ON THE NOTION OF LINEAR TIME COMPUTABILITY

1990 ◽  
Vol 01 (03) ◽  
pp. 295-307 ◽  
Author(s):  
ERICH GRÄDEL
Keyword(s):  
Linear Time ◽  
Random Access ◽  
Rational Numbers ◽  
Turing Machines ◽  
Machine Model ◽  
Class Time ◽  
Complete Problems ◽  
Time Bounds

To capture the informal concept of linear time computability, we propose and discuss the class TIME(n1+), consisting of all functions which are computable by a Successor RAM with exponent at most one; the exponent of a function is the infimum of all rational numbers r such that the function is computable in time O(nr). This class properly contains the class NLT (“Nearly Linear Time”)—proposed by Gurevich and Shelah—which contains all functions computable by a Successor RAM in time O(n(log n)k) for some k∈N. Both classes are very robust under changes of the underlying machine model: using the same time bounds they can be defined by Random Access Computers, Storage Modification Machines, Kolmogorov algorithms, Random Access Turing machines etc.; this distinguishes them from similar classes defined by ordinary Turing machines. However, we show that TIME(n1+) can be defined e.g. by multi-dimensional Turing machines or by Turing machines which can jump; this is probably not true for NLT. Furthermore we consider two notions of completeness for TIME(n1+) and construct complete problems. These problems are based on restrictions of Gurevich’s function logic for P.

scholarly journals On the Comparison Complexity of the String Prefix-Matching Problem

1995 ◽  
Vol 2 (46) ◽  
Author(s):  
Dany Breslauer ◽  
Livio Colussi ◽  
Laura Toniolo
Keyword(s):  
Linear Time ◽  
String Matching ◽  
Random Access ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Matching Problem ◽  
Machine Model ◽  
Worst Case ◽  
On Line ◽  
Sequential Comparison

In this paper we study the exact comparison complexity of the string<br />prefix-matching problem in the deterministic sequential comparison model<br />with equality tests. We derive almost tight lower and upper bounds on<br />the number of symbol comparisons required in the worst case by on-line<br />prefix-matching algorithms for any fixed pattern and variable text. Unlike<br />previous results on the comparison complexity of string-matching and<br />prefix-matching algorithms, our bounds are almost tight for any particular pattern.<br />We also consider the special case where the pattern and the text are the<br />same string. This problem, which we call the string self-prefix problem, is<br />similar to the pattern preprocessing step of the Knuth-Morris-Pratt string-matching<br />algorithm that is used in several comparison efficient string-matching<br />and prefix-matching algorithms, including in our new algorithm.<br />We obtain roughly tight lower and upper bounds on the number of symbol<br />comparisons required in the worst case by on-line self-prefix algorithms.<br />Our algorithms can be implemented in linear time and space in the<br />standard uniform-cost random-access-machine model.

Logical Characterizations of Bounded Query Classes I: Logspace Oracle Machines

1993 ◽  
Vol 18 (1) ◽  
pp. 65-92
Author(s):  
Iain A. Stewart
Keyword(s):  
Truth Table ◽  
Complexity Class ◽  
Polynomial Hierarchy ◽  
Complexity Classes ◽  
Turing Machines ◽  
Complete Problems

We consider three sub-logics of the logic (±HP)*[FOs] and show that these sub-logics capture the complexity classes obtained by considering logspace deterministic oracle Turing machines with oracles in NP where the number of oracle calls is unrestricted and constant, respectively; that is, the classes LNP and LNP[O(1)]. We conclude that if certain logics are of the same expressibility then the Polynomial Hierarchy collapses. We also exhibit some new complete problems for the complexity class LNP via projection translations (the first to be discovered: projection translations are extremely weak logical reductions between problems) and characterize the complexity class LNP[O(1)] as the closure of NP under a new, extremely strict truth-table reduction (which we introduce in this paper).

scholarly journals Linear-time simulation of multihead turing machines

1989 ◽  
Vol 81 (3) ◽  
pp. 353-363 ◽  
Author(s):  
W. Schnitzlein ◽  
H.-J. Stoss
Keyword(s):  
Linear Time ◽  
Turing Machines ◽  
Time Simulation

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

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 SAFE RECURSIVE SET FUNCTIONS

2015 ◽  
Vol 80 (3) ◽  
pp. 730-762 ◽  
Author(s):  
ARNOLD BECKMANN ◽  
SAMUEL R. BUSS ◽  
SY-DAVID FRIEDMAN
Keyword(s):  
Growth Rate ◽  
Polynomial Growth ◽  
Turing Machines ◽  
Finite Sets ◽  
Infinite Time ◽  
Machine Model ◽  
Set Functions ◽  
Rate Functions ◽  
Primitive Recursive ◽  
Recursive Set

AbstractWe introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments.We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

scholarly journals A Linear Parallel Algorithm to Compute Bisimulation and Relational Coarsest Partitions

2021 ◽  
pp. 115-133
Author(s):  
Jan Martens ◽  
Jan Friso Groote ◽  
Lars van den Haak ◽  
Pieter Hijma ◽  
Anton Wijs
Keyword(s):  
Parallel Algorithm ◽  
Time Complexity ◽  
Linear Time ◽  
Random Access ◽  
Time Algorithm ◽  
Transition System ◽  
Linear Time Algorithm

AbstractThe most efficient way to calculate strong bisimilarity is by finding the relational coarsest partition of a transition system. We provide the first linear-time algorithm to calculate strong bisimulation using parallel random access machines (PRAMs). More precisely, with n states, m transitions and $$| Act |\le m$$ | A c t | ≤ m action labels, we provide an algorithm for $$\max (n,m)$$ max ( n , m ) processors that calculates strong bisimulation in time $$\mathcal {O}(n+| Act |)$$ O ( n + | A c t | ) and space $$\mathcal {O}(n+m)$$ O ( n + m ) . The best-known PRAM algorithm has time complexity $$\mathcal {O}(n\log n)$$ O ( n log n ) on a smaller number of processors making it less suitable for massive parallel devices such as GPUs. An implementation on a GPU shows that the linear time-bound is achievable on contemporary hardware.

Lower Bounds for the Computational Power of Networks of Spiking Neurons

1996 ◽  
Vol 8 (1) ◽  
pp. 1-40 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Formal Model ◽  
Random Access ◽  
Spiking Neurons ◽  
Turing Machines ◽  
Computational Power ◽  
Computational Tool ◽  
Threshold Functions ◽  
Basic Assumptions ◽  
Threshold Circuits ◽  
Phase Differences

We investigate the computational power of a formal model for networks of spiking neurons. It is shown that simple operations on phase differences between spike-trains provide a very powerful computational tool that can in principle be used to carry out highly complex computations on a small network of spiking neurons. We construct networks of spiking neurons that simulate arbitrary threshold circuits, Turing machines, and a certain type of random access machines with real valued inputs. We also show that relatively weak basic assumptions about the response and threshold functions of the spiking neurons are sufficient to employ them for such computations.

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