We present a logical approach to complexity over the real numbers with respect to the model of Blum, Shub and Smale. The logics under consideration are interpreted over a special class of two-sorted structures, called R-structures: They consist of a finite structure together with the ordered field of reals and a finite set of functions from the finite structure into R. They are a special case of the metafinite structures introduced recently by Grädel and Gurevich. We argue that R-structures provide the right class of structures to develop a descriptive complexity theory over R. We substantiate this claim by a number of results that relate logical definability on R-structures with complexity of computations of BSS-machines.

We show that for most complexity classes of interest, all sets complete under first-order projections are isomorphic under first-order isomorphisms. That is, a very restricted version of the Berman-Hartmanis Conjecture holds.

The expressive power of first order logic with generalized quantifiers over finite ordered structures is studied. The following problem is addressed: Given a family Q of generalized quantifiers expressing a complexity class C, what is the expressive power of first order logic FO(Q) extended by the quantifiers in Q? From previously studied examples, one would expect that FO(Q) captures L C , i.e., logarithmic space relativized by an oracle in C. We show that this is not always true. However, we derive sufficient conditions on complexity class C such that FO(Q) captures L C . These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, by NP. As an application of this result, it follows that first order logic extended by Henkin quantifiers captures L NP . This answers a question raised by Blass and Gurevich. Furthermore we show that for many families Q of generalized quantifiers (including the family of Henkin quantifiers), each FO(Q...

. We describe a general way of building logics with Lindstrom quantifiers, which capture regular complexity classes on ordered structures with polysize reductions. We then extend this method so as to accommodate complexity classes based on oracle Turing machines. Our main result shows an equivalence between enhancing a logic with a Lindstr om quantifier and enhancing a complexity class with an oracle such that, if K is a set of structures, QK the associated Lindstrom quantifier and L a logic that captures a complexity class D, then the enhanced logic L[K] captures D K - the complexity class of machines in D using oracles for K. Our results are sensitive to the oracle computation model and hold in a natural modification of the unbounded model introduced by Buss [Bus88]. They do not hold in the, so called, space bounded oracle models or those that violate the `relativization thesis' of Buss. Our results generalize and extend previous results of Stewart [Ste93a, Ste93b] and Makowsky and...

We begin by proving that the class of problems accepted by the program schemes of NPS is exactly the class of problems defined by the sentences of transitive closure logic (program schemes of NPS are obtained by generalizing basic non-deterministic while-programs whose tests within while instructions are quantifier-free first-order formulae). We then show that our program schemes form a proper infinite hierarchy within NPS whose analogy in transitive closure logic is a proper infinite hierarchy, the union of which is full transitive closure logic but for which every level of the hierarchy has associated with it a first-order definable problem not in that level. We then proceed to add a stack to our program schemes, so obtaining the class of program schemes NPSS, and characterize the class of problems accepted by the program schemes of NPSS as the class of problems defined by the sentences of path system logic. We show that there is a proper infinite hierarchy within NPSS, with an analo...

We characterize the class of problems accepted by a class of program schemes with arrays, NPSA, as the class of problems defined by the sentences of a logic formed by extending firstorder logic with a particular uniform (or vectorized) sequence of Lindstrom quantifiers. A simple extension of a known result thus enables us to prove that our logic, and consequently our class of program schemes, has a zero-one law. However, we use another existing result to show that there are problems definable in a basic fragment of our logic, and so also accepted by basic program schemes, which are not definable in bounded-variable infinitary logic. As a consequence, the class of problems NPSA is not contained in the class of problems defined by the sentences of partial fixed-point logic even though in the presence of a built-in successor relation, both NPSA and partial fixed-point logic capture the complexity class PSPACE.

It is known, that over 1-dimensional strings, the expressive power of 2-way multihead (non) deterministic automata and (non) deterministic Transitive Closure formulas is (non) deterministic log space [Ib73, Im88]. However, the subset of formulas needed to simulate exactly k heads is unknown. It is also unknown if the automata and formulas have the same expressive power over more general structures such as multidimensional grids. We define a reduction from k-head automata to formulas of arity k, which works also for grids. The method used is a generalization of [Kl56], and the formulas obtained are a generalization of regular expressions to multihead automata and to grid languages. As simple applications, we use the reduction to show that the power of formulas of arity 1 over strings define (classical) regular languages, to give a simpler equivalent of the L=NL open problem, and to establish the equivalence of the automata and formulas over grids. Faculty of Computer Science, Technio...

. We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula and structure transformations. This allows us to re-examine the notion of dependency preserving decomposition and its generalization refinement. The most important aspect of this approach lies in laying the groundwork for a formulation of the Fundamental Problem of Database Design, namely to exhibit desirable differences between translation--equivalent presentations of data and to examine refinements of data presentations in a systematic way. The emphasis in this paper is not on results. The main line of thought is an exploration of the use of an old logical tool in addressing the Fundamental Problem. Translation schemes allow us to have a new look at normal forms of database schemes and to suggest a new line of search for other normal forms. Furthermore we give a characterization of the embedded implicational dependencies (EID'...

. We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula and structure transformations. This allows us to re-examine the notion of dependency preserving decomposition and its generalization refinement. We demonstrate the usefulness of this approach by recasting the theory of vertical and horizontal decompositions in our terminology. The most important aspect of this approach, however, lies in laying the groundwork for a formulation of the Fundamental Problem of Database Design, namely to exhibit desirable differences between translation equivalent presentations of data and to examine refinements of data presentations in a systematic way. The emphasis in this paper is not on results. The main line of thought is an exploration of the use of an old logical tool in addressing the Fundamental Problem. Translation schemes allow us also to have a new look at normal forms of database schemes an...

, April 1996 Abstract. We introduce a new point of view into database schemes by applying systematically an old logical technique: translation schemes, and their induced formula transformation. This allows us to reexamine the notion of dependency preserving decomposition and its generalization refinement. This logical tool allows us to have a new look a BCNF and show that every scheme specified by functional dependencies (FD's) has a dependency preserving refinement into BCNF. Furthermore we give a characterization of the embedded implicational dependencies (EID's) using FD's and inclusion dependencies (ID's) and a basic class of refinements consisting of projections and natural joins. Finally, we discuss how this technique allows us to define refinements of entity-relationship schemes. 1 Introduction Traditional design theory of relational databases comprises the following steps: -- Identify a set of entities; -- Identify, for each entity, a fixed set of attributes; -- Identify a...