In this paper, we describe a parallelization scheme for Collins' cylindrical algebraic decomposition algorithm for quantifier elimination in the theory of real closed fields. We first discuss a parallel implementation of the computer algebra system SAC2 in which a complete sequential implementation of Collins' algorithm already exists. We report some initial results on the speedup obtained, drawing on a suite of examples previously given by Arnon.

This paper describes the design of an algebra system Views implemented in Smalltalk. Views contains facilities for dynamic creation and manipulation of computational domains, for viewing these domains as various categories such as groups, rings, or fields, and for expressing algorithms generically at the level of categories. The design of Views has resulted in the addition of some new abstractions to Smalltalk that are quite useful in their own right. Parameterized classes provide a means for run-time creation of new classes that exhibit generally very similar behavior, differing only in minor ways that can be described by different instantiations of certain parameters. Categories allow the abstraction of the common behavior of classes that derives from the class objects and operations satisfying certain laws independently of the implementation of those objects and operations. Views allow the run-time association of classes with categories (and of categories with other categories), facilitating the use of code written for categories with quite different interpretations of operations. Together, categories and views provide an additional mechanism for code sharing that is richer than both single and multiple inheritance. The paper gives algebraic as well as non-algebraic examples of the above-mentioned features.

This note presents a practical algorithm for carrying out abstraction on combinatory terms. The well-known abstraction algorithms [1, pp. 188ff.] defining abstracts in terms of the combinator sets {S, K}, {B, C, K, W}, etc. operate on one variable at a time, and result in rather long abstracts when several variables are involved. These algorithms are not practical for the applications of the combinatory logic to the theory of computing which make much use of multi-variable abstraction (e.g., [2], [3]). The present algorithm performs the abstraction with respect to all specified variables in a single step, and yields abstracts in a concise form (with sizes proportional to those of given combinatory terms).

Based on a program-proving technique, a method for proving sequential machine designs is presented. The method associates with each state of the machine an assertion about the sequences taking the machine into that state. The design is proved by: showing all assertions true irrespective of state changes; and deriving the design specifications from the assertions at final states. The method is illustrated' on two finite automata.