Kevin J. Compton

A uniform method for proving lower bounds on the computational complexity of logical theories

Abstract A new method for obtaining lower bounds on the computational complexity of logical theories is presented. It extends widely used techniques for proving the undecidability of theories by interpreting models of a theory already known to be undecidable. New inseparability results related to the well known inseparability result of Trakhtenbrot and Vaught are the foundation of the method. Their use yields hereditary lower bounds (i.e., bounds which apply uniformly to all subtheories of a theory). By means of interpretations lower bounds can be transferred from one theory to another. Complicated machine codings are replaced by much simpler definability considerations, viz., the kinds of binary relations definable with short formulas on large finite sets. Numerous examples are given, including new proofs of essentially all previously known lower bounds for theories, and lower bounds for various theories of finite trees, which turn out to be particularly useful.

Regular languages in NC 1

Abstract We give several characterizations, in terms of formal logic, semigroup theory, and operations on languages, of the regular languages in the circuit complexity class AC 0 , thus answering a question of Chandra, Fortune, and Lipton. As a by-product, we are able to determine effectively whether a given regular language is in AC 0 and to solve in part an open problem originally posed by McNaughton. Using recent lower-bound results of Razborov and Smolensky, we obtain similar characterizations of the family of regular languages recognized by constant-depth circuit families that include unbounded fan-in mod p addition gates for a fixed prime p along with unbounded fan-in boolean gates. We also obtain logical characterizations for the class of all languages recognized by nonuniform circuit families in which mod m gates (where m is not necessarily prime) are permitted. Comparison of this characterization with our previous results provides evidence for a conjecture concerning the regular languages in this class. A proof of this conjecture would show that computing the bit sum modulo p , where p is a prime not dividing m , is not AC 0 -reducible to addition mod m , and thus that MAJORITY is not AC 0 -reducible to addition mod m .

A toolset for supporting UML static and dynamic model checking

The Unified Modeling Language has become widely accepted as a standard in software development. Several tools have been produced to support UML model validation. However most of them support either static or dynamic model checking; and no tools support to check both static and dynamic aspects of a UML model. But a UML model should include the static and dynamic aspects of a software system. Furthermore, these UML tools translate a UML model into a validation language such as PROMELA. But they have some shortcomings: there is no proof of correctness (with respect to the UML semantics) for these tools. In order to overcome these shortcomings, we present a toolset which can validate both static and dynamic aspects of a model; and this toolset is based on the semantic model using Abstract State Machines. Since the toolset is derived from the semantic model, the toolset is correct with respect to the semantic model.

Probabilistic Transforms for Combinatorial Urn Models

In this paper, we present several probabilistic transforms related to classical urn models. These transforms render the dependent random variables describing the urn occupancies into independent random variables with appropriate distributions. This simplifies the analysis of a large number of problems for which a function under investigation depends on the urn occupancies. The approach used for constructing the transforms involves generating functions of combinatorial numbers characterizing the urn distributions. We also show, by using Tauberian theorems derived in this paper, that under certain simple conditions the asymptotic expressions of target functions in the transform domain and in the inverse–transform domain are identical. Therefore, asymptotic information about certain statistics can be obtained without evaluating the inverse transform.

An algebra and a logic for NC

Abstract Presented here are an algebra and a logic characterizing the complexity class NC1, which consists of functions computed by uniform families of polynomial size, log depth circuits. In both characterizations, NC1 functions are regarded as functions from one class of finite relational structures to another. In the algebraic characterization a recursion scheme called upward tree recursion is applied to a class of simple functions. In the logical characterization, first-order logic is augmented by an operator for defining relations by primitive recursion where it is assumed that every structure has an underlying relation BIT giving the binary representations of integers.

Nonconvergence, undecidability, and intractability in asymptotic problems

Abstract Results delimiting the logical and effective content of asymptotic combinatorics are presented. For the class of binary relations with an underlying linear order, and the class of binary functions, there are properties, given by first-order sentences, without asymptotic probabilities; every first-order asymptotic problem (i.e., set of first-order sentences with asymptotic probabilities bounded by a given rational number between zero and one) for these two classes is undecidable. For the class of pairs of unary functions or permutations, there are monadic second-order properties without asymptotic probabilities; every monadic second-order asymptotic problem for this class is undecidable. No first-order asymptotic problem for the class of unary functions is elementary recursive.

The computational complexity of asymptotic problems 1: partial orders

The class of partial orders is shown to have Ol laws for first-order logic and for inductive fixed-point logic, a logic which properly contains first-order logic. This means that for every sentence in one of these logics the proportion of labeled (or unlabeled) partial orders of size n satisfying the sentence has a limit of either 0 or 1 as n goes to co. This limit, called the asymptotic probability of the sentence, is the same for labeled and unlabeled structures. The computational complexity of the set of sentences with asymptotic probability 1 is determined. For first-order logic, it is PSPACE-complete. For inductive fixed-point logic, it is EXPTIME-complete.

SPHIN: A model checker for reconfigurable hybrid systems based on SPIN

We present SPHIN, a model checker for reconfigurable hybrid systems based on the model checker SPIN. We observe that physical (analog) mobility can be modeled in the same way as logical (discrete) mobility is modeled in the @p-calculus by means of channel name passing. We chose SPIN because it supports channel name passing and can model reconfigurations. We extend the syntax of PROMELA and the verification algorithms based on the expected semantics. We demonstrate the tools capabilities by modeling and verifying a reconfigurable hybrid system.

Laws in Logic and Combinatorics

This is a survey of logical results concerning random structures. A class of relational structures on which a (finitely additive) probability measure has been defined has a 0–1 law for a particular logic if every sentence of that logic has probability either 0 or 1. The measure may be an asymptotic probability on finite structures or generated on a class of infinite structures by assigning fixed probabilities to independently occurring properties. Conditions under which all sentences of a logic have a probability, and under which 0–1 laws occur, are examined. Also, the complexity of computing probabilities of sentences is considered.

The third support weight enumerators of the doubly-even, self-dual [32,16,8] codes

We present combinatorial methods for computing the third support weight enumerators of the five doubly-even, self-dual [32,16,8] codes. The methods exploit relationships that exist between support weight enumerators and complete coset weight enumerators of a self-dual code.