Decidable Extensions of Church’s Problem
Alexander Rabinovich
The Blavatnik School of Computer Science, Tel Aviv University, Israel
rabinoa@post.tau.ac.il
Abstract. For a two-variable formula B(X,Y) of Monadic Logic of Or-
der (MLO) the Church Synthesis Problem concerns the existence and
construction of a finite-state operator Y=F(X) such that B(X,F(X)) is
universally valid over Nat.
B ¨uchi and Landweber (1969) proved that the Church synthesis problem
is decidable.
We investigate a parameterized version of the Church synthesis problem.
In this extended version a formula B and a finite-state operator F might
contain as a parameter a unary predicate P.
A large class of predicates P is exhibited such that the Church problem
with the parameter P is decidable.
Our proofs use Composition Method and game theoretical techniques.
1 Introduction
Two fundamental results of classical automata theory are decidability of the
monadic second-order logic of order (MLO) over ω = (N, <) and computability
of the Church synthesis problem. These results have provided the underlying
mathematical framework for the development of formalisms for the description of
interactive systems and their desired properties, the algorithmic verification and
the automatic synthesis of correct implementations from logical specifications,
and advanced algorithmic techniques that are now embodied in industrial tools
for verification and validation.
Decidable Expansions of ω B¨uchi [1] proved that the monadic theory of
ω = (N, <) is decidable. Even before the decidability of the monadic theory
of ω has been proved, it was shown that the expansions of ω by “interesting”
functions have undecidable monadic theory. In particular, the monadic theory
of (N, <, +) and the monadic theory of (N, <, λx.2 × x) are undecidable [15, 20].
Therefore, most efforts to find decidable expansions of ω deal with expansions
of ω by monadic predicates.
Elgot and Rabin [5] found many interesting predicates P for which MLO over
(N, <, P ) is decidable. Among these predicates are the set of factorial numbers
{n! | n ∈ N}, the sets of k-th powers {n
k
| n ∈ N} and the sets {k
n
| n ∈ N} (for
k ∈ N ).
The Elgot and Rabin method has been generalized and sharpened over the
years and their results were extended to a variety of unary predicates (see e.g.,
[18, 16, 3]). In [11, 14] we provided necessary and sufficient conditions for the
decidability of monadic (second-order) theory of expansions of the linear order
of the naturals ω by unary predicates.
Church’s Problem What is known as the “Church synthesis problem” was
first posed by A. Church in [4] for the case of (ω, <). The Church problem is
much more complicated than the decidability problem for M LO. Church uses
the language of automata theory. It was McNaughton (see [9]) who first observed
that the Church problem can be equivalently phrased in game-theoretic language
and in recent years many authors took up the generalizations of such games for
various applications of the algorithmic theory of infinite games (see e.g., [6, 10]).
McNaughton considered games over ω. We consider such games over expansions
of ω by unary predicates.
Let M = (N, <, P ) be the expansion of ω by a unary predicate P . Let
ϕ(X
1
, X
2
, Z) be a formula, where X
1
, X
2
and Z are set (monadic predicate)
variables. The McNaughton game G
M
ϕ
is defined as follows.
1. The game is played by two players, called Player I and Player II.
2. A pla y of the game has ω rounds.
3. At round i ∈ N: first, Player I chooses ρ
X
1
(i) ∈ {0, 1}; then, Player II chooses
ρ
X
2
(i) ∈ {0, 1}. Both players can observe whether i ∈ P .
4. By the end of the play two predicates ρ
X
1
, ρ
X
2
⊆ N have been constructed
1
5. Then, Player I wins the play if M |= ϕ(ρ
X
1
, ρ
X
2
, P ); otherwise, Player II
wins the play.
What we want to know is: Does either one of the players have a winning strategy
in G
M
ϕ
? If so, which one? That is, can Player I choose his moves so that, whatever
way Player II responds we have ϕ(ρ
X
1
, ρ
X
2
, P )? Or can Player II respond to
Player I’s moves in a way that ensures the opposite?
At round i, Player I has access only to ρ
X
1
(0) . . . ρ
X
1
(i−1), ρ
X
2
(0) . . . ρ
X
2
(i−
1) and P (0) . . . P (i).
Hence, a strategy of Player I can be defined as a function which assigns to
any finite sequence
(ρ
X
1
(0), ρ
X
2
(0), P (0)) . . . (ρ
X
1
(i − 1), ρ
X
2
(i − 1), P (i − 1)) (∗, ∗, P (i))
a value in {0, 1} which is taken to be ρ
X
1
(i).
At round i, Player II has access only to ρ
X
1
(0) . . . ρ
X
1
(i), ρ
X
2
(0) . . . ρ
X
2
(i−1)
and P (0) . . . P (i).
