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Existential second-order logic (ESO) and monadic second-order logic (MSO) have attracted much interest in logic and computer science. ESO is a much more expressive logic over word structures than MSO. However, little was known about the relationship between MSO and syntactic fragments of ESO. We shed light on this issue by completely characterizing this relationship for the prefix classes of ESO over strings, (i.e., finite word structures). Moreover, we determine the complexity of model checking over strings, for all ESO-prefix classes. We also give a precise characterization of those ESO-prefix classes which are equivalent to MSO over strings, and of the ESO-prefix classes which are closed under complementation on strings. 1 Introduction and Overview of Results Second-order logic (Sigma 1 1 ) over finite structures has attracted the interest of logicians, mathematicians, and computer scientists for a long time. In particular, several important results have been obtained which ...

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... A recent result of Eiter and al [8] , which says that every Existential Secondorder prefix class either describes only regular languages or describes an NPcomplete problem. I suggest to curious readers to see the expository papers of Thomas [25], or Pin's one [22] for more details and results. ...

... Logic. On the other hand, existential quantification over a simple binary relation express all context free languages and some NP −complete languages by the result of Eiter and al [8] , their result says that any prefix class of Existential Second-order Logic either expresses only regular languages or expresses some NP −complete language. ...

... But the undecidability of IMP 2 is in the sense that we can't decide if a given binary predicate, which is a matching, can be whether or not implicitly defined by a first-order formula. The result of Eiter and al [8] discouraged me to look for some more syntactic logic for all classes between N.P. and regular sets. ...

We give in this paper a logical characterization for unambiguous Context Free Languages, in the vein of descriptive complexity. A fragment of the logic characterizing context free languages given by Lautemann, Schwentick and Thérien [18] based on implicit definability is used for this aim. We obtain a new connection between two undecidable problems, a logical one and a language theoretical one.

... A star free set is of dot depth n if and only if it is definable in the boolean closure of Σ n , Where Σ n is the set of first-order formulas allowing n alternations of quantifiers (universal, existential). A recent result of Eiter and al [8] , which says that every Existential Second-order prefix class either describes only regular languages or describes an NP-complete problem. I suggest to curious readers to see the expository papers of Thomas [25], or Pin's one [22] for more details and results. ...

... Logic. On the other hand, existential quantification over a simple binary relation express all context free languages and some NP−complete languages by the result of Eiter and al [8] , their result says that any prefix class of Existential Second-order Logic either expresses only regular languages or expresses some NP−complete language. Moreover they proved that NP-hardness is present with a sentence ∃Rφ where R is a binary predicate and φ is first-order of the appropriate form. ...

... But the undecidability of IMP 2 is in the sense that we can't decide if a given binary predicate, which is a matching, can be whether or not implicitly defined by a first-order formula. The result of Eiter and al [8] discouraged me to look for some more syntactic logic for all classes between N.P. and regular sets. The result makes the link between two undecidable problems , a logical one and a language theoretic one. ...

We give in this paper a logical characterization for unambiguous Context Free Languages, in the vein of descriptive complexity. A fragment of the logic characterizing context free languages given by Lautemann, Schwentick and Thérien [18] based on implicit definability is used for this aim. We obtain a new connection between two undecidable problems, a logical one and a language theoretical one.

This survey article presents some standard and less standard methods used to prove that a language is regular or star-free.

This paper reports about the results achieved so far in the context of a research programme at the cutting point of logic,
formal language theory, and complexity theory. The aim of this research programme is to classify the complexity of evaluating
formulas from different prefix classes of second-order logic over different types of finite structures, such as strings, graphs,
or arbitrary structures. In particular, we report on classifications of second-order logic on strings and of existential second-order
logic on graphs.

By a well-known result due to Büchi and Trakhtenbrot, all monadic second-order sentences over words describe regular languages.
In this paper, we investigate prefix classes of general second-order logic. Such a prefix class is called regular, if each of its sentences describes a regular language, and nonregular otherwise. Recently, the regular and nonregular prefix classes of existential second order logic (Σ1
1) were exhaustively determined. We briefly recall these results and continue this line of research by systematically investigating
the syntactically more complex prefix classes Σ1
k(Q) of second-order logic for each integer k > 1 and for each first-order quantifier prefix Q. We give an exhaustive classification
of the regular and nonregular prefix classes of this form, and derive of complexity results for the corresponding model checking
problems. We also give a brief survey of recent results on the complexity of evaluating existential second-order logic over
graphs, and a list of interesting open problems.

In this paper, we survey results related to the model checking problem for second-order logic over classes of finite structures, including word structures (strings), graphs, and trees, with a focus on prefix classes, that is, where all quantifiers (both first- and second-order ones) are at the beginning of formulas. A complete picture of the prefix classes defining regular and non-regular languages over strings is known, which nearly completely coincides with the tractability frontier; some complexity issues remain to be settled, though. Over graphs and arbitrary relational structures, the tractability frontier is completely delineated for the existential second-order fragment, while it is less explored for trees. Besides surveying some of the results, we mention some open issues for research.

Lautemann et al. (1995) gave a descriptive characterisation of the class of context-free languages, showing that a language is context-free iff it is definable as the set of words satisfying some sentence of a particular logic (fragment) over words. The present notes discuss how to specialise this result to the class of linear languages. Somewhat surprisingly, what would seem the most straightforward specialisation actually fails, due to the fact that linear grammars fail to admit a Greibach normal form. We identify an alternative specialisation, based on an alternative characterisation of context-free languages, also noted by Lautemann et al. (1995).

