Paul Cull

Oregon State University, Corvallis, Oregon, United States

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Publications (36)14.91 Total impact

  • Paul Cull
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    ABSTRACT: At first glance, Biology and Computer Science are diametrically opposed sciences. Biology deals with carbon based life forms shaped by evolution and natural selection. Computer Science deals with electronic machines designed by engineers and guided by mathematical algorithms. In this brief paper, we review biologically inspired computing. We discuss several models of computation which have arisen from various biological studies We show what these have in common, and conjecture how biology can still suggest answers and models for the next generation of computing problems. We discuss computation and argue that these biologically inspired models do not extend the theoretical limits on computation. We suggest that, in practice, biological models may give more succinct representations of various problems, and we mention a few cases in which biological models have proved useful. We also discuss the reciprocal impact of computer science on biology and cite a few significant contributions to biological science.
    Bio Systems 01/2013; · 1.27 Impact Factor
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    Paul Cull
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    ABSTRACT: N. Rashevsky (1899-1972) was one of the pioneers in the application of mathematics to biology. With the slogan: mathematical biophysics : biology :: mathematical physics ; physics, he proposed the creation of a quantitative theoretical biology. Here, we will give a brief biography, and consider Rashevsky's contributions to mathematical biology including neural nets and relational biology. We conclude that Rashevsky was an important figure in the introduction of quantitative models and methods into biology.
    Biosystems 05/2007; 88(3):178-84. · 1.58 Impact Factor
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    Paul Cull
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    ABSTRACT: Some of the simplest models of population growth are one dimensional nonlinear difference equations. While such models can display wild behavior including chaos, the standard biological models have the interesting property that they display global stability if they display local stability. Various researchers have sought a simple explanation for this agreement of local and global stability. Here, we show that enveloping by a linear fractional function is sufficient for global stability. We also show that for seven standard biological models local stability implies enveloping and hence global stability. We derive two methods to demonstrate enveloping and show that these methods can easily be applied to the seven example models. Although enveloping by a linear fractional is a sufficient for global stability, we show by example that such enveloping is not necessary. We extend our results by showing that enveloping implies global stability even when f(x) is a discontinuous multi-function which might be a more reasonable description of real bilogical data. We show that our techniques can be applied to situations which are not population models. Finally, we give examples of population models which have local stability but not global stability.
    Bulletin of Mathematical Biology 05/2007; 69(3):989-1017. · 2.02 Impact Factor
  • Paul Cull
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    ABSTRACT: We are gathered here today to celebrate the life and scientific career of Prof. Eduardo Caianiello. It is my task to present and as-sess one small area of his work-neural nets. I first met Prof. Caianiello many years ago when he visited the University of Chicago. After writing my thesis on neural nets, I subsequently worked with Prof. Ca-ianiello on various occasions at the Labora-torio di Cibernetica and at the University of Salerno. Hence my discussion will be biased by personal recollections and hence should count more as scientific heritage rather than as strict history.
    07/2006: pages 47-61;
  • Paul Cull
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    ABSTRACT: Difference equations have been used in population biology from Fibonacci to May and Yorke and to the present day. They have also been used in other biological fields. Here, we describe a number of fairly well know examples. We give techniques and results about the analysis of linear and nonlinear difference equations. We also show that there are practical and theoretical limitations on the analysis of nonlinear models. 1 Introduction For almost 1000 years from Fibonacci's rabbits of 1200 to the present day, difference equations have been used as biological models. Here we will give a brief run through some of the rudiments of difference equations and describe some simple models used in biology. To keep the paper short, we will only discuss some models from population biology. We start with the story of Fibonacci's rabbits, where we find and solve a simple difference equation. We then generalize to kth order linear difference equations and outline the theory of these equations. We give stronger results about estimating and computing the solutions to nonnegative difference equations including those with inputs. We return to Fibonacci's model and show how it can be generalized to Leslie's matrix model. After a brief discussion of Leslie's model, we describe the generalization of these results contained in the Perron- Frobenius theorem. We then turn to a consideration of nonlinear models. We show that a number of these models can be analyzed quite easily, but that chaos and particularly sensitive dependence on initial conditions may make even simple appearing nonlinear models difficult to use in practice. We briefly show that enveloping can be used for many common population models. Finally, we consider universality and undecidability and argue that a complete theory of nonlinear difference equations is impossible. 1.1 Notation There are a few items of notation in this paper which may not be familiar. When we are talking about sequences, we often use the notation xn to mean the nth element of the sequence. But we also, ambiguously, use xn to mean the entire sequence. When we want to be careful, we use the notationxnto mean a whole sequence. There are two notions of bounding with special notation. (16) We say that
    01/2006;
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    MARGARET A. SANDERS, Paul Cull
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    ABSTRACT: Fibonacci numbers and difference equations show up in many counting prob- lems. Zeckendorf showed how to represent natural numbers in "binary" Fibonacci bases. Capocelli counted the number of 0 bits and 1 bits in such representation. Here we use the theory of difference equations to try to provide proofs for Capocelli's claims. We also inves- tigate generalization of the Fibonacci difference equation which may show behavior similar to that observed by Capocelli. In particular, we conjecture that "doubly non-negative" difference equations will have solutions whose ratios monotonically approach a limit.
