The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as the number of variables, quantifiers, operators, etc. A close correspondence has been observed between these two, with many natural logics corresponding exactly to independently defined complexity classes. For the complexity classes that are generally identified with feasible computation, such characterizations require the presence of a linear order on the domain of every structure, in which case the class PTIME is characterized by an extension of first-order logic by means of an inductive operator. No logical characterization of feasible computation is known for unordered structures. We approach this question from two directions. On the one hand, we seek to accurately characterize the expre...

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...

Abstract not found

We investigate the expressive power of the typed -calculus when expressing computations over finite structures, i.e., databases. We show that the simply typed -calculus can express various database query languages such as the relational algebra, fixpoint logic, and the complex object algebra. In our embeddings, inputs and outputs are -terms encoding databases, and a program expressing a query is a -term which types when applied to an input and reduces to an output.

Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

. We introduce the notion of a logic capturing a relativized complexity class uniformly by treating both generalized quantifiers and oracles as indeterminates and requiring that the correspondence be uniform. Besides reinterpreting previous results from this point of view, we show that Fixed Point Logic with inflationary fixed points IFPL captures P uniformly, whereas First Order Logic FOL[K1 ; : : : ; Km ] with any number of generalized quantifiers cannot capture uniformly a relativized complexity class which contains NC2 . This contrasts the fact that in the non--uniform approach both IFPL and FOL[ATC] capture P. 1 Introduction and Survey In this paper we continue our investigations, cf. [MP93, MP94], of the relationship between logics augmented with generalized quantifiers and oracle computations. Intuitively, we think of a logic as a descriptive language and of generalized quantifiers as library calls. The underlying logic allows us to compose such library calls into more comple...

We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilinear-time computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL),and present a variety of problems in these classes, including ql m -complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6= PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on first-order definability, as well as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

We de ne the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P, using quasilinear-time computation and quanti cation over strings of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy hasseveral alternative characterizations. We de ne both SBH (QL) and its corresponding hierarchy of function classes, ql and present a variety of problems in these classes, including m-complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that determining its precise relationship to deterministic time classes can imply P 6 = PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on rst-order de nability, aswell as recursion-theoretic characterizations of function classes corresponding to SBH (QL).

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

Abstract not found