Fixed-point Computation Research Articles

Novel fixed-point roundoff analysis of the decimation-in-time FHT

A least upper bound for the increasing factor of the magnitude of the decimation-in-time fast Hartley transform (FHT) in fixed-point arithmetic is developed and a new scaling model for the roundoff analysis in the fixed-point arithmetic computation is proposed. In this new scaling model, the input data for each computing stage of the decimation-in-time FHT only need to be divided by a constant of 2, and this can prevent overflow successfully. Hence, the novel approach would result in a higher noise-to-signal ratio for the fixed-point computation of FHT. >

CONSAT: A parallel constraint satisfaction system

In this paper we describe the parallelization of a medium-size symbolic fixed-point computation, CONSAT. CONSAT is a constraint satisfaction system that computes globally consistent solutions. The parallel version of CONSAT is implemented using abstractions from a parallel programming toolbox we developed. The toolbox is intended for novice parallel programmers, and programs based on abstractions from this toolbox may be executed on both uniprocessors and shared-memory multiprocessors without modifications. We explain how parallelism is introduced, and how concurrent accesses to shared data structures are handled. We will also describe the performance of CONSAT on sample inputs.

Symbolic model checking: 1020 States and beyond

Many different methods have been devised for automatically verifying finite state systems by examining state-graph models of system behavior. These methods all depend on decision procedures that explicitly represent the state space using a list or a table that grows in proportion to the number of states. We describe a general method that represents the state space symbolically instead of explicitly. The generality of our method comes from using a dialect of the Mu-Calculus as the primary specification language. We describe a model checking algorithm for Mu-Calculus formulas that uses Bryant's Binary Decision Diagrans (Bryant, R. E., 1986, IEEE Trans. Comput. C-35 ) to represent relations and formulas. We then show how our new Mu-Calculus model checking algorithm can be used to derive efficient decision procedures for CTL model checking, satisfiability of linear-time temporal logic formulas, strong and weak observational equivalence of finite transition systems, and language containment for finite ω-automata. The fixed point computations for each decision procedure are sometimes complex, but can be concisely expressed in the Mu-Calculus. We illustrate the practicality of our approach to symbolic model checking by discussing how it can be used to verify a simple synchronous pipeline circuit.

New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic

This paper introduces a new higher-order typed constructive predicate logic for fixpoint computations, which exploits the categorical semantics of computations introduced by Moggi ( in “Proceedings, 4th Annual Symposium on Logic in Computer Science,” pp. 14–23, IEEE Comput. Soc. Press, Washington, 1989) and contains a version of Martin-Löf's “iteration type” ( in “Proceedings, Workshop on Semantics in Programming Laguages,” Chalmers University, 1983) . The type system enforces a separation of computations from values. The logic contains a novel form of fixpoint induction and can express partial and total correctness statements about evaluation of computations to values. The constructive nature of the logic is witnessed by strong metalogical properties which are proved using a category-theoretic version of the “logical relations” method ( Plotkin, unpublished lecture notes from CSLI Summer School, 1985 ).

Polyvariant Analysis of the Untyped Lambda Calculus

<p>We present a polyvariant closure, safety, and binding time analysis for the untyped lambda calculus. The innovation is to analyze each abstraction afresh at all syntactic application points. This is achieved by a semantics-preserving program transformation followed by a novel monovariant analysis, expressed using type constraints. The constraints are solved in cubic time by a single fixed-point computation.</p><p>Safety analysis is aimed at determining if a term will cause an error during evaluation. We have recently proved that the monovariant safety analysis accepts strictly more terms than simple type inference. This paper demonstrates that the polyvariant transformation makes even more terms acceptable, even some without higher-order polymorphic types. Furthermore, polyvariant binding time analysis can improve the partial evaluators that base a polyvariant specialization on only monovariant binding time analysis.</p>

A syntactic approach to fixed point computation on finite domains

We propose a syntactic approach to performing fixed point computation on finite domains. Finding fixed points in finite domains for monotonic fuctions is an essential task when calculating abstract semantics of functional programs. Previous methods for fixed point finding have been mainly based on semantic approaches which may be very inefficient even for simple programs. We outline the development of a syntactic approach, and show that the syntactic approach is sound and complete with respect to semantics. A few examples are provided to illustrate this syntactic approach.

