5 research outputs found
Monadic Second-Order Logic with Arbitrary Monadic Predicates
We study Monadic Second-Order Logic (MSO) over finite words, extended with
(non-uniform arbitrary) monadic predicates. We show that it defines a class of
languages that has algebraic, automata-theoretic and machine-independent
characterizations. We consider the regularity question: given a language in
this class, when is it regular? To answer this, we show a substitution property
and the existence of a syntactical predicate.
We give three applications. The first two are to give very simple proofs that
the Straubing Conjecture holds for all fragments of MSO with monadic
predicates, and that the Crane Beach Conjecture holds for MSO with monadic
predicates. The third is to show that it is decidable whether a language
defined by an MSO formula with morphic predicates is regular.Comment: Conference version: MFCS'14, Mathematical Foundations of Computer
Science Journal version: ToCL'17, Transactions on Computational Logi
The Church Synthesis Problem with Parameters
For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO) the
Church Synthesis Problem concerns the existence and construction of an operator
Y=F(X) such that ψ(X,F(X)) is universally valid over Nat.
B\"{u}chi and Landweber proved that the Church synthesis problem is
decidable; moreover, they showed that if there is an operator F that solves the
Church Synthesis Problem, then it can also be solved by an operator defined by
a finite state automaton or equivalently by an MLO formula. We investigate a
parameterized version of the Church synthesis problem. In this version ψ
might contain as a parameter a unary predicate P. We show that the Church
synthesis problem for P is computable if and only if the monadic theory of
is decidable. We prove that the B\"{u}chi-Landweber theorem can be
extended only to ultimately periodic parameters. However, the MLO-definability
part of the B\"{u}chi-Landweber theorem holds for the parameterized version of
the Church synthesis problem