scholarly journals Fractal Dimension versus Process Complexity

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Joost J. Joosten ◽  
Fernando Soler-Toscano ◽  
Hector Zenil
Keyword(s):  
Fractal Dimension ◽  
Turing Machine ◽  
Time Complexity ◽  
Linear Time ◽  
Space Time ◽  
Polynomial Space ◽  
Turing Machines ◽  
Complexity Measures ◽  
Strong Relation ◽  
Process Complexity

We look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machineτand any particular inputx, we consider what we call thespace-timediagram which is basically the collection of consecutive tape configurations of the computationτ(x). In our setting, it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time, if and only if its dimension is 2, and its dimension is 1, if and only if it runs in superpolynomial time and it uses polynomial space. If a TM runs in timeO(xn), we have empirically verified that the corresponding dimension is(n+1)/n, a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on one side versus the time complexity of a computation on the other side.

Computational speed-up by effective operators

1972 ◽  
Vol 37 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Albert R. Meyer ◽  
Patrick C. Fischer
Keyword(s):  
Turing Machine ◽  
Computable Function ◽  
Turing Machines ◽  
Complexity Measures ◽  
Computational Speed ◽  
Partial Recursive Functions ◽  
Speed Up ◽  
Recursive Equations ◽  
Computable Functions ◽  
Effective Operators

The complexity of a computable function can be measured by considering the time or space required to compute its values. Particular notions of time and space arising from variants of Turing machines have been investigated by R. W. Ritchie [14], Hartmanis and Stearns [8], and Arbib and Blum [1], among others. General properties of such complexity measures have been characterized axiomatically by Rabin [12], Blum [2], Young [16], [17], and McCreight and Meyer [10].In this paper the speed-up and super-speed-up theorems of Blum [2] are generalized to speed-up by arbitrary total effective operators. The significance of such theorems is that one cannot equate the complexity of a computable function with the running time of its fastest program, for the simple reason that there are computable functions which in a very strong sense have no fastest programs.Let φi be the ith partial recursive function of one variable in a standard Gödel numbering of partial recursive functions. A family Φ0, Φ1, … of functions of one variable is called a Blum measure on computation providing(1) domain (φi) = domain (Φi), and(2) the predicate [Φi(x) = m] is recursive in i, x and m.Typical interpretations of Φi(x) are the number of steps required by the ith Turing machine (in a standard enumeration of Turing machines) to converge on input x, the space or number of tape squares required by the ith Turing machine to converge on input x (with the convention that Φi(x) is undefined even if the machine fails to halt in a finite loop), and the length of the shortest derivation of the value of φi(x) from the ith set of recursive equations.

scholarly journals SOLVING THE WORD PROBLEM IN REAL TIME

2001 ◽  
Vol 63 (3) ◽  
pp. 623-639 ◽  
Author(s):  
DEREK F. HOLT ◽  
SARAH REES
Keyword(s):  
Recent Result ◽  
Real Time ◽  
Word Problem ◽  
Turing Machine ◽  
Linear Time ◽  
Nilpotent Groups ◽  
Hyperbolic Groups ◽  
Turing Machines ◽  
Geometrically Finite ◽  
Word Hyperbolic Groups

The paper is devoted to the study of groups whose word problem can be solved by a Turing machine which operates in real time. A recent result of the first author for word hyperbolic groups is extended to prove that under certain conditions the generalised Dehn algorithms of Cannon, Goodman and Shapiro, which clearly run in linear time, can be programmed on real-time Turing machines. It follows that word-hyperbolic groups, finitely generated nilpotent groups and geometrically finite hyperbolic groups all have real-time word problems.

scholarly journals Complexity Results for Model Checking

1995 ◽  
Vol 2 (18) ◽  
Author(s):  
Allan Cheng
Keyword(s):  
Model Checking ◽  
Linear Time ◽  
Kripke Model ◽  
Upper Bounds ◽  
Polynomial Space ◽  
Turing Machines ◽  
Temporal Logics ◽  
Kripke Structures ◽  
Np Complete ◽  
Exponential Space

The complexity of model checking branching and linear time<br />temporal logics over Kripke structures has been addressed in e.g. [SC85,<br />CES86]. In terms of the size of the Kripke model and the length of the<br />formula, they show that the model checking problem is solvable in <br />polynomial time for CTL and NP-complete for L(F). The model checking<br />problem can be generalised by allowing more succinct descriptions of<br />systems than Kripke structures. We investigate the complexity of the<br />model checking problem when the instances of the problem consist of<br />a formula and a description of a system whose state space is at most<br />exponentially larger than the description. Based on Turing machines,<br />we define compact systems as a general formalisation of such system<br />descriptions. Examples of such compact systems are K-bounded Petri<br />nets and synchronised automata, and in these cases the well-known <br />algorithms presented in [SC85, CES86] would require exponential space in<br />term of the sizes of the system descriptions and the formulas; we present<br />polynomial space upper bounds for the model checking problem over<br />compact systems and the logics CTL and L(X,U,S). As an example of<br />an application of our general results we show that the model checking<br />problems of both the branching time temporal logic CTL and the linear<br />time temporal logics L(F) and L(X,U, S) over K-bounded Petri nets are<br />PSPACE-complete.