Hence, a strategy of Player II can be defined as a function which assigns to
any finite sequence
(ρ
X
1
(0), ρ
X
2
(0), P (0)) . . . (ρ
X
1
(i − 1), ρ
X
2
(i − 1), P (i − 1)) (ρ
X
1
(i), ∗, P (i))
a value in {0, 1} which is taken to be ρ
X
2
(i).
Since strategies are functions from finite strings (over a finite alphabet) to
{0, 1} we can classify them according to their complexity. The recursive strate-
gies, the finite-memory strategies, i.e., the strategies computable by finite-state
transducers are defined in a natural way (see Sect. 3).
1
We identify monadic predicates with their characteristic functions.
We investigate the following parameterized version of the Church synthesis
problem.
Synthesis Problems for M = (N, <, P ), where P ⊆ N
Input: an MLO formula ϕ(X
1
, X
2
, Z).
Task: Check whether Player I has a finite-memory winning strategy in G
M
ϕ
and if there is such a strategy - construct it.
To simplify notations, games and the synthesis problem were previously de-
fined for formulas with three free variables X
1
, X
2
and Z. It is easy to generalize
all definitions and results to formulas ψ(X
1
, . . . , X
m
, Y
1
, . . . Y
n
, Z
1
, . . . , Z
l
) with
many variables. In this generalization at r ound β, Player I chooses values for
X
1
(β), . . . , X
m
(β), then Player II replies by choosing the values to Y
1
(β), . . . , Y
n
(β)
and the structure M provides the interpretation for Z
1
, . . . Z
l
. Note that, strictly
speaking, the input to the synthesis problem is not only a formula, but a for-
mula plus a partition of its free-variables to Player I’s variables and Player II’s
variables and parameter’s variables.
In [2], B¨uchi and Landweber prove the computability of the synthesis problem
in ω = (N, <) (no parameters).
Theorem 1.1 (B¨uchi-Landweber, 1969). Let ϕ(
¯
X,
¯
Y ) be a formula, where
¯
X and
¯
Y are disjoint lists of variables. Then:
Determinacy: One o f the players has a winning strategy in the game G
ω
ϕ
.
Decidability: It is decidable which of the players has a winning strategy.
Finite-state strategy: The player who has a winning strategy, als o has a finite-
state winning strategy.
Synthesis algorithm: We can compute for the winning player in G
ω
ϕ
a finite-
state winning strategy.
The determinacy part of the theorem follows from the topological arguments.
In particular for every expansion M of ω by unary predicates, the game G
M
ϕ
is
determinate.
Let M be an expansion of ω by unary predicates. We proved in [12], that
there is an algorithm which for every MLO formula ϕ decides who wins G
M
ϕ
if
and only if the monadic theory of M is decidable. Moreover, we proved that if the
monadic theory of M is decidable, then the player who has a winning strategy in
G
M
ϕ
has a recursive MLO-definable winning strategy which is computable from
ϕ.
The finite-state strategy part of Theorem 1.1 fails for decidable expansions
of ω. For example, let Fac = {n! | n ∈ N} be the set of factorial numbers. The
monadic theory of M
fac
:= (N, <, Fac) is decidable by [5]. Let ϕ(X
1
, X
2
, Z) be
a formula which specifies that t ∈ X
1
iff t + 1 ∈ Z (hence for the game G
M
fac
ϕ
the
moves of Player II are irrelevant). It is easy to see that Player I has a winning
strategy in G
M
fac
ϕ
, yet Player I has no finite-state winning strategy in this game.
The results of this paper imply that the synthesis problem for (N, <, Fac) is
decidable.
Main Result Our main result describes a large class of predicates P such that
the synthesis problem for (N, <, P ) is decidable.
An ω-sequence a
i
is said to be ultimately periodic with lag l and period d if
a
i
= a
i+d
for i > l.
Definition 1.2. Let
¯
k = (k
1
< k
2
< . . . k
i
< . . . ) be an increasing ω-sequence
of integers.
1.
¯
k is sparse if for each d there is n such that k
i+1
− k
i
> d for each i > n.
¯
k is effectively sparse if there is an algorithm that for each d computes n
such that k
i+1
− k
i
> d for each i > n.
2.
¯
k is ultimately reducible if for every m > 1 the sequence k
i
mod m is ulti-
mately periodic.
¯
k is effectively ultimately reducible if there is an algorithm
that for each m computes a lag and a period of k
i
mod m.
Definition 1.3. Let ER be the class of increasing recursive ω-sequences of in-
tegers which are effectively sparse and effectively ultimately reducible.
