The Synthesis of Cyclic Combinatorial Circuits

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Exact Stochastic Simulation of Chemical Reactions with Cycle Leaping

The stochastic simulation algorithm (SSA), first proposed by Gillespie, has become the workhorse of computational biology. It tracks integer quantities of the molecular species, executing reactions at random based on propensity calculations. An estimate for the resulting quantities of the different species is obtained by averaging the results of repeated trials. Unfortunately, for models with many reaction channels and many species, the algorithm requires a prohibitive amount of computation time. Many trials must be performed, each forming a lengthy trajectory through the state space. With coupled or reversible reactions, the simulation often loops through the same sequence of states repeatedly, consuming computing time, but making no forward progress. We propose a algorithm that reduces the simulation time through cycle leaping: when cycles are encountered, the exit probabilities are calculated. Then, in a single bound, the simulation leaps directly to one of the exit states. The technique is exact, sampling the state space with the expected probability distribution. It is a component of a general framework that we have developed for stochastic simulation based on probabilistic analysis and caching

Cyclic Boolean circuits

A Boolean circuit is a collection of gates and wires that performs a mapping from Boolean inputs to Boolean outputs. The accepted wisdom is that such circuits must have acyclic (i.e., loop-free or feed-forward) topologies. In fact, the model is often defined this way – as a directed acyclic graph (DAG). And yet simple examples suggest that this is incorrect. We advocate that Boolean circuits should have cyclic topologies (i.e., loops or feedback paths). In other work, we demonstrated the practical implications of this view: digital circuits can be designed with fewer gates if they contain cycles. In this paper, we explore the theoretical underpinnings of the idea. We show that the complexity of implementing Boolean functions can be lower with cyclic topologies than with acyclic topologies. With examples, we show that certain Boolean functions can be implemented by cyclic circuits with as little as one-half the number gates that are required by equivalent acyclic circuits

Algorithmic Aspects of Cyclic Combinational Circuit Synthesis

Digital circuits are called combinational if they are memoryless: if they have outputs that depend only on the current values of the inputs. Combinational circuits are generally thought of as acyclic (i.e., feed-forward) structures. And yet, cyclic circuits can be combinational. Cycles sometimes occur in designs synthesized from high-level descriptions, as well as in bus-based designs [16]. Feedback in such cases is carefully contrived, typically occurring when functional units are connected in a cyclic topology. Although the premise of cycles in combinational circuits has been accepted, and analysis techniques have been proposed [7], no one has attempted the synthesis of circuits with feedback at the logic level. We have argued the case for a paradigm shift in combinational circuit design [10]. We should no longer think of combinational logic as acyclic in theory or in practice, since most combinational circuits are best designed with cycles. We have proposed a general methodology for the synthesis of multilevel networks with cyclic topologies and incorporated it in a general logic synthesis environment. In trials, benchmark circuits were optimized significantly, with improvements of up to 30%I n the area. In this paper, we discuss algorithmic aspects of cyclic circuit design. We formulate a symbolic framework for analysis based on a divide-and-conquer strategy. Unlike previous approaches, our method does not require ternary-valued simulation. Our analysis for combinationality is tightly coupled with the synthesis phase, in which we assemble a combinational network from smaller combinational components. We discuss the underpinnings of the heuristic search methods and present examples as well as synthesis results for benchmark circuits. In this paper, we discuss algorithmic aspects of cyclic circuit design. We formulate a symbolic framework for analysis based on a divide-and-conquer strategy. Unlike previous approaches, our method does not require ternary-valued simulation. Our analysis for combinationality is tightly coupled with the synthesis phase, in which we assemble a combinational network from smaller combinational components. We discuss the underpinnings of the heuristic search methods and present examples as well as synthesis results for benchmark circuits

The Death Penalty and Discretion: Implications of the Furman Decision for Criminal Justice

Whether the deatn penalty should remain as a penalty available in American criminal law continues to be a subject of controversy among social scientists, lawyers, the judiciary and the public. While the traditional areas of debate over whether the death penalty is a deterrent and whether it is imposed ina discriminatory manner continue to be important issues, the recent Supreme Court decision (Furman v Georgia, 1972) and subsequent legislation has introduced another dimension: the nature and use of discretion. Current litigation on the death penalty (Fowler v North Carolina, 1974) is directed toward a resolution of issues raised by Furman. However, it is our contention that the results of such efforts will raise a range of policy questions regarding how discretion can be exercised not only in other parts of the criminal justice process, but for non-death penalty offenses as well. The purpose of this paper is to examine the Furman decision and subsequent legislation and show that the questions raised about discretionary decisions are questions that are equally applicable to processing all criminal offenses

Rate-Independent Constructs for Chemical Computation

This paper presents a collection of computational modules implemented with chemical reactions: an inverter, an incrementer, a decrementer, a copier, a comparator, a multiplier, an exponentiator, a raise-to-a-power operation, and a logarithm in base two. Unlike previous schemes for chemical computation, this method produces designs that are dependent only on coarse rate categories for the reactions (“fast” vs. “slow”). Given such categories, the computation is exact and independent of the specific reaction rates. The designs are validated through stochastic simulations of the chemical kinetics

A Scalable Approach to Performing Multiplication and Matrix Dot-Products in Unary

Stochastic computing is a paradigm in which logical operations are performed on randomly generated bit streams. Complex arithmetic operations can be executed by simple logic circuits, resulting in a much smaller area footprint compared to conventional binary counterparts. However, the random or pseudorandom sources required for generating the bit streams are costly in terms of area and offset the advantages. Additionally, due to the inherent randomness, the computation lacks precision, limiting the applicability of this paradigm. Importantly, achieving reasonable accuracy in stochastic computing involves high latency. Recently, deterministic approaches to stochastic computing have been proposed, demonstrating that randomness is not a requirement. By structuring the computation deterministically, exact results can be obtained, and the latency greatly reduced. The bit stream generated adheres to a "unary" encoding, retaining the non-positional nature of the bits while discarding the random bit generation of traditional stochastic computing. This deterministic approach overcomes many drawbacks of stochastic computing, although the latency increases quadratically with each level of logic, becoming unmanageable beyond a few levels. In this paper, we present a method for approximating the results of the deterministic method while maintaining low latency at each level. This improvement comes at the cost of additional logic, but we demonstrate that the increase in area scales with the square root of n, where n represents the equivalent number of binary bits of precision. Our new approach is general, efficient, composable, and applicable to all arithmetic operations performed with stochastic logic. We show that this approach outperforms other stochastic designs for matrix multiplication (dot-product), which is an integral step in nearly all machine learning algorithms.Comment: 25 pages, 8 figure

The Application of Missing Data Estimation Models to the Problem of Unknown Victim/Offender Relationships in Homicide Cases.

Homicide cases suffer from substantial levels of missing data, a problem largely ignored by criminological researchers. The present research seeks to address this problem by imputing values for unknown victim/offender relationships using the EM algorithm. The analysis is carried out first using homicide data from the Los Angeles Police Department (1994-1998), and then compared with imputations using homicide data for Chicago (1991-1995), using a variety of predictor variables to assess the extent to which they influence the assignment of cases to the various relationship categories. The findings indicate that, contrary to popular belief, many of the unknown cases likely involve intimate partners, other family, and friends/acquaintances. However, they disproportionately involve strangers. Yet even after imputations, stranger homicides do not increase more than approximately 5%. The paper addresses the issue of whether data on victim/offender relationships can be considered missing at random (MAR), and the im-plications of the current findings for both existing and future research on homicide