Monadic Second-Order Logic with Arbitrary Monadic Predicates (original) (raw)

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.

First order quantifiers in monadic second order logic

The Journal of Symbolic Logic, 2004

This paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].

Quantitative Monadic Second-Order Logic

2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, 2013

While monadic second-order logic is a prominent logic for specifying languages of finite words, it lacks the power to compute quantitative properties, e.g. to count. An automata model capable of computing such properties are weighted automata, but logics equivalent to these automata have only recently emerged. We propose a new framework for adding quantitative properties to logics specifying Boolean properties of words. We use this to define Quantitative Monadic Second-Order Logic (QMSO). In this way we obtain a simple logic which is equally expressive to weighted automata. We analyse its evaluation complexity, both data and combined complexity, and show completeness results for combined complexity. We further refine the analysis of this logic and obtain fragments that characterise exactly subclasses of weighted automata defined by the level of ambiguity allowed in the automata. In this way, we define a quantitative logic which has good decidability properties while being resonably expressive and enjoying a simple syntactical definition. I. INTRODUCTION Using logics as specification languages for properties of finite and infinite words or trees has a long history in computer science. Of particular importance in this context is Monadic Second-Order Logic (MSO), the extension of first-order logic by quantification over sets of positions in the input word (see e.g. [23], [10]). The prominence of MSO as logic over words has many causes: It is a very expressive and yet simple logic in which many properties can be expressed very naturally. In fact, Büchi's classical theorem [6] states that a language is recognisable by a finite state automaton if, and only if, it is definable in MSO. Hence, MSO can define precisely the regular languages and provides an elegant specification mechanism for regular properties. Furthermore , the proof of the theorem is algorithmic which implies that MSO formulas can effectively be compiled into finite automata which can then be run in linear time on any input word. Finally, MSO has very good decidability properties and standard problems such as satisfiability and therefore equivalence and containment of formulas are decidable over finite words. While MSO is an elegant and highly successful mechanism for specifying word languages, there are many ap

Fragments of Monadic Second-Order Logics Over Word Structures

Electronic Notes in Theoretical Computer Science, 2005

In this paper, we explore the expressive power of fragments of monadic second-order logic enhanced with some generalized quantifiers of comparison of cardinality over finite word structures. The full monadic second-order fragment of the logics that we study correspond to the famous linear hierarchy, see , and their existential fragments characterize some sequential recognizers. We prove that the first-order closure of the existential fragments of these logics is strictly beyond the existential fragments.

On the expressive power of monadic least fixed point logic

Theoretical Computer Science, 2006

Monadic least fixed point logic MLFP is a natural logic whose expressiveness lies between that of first-order logic FO and monadic second-order logic MSO. In this paper we take a closer look at the expressive power of MLFP. Our results are 1. MLFP can describe graph properties beyond any fixed level of the monadic second-order quantifier alternation hierarchy.

On the Expressive Power of Monadic Least Fixed Point Logic (Full Version

Monadic Second Order Logic and Automata on Infinite Words : Büchi ’ s Theorem

2007

Büchi’s theorem establishes the equivalence of the satisfiability relation for monadic second-order logic, and the acceptance relation for Büchi automata. The development of the theories of monadic second-order logic and Büchi automata follows Thomas’s survey[6] closely, in that all of the concepts and results found in this report are also in [6]. However, because the scope of Thomas’s survey is much greater, he develops the theories in a more general (and more complicated) way than is necessary to understand Büchi’s theorem, and he only sketches the proof of Büchi’s theorem, which is given in detail here.

Expressiveness of Monadic Second-Order Logics on Infinite Trees of Arbitrary Branching Degree

2012

"In this thesis we study the expressive power of variants of monadic second-order logic (MSO) on infinite trees by means of automata. In particular we are interested in weak MSO and well-founded MSO, where the second-order quantifiers range respectively over finite sets and over subsets of well-founded trees. On finitely branching trees, weak and well-founded MSO have the same expressive power and are both strictly weaker than MSO. The associated class of automata (called weak MSO-automata) is a restriction of the class characterizing MSO-expressivity. We show that, on trees with arbitrary branching degree, weak MSO-automata characterize the expressive power of well-founded MSO, which turns out to be incomparable with weak MSO. Indeed, in this generalized setting, weak MSO gives an account of properties of the ‘horizontal dimension’ of trees, which cannot be described by means of MSO or well-founded MSO formulae. In analogy with the result of Janin and Walukiewicz for MSO and the modal μ-calculus, this raises the issue of which modal logic captures the bisimulation-invariant fragment of well-founded MSO and weak MSO. We show that the alternation-free fragment of the modal μ-calculus and the bisimulation-invariant fragment of well-founded MSO have the same expressive power on trees of arbitrary branching degree. We motivate the conjecture that weak MSO modulo bisimulation collapses inside MSO and well-founded MSO."

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.

Monadic Second-Order Logic with Arbitrary Monadic Predicates (original) (raw)

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