Clifford Bergman

Complexity of Some Problems Concerning Varieties and Quasi-Varieties of Algebras

In this paper we consider the complexity of several problems involving finite algebraic structures. Given finite algebras A and B, these problems ask the following. (1) Do A and B satisfy precisely the same identities? (2) Do they satisfy the same quasi-identities? (3) Do A and B have the same set of term operations? In addition to the general case in which we allow arbitrary (finite) algebras, we consider each of these problems under the restrictions that all operations are unary and that A and B have cardinality two. We briefly discuss the relationship of these problems to algebraic specification theory.

COMPUTATIONAL COMPLEXITY OF TERM-EQUIVALENCE

Two algebraic structures with the same universe are called term-equivalent if they have the same clone of term operations. We show that the problem of determining whether two finite algebras of finite similarity type are term-equivalent is complete for deterministic exponential time.

Algebraic Logic and Universal Algebra in Computer Science

Algebraic methods, in particular those of universal algebra and algebraic logic, are playing an increasingly important role in computer science, especially in the areas of algebraic specification of data types, relational data types, relational database theory, logic of programmes, functional and logic programming, and semantics of programming languages. To a large extent this work has been carried forward by computer scientists independent of the very active group of mathematicians who work in universal algebra and algebraic logic. A conference was held at Iowa State University in June 1988 to bring together leading researchers from both groups to identify areas of common interest. Addresses were given by Joel Berman, H.Peter Gumm, Bjarni Jonsson, Dexter Kozen, Istvan Nemeti, Vaughan Pratt, Dana Scott and Eric Wagner. The programme also included contributed papers and a round-table discussion of the role of algebra and logic in computer science.

Amalgamation classes of some distributive varieties

This paper investigates the amalgamation classes of finitely generated varieties with distributive congruence lattices. Necessary and sufficient conditions are given for an algebra to be a member of the amalgamation class of a variety generated by a finite modular lattice or pseudocomplemented distributive lattice and of a filtral variety.

Morita equivalence of almost-primal clones☆

Abstract Two algebraic structures A and B are called categorically equivalent if there is a functor from the variety generated by A to the variety generated by B, carrying A to B, that is an equivalence of the varieties when viewed as categories. We characterize those algebras categorically equivalent to A when A is an algebra whose set of term operations is as large as possible subject to constraints placed on it by the subalgebra or congruence lattice of A, or the automorphism group of A.

An automatic, time-based, secure pairing protocol for passive RFID

This paper introduces the Adopted-Pet (AP) protocol, an automatic (i.e. requiring no human interaction) secure pairing protocol, adequate for the pairing between a passive RFID tag and a reader. Most pairing protocols rely for their security on a certain advantage that the legitimate devices have over any malicious users. Such advantages include proximity (employing near-field communication) or secret keys that are either produced with the assistance of, or verified by, the legitimate user. The advantage exploited by our novel AP protocol is the amount of uninterrupted time spent by the two devices in the proximity (although not requiring near-field communication) of each-other. We discuss several implementation configurations, all based on pseudo-random bit generators, employing short-length LFSRs, and requiring no more than 2000 transistors. This makes the protocol ideally suited for low-cost passive RFID tags. For each configuration we show that the AP protocol is highly secure against occasional malicious entities.

An artificial neural network for wavelet steganalysis

Hiding messages in image data, called steganography, is used for both legal and illicit purposes. The detection of hidden messages in image data stored on websites and computers, called steganalysis, is of prime importance to cyber forensics personnel. Automating the detection of hidden messages is a requirement, since the shear amount of image data stored on computers or websites makes it impossible for a person to investigate each image separately. This paper describes research on a prototype software system that automatically classifies an image as having hidden information or not, using a sophisticated artificial neural network (ANN) system. An ANN software package, the ISU ACL NetWorks Toolkit, is trained on a selection of image features that distinguish between stego and nonstego images. The novelty of this ANN is that it is a blind classifier that gives more accurate results than previous systems. It can detect messages hidden using a variety of different types of embedding algorithms. A Graphical User Interface (GUI) combines the ANN, feature selection, and embedding algorithms into a prototype software package that is not currently available to the cyber forensics community.

Subquasivarieties of regularized varieties

This paper considers the lattice of subquasivarieties of a regular variety. In particular we show that if V is a strongly irregular variety that is minimal as a quasivariety, then the smallest quasivariety containing both V and SI (the variety of semilattices) is never equal to the regularization V of V.We use this result to describe the lattice of subquasivarieties of V in several special but quite common, cases and give a number of applications and examples.

COMPUTATIONAL COMPLEXITY OF GENERATORS AND NONGENERATORS IN ALGEBRA

We discuss the computational complexity of several problems concerning subsets of an algebraic structure that generate the structure. We show that the problem of determining whether a given subset X generates an algebra A is P-complete, while determining the size of the smallest generating set is NP-complete. We also consider several questions related to the Frattini subuniverse, Φ(A), of an algebra A. We show that the membership problem for Φ(A) is co-NP-complete, while the membership problems for Φ(Φ(A)), Φ(Φ(Φ(A))),… all lie in the class P‖(NP).

Non-Axiomatizability of the amalgamation class of modular lattice varieties

We prove that if v is the variety generated by a finite modular lattice, then v is not an elementary class. We also consider the same question for the variety generated by N5.