Weakly maximal decidable structures (original) (raw)

Nonmaximal decidable structures

Journal of Mathematical Sciences, 2009

Given any infinite structure M with a decidable first-order theory, we give a sufficient condition in terms of the Gaifman graph of M that ensures that M can be expanded with some nondefinable predicate in such a way that the first-order theory of the expansion is still decidable. Bibliography: 10 titles.

Decidability of Monadic Theories

Mathematical Foundations of Computer Science, Praha, Czechoslovakia, September 3–7, 1984. Proceedings. Lecture Notes in Computer Science. V. 176., 1984

In this survey article the author describes recent results on monadic theories. The first part concerns the monadic theory of < N, ≤, f > MTf, where N is the set of nonnegative integers, ≤ is the usual order on N, and f is a function from N into N with a finite range. f can be thought of as an ω-word in some finite alphabet. The author gives a general theorem characterizing the decidability of MTf. In the case where f is almost periodic one gets MTF is decidable if and only if f is computable and effectively almost periodic. Results on the monadic theory of < Z, ≤, f > where Z is the set all integers are also presented and their relation to symbolic dynamics indicated. The notion of a minimal transducer is described and a uniformization theorem for the monadic theory of < N, ≤> is given in terms of these automata. The second part of the paper describes a proof due to Muchnik of Rabin’s theorem on the decidability of the monadic theory of S2S. Muchnik’s proof uses a notion of automaton but avoids transfinite induction. The paper ends with brief remarks on weak monadic theories.

On countable chains having decidable monadic theory

The Journal of Symbolic Logic, 2012

Rationals and countable ordinals are important examples of structures with decidable monadic second-order theories. A chain is an expansion of a linear order by monadic predicates. We show that if the monadic second-order theory of a countable chain C is decidable then C has a non-trivial expansion with decidable monadic second-order theory.

On the Undecidability of Second-Order Unification

Information and Computation/information and Control, 2000

There is a close relationship between word unification and secondorder unification. This similarity has been exploited, for instance, in order to prove decidability of monadic second-order unification and decidability of linear second-order unification when no second-order variable occurs more than twice. The attempt to prove the second result for (nonlinear) second-order unification failed and led instead to a natural reduction from simultaneous rigid E-unification to this second-order unification. This reduction is the first main result of this paper, and it is the starting point for proving some novel results about the undecidability of second-order unification presented in the rest of the paper. We prove that second-order unification is undecidable in the following three cases: (1) each secondorder variable occurs at most twice and there are only two second-order variables; (2) there is only one second-order variable and it is unary; (3) the following conditions (i) (iv) hold for some fixed integer n: (i) the arguments of all second-order variables are ground terms of size <n, (ii) the arity of all second-order variables is <n, (iii) the number of occurrences of second-order variables is 5, (iv) there is either a single second-order variable or there are two second-order variables and no first-order variables. ]

Monadic Second-Order Logic with Arbitrary Monadic Predicates

Lecture Notes in Computer Science, 2014

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, automatatheoretic 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.

Decidability of the Theory of the Totally Unbounded omega-Layered Structure

Workshops, 2004

In this paper, we address the decision problem for a sys- tem of monadic second-order logic interpreted over an!- layered temporal structure devoid of both a finest layer and a coarsest one (we call such a structure totally unbounded). We propose an automaton-theoretic method that solves the problem in two steps: first, we reduce the considered prob- lem to the

Decidability of MSO Theories of Tree Structures

Lecture Notes in Computer Science, 2004

In this paper we provide an automaton-based solution to the decision problem for a large set of monadic second-order theories of deterministic tree structures. We achieve it in two steps: first, we reduce the considered problem to the problem of determining, for any Rabin tree automaton, whether it accepts a given tree; then, we exploit a suitable notion of tree equivalence to reduce (a number of instances of) the latter problem to the decidable case of regular trees. We prove that such a reduction works for a large class of trees, that we call residually regular trees. We conclude the paper with a short discussion of related work.

Monadic Monadic Second Order Logic

2022

One of the main reasons for the correspondence of regular languages and monadic second-order logic is that the class of regular languages is closed under images of surjective letter-to-letter homomorphisms. This closure property holds for structures such as finite words, finite trees, infinite words, infinite trees, elements of the free group, etc. Such structures can be modelled using monads. In this paper, we study which structures (understood via monads in the category of sets) are such that the class of regular languages (i.e. languages recognized by finite algebras) are closed under direct images of surjective letter-to-letter homomorphisms. We provide diverse sufficient conditions for a monad to satisfy this property. We also present numerous examples of monads, including positive examples that do not satisfy our sufficient conditions, and counterexamples where the closure property fails.