# Decidable extensions of Church's problem

TL;DR: 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, such that the Church problem with the parameter P is decidable.

Abstract: For a two-variable formula B(X,Y) of Monadic Logic of Order (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. Buchi 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.

## Summary (2 min read)

### 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.
- In [11, 14] the authors 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.
- The authors can compute for the winning player in Gωϕ a finite- state winning strategy.
- The authors main results show that the synthesis problem for each predicate P ∈ ER is decidable.

### 2 Preliminaries and Background

- The authors use N for the set of natural numbers and ω for the first infinite ordinal.
- The authors use the expressions “chain” and “linear order” interchangeably.
- 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 secondorder quantifiers ∃X and ∀X.

### 2.2 Elements of the composition method

- The authors proofs make use of the technique known as the composition method developed by Feferman-Vaught and Shelah [8, 17].
- So, there are finitely many ≡n-equivalence classes of l-structures.
- The authors will use only special cases of this definition in which the index chain I and the summand chains.
- The function which maps the pairs of characteristic formulas to their sum is a recursive function.
- The authors often use the following well-known lemmas: Lemma 2.8.

### 3 Game types

- The authors define games on game types and show that these game are reducible to the McNaughton games.
- But first the authors introduce a terminology, define finite-memory strategies and fix some notational conventions.
- Whenever M is clear from the context the authors write typem(ρ) for typem(M ⌢ρ).
- Notational Conventions 1. In Hintikka’s Lemma the authors considered formulas with the free variables among X1, . . . ,Xl. Definition 3.1.
- The authors consider the following ω-game Game(F,G). Game(F,G): The following proposition plays an important role in their proofs: Proposition 3.4.

### 4 Winning strategies over classes of finite chains

- In this subsection the authors will consider the games over expansions of finite chains.
- The only difference is that these games are of finite length.
- The games over an l-chains with m elements have m rounds.
- ⊓⊔ Proposition 4.6 is crucial for the design of their algorithm, due the decidability of (4).

### 5 Algorithm

- For every MLO formula ϕ(X1,X2, P ), first construct a set of the characteristic formulas G such that ϕ is equivalent to their disjunction and then use the following algorithm.
- From U and {V l}∞l=0 the authors can compute the desirable Out.
- Usually, this property fails for ER predicates; however, the sequence of the distances modulo n behaves periodically.
- The authors algorithm is more subtle than the above sketch for Pex and relies on this periodicity.
- The rest of the game Player I will play according to his finite-memory strategy st3(F,G2) computed in the Step 3 Clearly, the described strategy is a finite-memory strategy.

### 6 Further Results and Open Questions

- The authors proved that the finite-memory synthesis problem is decidable for the expansions of ω by the predicates from ER.
- The game GMϕ (h1, h2) with look-ahead h1 for Player I and look-ahead h2 for Player II is defined as follows.
- The proof of the next proposition is similar to the proof of Theorem 1.4.
- There is an algorithm that for every MLO formula ϕ(X1,X2, Z) decides whether Player I has a finite-memory winning strategy in GMϕ (h1, h2), and if so, constructs such a strategy.
- It is plausible that in their proofs the compositional methods can be hidden and a presentation can be given based on automata theoretic concepts.

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##### References

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### "Decidable extensions of Church's pr..." refers background in this paper

...Theorem 1.1 (Büchi-Landweber, 1969)....

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...Decidable Expansions of ω Büchi [1] proved that the monadic theory of ω = (N, <) is decidable....

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...Decidable Expansions of ω Büchi [1] proved that the monadic theory of ω = (N, ) is decidable....

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...In [2], Büchi and Landweber prove the computability of the synthesis problem in ω = (N, ) (no parameters)....

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### "Decidable extensions of Church's pr..." refers background in this paper

...In [2], Büchi and Landweber prove the computability of the synthesis problem in ω = (N, <) (no parameters)....

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