Playing Games in the Baire Space
Benedikt Brütsch
Wolfgang Thomas
RWTH Aachen University
Cassting meeting – Cachan
October 28, 2015
Outline
Recall
Cantor space: 2ω (or Σω for finite Σ)
Baire space: Nω
Plan:
Lifting Church’s Problem to the Baire Space
Automata for ω-Words over N
Solving Games in the Baire Space
Lifting Church’s Problem to the Baire Space
Automata for ω-Words over N
Solving Games in the Baire Space
Church’s Problem
Gale-Stewart Game Γ(L) for L ⊆ 2ω
Player 0 and Player1 choose bits in alternation. Resulting play:
α
=
a0 b0 a1 b1 a2 b2 . . .
with ai , bi ∈ {0, 1}
Player 0 wins the play α if α ∈ L (otherwise Player 1 wins).
Player 0 wins the game Γ(L) if (s)he has a winning strategy.
Church’s Problem
Given a regular ω-language L ⊆ {0, 1}ω , decide who wins Γ(L) and
construct winning strategy for the winner.
In this talk N replaces {0, 1}, so plays are in the Baire space Nω
Solving Gale-Stewart Games
Classical approach for solving regular games:
1. Transform definition of L (say by an MSO-formula or a Büchi
automaton) into a deterministic parity automaton.
2. Convert the automaton into a parity game (over a finite
arena) with designated start vertex.
3. Solve the parity game (finding the winner and computing a
memoryless winning strategy).
Three steps needed for games in the Baire space
1. Introduce deterministic automata for ω-languages over N.
2. Define a corresponding model of transducer.
3. Solve Church’s Problem in this setting.
Lifting Church’s Problem to the Baire Space
Automata for ω-Words over N
Solving Games in the Baire Space
How to Define ω-Automata over N?
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Straightforward approach: Banach-Mazur game
m0 m1 m2 . . . coded by
0m0 +1 1m1 +1 0m2 +1 . . .
Unsatisfactory: no interesting comparison of successive letters
(equality, +1, etc.)
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Known models: Register automata (Kaminski-Francez),
data word automata (Bojanczyk, Segoufin et al.)
only allow equality tests between numbers.
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Our model: Progressive grid walking automata (PGAs)
allowing description of “incremental change” (and equality) in
a sequence of numbers,
introduced in DLT 2015 (Czyba, Spinrath, Th.), refined here.
Progressive Grid Walking Automata
(PGAs)
For example,
the word
4 2 0 4 3 . . . is represented as
..
.
⊥
1
1
1
1
#
..
.
⊥
⊥
⊥
1
1
#
..
.
⊥
⊥
⊥
⊥
⊥
#
..
.
⊥
1
1
1
1
#
..
.
⊥
⊥
1
1
1
#
···
PGA enters a column equipped
with a memory token (in first
column at position 0), works
in 2-way mode on the column,
puts a memory update token
say at position j,
starts on next column with
memory token at position j,
etc.
(Memory token stays, update
token may be moved before
reaching final position on column)
Two Versions of PGAs
Version 1: “Micro-step PGA”
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Explicit transitions for two-way processing of columns
Version 2: “Macro-step PGA”
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Use only macro transitions of the form (p, x), z 7→ (q, y ).
“Starting in state p with memory token at position x on a
column of height z, the computation terminates in state q
with memory update token at position y .”
Transition function is represented by MSO formulas
ϕp,q (x, z, y ) such that for each p:
∀x∀z∃=1 y ϕp,q (x, z, y ) where q is unique
Theorem
Version 1 and Version 2 are equivalent.
We call this model N-memory (parity) automata.
ω-Languages Recognizable by PGA’s
1. {1 2 3 4 . . .}
2. {α ∈ Nω | α is unbounded}
3. {m0 m1 m2 . . . | mi+1 = mi + 1 or mi+1 = mi − 1}
4. {m0 m1 m2 . . . | mi+1 even iff mi odd}
5. {m0 m1 m2 . . . | m2i+2 = m2i + 1 if m2i+1 even
m2i+2 = m2i − 1 if m2i+1 odd }
Theorem
Deterministic PGA’s are strictly weaker than nondeterministic ones.
The emptiness problem for N-memory parity automata is decidable.
This fails when the automata are equipped with two memory and
memory update tokens.
N-Memory Transducers
An N-memory transducer is defined in analogy to a PGA, with the
following modification:
Reading a finite word m0 . . . m`−1 , it produces a number
m` on the initially empty next column:
When given a starting state p and the memory token on
the empty column, it places two tokens:
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the output token, specifying the output number m`
the memory update token at a position j, referring
to this m` (as for standard PGA)
plus a state q at termination.
