2
• e/2 ≅ one_hour_trail/2 => q/2 ≅ three_hour_hike/2

What’s this? (a harder query puzzle ..)
• win(X) :- move(X,Y), not win(Y).
• defeated(X) :- attacks(Y,X), not defeated(Y).
• defeated(X) :- attacked_by(X,Y), not defeated(Y).
• kerC(X) :- edge(X,Y), not kerC(Y).
• One query to rule them all: q(X) :- e(X,Y), not q(Y).
• Semantics: stratified, well-founded, stable models, …

More Practice: Family Relations
% ancestor := parent+
ancestor(X,Y) :- parent(X,Y).
ancestor(X,Y) :- parent(X,Z),ancestor(Z,Y).
% great_grandparent := parent.parent.parent
gg_p(X,Y) :- p(X,A), p(A,B), p(B,Y).
% child := parent-1
c(P,C) :- p(C,P).
% great_grandchild := child.child.child
gg_c(X,Y) :- c(X,A), c(A,B), c(B,Y).

Game Example: move(X,Y) relation
a k
b c l
d e m
g h n
f
6

Solving the Example
a k
b c l
d e m
g h n
f
7

Solving the Example
a k
b c l
d e m
g h n
f
8

Solving the Example
a k
b c l
d e m
g h n
f
9

Solving the Example
a k
b c l
d e m
g h n
f
10
win(X) :- move(X,Y), not win(Y).
One rule
… to rule them all!

A Claim: Stratified Datalog = FIXPOINT

Kolaitis’88: .. not so fast!
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A question from the DB-Theory “bible” [AHV95]
13
Well-founded (WF-)Datalog queries
have 3-valued models in general.
Can every query Q in WF-Datalog-3
be rewritten into an equivalent Q’ in
WF-Datalog-2?
=> Total WF-Datalog-2 =?=
Partial WF-Datalog-3?
Example:
Can we detected draws for GAME?
win(X) :- move(X,Y), not win(Y).

… answering the question! [FKL-ICDT’97]
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All you need is GAME!
(i.e., the win-move / GAME query)

… answering the question!
15
The tricky bit!
Useful notion: Length of a position!
All you need is DRAW-FREE GAMEs!
(i.e., the win-move / GAME query,
… but draws can be detected and avoided!)

Win-Move vs Argumentation Frameworks
% We understand this now:
% This is the mother of AF rules:
• defeated(X) :- attacks(Y,X), not defeated(Y).
% But they are both equivalent to this:
• q(X) :- edge(X,Y), not q(Y).
• GAME: q = win edge = move
• AF: q = defeated edge = attacks-1 (= attacked_by)

The Correspondence: GAME ~ AF
WF and Stable Semantics
17

move(X,Y)
18
a
b c
d e
f g h
m
k
l
n

attacks(Y,X)
19
a
b c
d e
f g h
m
k
l
n

Win-Move GAME
20
a
b c
d e
f g h
m
k
l
n

Argumentation Framework
21
a
b c
d e
f g h
m
k
l
n

Harvesting Time!
• Notions from games
translate to AF via the
natural correspondence!
• Length of a position (i.e.,
argument)
• Type of an edge (not all
edges are created equal)
• winning, delaying,
drawing, bad

Harvesting Time!
• Provenance of a position (i.e., argument)
• ... = Explanations of the labeling
• … can be computed via Regular Path Queries (RPQs):
• prov(X,Y):-
path(X, green(red.green).*, Y)
• Question:
• What is the provenance games?
• Answer:
• Solve the game (AF) and look!
• Provenance/Explanations for free!

Harvesting Time!
W
bad D
bad
L
winning
bad
drawing
n/a
delaying
n/a
n/a

Summary: Time for a Family Reunion!
1. Identical LP Twins (triplets)
• Game- & DB-Theory: win-move
• Argumentation: defeated-attacked_by
• Semantics: Well-founded, Stable, …
2. Harvesting Time (translational research)
• Not all edges are created equal! (types)
• Length of positions/arguments
• Decomposition Theorem (Fraenkel)
• Provenance & Explainability
Join the reunion!

Whose turn is it? G(G => 2G)

Games ~ AF ~ Kernels è Family Reunion
è Teaching the Next Generation!
• Notions from LP, Game Theory & Graph Kernels,
Argumentation Frameworks are all closely related, but
terminology and details differ!
• Winning strategies are explanations!
• Let’s have a family reunion and take another look!

Edge Types => New explanations for
Argumentation Frameworks ... !?
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Applying this to AF (coming from GAME and provenance) seems new!?

More on Queries as Games
• Games & argumentation have been around for a while …
• Socratic dialogues …
• ... Lorenzen & Lorenz (Dialogical Logic)
• ... Hintikka (game-oriented FO semantics)
• … [KLZ13] (FO Provenance Games)
• ... as a universal query-evaluation machine!
• … SLD(NF) game … (and others)
• … as a universal argumentation solver ..
• Eureka!
• Solved (= evaluated) games contain their own provenance!

First-Order Provenance Games

SLD(NF) / FO Provenance Game
7th Workshop on Advances in Argumentation in Artificial
Intelligence (AI3)
34

Games, Queries, and Argumentation Frameworks: Time for a Family Reunion!

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