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Term-rewriting systems, that is, sets of directed equations, provide a paradigm of computation with particularly simple syntax and semantics. Rewrite systems may be used for straightforward computation by simplifying terms. We show how, in addition, restricted forms of the Knuth—Bendix “completion” procedure may be used to interpret logic programs written as a set of equivalence-preserving rewrite rules. We discuss verification issues and also illustrate the use of the full completion procedure to synthesize rewrite programs from specifications.

This paper is concerned with the semantics (or computational power) of very simple loop programs over different sets of primitive instructions. Recently, a complete and consistent Hoare axiomatics for the class of {x ← 0, x ← y, x ← x + 1, x ← x ∸ 1, do x … end} programs which contain no nested loops, was given, where the allowable assertions were those formulas in the logic of Presburger arithmetic. The class of functions computable by such programs is exactly the class of Presburger functions. Thus, the resulting class of correctness formulas has a decidable validity problem. In this paper, we present simple loop programming languages which are, computationally, strictly more powerful, i.e., which can compute more than the class of Presburger functions. Furthermore, using a logical assertion language that is also more powerful than the logic of Presburger arithmetic, we present a class of correctness formulas over such programs that also has a decidable validity problem.

This paper is concerned with the expressive power (or computational power) of loop programs over different sets of primitive instructions. In particular, we show that an {x ← 0, x ← y, x ← x + 1, do x … end, if x = 0 then y ← z}-program which contains no nested loops can be transformed into an equivalent {x ← 0, x ← y, x ← x + 1, do x … end}-program (also without nested loops) in exponential time and space. This translation was earlier claimed, in the literature, to be obtainable in polynomial time, but then this was subsequently shown to imply that PSPACE = PTIME. Consequently, the question of translatability was left unanswered. Also, we show that the class of functions computable by {x ← 0, x ← y, x ← x + 1, x − 1, do x … end, if x = 0 then x ← c}-programs is exactly the class of Presburger functions.

We show how to use both horizontal and vertical decomposition to normalize a database schema which contains numerical dependencies. We present a finite set of inference rules for numerical dependencies which is a generalization of the Armstrong axioms. We prove that this set is sound and complete for some special cases.

We show that any countable distributive lattice can be embedded in any interval of polynomial time degrees. Furthermore the embeddings can be chosen to preserve the least or the greatest element. This holds for both polynomial time bounded many-one and Turing reducibilities, as well as for all of the common intermediate reducibilities.

This paper considers the problem of granting a dynamic data structure the capability of remembering the situation it held at previous times. We present a new scheme for recording a history of h updates over an ordered set S of n objects, which allows fast neighbor computation at any time in the history. The novelty of the method is to allow the set S to be only partially ordered with respect to queries and the time measure to be multi-dimensional. The generality of the method makes it useful for a number of problems in 3-dimensional geometry. For example, we are able to give fast algorithms for locating a point in a 3-dimensional complex, using linear space, or for finding which of n given points is closest to a query plane. Using a simpler, yet conceptually similar technique, we show that with O(n²) preprocessing, it is possible to determine in O(log² n) time which of n given points in E³ is closest to an arbitrary query point.

In C. A. R. Hoare, S. D. Brookes, and A. D. Roscoe (1984, J. Assoc. Comput. Mach. 31(3), 560) an abstract version of Hoare's CSP is defined and a denotational semantics based on the possible failures of processes is given for it. This semantics induces a natural preorder on processes. We define formally this preorder and prove that it can be characterized as the smallest relation satisfying a particular set of axioms. The characterization sheds lights on problems arising from the way divergence and underspecification are handled. After small changes to the semantic domains we propose a new semantics which is closer to the operational intuitions and suggests a possible solution to the above problems. Finally we give an axiomatic characterization for the equivalence induced by the new semantics which leads to fully abstract models in the sense of Scott.

The edit distance between strings a₁ … a_m and b₁ … b_n is the minimum cost s of a sequence of editing steps (insertions, deletions, changes) that convert one string into the other. A well-known tabulating method computes s as well as the corresponding editing sequence in time and in space O(mn) (in space O(min(m, n)) if the editing sequence is not required). Starting from this method, we develop an improved algorithm that works in time and in space O(s · min(m, n)). Another improvement with time O(s · min(m, n)) and space O(s · min(s, m, n)) is given for the special case where all editing steps have the same cost independently of the characters involved. If the editing sequence that gives cost s is not required, our algorithms can be implemented in space O(min(s, m, n)). Since s = O(max(m, n)), the new methods are always asymptotically as good as the original tabulating method. As a by-product, algorithms are obtained that, given a threshold value t, test in time O(t · min(m, n)) and in space O(min(t, m, n)) whether s ⩽ t. Finally, different generalized edit distances are analyzed and conditions are given under which our algorithms can be used in conjunction with extended edit operation sets, including, for example, transposition of adjacent characters.