    01/2006;
  • Tai Hsu, Paul Cull
    Proceedings of the ISCA 18th International Conference on Computer Applications in Industry and Engineering, November 9-11, 2005, Sheraton Moana Surfrider, Honolulu, Hawaii, USA; 01/2005
  • Paul Cull
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    ABSTRACT: Convergence is a central problem in both computer science and in population biology. Will a program terminate? Will a population go to an equilibrium? In general these questions are quite di cult-even unsolvable. In this paper we will concentrate on very simple iterations of the form
    Computer Aided Systems Theory - EUROCAST 2005, 10th International Conference on Computer Aided Systems Theory, Las Palmas de Gran Canaria, Spain, February 7-11, 2005, Revised Selected Papers; 01/2005
  • Tai Hsu, Paul Cull
    Proceedings of the ISCA 17th International Conference on Parallel and Distributed Computing Systems, September 15-17, 2004, The Canterbury Hotel, San Francisco, California, USA; 01/2004
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    ABSTRACT: Genomic sequence data is available for an ever-increasing number of organisms, but the full meaning of this data remains an enigma. String alignment is one approach for deciphering the information contained in genetic strings. Sequences which are conserved across species will help identify genes and other important structures. Similarity between species can be scored by measuring how well their sequences align. The walking tree method is an approximate string alignment method that can handle insertions, deletions, substitutions, translocations, and more than one level of inversion. We will describe this method and recent improvements which allow fast alignment of megabase strings. We will show examples in which the method located or discovered genes. We show how the method can be used to construct phylogenetic trees. We also show that the method can be used to identify essential regions for protein function.
    Mathematical Biosciences 01/2004; 188:207-19. · 1.45 Impact Factor
  • Paul Cull, Tai Hsu
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    ABSTRACT: The meaning of biological sequences is a central problem of modern biology. Although string matching is well-understood in the edit-distance model, biological strings with transpositions and inversions violate this model’s assumptions. To align biologically reasonable strings, we proposed the Walking Tree Method [4,5,6,7,8]; an approximate string alignment method that can handle insertion, deletions, substitutions, translocations, and more than one level of inversions. Our earlier versions were able to align whole bacterial genomes (1 Mbps) and discover and verify genes. As extremely long sequences can now be deciphered rapidly and accurately without amplification [2,3,15], speeding up the method becomes necessary. Via a technique that we call recurrence reduction in which some computations can be looked up rather than recomputed, we are able to significantly improve the performance, e.g. 400% for a 1-million base pair alignment. In theory, our method can align a length |P| string with a length |T| string in time |P||T|/(nlog |P|) using n processors in parallel. In practice, we can align 10 Mbps strings within a week using 30 processors.
    Computer Aided Systems Theory - EUROCAST 2003, 9th International Workshop on Computer Aided Systems Theory, Las Palmas de Gran Canaria, Spain, February 24-28, 2003, Revised Selected Papers; 01/2003
  • P. Cull
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    ABSTRACT: Since the seminal papers of Li and Yorke [5] and May [6] [7] the importance of chaos in science has been clear. But, the impressive theorems leading to an understanding of chaos have been largely proved for one-dimensional continuous maps and only conjectured to hold for more complicated systems. In this paper, we want to consider a simple but discontinuous family of maps, and ask how closely these maps, the linear fractionals, come to being chaotic. Our basic answers are that linear fractionals are easy to understand; that they do not show fully chaotic behavior in the technical sense; and that they do show behavior that would intuitively be considered chaotic. In particular, we show that many linear fractionals exhibit global asymptotic stability and do not show chaos; that rational coe cient linear fractionals may be periodic, but the the periods are restricted to be 1, 2, 3, 4, or 6; that more general periodicity is possible with irrational coe cients; that some linear fractionals are aperiodic but have chaotic-like orbits; that these aperiodic maps have invariant distributions, but such distributions are not attractive. We also show, following Cull and Cha ee [2], that linear fractionals may be used to show global stability of other nonlinear maps. In summary, linear fractionals are easy-to-understand nonlinear maps that have a variety of appli-cations, and these maps can display complex chaotic-like behavior.
