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12
Dynamical Recognizers: Realtime Language Recognition by Analog Computers
 Theoretical Computer Science
, 1996
"... We consider a model of analog computation which can recognize various languages in real time. We encode an input word as a point in R d by composing iterated maps, and then apply inequalities to the resulting point to test for membership in the language. Each class of maps and inequalities, suc ..."
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We consider a model of analog computation which can recognize various languages in real time. We encode an input word as a point in R d by composing iterated maps, and then apply inequalities to the resulting point to test for membership in the language. Each class of maps and inequalities, such as quadratic functions with rational coefficients, is capable of recognizing a particular class of languages; for instance, linear and quadratic maps can have both stacklike and queuelike memories. We use methods equivalent to the VapnikChervonenkis dimension to separate some of our classes from each other, e.g. linear maps are less powerful than quadratic or piecewiselinear ones, polynomials are less powerful than elementary (trigonometric and exponential) maps, and deterministic polynomials of each degree are less powerful than their nondeterministic counterparts. Comparing these dynamical classes with various discrete language classes helps illuminate how iterated maps can...
Purely Functional, RealTime Deques with Catenation
 Journal of the ACM
, 1999
"... We describe an efficient, purely functional implementation of deques with catenation. In addition to being an intriguing problem in its own right, finding a purely functional implementation of catenable deques is required to add certain sophisticated programming constructs to functional programming ..."
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We describe an efficient, purely functional implementation of deques with catenation. In addition to being an intriguing problem in its own right, finding a purely functional implementation of catenable deques is required to add certain sophisticated programming constructs to functional programming languages. Our solution has a worstcase running time of O(1) for each push, pop, inject, eject and catenation. The best previously known solution has an O(log k) time bound for the k deque operation. Our solution is not only faster but simpler. A key idea used in our result is an algorithmic technique related to the redundant digital representations used to avoid carry propagation in binary counting.
On computation with pulses
 Information and Computation
, 1999
"... We explore the computational power of formal models for computation with pulses. Such models are motivated by realistic models for biological neurons, and by related new types of VLSI (\pulse stream VLSI"). In preceding work it was shown that the computational power of formal models for com ..."
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Cited by 15 (1 self)
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We explore the computational power of formal models for computation with pulses. Such models are motivated by realistic models for biological neurons, and by related new types of VLSI (\pulse stream VLSI&quot;). In preceding work it was shown that the computational power of formal models for computation with pulses is quite high if the pulses arriving at a computational unit have an approximately linearly rising or linearly decreasing initial segment. This property is satis ed by common models for biological neurons. On the other hand several implementations of pulse stream VLSI employ pulses that are approximately piecewise constant (i.e. step functions). In this article we investigate the relevance of the shape of pulses in formal models for computation with pulses. It turns out that the computational power drops signi cantly if one replaces pulses with linearly rising or decreasing initial segments by piecewise constant pulses. We provide an exact characterization of the latter model in terms of a weak version of a random access machine (RAM). We also compare the language recognition capability of a recurrent version of this model with that of deterministic nite automata and Turing machines. 1
RealTime Deques, Multihead Turing Machines, and Purely Functional Programming
 In Conference on Functional Programming Languages and Computer Architecture
, 1993
"... We answer the following question: Can a deque (double ended queue) be implemented in a purely functional language such that each push or pop operation on either end of a queue is accomplished in O(1) time in the worst case? The answer is yes, thus solving a problem posted by Gajewska and Tarjan [1 ..."
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We answer the following question: Can a deque (double ended queue) be implemented in a purely functional language such that each push or pop operation on either end of a queue is accomplished in O(1) time in the worst case? The answer is yes, thus solving a problem posted by Gajewska and Tarjan [14] and by Ponder, McGeer, and Ng [25], and refining results of Sarnak [26] and Hoogerwoord [18]. We term such a deque realtime, since its constant worstcase behavior might be useful in real time programs (assuming realtime garbage collection [3], etc.) Furthermore, we show that no restriction of the functional language is necessary, and that push and pop operations on previous versions of a deque can also be achieved in constant time. We present a purely functional implementation of real time deques and its complexity analysis. We then show that the implementation has some interesting implications, and can be used to give a realtime simulation of a multihead Turing machine in a purel...
Two Heads are Better than Two Tapes
, 1994
"... . We show that a Turing machine with two singlehead onedimensional tapes cannot recognize the set f x2x 0 j x 2 f0; 1g and x 0 is a prefix of x g in real time, although it can do so with three tapes, two twodimensional tapes, or one twohead tape, or in linear time with just one tape. In ..."
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. We show that a Turing machine with two singlehead onedimensional tapes cannot recognize the set f x2x 0 j x 2 f0; 1g and x 0 is a prefix of x g in real time, although it can do so with three tapes, two twodimensional tapes, or one twohead tape, or in linear time with just one tape. In particular, this settles the longstanding conjecture that a twohead Turing machine can recognize more languages in real time if its heads are on the same onedimensional tape than if they are on separate onedimensional tapes. 1. Introduction The Turing machines commonly used and studied in computer science have separate tapes for input/output and for storage, so that we can conveniently study both storage as a dynamic resource and the more complex storage structures required for efficient implementation of practical algorithms [HS65]. Early researchers [MRF67] asked specifically whether twohead storage is more powerful if both heads are on the same onedimensional storage tape than if t...
