26,733 research outputs found
Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length
We provide an implicit characterization of polynomial time computation in
terms of ordinary differential equations: we characterize the class
of languages computable in polynomial time in terms of
differential equations with polynomial right-hand side.
This result gives a purely continuous (time and space) elegant and simple
characterization of . This is the first time such classes
are characterized using only ordinary differential equations. Our
characterization extends to functions computable in polynomial time over the
reals in the sense of computable analysis. This extends to deterministic
complexity classes above polynomial time.
This may provide a new perspective on classical complexity, by giving a way
to define complexity classes, like , in a very simple
way, without any reference to a notion of (discrete) machine. This may also
provide ways to state classical questions about computational complexity via
ordinary differential equations, i.e.~by using the framework of analysis
The Light Lexicographic path Ordering
We introduce syntactic restrictions of the lexicographic path ordering to
obtain the Light Lexicographic Path Ordering. We show that the light
lexicographic path ordering leads to a characterisation of the functions
computable in space bounded by a polynomial in the size of the inputs
Unbounded Recursion and Non-size-increasing Functions
We investigate the computing power of function algebras defined by means of unbounded recursion on notation. We introduce two function algebras which contain respectively the regressive logspace computable functions and the non-size-increasing logspace computable functions. However, such algebras are unlikely to be contained in the set of logspace computable functions because this is equivalent to L=P . Finally, we introduce a function algebra based on simultaneous recursion on notation for the non-size-increasing functions computable in polynomial time and linear space
A Characterisation of Functions Computable in Polynomial Time and Space over the Reals with Discrete Ordinary Differential Equations: Simulation of Turing Machines with Analytic Discrete ODEs
We prove that functions over the reals computable in polynomial time can be characterised using discrete ordinary differential equations (ODE), also known as finite differences. We also provide a characterisation of functions computable in polynomial space over the reals. In particular, this covers space complexity, while existing characterisations were only able to cover time complexity, and were restricted to functions over the integers, and we prove that no artificial sign or test function is needed even for time complexity. At a technical level, this is obtained by proving that Turing machines can be simulated with analytic discrete ordinary differential equations. We believe this result opens the way to many applications, as it opens the possibility of programming with ODEs, with an underlying well-understood time and space complexity
Simulation of Turing machines with analytic discrete ODEs: FPTIME and FPSPACE over the reals characterised with discrete ordinary differential equations
We prove that functions over the reals computable in polynomial time can be
characterised using discrete ordinary differential equations (ODE), also known
as finite differences. We also provide a characterisation of functions
computable in polynomial space over the reals.
In particular, this covers space complexity, while existing characterisations
were only able to cover time complexity, and were restricted to functions over
the integers. We prove furthermore that no artificial sign or test function is
needed even for time complexity.
At a technical level, this is obtained by proving that Turing machines can be
simulated with analytic discrete ordinary differential equations. We believe
this result opens the way to many applications, as it opens the possibility of
programming with ODEs, with an underlying well-understood time and space
complexity.Comment: arXiv admin note: text overlap with arXiv:2209.1340
Polynomial Running Times for Polynomial-Time Oracle Machines
This paper introduces a more restrictive notion of feasibility of functionals on Baire space than the established one from second-order complexity theory. Thereby making it possible to consider functions on the natural numbers as running times of oracle Turing machines and avoiding second-order polynomials, which are notoriously difficult to handle. Furthermore, all machines that witness this stronger kind of feasibility can be clocked and the different traditions of treating partial functionals from computable analysis and second-order complexity theory are equated in a precise sense. The new notion is named "strong polynomial-time computability", and proven to be a strictly stronger requirement than polynomial-time computability. It is proven that within the framework for complexity of operators from analysis introduced by Kawamura and Cook the classes of strongly polynomial-time computable functionals and polynomial-time computable functionals coincide
Computational complexity of real functions
AbstractRecursive analysis, the theory of computation of functions on real numbers, has been studied from various aspects. We investigate the computational complexity of real functions using the methods of recursive function theory. Partial recursive real functions are defined and their domains are characterized as the recursively open sets. We define the time complexity of recursive real continuous functions and show that the time complexity and the modulus of uniform continuity of a function are closely related. We study the complexity of the roots and the differentiability of polynomial time computable real functions. In particular, a polynomial time computable real function may have a root of arbitrarily high complexity and may be nowhere differentiable. The concepts of the space complexity and nondeterministic computation are used to study the complexity of the integrals and the maximum values of real functions. These problems are shown to be related to the “P=?NP” and the “P=?PSPACE” questions
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