We exhibit an NP-complete problem defined by an existential monadic second-order (EMSO) formula over functional structures that is:
1
minimal under several syntactic criteria (i.e., any EMSO formula that further strengthens any criterion defines a PTIME problem even if all other criteria are weakened);
2
unique for such restrictions, up to renamings and symmetries.
Our reductions and proofs are surprisingly very elementary and simple in comparison with some recent similar results classifying existential second-order formulas over relational structures according to their ability either to express NP-complete problems or to express only PTIME ones.

Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objects with bounded resources. This chapter consists of two independent parts. The first part is devoted to finite model theory; it is mostly a survey of logics tailored for computational complexity. The second part is devoted to dynamic structures with bounded resources. In particular, we use dynamic structures with bounded resources to model Pascal.

The spectrum of a first-order sentence is the set of cardinalities of its finite models. Jones and Selman showed that a set C of numbers
(written in binary) is a spectrum if and only if C is in the complexity class NEXP (nondeterministic exponential time). An alternative viewpoint of a spectrum is to consider the spectrum of sigma to be the class of finite models of the existential second-order sentence “there exists Q (sigma(Q))”, where Q is the similarity type (set of relational symbols) of sigma. A generalized spectrum is the class of finite models of an existential second-order sentence “there exists Q (sigma(P,Q))”, where sigma is first-order with similarity type P union Q, with P and Q disjoint. Let C be a class of finite structures with
similarity type P, where C is closed under isomorphism. If P is nonempty, we show that C is a generalized spectrum if and only if the set of encodings of members of C is in NP. We unify this result with that of Jones and Selman by encoding numbers in unary rather than binary, so that C is a spectrum if and only if C is in NP. We then have that C is a generalized spectrum if and only if the set of encodings of members of C is in NP, whether or not P is empty. Using this connection between logic and complexity, we take results from complexity theory and convert them into results in logic. We now mention some of our other results. We show that P = NP if and only if the following apparently much stronger condition holds: there is a constant k such that if T is a “countable" function (a standard notion in automata theory), then every set recognizable nondeterministically in time T can be recognized deterministically in time T^k (analogous to Savitch's Theorem for nondeterministic vs. deterministic space complexity). We show that there is a spectrum S such that {n: 2^n is in S} is not a spectrum. In fact, we show that there is such a spectrum S definable using only a single binary relation symbol. This contrasts with the simple result that if S is a spectrum, and if p is a polynomial, then {n: p(n) is in S} is a spectrum. Let us say that a generalized spectrum S is complete if the following condition holds: the complement of every generalized spectrum is a generalized spectrum if and only if the complement of S is a generalized spectrum. We show that there is a complete generalized spectrum defined by “there exists Q (sigma(P,Q))”, where Q consists of a single unary relation symbol, and where P consists of a single binary relation symbol. We show that if we define a complete spectrum similarly, then there is a complete spectrum definable using only a single binary relation symbol. These latter two results are best possible, in terms of minimizing the arity and the number of relation symbols.

Part 1 Mathematical preliminaries: words and languages automata and regular languages semigroups and homomorphisms. Part 2 Formal languages and formal logic: examples definitions. Part 3 Finite automata: monadic second-order sentences and regular languages regular numerical predicates infinite words and decidable theories. Part 4 Model-theoretic games: the Ehrenfeucht-Fraisse game application to FO [decreasing] application to FO [+1]. Part 5 Finite semigroups: the syntactic monoid calculation of the syntactic monoid application to FO [decreasing] semidirect products categories and path conditions pseudovarieties. Part 6 First-order logic: characterization of FO [decreasing] a hierarchy in FO [decreasing] another characterization of FO [+1] sentences with regular numerical predicates. Part 7 Modular quantifiers: definition and examples languages in (FO + MOD(P))[decreasing] languages in (FO + MOD)[+1] languages in (FO + MOD)[Reg] summary. Part 8 Circuit complexity: examples of circuits circuits and circuit complexity classes lower bounds. Part 9 Regular languages and circuit complexity: regular languages in NC1 formulas with arbitrary numerical predicates regular languages and non-regular numerical predicates special cases of the central conjecture. Appendices: proof of the Krohn-Rhodes theorem proofs of the category theorems.

It has been conjectured [FSV93] that an existential secondoder formula, in which the second-order quantification is restricted to unary relations (i.e. a Monadic NP formula), cannot express Graph Connectivity even in the presence of arbitrary built-in relations.
In this paper it is shown that Graph Connectivity cannot be expressed by Monadic NP formulas in the presence of arbitrary built-in relations of degree n0(1). The result is obtained by using a simplified version of a method introduced in [Sch94] that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy.

This chapter focuses on finite automata on infinite sequences and infinite trees. The chapter discusses the complexity of the complementation process and the equivalence test. Deterministic Muller automata and nondeterministic Buchi automata are equivalent in recognition power. Any nonempty Rabin recognizable set contains a regular tree and shows that the emptiness problem for Rabin tree automata is decidable. The chapter discusses the formulation of two interesting generalizations of Rabin's Tree Theorem and presents some remarks on the undecidable extensions of the monadic theory of the binary tree. A short overview of the work that studies the fine structure of the class of Rabin recognizable sets of trees is also presented in the chapter. Depending on the formalism in which tree properties are classified, the results fall in three categories: monadic second-order logic, tree automata, and fixed-point calculi.