Two player corridor tiling game
Tiling system T: finite set of tile types with horizontal and verical
adjacency conditions
ExpTime Hardness of ALC
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Declarative Knowledge Processing
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A corridor tiling is a tiling of a corridor of width n with tiles of T
respecting adjacency conditions
Magdalena Ortiz
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Knowledge Base Systems Group
Institute of Information Systems
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[email protected]
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WS 2012/2013
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Complexity of ALC
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Two player corridor tiling game
∀belard and ∃loise alternatively place a tile, row by row, from left to
right, respecting adjacency conditions
Theorem:
Concept subsumption w.r.t. an ALC TBox (and KB satisfiability)
is ExpTime-hard
∃loise wins if
• she can place a special “winning tile” in the second position of a row,
or
• she can play in such a way that ∀belard can no longer place a tile
We look at a proof based on encoding the two player corridor tiling
problem
(i.e., ∃loise loses if she cannot place a tile, or if the game goes on forever)
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Encoding of two player corridor tiling in ALC (2)
Two player corridor tiling problem
We introduce in TT the following GCIs to ensure that tilings are correctly
represented:
Instance:
a tiling system expressed as T = (k, H, V ), where
To encode that each column has exactly one tile last placed into it:
• 0, 1, . . . , k are the tile types, with k being the winning tile
• H ⊆ [0..k] × [0..k] is the horizontal adjacency relation
> v Ri0 t · · · t Rik
0
Rit v ¬Rit
• V ⊆ [0..k] × [0..k] is the vertical adjacency relation
an initial row of tiles t1 t2 · · · tn of length n
for i ∈ [1..n]
for i ∈ [1..n],
t, t0 ∈ [0..k],
t 6= t0
To encode that each move occurs in exactly one column in the
corridor:
Question: Does ∃loise have a winning strategy?
> v Q1 t · · · t Qn
Qi v ¬Qj
i.e., for every move ∀belard makes, is there a move ∃loise can counter
with in such a way that she wins?
for i, j ∈ [1..n],
i 6= j
To encode that the tiles are placed in the correct left-to-right order:
Qi v ∀Next.Qi+1
Qn v ∀Next.Q1
Theorem:
Two player corridor tiling is ExpTime-complete
for i ∈ [1..n−1]
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Encoding of two player corridor tiling in ALC
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Encoding of two player corridor tiling in ALC (3)
We introduce in TT the following GCIs to encode the adjacency
conditions:
The intention is to represent each placed tile by an object;
the object carries the information about the last n moves made
To encode the vertical adjacency relation V :
We use an atomic role Next to connect individuals representing
successive tiles
Qi u Rit v ∀Next.(
t
t0 |(t,t0 )∈V
0
Rit )
for i ∈ [1..n],
t ∈ [0..k]
We use the following atomic concepts:
• Rit , for each i ∈ [1..n] and each t ∈ [0..k], denoting that the last tile
placed in column i has been tile t
• Qi , for i ∈ [1..n], denoting that the next tile will be placed in column
i
• A, denoting that it is ∀belard’s turn to place the next tile
• W , denoting that ∃loise wins
To encode the horizontal adjacency relation H:
t
Qi u Ri−1
v ∀Next.(
t
t0 |(t,t0 )∈H
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for i ∈ [2..n],
t ∈ [0..k]
To encode that in columns where no move is made nothing changes:
¬Qi u Rit v ∀Next.Rit
¬Qi u ¬Rit v ∀Next.¬Rit
We now show how to build an ALC TBox TT using these concepts and
roles
0
Rit )
for i ∈ [1..n],
for i ∈ [1..n],
t ∈ [0..k]
t ∈ [0..k]
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Encoding of two player corridor tiling in ALC (4)
Complexity of ALC
We introduce in TT the following GCIs to encode the game:
To encode the existence of all possible moves in the game tree,
provided ∃loise hasn’t already won:
0
t
u Rit
¬R2k u Qi u Ri−1
v
t00
|
u
(t,t00 )∈H
Since the size of TT is polynomial in T and n, this shows that
concept subsumption w.r.t. to an ALC TBox is ExpTime-complete
00
∧
(t0 ,t00 )∈V
for i ∈ [2..n],
∃Next.Rit
We can easily adapt the reduction to KB sastisfiability: simply add
an ABox
{A u Q1 u R1t1 u · · · u Rntn u ¬W (a)}
t, t0 ∈ [0..k]
To encode the alternation of moves:
A v ∀Next.¬A
¬A v ∀Next.A
With some modifications, we can show even stornger results: the
fragment of ALC that has no qualified existentials and no
disjunctions is also ExpTime-hard
To encode the winning of ∃loise:
W
≡ (A u R2k ) t (A u ∀Next.W ) t (¬A u ∃Next.W )
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Encoding of two player corridor tiling in ALC (4)
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Tiling systems and TMs
Tiling problems are a very closely related to Turing Machine.
Observations:
This particular form of tiling is related to Alternating Turing Machines
if ∃loise cannot move when it is her turn, then W is false for the
object representing that tile
An ATM has the form M = (Σ, Q∀ , Q∃ , q0 , δ, qf )
if ∀belard can force the game to go on forever, then there will be
models of TT in which W is false
As usual, Σ is the alphabet, q0 is the initial state, qf the
final one, and δ : Q × Σ → Q × Σ × {right, left} is the
transition function
Q∀ are universal states, and Q∃ existential ones
Theorem:
∃loise has a winning strategy for tiling system T with initial row t1 · · · tn
iff
TT |= A u Q1 u R1t1 u · · · u Rntn v W
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From an existential state, the machine moves to some
successive configuration
From a universal state, the machine moves to all
successive configutrations
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Tiling systems and TMs (2)
Tiling problems are a very closely related to Turing Machines.
In fact, this form of tiling is a simple ‘disguise’ for an Alternating Turing
Machine that runs in polynomial space, and we want to decide acceptance
of a word
Think of each sequence of last n-tiles (i.e., each row) as the tape
contents
Configurations evolving in time appear along the corridor
The two players correspond to existential and universal states
The winning tile is (the only) final state
The initial configuration is the word initially written on the tape
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Hardness proofs using tilings
Tiling problems are a very useful tool for showing complexity results in
Modal Logics and fragments of FOL
In DLs, they are very frequently used to:
Show NExpTime-hardness (e.g., for extensions of ALCIOF):
Deciding the existence of a tiling for a
2n × 2n torus is NExpTime-complete
Show undecidability (e.g., for DLs with transitive roles in the number
restrictions, role value maps, etc.)
Deciding the existence of a tiling for an
unbounded grid is undecidable
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