A reduced complexity multipulse compression system

A reduced complexity multipulse coder is proposed in this paper. A 33% complexity reduction affects the overall speech output quality only slightly as is demonstrated in both informal listening and SNR computation. On the other hand, it makes possible the implementation of the coder with cheap commercial signal processing chips. The TMS32020 has been used for real time implementation of the proposed algorithm at 16 kbit/s with only fixed point computations. The method studied here can be considered as an alternative to the regular pulse excitation technique.

Convergence of chaotic iterative least fixed point computations

Iterative methods for computing fixed points of functions could usually be accommodated with some kinds of chaotic mechanisms for decreasing computational complexity. In this paper, chaotic iterative least fixed point computations of multidimensional functions on partially ordered sets are investigated, representative functions are devised to describe these computation processes, and based on this representation, a necessary and sufficient condition judging the convergence of all these computations is given.

A numerically stable relay structure for fast RLS adaptive filtering

Long-term numerical instability is a critical problem in fast recursive least squares (RLS) adaptive algorithms. The author presents a numerically stable fast RLS relay structure (FRLS-RS), in which a conventional fast transversal RLS algorithm and a mixed time-and-order updating procedure are combined in a relay form to provide, with O(M) operations, the instantaneous filter-coefficient solution for each time step, where M is the order of the filter. Since continuous propagation of the FRLS-RS is limited to 2M+1 time steps, accumulation of the numerical errors is isolated. Both fixed-point analysis and computer simulations show that in the case of moderate precision, and when the order is not very high, the present structure is long-term numerically stable. Efficient implementation and exact initialization of the FRLS-RS are also considered. >

High speed architectures for approximate fixed-point division and square root realization

High speed architectures for fixed-point division and square root calculations are developed based on an approximate fast log and antilog algorithm. The architectures can be utilized in applications where high speed is required, and a low percentage error can be tolerated. Error analysis and an application example where the divider architecture is utilized are also presented.

Program derivation by fixed point computation

This paper develops a transformational paradigm by which nonnumerical algorithms are treated as fixed point computations derived from very high level problem specifications. We begin by presenting an abstract functional problem specification language SQ +, which is shown to express any partial recursive function in a fixed point normal form. Next, we give a nondeterministic iterative schema that in the case of finite iteration generalizes the “chaotic iteration” of Cousot and Cousot for computing fixed points of monotone functions efficiently. New techniques are discussed for recomputing fixed points of distributive functions efficiently. Numerous examples illustrate how these techniques for computing and recomputing fixed points can be incorporated within a transformational programming methodology to facilitate the design and verification of nonnumerical algorithms.

A front-end processor for modems

The authors describe the front-end processor (FEP) which permits the construction of a multistandard modem system, incorporating the CCITT V.32 echo canceling scheme, using a general-purpose signal processor. The FEP is divided into a digital signal processing (DSP) block and an analog-digital/digital-analog conversion block. The DSP is designed to handle biquad filtering, tone generation, automatic gain control, call-progress-tone detection, and carrier detection. The multiplier, ALU (arithmetic and logic unit), and RAM are adapted to handle 22-bit data operation and to obtain high resolution and a 16-bit dynamic range operation in fixed-point calculations. The measured bit error rate of 14.4-kb/s trellis coding modem using the FEP is less than 10/sup -5/ at a signal-to-noise ratio of 27 dB under the US unconditioned 3002 line. The chip is fabricated in a 1.5- mu m double-poly double-metal CMOS process and designed using an automatic layout tool. >

A framework for testing safety and effective computability of extended datalog

This paper presents a methodology for testing a general logic program containing function symbols and built-in predicates for safety and effective computability . Safety is the property that the set of answers for a given query is finite. A related issues is whether the evaluation strategy can effectively compute all answers and terminate. We consider these problems under the assumption that queries are evaluated using a bottom-up fixpoint computation. We also approximate the use of function symbols by considering Datalog programs with infinite base relations over which finiteness constraints and monotonicity constraints are considered. One of the main results of this paper is a recursive algorithm, check_clique , to test the safety and effective computability of predicates in arbitrarily complex cliques. This algorithm takes certain procedures as parameters, and its applicability can be strengthened by making these procedures more sophisticated. We specify the properties required of these procedures precisely, and present a formal proof of correctness for algorithm check_clique . This work provides a framework for testing safety and effective computability of recursive programs, and is based on a clique by clique analysis. The results reported here form the basis of the safety testing for the LDL language, being implemented at MCC.

