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Pcompleteness of cellular automaton Rule 110
 In International Colloquium on Automata Languages and Programming (ICALP), volume 4051 of LNCS
, 2006
"... We show that the problem of predicting t steps of the 1D cellular automaton Rule 110 is Pcomplete. The result is found by showing that Rule 110 simulates deterministic Turing machines in polynomial time. As a corollary we find that the small universal Turing machines of Mathew Cook run in polyn ..."
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We show that the problem of predicting t steps of the 1D cellular automaton Rule 110 is Pcomplete. The result is found by showing that Rule 110 simulates deterministic Turing machines in polynomial time. As a corollary we find that the small universal Turing machines of Mathew Cook run in polynomial time, this is an exponential improvement on their previously known simulation time overhead.
Small weakly universal Turing machines
"... Abstract. We give small universal Turing machines with statesymbol pairs of (6, 2), (3,3) and (2,4). These machines are weakly universal, which means that they have an infinitely repeated word to the left of their input and another to the right. They simulate Rule 110 and are currently the smallest ..."
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Cited by 7 (4 self)
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Abstract. We give small universal Turing machines with statesymbol pairs of (6, 2), (3,3) and (2,4). These machines are weakly universal, which means that they have an infinitely repeated word to the left of their input and another to the right. They simulate Rule 110 and are currently the smallest known weakly universal Turing machines. Despite their small size these machines are efficient polynomial time simulators of Turing machines. 1
An Associative Processor for Multicomparand Parallel Searching and Its Selected Applications
"... In this paper a multicomparand associative processor is presented. The structure of the processor and its functions are described in detail. The processor works in a combined bitserial/bitparallel mode. Its main component is a multicomparand associative memory with programmable prescription functi ..."
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Cited by 6 (3 self)
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In this paper a multicomparand associative processor is presented. The structure of the processor and its functions are described in detail. The processor works in a combined bitserial/bitparallel mode. Its main component is a multicomparand associative memory with programmable prescription functions. The multicomparand associative search paradigm is shown to be effective in processing complex search problems from many application areas including computational geometry, graph theory and list/matrix computations. Several representative problems, belonging to different complexity classes, and algorithms for them are presented and discussed.
Parallel and sequential optical computing
, 2008
"... We present a number of computational complexity results for an optical model of computation called the continuous space machine. We also describe an implementation for an optical computing algorithm that can be easily defined within the model. Our optical model is designed to model a wide class of ..."
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Cited by 3 (1 self)
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We present a number of computational complexity results for an optical model of computation called the continuous space machine. We also describe an implementation for an optical computing algorithm that can be easily defined within the model. Our optical model is designed to model a wide class of optical computers, such as matrix vector multipliers and pattern recognition architectures. It is known that the model solves intractable PSPACE problems in polynomial time, and NC problems in polylogarithmic time. Both of these results use large spatial resolution (number of pixels). Here we look at what happens when we have constant spatial resolution. It turns out that we obtain similar results by exploiting other resources, such as dynamic range and amplitude resolution. However, with certain other restrictions we essentially have a sequential device. Thus we are exploring the border between parallel and sequential computation in optical computing. We describe an optical architecture for the unordered search problem of finding a one in a list of zeros. We argue that our algorithm scales well, and is relatively straightforward to implement. This problem is easily parallelisable and is from the class NC. We go on to argue that the optical computing community should focus their attention on problems within P (and especially NC), rather than developing systems for tackling intractable problems. 1
On the complexity of the equivalence problem for probabilistic automata
 In Proc. of FoSSaCS’12, volume 7213 of LNCS
, 2012
"... Abstract. Deciding equivalence of probabilistic automata is a key problem for establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this pa ..."
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Abstract. Deciding equivalence of probabilistic automata is a key problem for establishing various behavioural and anonymity properties of probabilistic systems. In recent experiments a randomised equivalence test based on polynomial identity testing outperformed deterministic algorithms. In this paper we show that polynomial identity testing yields efficient algorithms for various generalisations of the equivalence problem. First, we provide a randomized NC procedure that also outputs a counterexample trace in case of inequivalence. Second, we consider equivalence of probabilistic cost automata. In these automata transitions are labelled with integer costs and each word is associated with a distribution on costs, corresponding to the cumulative costs of the accepting runs on that word. Two automata are equivalent if they induce the same cost distributions on each input word. We show that equivalence can be checked in randomised polynomial time. Finally we show that the equivalence problem for probabilistic visibly pushdown automata is logspace equivalent to the problem of whether a polynomial represented by an arithmetic circuit is identically zero. 1
OntheSublinearProcessorGap for Parallel Architectures
"... Abstract. In the past, parallel algorithms were developed, for the most part, under the assumption that the number of processors is Θ(n) (where n is the size of the input) and that if in practice the actual number was smaller, this could be resolved using Brent’s Lemma to simulate the highly paralle ..."
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Abstract. In the past, parallel algorithms were developed, for the most part, under the assumption that the number of processors is Θ(n) (where n is the size of the input) and that if in practice the actual number was smaller, this could be resolved using Brent’s Lemma to simulate the highly parallel solution on a lowerdegree parallel architecture. In this paper, however, we argue that design and implementation issues of algorithms and architectures are significantly different—both in theory and in practice—between computational models with high and low degrees of parallelism. We report an observed gap in the behavior of a parallel architecture depending on the number of processors. This gap appears repeatedly in both empirical cases, when studying practical aspects of architecture design and program implementation as well as in theoretical instances when studying the behaviour of various parallel algorithms. It separates the performance, design and analysis of systems with a sublinear number of processors and systems with linearly many processors. More specifically we observe that systems with either logarithmically many cores or with O(n α)cores(withα<1) exhibit a qualitatively different behavior than a system with a linear number of cores on the size of the input, i.e., Θ(n).Theevidencewepresentsuggeststheexistence of a sharp theoretical gap between the classes of problems that can be efficiently parallelized with o(n) processors and with Θ(n) processors unless P = NC. 1
Backward Induction is PTIMEcomplete
"... Abstract. We prove that the computational problem of finding backward induction outcome is PTIMEcomplete. Key words: gametheory, backward induction, computational complexity, finite extensive games 1 ..."
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Abstract. We prove that the computational problem of finding backward induction outcome is PTIMEcomplete. Key words: gametheory, backward induction, computational complexity, finite extensive games 1