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The complexity of the normal form algorithms which transform a given polynomial ideal basis into a Gröbner basis or a given commutative Thue system into a Church-Rosser system is presently unknown. In this paper we derive a double-exponential lower bound (2^2nC) for the production length and cardinality of Church-Rosser commutative Thue systems, and the degree and cardinality of Gröbner bases.

A distributed algorithm is presented, for allocating a large number of identical resources (such as airline tickets) to requests which can arrive anywhere in a distributed network. Resources, once allocated, are never returned. The algorithm searches sequentially, exhausting certain neighborhoods of the request origin before proceeding to search at greater distances. Choice of search direction is made non-deterministically. Analysis of expected response time is simplified by assuming that the search direction is chosen probabilistically, that messages require constant time, that the network is a tree with all leaves at the same distance from the root, and that requests and resources occur only at leaves. It is shown that the response time is approximated by the number of messages of one type that are sent during the execution of the algorithm, and that this number of messages is a nondecreasing function of the interarrival time for requests. Therefore, the worst case occurs when requests come in so far apart that they are processed sequentially. The expected time for the sequential case of the algorithm is analyzed by standard techniques. This time is shown to be bounded by a constant, independent of the size of the network. It follows that the expected response time for the algorithm is bounded in the same way.

The time, space, and data complexity of an optimally data efficient isomorphism identification algorithm are presented. The data complexity, the amount of data required for an inference algorithm to terminate, is analyzed and shown to be the minimum possible for all possible isomorphism inference algorithms. The minimum data requirement is shown to be ⌈log₂ (n)⌉, and a method for constructing this minimal sequence of data is presented. The average data requirement is shown to be approximately 2 log₂(n). The time complexity is O(n²log₂(n)) and the space requirement is O(n²)

We present a logic, called Synchronization Tree Logic (STL), for the specification and proof of programs described in a simple term language obtained from a constant Nil by using a set A of unary operators, a binary operator + and recursion. The elements of A represent names of actions, + represents non-deterministic choice, and Nil is the program preforming no action. The language of formulas of the logic proposed, contains the term language used for the description of programs, i.e., programs are formulas of the logic. This provides a uniform frame to deal with programs and their properties as the verification of anassertion t ⊨ f (t is a program, f is a formula) is reduced to the proof of the validity of the formula t ⊃ f. We propose a sound and under some conditions complete deductive system for synchronization tree logics and discuss their relation with modal logics used for the specification of programs.

Non-uniform complexity measures originated in automata and formal languages theory are characterized in terms of well-known uniform complexity classes. The initial index of languages is introduced by means of several computational models. It is shown to be closely related to context-free cost, boolean circuits, straight line programs, and Turing machines with sparse oracles and time or space bounds.

CREW-PRAM's are a powerful model of parallel computers. Lower bounds for this model are rather general. Cook, Dwork, and Reischuk upper and lower time bounds for parallel random access machines without simultaneous writes, SIAM J. Comput. (in press) proved that the CREW-PRAM complexity of Boolean functions is bounded by log_b c(f), where b ≈ 4.79 and c(f) is the critical complexity of f. This lower bound is often even tight. For a class of functions F the critical complexity c(F), the minimum of all c(f) where f ∈ F, is the best general lower bound on the critical complexity of all f ∈ F. We determine the critical complexity of the set of all nondegenerate Boolean functions and all monotone nondegenerate Boolean functions up to a small additive term. And we compute exactly the critical complexity of the class of all monotone graph properties, proving partially a conjecture of Turán (1984).

The logic obtained by adding the least-fixed-point operator to first-order logic was proposed as a query language by Aho and Ullman (in “Proc. 6th ACM Sympos. on Principles of Programming Languages,” 1979, pp. 110–120) and has been studied, particularly in connection with finite models, by numerous authors. We extend to this logic, and to the logic containing the more powerful iterative-fixed-point operator, the zero-one law proved for first-order logic in (Glebskii, Kogan, Liogonki, and Talanov (1969), Kibernetika 2, 31–42; Fagin (1976), J. Symbolic Logic 41, 50–58). For any sentence φ of the extend logic, the proportion of models of φ among all structures with universe {1,2,…, n} approaches 0 or 1 as n tends to infinity. We also show that the problem of deciding, for any φ, whether this proportion approaches 1 is complete for exponential time, if we consider only φ's with a fixed finite vocabulary (or vocabularies of bounded arity) and complete for double-exponential time if φ is unrestricted. In addition, we establish some related results.