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In this paper a generalization of a certain theorem of Lipton (“Proc. 18th IEEE Sympos. Found. of Comput Sci.” (1977), pp. 1–6) is presented. Namely, we show that for a wide class of programming languages the following holds: the set of all partial correctness assertions true in an expressive interpretation I is uniformly dedicable (in I) in the theory of I iff the halting problem is decidable for finite interpretations. In the effect we show that such limitations as effectiveness or Herbrand-definability of interpretation (they are relevant in the previous proofs) can be removed in the case of partial correctness.

A graph is cubical if it is a subgraph of a hypercube; the dimension of the smallest such hypercube is the dimension of the graph. We show several results concerning this class of graphs. We use a characterization of cubical graphs in terms of edge coloring to show that the dimension of biconnected cubical graphs is at most half the number of nodes. We also show that telling whether a graph is cubical is NP-complete. Finally, we propose a heuristic for minimizing the dimension of trees, which yields an embedding of the tree in a hypercube of dimension at most the square of the true dimension of the tree.

Several new data structures are presented for dictionaries containing elements with different weights (access probabilities). The structures use just one location in addition to those required for the values of the elements. The first structure supports a worst-case search time that is within a constant multiplicative factor of optimal, in terms of the rank of the weight of the desired element with respect to the multiset of weights. If the values of the elements that comprise the dictionary have been drawn from a uniform distribution, then a variation of this structure achieves average search times that are asymptotically very good. Similar results are established for data structures which handle the case in which the intervals between consecutive dictionary values also have access probabilities. Lower bounds are presented for the worst-case search complexity.

We present a proof rule for fairly terminating guarded commands based on a well-foundedness argument. The rule is applied to several examples, and proved to be sound and (semantically) complete w.r.t. an operational semantics of computation trees. The rule is related to another rule suggested by Lehmann, Pnueli, and Stavi (in “Proc. Internat. Colloq. Automata Lang. and Programming, '81,” Acre, July 1981), by showing that the (semantic) completeness of the LPS-rule follows from the completeness or ours.

I show in this note that for any integer k and any k page graph G, there is an easily constructed 3-page graph G′ (called the unraveling of G) such that the minimum separator sizes of G and G′ are within a factor of k of each other. Further the maximum degree of a vertex of G′ is at most 2 plus the maximum degree of G.

This paper investigates the structural properties of sets in NP-P and shows that the computational difficulty of lower density sets in NP depends explicitly on the relations between higher deterministic and nondeterministic time-bounded complexity classes. The paper exploits the recently discovered upward separation method, which shows for example that there exist sparse sets in NP-P if and only if EXPTIME ≠ NEXPTIME. In addition, the paper uses relativization techniques to determine logical possibilities, limitations of these proof techniques, and exhibits one of the first natural structural differences between relativized NP and CoNP.

We make explicit a connection between the “unwind property” and first-order logics of programs. Using known results on the unwind property, we can then quickly compare various logics of programs. In Section 1, we give a sample of these comparative results, which are already known but established differently in this paper. In Sections 2 and 3, given an arbitrary deterministic regular program S (with or without parameterless recursive calls), we show how to construct a first-order structure where S will unwind. Based on this construction, we then prove that the logic of regular programs (with or without parameterless recursive calls) is more expressive than the logic of deterministic regular programs (with or without parameterless recursive calls, respectively).

We present semantic methods for showing that a term-rewriting system is confluent. We also present methods for completing a given term-rewriting system to obtain an equivalent confluent system. These methods differ from the well-known and widely studied Knuth-Bendix method in that they emphasize semantics rather than syntax. Also, they often require more user interaction than the purely syntactic Knuth-Bendix method. The concept of “ground confluence” is discussed; methods for demonstrating ground confluence are also given. We give decision procedures for some sub-problems that arise in this method.