    01/2002;
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    ABSTRACT: Tree patterns are natural candidates for representing rules and hypotheses in many tasks such as information extraction and symbolic mathematics. A tree pattern is a tree with labeled nodes where some of the leaves may be labeled with variables, whereas a tree instance has no variables. A tree pattern matches an instance if there is a consistent substitution for the variables that allows a mapping of subtrees to matching subtrees of the instance. A finite union of tree patterns is called a forest. In this paper, we study the learnability of tree patterns from queries when the subtrees are unordered. The learnability is determined by the semantics of matching as defined by the types of mappings from the pattern subtrees to the instance subtrees. We first show that unordered tree patterns and forests are not exactly learnable from equivalence and subset queries when the mapping between subtrees is one-to-one onto, regardless of the computational power of the learner. Tree and forest patterns are learnable from equivalence and membership queries for the one-to-one into mapping. Finally, we connect the problem of learning tree patterns to inductive logic programming by describing a class of tree patterns called Clausal trees that includes non-recursive single-predicate Horn clauses and show that this class is learnable from equivalence and membership queries.
    Machine Learning 08/2001; 44(3):211-243. · 1.47 Impact Factor
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    T Hsu, P Cull
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    ABSTRACT: The Walking Tree Method [3, 4, 5, 18] is an approximate string alignment method that can handle insertions, deletions, substitutions, translocations, and more than one level of inversions all together. Moreover, it tends to highlight gene locations, and helps discover unknown genes. Its recent improvements in runtime and space use extends its capability in exploring large strings. We will briefly describe the Walking Tree Method with its recent improvements [18], and demonstrate its speed and ability to align real complete genomes such as Borrelia burgdorferi (910724 base pairs of its single chromosome) and Chlamydia trachomatis (1042519 base pairs) in reasonable time, and to locate and verify genes.
    Pacific Symposium on Biocomputing. Pacific Symposium on Biocomputing 02/2001;
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    P. Cull, J. Chaffee
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    ABSTRACT: One dimensional nonlinear difference equations have been used to model population growth. The standard biological models have the interesting characteristic that they display global stability if they display local stability. Various researchers have sought a simple explanation for this agreement of local and global stability. Here, we show that enveloping by a linear fractional function is sufficient for global stability. We also show that for seven standard biological models local stability implies enveloping and hence global stability. We derive two methods to demonstrate enveloping and show that these methods can easily be applied to the seven example models. © 2000 American Institute of Physics.
    AIP Conference Proceedings. 05/2000; 517(1):263-276.
  • R D Hangartner, P Cull
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    ABSTRACT: In this paper, we address the question, can biologically feasible neural nets compute more than can be computed by deterministic polynomial time algorithms? Since we want to maintain a claim of plausibility and reasonableness we restrict ourselves to algorithmically easy to construct nets and we rule out infinite precision in parameters and in any analog parts of the computation. Our approach is to consider the recent advances in randomized algorithms and see if such randomized computations can be described by neural nets. We start with a pair of neurons and show that by connecting them with reciprocal inhibition and some tonic input, then the steady-state will be one neuron ON and one neuron OFF, but which neuron will be ON and which neuron will be OFF will be chosen at random (perhaps, it would be better to say that microscopic noise in the analog computation will be turned into a megascale random bit). We then show that we can build a small network that uses this random bit process to generate repeatedly random bits. This random bit generator can then be connected with a neural net representing the deterministic part of randomized algorithm. We, therefore, demonstrate that these neural nets can carry out probabilistic computation and thus be less limited than classical neural nets.