Machine Models and Linear Time Complexity
 SIGACT News
, 1993
"... wer bounds. Machine models. Suppose that for every machine M 1 in model M 1 running in time t = t(n) there is a machine M 2 in M 2 which computes the same partial function in time g = g(t; n). If g = O(t)+O(n) we say that model M 2 simulates M 1 linearly. If g = O(t) the simulation has constantf ..."
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Cited by 5 (3 self)
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wer bounds. Machine models. Suppose that for every machine M 1 in model M 1 running in time t = t(n) there is a machine M 2 in M 2 which computes the same partial function in time g = g(t; n). If g = O(t)+O(n) we say that model M 2 simulates M 1 linearly. If g = O(t) the simulation has constantfactor overhead ; if g = O(t log t) it has a factorofO(log t) overhead , and so on. The simulation is online if each step of M 1 i
The Computational Power of Spiking Neurons Depends on the Shape of the Postsynaptic Potentials
, 1996
"... Recently one has started to investigate the computational power of spiking neurons (also called "integrate and fire neurons"). These are neuron models that are substantially more realistic from the biological point of view than the ones which are traditionally employed in artificial neu ..."
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Cited by 4 (0 self)
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Recently one has started to investigate the computational power of spiking neurons (also called "integrate and fire neurons"). These are neuron models that are substantially more realistic from the biological point of view than the ones which are traditionally employed in artificial neural nets. It has turned out that the computational power of networks of spiking neurons is quite large. In particular they have the ability to communicate and manipulate analog variables in spatiotemporal coding, i.e. encoded in the time points when specific neurons "fire" (and thus send a "spike" to other neurons). These preceding results have motivated the question which details of the firing mechanism of spiking neurons are essential for their computational power, and which details are "accidental" aspects of their realization in biological "wetware". Obviously this question becomes important if one wants to capture some of the advantages of computing and learning with spatiotemporal c...
On superlinear lower bounds in complexity theory
 In Proc. 10th Annual IEEE Conference on Structure in Complexity Theory
, 1995
"... This paper first surveys the neartotal lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the mode ..."
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Cited by 1 (1 self)
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This paper first surveys the neartotal lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the model allows nonlocal communication with memory at unit cost. We study a model that imposes a “fair cost ” for nonlocal communication, and obtain modest superlinear lower bounds for some problems via a Kolmogorovcomplexity argument. Then we look to the larger picture of what it will take to prove really striking lower bounds, and pull from ours and others’ work a concept of information vicinity that may offer new tools and modes of analysis to a young field that rather lacks them.
On TwoTape RealTime Computation and Queues
"... This is a draft version of [J. Computer and System Sciences, 29 (1984) 303  311]. A Turing machine with two storage tapes can not simulate a queue in both realtime and with at least one storage tape head always within o (n) squares from the start square. This fact may be useful for showing that a ..."
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This is a draft version of [J. Computer and System Sciences, 29 (1984) 303  311]. A Turing machine with two storage tapes can not simulate a queue in both realtime and with at least one storage tape head always within o (n) squares from the start square. This fact may be useful for showing that a twohead tape unit is more powerful in realtime than two onehead tape units, as is commonly conjectured. Keywords & Phrases: multitape Turing machine, multihead Turing machine, realtime computation, two heads versus two tapes, storage retrieval, queue, incompressible string, Kolmogorov complexity Proposed Running Head: TWOTAPE REALTIME COMPUTATION ################## * This work was supported by the Stichting Mathematisch Centrum.    2  1. Introduction. Realtime computation in the world of Turing machine like devices is the analogon to realtime computation in concrete computer systems. To compare the relative computation power of two storage devices a fine distinction can be...
Information and Computation 148, 202218 (1999) On Computation with Pulses
"... We explore the computational power of formal models for computation with pulses. Such models are motivated by realistic models for biological neurons and by related new types of VLSI (‘‘pulse stream VLSI’’). In preceding work it was shown that the computational power of formal models for computation ..."
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We explore the computational power of formal models for computation with pulses. Such models are motivated by realistic models for biological neurons and by related new types of VLSI (‘‘pulse stream VLSI’’). In preceding work it was shown that the computational power of formal models for computation with pulses is quite high if the pulses arriving at a computational unit have an approximately linearly rising or linearly decreasing initial segment. This property is satisfied by common models for biological neurons. On the other hand, several implementations of pulse stream VLSI employ pulses that are approximately piecewise constant (i.e., step functions). In this article we investigate the relevance of the shape of pulses in formal models for computation with pulses. The results show that the computational power drops significantly if one replaces pulses with linearly rising or decreasing initial segments by piecewise constant pulses. We provide an exact characterization of the latter model in terms of a weak version of a random access machine (RAM). We also compare the language recognition capability of a recurrent version of this model with that of deterministic finite automata and Turing machines.] 1999 Academic Press 1.