Parallel Beta Reduction Is Not Elementary Recursive

AbstractWe analyze the inherent complexity of implementing Lévy's notion of optimal evaluation for the λ-calculus, where similar redexes are contracted in one step via so-called parallel β-reduction. Optimal evaluation was finally realized by Lamping, who introduced a beautiful graph reduction technology for sharing evaluation contexts dual to the sharing of values. His pioneering insights have been modified and improved in subsequent implementations of optimal reduction. We prove that the cost of parallel β-reduction is not bounded by any Kalmár-elementary recursive function. Not only do we establish that the parallel β-step cannot be a unit-cost operation, we demonstrate that the time complexity of implementing a sequence of n parallel β-steps is not bounded as O(2n), O(22n), O(222n), or in general, O(Kl(n)), where Kl(n) is a fixed stack of l 2's with an n on top. A key insight, essential to the establishment of this non-elementary lower bound, is that any simply typed λ-term can be reduced to normal form in a number of parallel β-steps that is only polynomial in the length of the explicitly typed term. The result follows from Statman's theorem that deciding equivalence of typed λ-terms is not elementary recursive. The main theorem gives a lower bound on the work that must be done by any technology that implements Lévy's notion of optimal reduction. However, in the significant case of Lamping's solution, we make some important remarks addressing how work done by β-reduction is translated into equivalent work carried out by his bookkeeping nodes. In particular, we identify the computational paradigms of superposition of values and of higher-order sharing, appealing to compelling analogies with quantum mechanics and SIMD-parallelism

Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe

In this dissertation we explore variations on Tic-Tac-Toe. We consider positional games played using a new type of move called a hop. A hop involves two parts: move and replace. In a hop the positions occupied by both players will change: one will move a piece to a new position and one will gain a piece in play. We play hop-positional games on the traditional Tic-Tac-Toe board, on the finite planes AG(2, q) and PG(2, q) as well as on a new class of boards which we call nested boards. A nested board is created by replacing the points of one board with copies of a second board. We also consider the traditional positional game played on nested boards where players alternately occupy open positions. We prove that the second player has a drawing strategy playing the hop-positional game on AG(2, q) for q ≥ 5 as well as on PG(2, q) for q ≥ 3. Moreover we provide an explicit strategy for the second player involving weight functions. For four classes of nested boards we provide a strategy and thresholds for the second player to force a draw playing a traditional positional game as well as the new hop-positional game. For example we show that the second player has a drawing strategy playing on the nested board [AG(2, q1 ) : PG(2, q2 )] for all q2 ≥ 7. Other bounds are also considered for this and other classes of nested boards

Distributed Load Testing by Modeling and Simulating User Behavior

Modern human-machine systems such as microservices rely upon agile engineering practices which require changes to be tested and released more frequently than classically engineered systems. A critical step in the testing of such systems is the generation of realistic workloads or load testing. Generated workload emulates the expected behaviors of users and machines within a system under test in order to find potentially unknown failure states. Typical testing tools rely on static testing artifacts to generate realistic workload conditions. Such artifacts can be cumbersome and costly to maintain; however, even model-based alternatives can prevent adaptation to changes in a system or its usage. Lack of adaptation can prevent the integration of load testing into system quality assurance, leading to an incomplete evaluation of system quality. The goal of this research is to improve the state of software engineering by addressing open challenges in load testing of human-machine systems with a novel process that a) models and classifies user behavior from streaming and aggregated log data, b) adapts to changes in system and user behavior, and c) generates distributed workload by realistically simulating user behavior. This research contributes a Learning, Online, Distributed Engine for Simulation and Testing based on the Operational Norms of Entities within a system (LODESTONE): a novel process to distributed load testing by modeling and simulating user behavior. We specify LODESTONE within the context of a human-machine system to illustrate distributed adaptation and execution in load testing processes. LODESTONE uses log data to generate and update user behavior models, cluster them into similar behavior profiles, and instantiate distributed workload on software systems. We analyze user behavioral data having differing characteristics to replicate human-machine interactions in a modern microservice environment. We discuss tools, algorithms, software design, and implementation in two different computational environments: client-server and cloud-based microservices. We illustrate the advantages of LODESTONE through a qualitative comparison of key feature parameters and experimentation based on shared data and models. LODESTONE continuously adapts to changes in the system to be tested which allows for the integration of load testing into the quality assurance process for cloud-based microservices

Unification and Matching on Compressed Terms

Term unification plays an important role in many areas of computer science, especially in those related to logic. The universal mechanism of grammar-based compression for terms, in particular the so-called Singleton Tree Grammars (STG), have recently drawn considerable attention. Using STGs, terms of exponential size and height can be represented in linear space. Furthermore, the term representation by directed acyclic graphs (dags) can be efficiently simulated. The present paper is the result of an investigation on term unification and matching when the terms given as input are represented using different compression mechanisms for terms such as dags and Singleton Tree Grammars. We describe a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. For the same problem, NP-completeness is obtained when the terms are represented using the more general formalism of Singleton Tree Grammars. For first-order unification and matching polynomial time algorithms are presented, each of them improving previous results for those problems.Comment: This paper is posted at the Computing Research Repository (CoRR) as part of the process of submission to the journal ACM Transactions on Computational Logic (TOCL)