    Biosystems 01/2000; 58(1-3):167-76. · 1.58 Impact Factor
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    P. Cull, Tai-Ching Hsu
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    ABSTRACT: Approximate string matching is commonly used to align genetic sequences (DNA or RNA) to determine their shared characteristics. Most genetic string matching methods are based on the edit-distance model, which does not provide alignments for inversions and translocations. Recently, a heuristic called the Walking Tree Method [2, 3] has been developed to solve this problem. Unlike other heuristics, it can handle more than one level of inversion, i.e., inversions within inversions. Furthermore, it tends to capture the matched strings' genes while other heuristics fail. There are two versions of the original walking tree heuristics: the score version gives only the alignment score, the alignment version gives both the score and the alignment mapping between the strings. The score version runs in quadratic time and uses linear space while the alignment version uses an extra log factor for time and space. In this paper, we will briefly describe the walking tree method and the original sequential and parallel algorithms. We will explain why different parallel algorithms are needed for a network of workstations rather than the original algorithm which worked well on a symmetric multi-processor. Our improved parallel method also led to a quadratic time sequential algorithm that uses less space. We give an example of our parallel method. We describe several experiments that show speedup linear in the number of processors, but eventual drop off in speedup as the communication network saturates. For big enough strings, we found linear speedup for all processors we had available. These results suggest that our improved parallel method will scale up as both the size of the problem and the number of processors increase. We include two figures that use real biological data and show that the walking tree methods can find translocations and inversions in DNA sequences and also discover unknown genes.
    Supercomputing, ACM/IEEE 1999 Conference; 12/1999
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    ABSTRACT: We consider learning tree patterns from queries extending our preceding work [Amoth, Cull, & Tadepalli, 1998] . The instances in this paper are unordered trees with nodes labeled by constant identifiers. The concepts are tree patterns and unions of tree patterns (unordered forests) with leaves labeled with constants or variables. A tree pattern matches any tree with its variables replaced with constant subtrees. A negative result for learning with equivalence and membership/subset queries is shown for unordered trees where a successful match requires the number of children in the pattern and instance to be the same. Unordered trees and forests are shown to be learnable with an alternative matching semantics that allows an instance to have extra children at each node. 1 INTRODUCTION Many applications in mathematics and language processing represent data more naturally as trees (or as unions of these) than as vectors of features. Tree patterns also provide more information than simple s...
    06/1999;
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    ABSTRACT: Basic biological information is stored in strings of nucleic acids (DNA, RNA) or amino acids (proteins). Teasing out the meaning of these strings is a central problem of modern biology. Matching and aligning strings brings out their shared characteristics. Although string matching is well-understood in the edit-distance model, biological strings with transpositions and inversions violate this model’s assumptions. We propose a family of heuristics called walking trees to align biologically reasonable strings. Both edit-distance and walking tree methods can locate specific genes within a large string when the genes’ sequences are given. When we attempt to match whole strings, the walking tree matches most genes, while the edit-distance method fails. We also give examples in which the walking tree matches substrings even if they have been moved or inverted. The edit-distance method was not designed to handle these problems. We include an example in which the walking tree “discovered” a gene. Calculating scores for whole genome matches gives a method for approximating evolutionary distance. We show two evolutionary trees for the picornaviruses which were computed by the walking tree heuristic. Both of these trees show great similarity to previously constructed trees. The point of this demonstration is that WHOLE genomes can be matched and distances calculated. The first tree was created on a Sequent parallel computer and demonstrates that the walking tree heuristic can be efficiently parallelized. The second tree was created using a network of work stations and demonstrates that there is suffient parallelism in the phylogenetic tree calculation that the sequential walking tree can be used effectively on a network. © 1999 American Institute of Physics.
    AIP Conference Proceedings. 03/1999; 465(1):201-215.
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    Paul Cull, Ingrid Nelson
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    ABSTRACT: A perfect one-error-correcting code on a graph is a subset of the vertices so that no two vertices in the subset are adjacent and each vertex not in the subset is adjacent to exactly one vertex in the subset. We show that the Towers of Hanoi puzzle defines an infinite family of graphs, and that each such graph supports a perfect one-error-correcting code. We show that these codes are essentially unique. Our characterization of the codewords as those ternary strings with an even number of 1's and an even number of 2's, makes generation and decoding computationally easy. In particular, decoding can be carried out by a two-pass finite state machine. We also show that determining if a graph can support a perfect one-error-correcting code is an NP-complete problem.
    Discrete Mathematics 01/1999; · 0.58 Impact Factor