Weakness of N-Memory Transducers
A pumping argument shows:
Lemma (Boundedness Lemma)
There is a bound B such that for each i, the uniquely determined
m is in [0, B] ∪ [i − B, i + B] and the uniquely determined j is in
[0, B] ∪ [i − B, i + B] ∪ [m − B, m + B]
Lifting Church’s Problem to the Baire Space
Automata for ω-Words over N
Solving Games in the Baire Space
Main Result
Theorem
For any Baire space game Γ(L(A)), defined by a N-memory parity
automaton A, one can
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decide who wins Γ(L(A)), and
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construct a winning strategy for the winner, realized by an
N-memory transducer.
From a Parity PGA to a Game Arena
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Given the parity PGA A with state set Q, introduce states
(q, 0) and (q, 1) with q ∈ Q for the two players;
write Q0 , and Q1 for the resulting two copies of Q.
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Construct a game arena GA with domain (Q0 ∪ Q1 ) × N and
edges
m
(r , i) −
→ (s, j)
according to the macro transitions of A.
The Skeleton of GA
GA is coded in a product: Skeleton(GA ) := (Q0 ∪ Q1 ) × (N, Succ)
Skeleton(GA) in Detail
Domain: (Q0 ∪ Q1 ) × N
Available relations:
I Succ =
(r , n), (r , n + 1) | r ∈ Q0 ∪ Q1 , n ∈ N
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Pr = {(r , n) | n ∈ N} (fixing membership in a column)
SameRow = (r , n), (s, n) | r , s ∈ Q0 ∪ Q1 , n ∈ N
Variables now range over positions in this grid.
We can extract from an element x the two components:
state(x) (in Q0 ∪ Q1 ) and number(x) (the number component).
From Skeleton(GA) to GA
Theorem (Büchi + Folklore,
see e.g. Blumensath, Colcombet, Löding 2007)
For any finite sets Q0 , Q1 , the MSO-theory of
Skeleton(GA ) = (Q0 ∪ Q1 ) × (N, Succ) is decidable.
Lemma
GA is MSO-interpretable in Skeleton(GA ).
(Straightforward proof by describing the edge relation of GA in
Skeleton(GA ))
Consequence:
Proposition
The MSO-theory of GA is decidable.
Deciding the Winner
The classical proof of positional determinacy of parity games
shows:
Proposition (Definability Lemma)
Given a parity game graph (Q0 ∪ Q1 , E )
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with MSO-definable priorities,
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with an MSO-definable well-ordering on Q0 ∪ Q1
the winning regions of the two players and edge relations defining
respective winning memoryless strategies are MSO-definable.
Now we can check whether Player 0 wins Γ(L(A)):
Describe the initial vertex ((q0 , 0), 0) of GA by a formula ψinit (x),
let ϕ0 (x) be formula defining the winning region of Player 0.
Check whether
GA |= ∃x ψinit (x) ∧ ϕ0 (x)
Towards Constructing a Transducer
Proposition
Given an MSO-definable winning strategy of player 0 in the parity
game on GA , there is an N-memory transducer realizing a winning
strategy in Γ(L(A)).
Proof: Assume Player 0 wins Γ(L(A)).
Use the Definability Lemma to obtain an MSO-formula ϕ(x, z, y )
defining a winning strategy on the winning region W0 of GA .
We have:
For each x = (r , i) in W0 there is a unique z = m and a unique
y = (s, j) satisfying ϕ.
The Transition Test Lemma
Lemma
ϕ
We can construct a 2-way automaton Crs
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working over ω-words which are unlabelled except for tokens
at positions i, m, j,
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which terminates,
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and checks whether the MSO-formula ϕ is true for (r , i), m,
and (s, j).
The Transducer
The task is to find, for given (r , i),
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the output m
the updated memory (s, j)
such that ϕ is satisfied for (r , i), m, (s, j).
This is done as follows, relying on the Boundedness Lemma:
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Given r and the memory token on i,
move the output token through [0, B] ∪ [i − B, i + B], ranging
over positions m,
and in each case move the memory update token (for j)
through [0, B] ∪ [i − B, i + B] ∪ [m − B, m + B] for all choices
of s,
for each choice invoking the automaton Crs ,
until a combination of token positions m, j and a state s is
detected such that (r , i), m, (s, j) satisfy ϕ.
Summary and Perspectives
It seems we have a first example of an algorithmic solution of
games in the Baire space.
Some open issues:
1.
2.
3.
4.
5.
6.
7.
8.
Find a more direct construction.
Supply a complexity analysis.
Clarify which logic may correspond to PGA’s.
Strengthen the automaton model
(say by combining with register automata)
Replace plays over N by plays over Σ∗ for finite Σ.
Related: Find such results relying on decidability of the
MSO-theory of the infinite binary tree rather than of
(N, Succ).
Extend to a timed setting.
etc. etc.