Some algorithms for approximating convolutions

This paper presents some algorithms for approximating two-dimensional convolution operators of size n × n, n odd, by a product, or sum of products, of 3 × 3 convolutions. Inaccuracies resulting from the approximation as well as from fixed point computation are discussed and examples are given.

Magic counting methods

The problem considered is that of implementing recursive queries, expressed in a logic-based language, by efficient fixpoint computations. In particular, the situation is studied where the initial bindings in the recursive predicate can be used to restrict the search space and ensure safety of execution. Two key techniques previously proposed to solve this problem are (i) the highly efficient counting method, and (ii) the magic set method which is safe in a wider range of situations than (i). In this paper, we present a family of methods, called the magic counting methods, that combines the advantages of (i) and (ii). This is made possible by the similarity of the strategies used by the counting method and the magic set method for propagating the bindings. This paper introduces these new methods, examines their computational complexity, and illustrates the trade-offs between the family members and their superiority with respect to the old methods.

Real-time DKS on a single chip

The desire to add more intelligent functionality and enhanced real-time Control capability to robotic systems places a heavy demand on computational resources. Nascent application-specific VLSI technology offers an unprecedented opportunity to relieve this computational burden by implementing specific robotic functions directly in hardware. A VLSI architecture, designed to compute the direct kinematic solution (DKS) on a single chip, is described. The DKS chip features fixed-point calculation and on-chip generation of trigonometric functions. The calculation achieves the resolution required for most industrial applications. Innovative modifications of the conventional adder control and the form of the homogeneous transformation matrix have improved throughput. Simulation results indicate that speed improvements of several orders of magnitude in the kinematic computation are achievable with application-specific integrated circuit technology, compared to microprocessor implementation.

Using Digitised X-Ray Powder Diffraction Scans as Input for a New Pc-At Search/Match Program

AbstractA Search/Matcti program lias 'beea written for the IBM PC AT computer that is capable of -using "background - subtracted, digitized 2-ray powder diffraction scans as inputs in addition to the d/I data traditionally used. This novel procedure has proved especially effective when numerous unresolved lines are present in the pattern. The method is also less demanding of data quality thaii the peak location programs. The program may he extended to searching & data "base of digitized standard patterns.The program, has several parameters that can- "be adjusted, including chemistry. The results from the Johnson/Vand list type of output are directly accessible to the interactive graphics program. This gives the diffraction!st a fast method for verifying the phase identification. Because of the speed of fixed point computation techniques, the 52,791 pattern file can be scanned in about 90 seconds.This paper will illustrate the utility of the program.

The generalized counting method for recursive logic queries

This paper treats the problem of implementing efficiently recursive Horn clauses queries, including those with function symbols. In particular, the situation is studied where the initial bindings of the arguments in the recursive query goal can be used in the top-down (as in backward chaining) execution phase to improve the efficiency and, often, to guarantee the termination of the forward chaining execution phase that implements the fixpoint computation for the recursive query. A general method is given for solving these queries; the method performs an analysis of the binding-passing behavior of the query, and then reschedules the overall execution as two fixpoint computations derived as results of this analysis. The first such computation emulates the propagation of bindings in the top-down phase; the second generates the desired answer by proving the goals left unsolved during the previous step. Finally, sufficient conditions for safety are derived to ensure that the fixpoint computations are completed in a finite number of steps.

Combinatorial Theorems on the Simplotope that Generalize Results on the Simplex and Cube

In the context of the theory and computation of fixed points of continuous mappings, researchers have developed combinatorial analogs of Brouwer's fixed-point theorem on the simplex and on the n-cu...

Nonlinear Saturation of the E × B-Instability in a Plasma Slab

The saturation of the E × B-instability is investigated in the non-linear regime. The governing equations are studied analytically and numerically by using a spectral method with mode truncation. The nonlinear stabilization is due to modifications of the background density- and potential-profiles. In the time asymptotic limit a stationary solution, which is independent of the initial conditions is obtained. The asymptotic state is characterized by a splitting of the interacting modes into two almost non-interacting groups, where the modes with even mode number sum; i.e. the modes driven by the linearly most unstable mode, is found to dominate the system. For this group fixed point calculations are performed analytically with six interacting modes. Comparison with numerical calculations indicates excellent agreement far into the unstable region.