631 research outputs found
Real functions computable by finite automata using affine representations
AbstractThis paper tries to classify the functions of type InâI (for some interval IâR) that can be exactly realized by a finite transducer. Such a class of functions strongly depends on the choice of real number representation. This paper considers only the so-called affine representations where numbers are represented by infinite compositions of affine contracting functions on I. Affine representations include the radix (e.g. decimal, signed binary) representations. The first result is that all piecewise affine functions of n variables with rational coefficients are computable by a finite transducer which uses the signed binary representation. The second and main result is that any function computable by a finite transducer using an affine representation is affine on any open connected subset of In on which it is continuously differentiable. This limitation theorem shows that the set of finitely computable functions is very restricted
Kleene Algebras and Semimodules for Energy Problems
With the purpose of unifying a number of approaches to energy problems found
in the literature, we introduce generalized energy automata. These are finite
automata whose edges are labeled with energy functions that define how energy
levels evolve during transitions. Uncovering a close connection between energy
problems and reachability and B\"uchi acceptance for semiring-weighted
automata, we show that these generalized energy problems are decidable. We also
provide complexity results for important special cases
Semantics and computation of the evolution of hybrid systems with ariadne
In this talk we will present material on the semantics, computability, and algorithms for the evolution of hybrid dynamical systems, and an overview of the tool Ariadne for verification of hybrid systems [1]. Hybrid systems are characterised by undergoing continuous evolution interspersed by discrete jumps. They exhibit all the complexities of finite automata, nonlinear dynamic systems and differential equations, and are extremely difficult to analyze. We will consider hybrid systems in which the continuous dynamics is given by a differential equation x = f(x), with discrete jumps x' = ri(x) which occur as soon as a guard condition gi(x) = 0 is activated. It is clear that the evolution of a hybrid system undergoes discontinuities, but since only continuous functions are computable, it is not clear to what extent, if any, it is possible to perform a rigorous analysis of a hybrid system. We will first show that we can define lower and upper semantics of evolution under which it is possible to compute reachable sets, and that away from discontinuity points (such as grazing or corner collision points), these semantics agree [2]. In order to perform reachability analysis, it is necessary to define the evolution over bounded initial sets of states. We show that this can be done using the operations of range, compose, flow and solve operations on functions. We will see that constrained image sets of the form {f(x) | x ? D | g(x) ? C}, are sufficient to express the evolution exactly, except for the case of degenerate (non-transverse) cross- ings [3]. The flow operation is the most computationally demanding, and we will give some details of the implementation and efficiency considerations [4]. We will give examples of reachability analysis in Ariadne, including electrical power converters and heating systems. Finally, we will outline some areas of active research, including differential inclusions [5] and modular reasoning
Reachability problems for PAMs
Piecewise affine maps (PAMs) are frequently used as a reference model to show
the openness of the reachability questions in other systems. The reachability
problem for one-dimentional PAM is still open even if we define it with only
two intervals. As the main contribution of this paper we introduce new
techniques for solving reachability problems based on p-adic norms and weights
as well as showing decidability for two classes of maps. Then we show the
connections between topological properties for PAM's orbits, reachability
problems and representation of numbers in a rational base system. Finally we
show a particular instance where the uniform distribution of the original orbit
may not remain uniform or even dense after making regular shifts and taking a
fractional part in that sequence.Comment: 16 page
Algorithmic Verification of Continuous and Hybrid Systems
We provide a tutorial introduction to reachability computation, a class of
computational techniques that exports verification technology toward continuous
and hybrid systems. For open under-determined systems, this technique can
sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661
Universal neural field computation
Turing machines and G\"odel numbers are important pillars of the theory of
computation. Thus, any computational architecture needs to show how it could
relate to Turing machines and how stable implementations of Turing computation
are possible. In this chapter, we implement universal Turing computation in a
neural field environment. To this end, we employ the canonical symbologram
representation of a Turing machine obtained from a G\"odel encoding of its
symbolic repertoire and generalized shifts. The resulting nonlinear dynamical
automaton (NDA) is a piecewise affine-linear map acting on the unit square that
is partitioned into rectangular domains. Instead of looking at point dynamics
in phase space, we then consider functional dynamics of probability
distributions functions (p.d.f.s) over phase space. This is generally described
by a Frobenius-Perron integral transformation that can be regarded as a neural
field equation over the unit square as feature space of a dynamic field theory
(DFT). Solving the Frobenius-Perron equation yields that uniform p.d.f.s with
rectangular support are mapped onto uniform p.d.f.s with rectangular support,
again. We call the resulting representation \emph{dynamic field automaton}.Comment: 21 pages; 6 figures. arXiv admin note: text overlap with
arXiv:1204.546
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