Generalizations of Checking Stack Automata: Characterizations and Hierarchies

2021 ◽  
pp. 1-28
Author(s):  
Oscar H. Ibarra ◽  
Ian McQuillan
Keyword(s):  
Time Complexity ◽  
Finite Automata ◽  
Space Time ◽  
Turing Machines ◽  
Computing Power ◽  
Multiple Input

We examine different generalizations of checking stack automata by allowing multiple input heads and multiple stacks, and characterize their computing power in terms of two-way multi-head finite automata and space-bounded Turing machines. For various models, we obtain hierarchies in terms of their computing power. Our characterizations and hierarchies expand or tighten some previously known results. We also discuss some decidability questions and the space/time complexity of the models.

TWO-DIMENSIONAL THREE-WAY ARRAY GRAMMARS AND THEIR ACCEPTORS

1989 ◽  
Vol 03 (03n04) ◽  
pp. 353-376 ◽  
Author(s):  
KENICHI MORITA ◽  
YASUNORI YAMAMOTO ◽  
KAZUHIRO SUGATA
Keyword(s):  
Real Time ◽  
Turing Machine ◽  
Linear Time ◽  
Two Dimensional ◽  
Turing Machines ◽  
One Dimensional ◽  
Context Sensitive

Two kinds of three-way isometric array grammars arc proposed as subclasses of an isometric monotonic array grammar. They are a three-way horizontally context-sensitive array grammar (3HCSAG) and a three-way immediately terminating array grammar (3ITAG). In these three-way grammars, patterns of symbols can grow only in the leftward, rightward and downward directions. We show that their generating abilities of rectangular languages are precisely characterized by some kinds of three-way two-dimensional Turing machines or related acceptors. In this paper. the following results are proved. First, 3HCSAG is characterized by a nondeterministic two-dimensional three-way Turing machine with space-bound n (n is the width of a rectangular input) and a nondeterministic one-way parallel/sequential array acceptor. Second, 3ITAG is characterized by a nondeterministic two-dimensional three-way real-time (or linear-time) restricted Turing machine, a nondeterministic one-dimensional bounded cellular acceptor and a nondeterministic two-dimensional one-line tessellation acceptor.

AGE-BASED VARIATIONS OF FRACTAL STRUCTURE OF EEG SIGNAL IN PATIENTS WITH EPILEPSY

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850051 ◽  
Author(s):  
HAMIDREZA NAMAZI ◽  
SAJAD JAFARI
Keyword(s):  
Fractal Dimension ◽  
Brain Research ◽  
Age Groups ◽  
Eeg Signal ◽  
Brain Disorders ◽  
Eeg Signals ◽  
Important Challenge ◽  
Patients With Epilepsy ◽  
Process Complexity ◽  
Electroencephalogram Eeg

It is known that aging affects neuroplasticity. On the other hand, neuroplasticity can be studied by analyzing the electroencephalogram (EEG) signal. An important challenge in brain research is to study the variations of neuroplasticity during aging for patients suffering from epilepsy. This study investigates the variations of the complexity of EEG signal during aging for patients with epilepsy. For this purpose, we employed fractal dimension as an indicator of process complexity. We classified the subjects in different age groups and computed the fractal dimension of their EEG signals. Our investigations showed that as patients get older, their EEG signal will be more complex. The method of investigation that has been used in this study can be further employed to study the variations of EEG signal in case of other brain disorders during aging.

An Area Preserving Transformation Algorithm for Press Forming Blank Development

1993 ◽  
Author(s):  
Nirmal K. Nair ◽  
James H. Oliver
Keyword(s):  
Time Complexity ◽  
Linear Time ◽  
Geometric Interpretation ◽  
Surface Geometry ◽  
Forming Process ◽  
Surface Design ◽  
Design Assessment ◽  
Press Forming ◽  
Blank Shape ◽  
Area Preserving

Abstract An efficient algorithm is presented to determine the blank shape necessary to manufacture a surface by press forming. The technique is independent of material properties and instead uses surface geometry and an area conservation constraint to generate a geometrically feasible blank shape. The algorithm is formulated as an approximate geometric interpretation of the reversal of the forming process. The primary applications for this technique are in preliminary surface design, assessment of manufacturability, and location of binder wrap. Since the algorithm exhibits linear time complexity, it is amenable to implementation as an interactive design aid. The algorithm is applied to two example surfaces and the results are discussed.

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