Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computation. A mathematician might ask, "What is the complexity of expressing the property S?" It should not be surprising that these two questions -- that of checking and that of expressing -- are related. However it is startling how closely tied they are when the second question refers to expressing the property in first-order logic. Many complexity classes originally defined in terms of time or space resources have precise definitions as classes in first-order logic. In 1974 Fagin gave a characterization of nondeterministic polynomial time (NP) as the set of properties expressible in second-order existential logic

We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1st-order operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have the complexity classes between P and EXPTIME. Our parameterized fixpoint logics capture the complexity classes P, NP, PSPACE, and EXPTIME, but equality is achieved only over ordered structures. There is, however, an inherent mismatch between complexity and logic -- while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures. To overcome this mismatch, we develop a theory of relational complexity, which bridges tha gap between standard complexity and fixpoint logic. On one hand, we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complex...

The spectrum of a first-order sentence is the set of cardinalities of its finite models. This paper is concerned with spectra of sentences over languages that contain only unary function symbols. In particular, it is shown that a set S of natural numbers is the spectrum of a sentence over the language of one unary function symbol precisely if S is an eventually periodic set. 1 Introduction The spectrum of a first-order sentence is the set of cardinalities of its finite models. That is, if ' is a first-order sentence, and if n is a natural number, then n is in the spectrum of ' precisely if there is a structure A that satisfies ' where the cardinality of the universe of A is n. The notion of a spectrum was introduced by Scholz [Sc52]. As an example, if ' is a first-order sentence that gives the conjunction of the field axioms (' says that + and \Theta are associative and commutative, that \Theta distributes over +, etc.), then it is well-known that the spectrum of ' is the set of po...

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 successor 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 successor structures). Moreover, we determine the complexity of model checking over strings, for all ESO-prefix classes. Let ESO(Q) denote the prefix class containing all sentences of the shape 9RQ' where R is a list of predicate variables, Q is a first-order quantifier prefix from the prefix set Q, and ' is quantifier-free. We show that ESO(9 88) are the maximal standard ESO-prefix classes contained in MSO, thus expressing only regular languages.

this paper various devices operating directly on structures, without encoding. The motivation and benefits for doing this are manyfold. On a fundamental level, encodings of structures seem to be a technical device rather than an intrinsic feature. This point has already been made by several mathematicians such as Tarski [Tar86], and Harvey Friedman [Fri71] (see Section 5). It has come up more recently in the context of databases, where devices computing on structures model more acurately database computation carried out against an abstract interface hiding the internal representation of data. Thus, the primary benefit of studying devices and languages computing on structures is that they clarify issues which are obscured in classical devices such as Turing machines. For example, they yield new notions of complexity, quite different from classical computational complexity. They reflect more acurately the actual complexity of computation, which, like database computation, cannot take advantage of encodings of structures. An example is provided by the query even on a set

We show that the set of square numbers is the spectrum of an F \Sigma 1 1;1 - sentence (existential second order logic with quantification over unary function variables) with two unary function variables, but it is not the spectrum of F \Sigma 1 1;1 -sentences with only one unary function variable. keywords: existential second order logic; finite model theory; computational complexity; formal languages. 1 Introduction The main motivation for Finite Model Theory is to connect Logic to Computational Complexity Theory. The definition of complexity classes by means of definability in various logical calculi establishes a Descriptive Complexity Theory. Its complexity measures are completely machine-independent; instead of the time or space needed by some machine to calculate a query, the query's complexity is measured by means of the logical ressources required to define it. By establishing correspondances between computational and descriptive complexity classes, some (old) logical techn...

. A set of finite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quantification is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quantifiers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a three-level linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively. In this paper it is shown that -- Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the hierarchy coincide; -- the set of graphs with exactly as many vertices as arcs is expressible by unary functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels. 1 Introduction Fagin [Fag74] showed that NP coincides with the class of sets of finite structures that are characteri...

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