And-or Parallelism Research Articles

On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system.

Exploiting and-or parallelism in Prolog: The OASys computational model and abstract architecture

Different forms of parallelism have been extensively investigated over the last few years in logic programs and a number of systems have been proposed. Or/And System (OASys) is an experimental parallel Prolog system that exploits and-or-parallelism and comprises a computational model, a compiler, an abstract machine and an emulator. OASys computational model combines the two types of parallelism considering each alternative path as a totally independent computation which consists of a conjunction of determinate subgoals. It is based on distributed scheduling and supports recomputation of paths as well as stack copying. The system features modular design, high distribution and minimal inter-processor communication. This paper presents briefly the computational model and describes the abstract machine discussing data representation, memory organization, instruction set, operation and synchronization. Finally performance results obtained by the single-processor implementation and the multiple-processor emulation are discussed.

Computational Power of Quantum Machines, Quantum Grammars and Feasible Computation

This paper studies the computational power of quantum computers to explore as to whether they can recognize properties which are in nondeterministic polynomial-time class (NP) and beyond. To study the computational power, we use the Feynman's path integral (FPI) formulation of quantum mechanics. From a computational point of view the Feynman's path integral computes a quantum dynamical analogue of the k-ary relation computed by an Alternating Turing machine (ATM) using AND-OR Parallelism. Hence, if we can find a suitable mapping function between an instance of a mathematical problem and the corresponding interference problem, using suitable potential functions for which FPI can be integrated exactly, the computational power of a quantum computer can be bounded to that of an alternating Turing machine that can solve problems in NP (e.g, factorization problem) and in polynomial space. Unfortunately, FPI is exactly integrable only for a few problems (e.g., the harmonic oscillator) involving quadratic potentials; otherwise, they may be only approximately computable or noncomputable. This means we cannot in general solve all quantum dynamical problems exactly except for those special cases of quadratic potentials, e.g., harmonic oscillator. Since there is a one to one correspondence between the quantum mechanical problems that can be analytically solved and the path integrals that can be exactly evaluated, we can say that the noncomputability of FPI implies quantum unsolvability. This is the analogue of classical unsolvability. The Feynman's path graph can be considered as a semantic parse graph for the quantum mechanical sentence. It provides a semantic valuation function of the terminal sentence based on probability amplitudes to disambiguate a given quantum description and obtain an interpretation in a linear time. In Feynman's path integral, the kernels are partially ordered over time (different alternate paths acting concurrently at the same time) and multiplied. The semantic valuation is computable only if the FPI is computable. Thus both the expressive power and complexity aspects quantum computing are mirrored by the exact and efficient integrability of FPI.

Cuts and side-effects in and-or parallel Prolog

Practical Prolog programs usually contain extra-logical features like cuts, side-effects, and database manipulating predicates. In order to exploit implicit parallelism from real applications while preserving sequential Prolog semantics, a parallel logic programming system should necessarily support these features. In this paper we show how Prolog's extra-logical features can be supported in an and-or parallel logic programming system. We show that to support extra-logical features an and-or parallel logic programming system should recompute the solutions to independent goals instead of sharing them. We describe an abstraction called the composition tree for representing and-or parallel execution with recomputation. We introduce the notion of “focal-leftmostness” in the composition tree and use it for deriving complete and efficient methods for supporting extra-logical predicates in and-or parallel logic programming systems based on the composition tree abstraction.

Optimal implementation of and-or parallel Prolog

Most models that have been proposed, or implemented, so far for exploiting both or-parallelism and independent and-parallelism have only considered pure logic programs (pure Prolog). We present an abstract model, called the Composition-Tree, for representing and-or parallelism in full Prolog. The Composition-Tree recomputes independent goals to ensure that Prolog semantics is preserved. We combine the idea of Composition-Tree with ideas developed earlier, to develop an abstract execution model that supports full Prolog semantics while at the same time avoiding redundant inferences when computing solutions to (purely) independent and-parallel goals. This is accomplished by sharing solutions of independent goals when they are pure (i.e. have no side-effects or cuts in them). The Binding Array scheme is extended for and-or parallel execution based on this abstract execution model. This extension enables the Binding Array scheme to support or-parallelism in the presence of independent and-parallelism, both when solutions to independent goals are recomputed as well as when they are shared. We show how extra-logical predicates, such as cuts and side-effects, are supported in this model.

And-Or parallel Prolog: A recomputation based approach

We argue that in order to exploit both Independent And-and Or-parallelism in Prolog programs there is advantage in recomputing some of the independent goals, as opposed to all their solutions being reused. We present an abstract model, called the Composition-tree, for representing and-or parallelism in Prolog programs. The Composition-tree closely mirrors sequential Prolog execution by recomputing some independent goals rather than fully re-using them. We also outline two environment representation techniques for And-Or parallel execution offull Prolog based on the Composition-tree model abstraction. We argue that these techniques have advantages over earlier proposals for exploiting and-or parallelism in Prolog.

A theorem about running round a triangle and two applications

Diana runs from A to B with speed u and then from B to C with speed v. Eric runs from A to C with speed w. They start together and finish together. If we now draw three lines from a point O, with OP parallel to AB and of length u, OQ parallel to BC and of length v and OR parallel to AC and of length w, then we find P, Q and R are collinear. It is not difficult to construct a proof of the theorem so that is left as an exercise for the reader.