Monadic logical definability of NP-complete problems

2000

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is de nable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quanti cation. Such quanti cations are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this a ects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL de nable graph properties. Finally, our results are also applicable to SAT and ]SAT . ? 2001 Elsevier Science B.V. All rights reserved.

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.

2017

In the nineties Immerman and Medina initiated the search for syn- tactic tools to prove NP-completeness. In their work, amongst several results, they conjecture that the NP-completeness of a problem defined by the conjunction of a sentence in Existential Second Order Logic with a First Order sentence, necessarily imply the NP-completeness of the problem defined by the Existential Second Order sentence alone. This is interesting because if true it would justify the restriction heuristic pro- posed in Garey and Johnson in his classical book on NP completeness, which roughly says that in some cases one can prove NP- complete a problem A by proving NP-complete a problem B contained in A. Borges and Bonet extend some results from Immerman and Medina and they also prove for a host of complexity classes that the Immerman- Medina conjecture is true when the First Order sentence in the conjunc- tion is universal. Our work extends that result to the Second Level of the Polynomial-Time Hierarchy.

Theoretical Computer Science, 2004

In this paper, we introduce and study some syntactical fragments of monadic second-order and ÿrst-order (PEANO) arithmetic which we will prove the connection to famous complexity classes. Starting from descriptive complexity results, and giving an e ective method for translating formulas between di erent logical structures representing encodings of integers, we give some new arithmetical characterizations of NP, PH, NL, and P.

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.

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

The finite satisfiability problem of monadic second order logic is decidable only on classes of structures of bounded tree-width by the classic result of Seese (1991). We prove the following problem is decidable: Input: (i) A monadic second order logic sentence $\alpha$, and (ii) a sentence $\beta$ in the two-variable fragment of first order logic extended with counting quantifiers. The vocabularies of $\alpha$ and $\beta$ may intersect. Output: Is there a finite structure which satisfies $\alpha\land\beta$ such that the restriction of the structure to the vocabulary of $\alpha$ has bounded tree-width? (The tree-width of the desired structure is not bounded.) As a consequence, we prove the decidability of the satisfiability problem by a finite structure of bounded tree-width of a logic extending monadic second order logic with linear cardinality constraints of the form $|X_{1}|+\cdots+|X_{r}|<|Y_{1}|+\cdots+|Y_{s}|$, where the $X_{i}$ and $Y_{j}$ are monadic second order variable...

Journal of Computer and System Sciences, 2014

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier Q 1 is definable in terms of another quantifier Q 2 , the base logic being monadic second-order logic, reduces to the question if a quantifier Q 1 is definable in FO(Q 2 , <, +, ×) for certain first-order quantifiers Q 1 and Q 2 . We use our characterization to show new definability and non-definability results for second-order generalized quantifiers. We also show that the monadic second-order majority quantifier Most 1 is not definable in second-order logic.

Lecture Notes in Computer Science, 2004

Unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NP-complete. We also prove that Monadic Second-Order Matching is also NP-complete.

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.