Robert Paige

Three partition refinement algorithms

We present improved partition refinement algorithms for three problems: lexicographic sorting, relational coarsest partition, and double lexical ordering. Our double lexical ordering algorithm uses a new, efficient method for unmerging two sorted sets.

Geometric constraint solver

Abstract The paper reports on the development of a 2D geometric constraint solver. The solver is a major component of a new generation of cad systems based on a high-level geometry representation. The solver uses a graph-reduction directed algebraic approach, and achieves interactive speed. The paper describes the architecture of the solver and its basic capabilities. Then, it discusses in detail how to extend the scope of the solver, with particular emphasis on the theoretical and human factors involved in finding a solution, in an exponentially large search space, so that the solution is appropriate to the application, and so that the way of finding it is intuitive for an untrained user.

Program derivation by fixed point computation

Abstract This paper develops a transformational paradigm by which nonnumerical algorithms are treated as fixed point computations derived from very high level problem specifications. We begin by presenting an abstract functional problem specification language SQ+, which is shown to express any partial recursive function in a fixed point normal form. Next, we give a nondeterministic iterative schema that in the case of finite iteration generalizes the “chaotic iteration” of Cousot and Cousot for computing fixed points of monotone functions efficiently. New techniques are discussed for recomputing fixed points of distributive functions efficiently. Numerous examples illustrate how these techniques for computing and recomputing fixed points can be incorporated within a transformational programming methodology to facilitate the design and verification of nonnumerical algorithms.

A linear time solution to the single function coarsest partition problem

Abstract The problem of finding the coarsest partition of a set S with respect to another partition of S one or more functions on S has several applications, one of which is the state minimization of finite state automata. In 1971, Hopcroft presented an algorithm to solve the many function coarsest partition problem for sets of n elements in O( n log n ) time and O( n ) space. In 1974, Aho, Hopcroft and Ullman presented an O( n log n ) algorithm that solves the special case of this problem for only one function. Both these algorithms use a negative strategy that repeatedly refines the original partition until a solution is found. We present a new algorithm to solve the single function coarsest partition problem in O( n ) time and space using a different, constructive approach. Our algorithm can be applied to the automated manufacturing of woven fabric.

From Regular Expressions to DFA's Using Compressed NFA's

We show how to turn a regular expression R of length r into an O(s) space representation of McNaughton and Yamadas NFA, where s is the number of occurrences of alphabet symbols in R, and s+1 is the number of NFA states. The standard adjacency list representation of McNaughton and Yamadas NFA takes up s + s2 space in the worst case. The adjacency list representation of the NFA produced by Thompson takes up between 2r and 6r space, where r can be arbitrarily larger than s. Given any set V of NFA states, our representation can be used to compute the set U of states one transition away from the states in V in optimal time O(¦V¦+¦U¦). McNaughton and Yamadas NFA requires Θ(¦V¦ × ¦U¦) time in the worst case. Using Thompsons NFA, the equivalent calculation requires Θ(r) time in the worst case.

Transformational design and implementation of a new efficient solution to the ready simulation problem

A transformational methodology is described for simultaneously designing algorithms and developing programs. The methodology makes use of three transformational tools — dominated convergence, finite differencing, and real-time simulation of a set machine on a RAM. We ilustrate the methodology to design a new O(mn + n2)-time algorithm for deciding when n-state, m-transition processes are ready similar, which is a substantial improvement on the ⊖(mn6) algorithm presented in Bloom (1989). The methodology is also used to derive a program whose performance, we believe, is competitive with the most efficient hand-crafted implementation of our algorithm. Ready simulation is the finest fully abstract notion of process equivalence in the CCS setting.

Programming with Invariants

The use of a restricted class of invariants as part of a language supports both the accurate synthesis of high-level programs and their translation into efficient implementations.

From regular expressions to DFA's using compressed NFA's

We show how to turn a regular expression R of length r into an O(s) space representation of McNaughton and Yamadas NFA, where s is the number of occurrences of alphabet symbols in R, and s + 1 is the number of NFA states. The standard adjacency list representation of McNaughton and Yamadas NFA takes up 1 + 2s +

Mona: Monadic Second-Order Logic in Practice

s\sp2

APPLICATIONS OF FINITE DIFFERENCING TO DATABASE INTEGRITY CONTROL AND QUERY/TRANSACTION OPTIMIZATION

space in the worst case. The adjacency list representation of the NFA produced by Thompson takes up between 2r and 6r space, where r can be arbitrarily larger than s. Given any subset V of states in McNaughton and Yamadas NFA, our representation can be used to compute the set U of states one transition away from the states in V in optimal time O(