Let P ⊆ N be a predicate. We denote by Enum(P ) the sequence (k
1
, k
2
. . . k
i
. . . )
which enumerates the elements of P in the increasing order. Often we do not
distinguish between P and Enum(P ), In particular we say that a predicate is
ER predicate if Enum(P ) is in ER. The class ER contains many interesting
predicates. It contains the set Fact={n! | n ∈ N} of factorial numbers, the sets
{k
n
| n ∈ N}, the sets {n
k
| n ∈ N}. It has nice closure properties, e.g. if
¯
k and
¯
l are in ER then {k
i
+ l
i
| i ∈ N}, {k
i
× l
i
| i ∈ N}, and {k
l
i
i
| i ∈ N} are in ER.
In [18], Siefkes introduced ER predicates and generalized Elgot-Rabin con-
traction method to prove that for every ER predicate P the monadic theory of
M = (N, <, P ) is decidable. Our main results show that the synthesis problem
for each predicate P ∈ ER is decidable.
Theorem 1.4 (Main). Let P be an ER predicate and let M = (N, <, P ).
There is an algorithm that for every MLO formula ϕ(X
1
, X
2
, Z) decides whether
Player I has a finite-memory winning strategy in G
M
ϕ
, and if so constructs such
a st rategy.
Our algorithm is based on game theoretical techniques and the composition
method developed by Feferman-Vaught, Shelah and others.
Organization of the paper The article is organized as follows. The next sec-
tion recalls standard definitions about the monadic second-order logic of order,
and summarizes elements of the composition method. In Section 3, we introduce
game-types, define games on game types and show that these game are reducible
to the McNaughton games. Section 4 consider games over finite chains. Sufficient
conditions are provided for existence of a finite state strategies which uniformly
wins over a class of finite chains.
Section 5 describes an algorithm for the synthesis problem over the expan-
sions of ω by ER predicates, and proves the soundness of the algorithm, i.e., if
the algorithm outputs a strategy for G
M
ϕ
, then it is a finite state strategy which
wins ϕ over M. The proof of completeness appears in the full version of this
paper [13]. Further results and open questions are discussed in Sect. 6.
2 Preliminaries and Background
We use i, j, n, k, l, m, p, q for natural numbers. We use N for the set of natural
numbers and ω for the first infinite ordinal. We use the expressions “chain” and
“linear order” interchangeably. A chain with m elements will be denoted by m.
We use P(A) for the set of subsets of A.
2.1 The Monadic Logic of Order (MLO )
Syntax The syntax of the monadic second-order logic of order - MLO has in
its vocabulary individual (first order ) variables t
1
, t
2
. . ., monadic second-order
variables X
1
, X
2
. . . and one binary relation < (the order).
Atomic formulas are of the form X(t) and t
1
< t
2
. Well formed formulas
of the monadic logic MLO are obtained from atomic formulas using Boolean
connectives ¬, ∨, ∧, → and the first-order quantifiers ∃t and ∀t, and the second-
order quantifiers ∃X and ∀X. The quantifier depth of a formula ϕ is denoted by
qd(ϕ).
We use upper case letters X, Y , Z,... to denote second-order variables; with
an overline,
¯
X,
¯
Y , etc., to denote finite tuples of variables.
Semantics A structure is a tuple M := (A, <
M
,
¯
P
M
) where: A is a non-empty
set, <
M
is a binary relation on A, and
¯
P
M
:=
P
M
1
, . . . , P
M
l
is a finite tuple
of subsets of A.
If
¯
P
M
is a tuple of l sets, we call M an l-structure. If <
M
linearly orders A,
we call M an l-chain. When the specific l is unimportant, we simply say that
M is a labeled chain.
Suppose M is an l-structure and ϕ a formula with free-variables among
X
1
, . . . , X
l
. We define the relation M |= ϕ (read: M satisfies ϕ) as usual, un-
derstanding that the s econd-order quantifiers range over subsets of A.
Let M be an l-structure. The monadic theory of M, MTh(M), is the set of
all formulas with free-variables among X
1
, . . . , X
l
satisfied by M.
From now on, we omit the superscript in ‘<
M
’ and ‘
¯
P
M
’. We often write
(A, <) |= ϕ(
¯
P ) meaning (A, <,
¯
P ) |= ϕ.
For a chain M = (A, <,
¯
P ) and a subset I of A, we denote by M
I the
subchain of M over the set I.
2.2 Elements of the composition method
Our proofs make use of the technique known as the composition method de-
veloped by Feferman-Vaught and Shelah [8, 17]. To fix notations and to aid the
reader unfamiliar with this technique, we briefly review the definitions and re-
sults that we require. A more detailed presentation can be found in [19] or [7].
Let n, l ∈ N. We denote by Form
n
l
the set of MLO formulas with free variables
among X
1
, . . . , X
l
and of quantifier depth ≤ n.
Definition 2.1. Let n, l ∈ N and let M, N be l-structures. The n-theory of M
is Th
n
(M) := {ϕ ∈ Form
n
l
| M |= ϕ}. If Th
n
(M) = Th
n
(N ), we say that M
and N are n-equivalent and write M ≡
n
N .