Steven Lindell

A logspace algorithm for tree canonization (extended abstract)

We present a solution to the problem of assigning to each directed tree <italic>T</italic> of size <italic>n</italic> a unique isomorphism invariant name for <italic>T</italic>, using only work space <italic>O</italic>(log <italic>n</italic>). Hence, tree isomorphism is computable in logspace. As another consequence, we obtain the corollary that the set of logspace computable queries (<italic>Lspace</italic>) on trees is recursively enumerable. Our results extend easily to undirected trees and even forests.

An analysis of fixed-point queries on binary trees

Lindell, S., An analysis of fixed-point queries on binary trees, Theoretical Computer Science 85 (1991) 75-95. The presence of ordering appears to play an essential role in the logical expressibility of polynomialtime queries on finite structures. By examining the expressibility and complexity of inductive queries on the class of complete unordered binary trees, we are able to show that the ability to calculate cardinality is strictly less powerful than the assumption of order.

A purely logical characterization of circuit uniformity

Utilizing the connection between uniform constant-depth circuits and first-order logic with numerical predicates, the author provides a purely logical characterization of uniformity based on the intrinsic properties of these predicates. By requiring a numerical predicate R to satisfy a natural extensibility condition-that it can be translated to a polynomially magnified domain based on tuple constructions-he shows that R must already be elementarily definable from < and bit (both of which satisfy the extensibility condition). The answer is motivated by, and coincides with, DLOGTIME uniformity.<<ETX>>

Elementary Properties of the Finite Ranks

This note investigates the class of finite initial segments of the cumulative hierarchy of pure sets. We show that this class is first-order definable over the class of finite directed graphs and that this class admits a first-order definable global linear order. We apply this last result to show that FO(<, BIT) = FO(BIT).

First Order Logic, Fixed Point Logic and Linear Order

The Ordered conjecture of Kolaitis and Vardi asks whether fixed-point logic differs from first-order logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive definability and infinitary logic on proficient classes of finite structures (those admitting an unbounded induction). In particular, for a class of finite structures, we introduce a compactness notion which yields a new proof of a ramified version of McColms second conjecture. Furthermore, we show a connection between a model-theoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). We also elaborate on complexity-theoretic implications of this line of research.

A NORMAL FORM FOR FIRST-ORDER LOGIC OVER DOUBLY-LINKED DATA STRUCTURES

We use singulary vocabularies to analyze first-order definability over doubly-linked data structures. Singulary vocabularies contain only monadic predicate and monadic function symbols. A class of mathematical structures in any vocabulary can be elementarily interpreted in a singulary vocabulary, while preserving notions of total size and degree. Doubly-linked data structures are a special case of bounded-degree finite structures in which there are reciprocal connections between elements, corresponding closely with physically feasible models of information storage. They can be associated with logical models involving unary relations and bijective functions in what we call an invertible singulary vocabulary. Over classes of these models, there is a normal form for first-order logic which eliminates all quantification of dependent variables. The paper provides a syntactically based proof using counting quantifiers. It also makes precise the notion of implicit calculability for arbitrary arity first-order formulas. Linear-time evaluation of first-order logic over doubly-linked data structures becomes a direct corollary. Included is a discussion of why these special data structures are appropriate for physically realizable models of information.

The invariant problem for binary string structures and the parallel complexity theory of querie2

Abstract We define the isomorphism and canonical invariant problems as queries on finite structures, and show that they are first-order definable on binary string structures that include the bit predicate. Applying our results to the parallel complexity theory of queries, we prove a unique correspondence between complexity-derived query classes and parallel complexity classes closed under constant parallel time reducibility. This directly extends a similar theorem of Chandra and Harel originally proved for sequential complexity classes closed under logarithmic space reducibility.

Generalized Implicit Definitions on Finite Structures

We propose a natural generalization of the concept of implicit definitions over finite structures, allowing non-determinism at an intermediate level of a (deterministic) definition. These generalized implicit definitions offer more expressive power than classical implicit definitions. Moreover, their expressive power can be characterized over unordered finite structures in terms of the complexity class NP ∩ co-NP. Finally, we investigate a subclass of these where the non-determinism is restricted to the choice of a unique relation with respect to an implicit linear order, and prove that it captures UP ∩ co-UP also over the class of all finite structures. These results shed some light on the expressive power of non-deterministic primitives.

A constant-space sequential model of computation for first-order logic

We define and justify a natural sequential model of computation with a constant amount of read/write work space, despite unlimited (polynomial) access to read-only input and write-only output. The model is both deterministic, uniform, and sequential. The constant work space is modeled by a finite number of destructive read boolean variables, assignable by formulas over the canonical boolean operations. We then show that computation on this model is equivalent to expressibility in first-order logic, giving a duality between (read-once) constant-space serial algorithms and constant-time parallel algorithms.

The role of decidability in first order separations over classes of finite structures

We establish that the decidability of the first order theory of a class of finite structures C is a simple and useful condition for guaranteeing that the expressive power of FO+LFP properly extends that of FO on C, unifying separation results for various classes of structures that have been studied. We then apply this result to show that it encompasses certain constructive pebble game techniques which are widely used to establish separations between FO and FO+LFP, and demonstrate that these same techniques cannot succeed in performing separations from any complexity class that contains DLOGTIME.