scholarly journals Recursive Definitions of Monadic Functions

10.29007/1mdt ◽  
2018 ◽  
Author(s):  
Alexander Krauss
Keyword(s):  
Fixed Points ◽  
The State ◽  
Recursive Functions ◽  
Recursion Equation ◽  
Recursive Definitions ◽  
Standard Domain

Using standard domain-theoretic fixed-points, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the state-exception monad of Isabelle/HOL's imperative programming extension, which results in a convenient definition principle for imperative programs, which were previously hard to define.For such monadic functions, the recursion equation can always be derived without preconditions, even if the function is partial. The construction is easy to automate, and convenient induction principles can be derived automatically.

scholarly journals Final coalgebras as greatest fixed points in ZF set theory

1999 ◽  
Vol 9 (5) ◽  
pp. 545-567 ◽  
Author(s):  
LAWRENCE C. PAULSON
Keyword(s):  
Fixed Points ◽  
Set Theory ◽  
Theorem Prover ◽  
Infinite Length ◽  
Special Cases ◽  
Definition Of ◽  
Recursive Definitions ◽  
Ordered Pairs

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).

scholarly journals Recursive Definitions and Fixed-Points

2009 ◽  
Vol 247 ◽  
pp. 19-37 ◽  
Author(s):  
Francicleber Martins Ferreira ◽  
Ana Teresa Martins
Keyword(s):  
Fixed Points ◽  
Recursive Definitions

scholarly journals Construcción de un sensor electroquímico para determinación de gases contaminantes en el aire (CO, CO2 y O3), empleado una tarjeta Arduino

2020 ◽  
pp. 1-5
Author(s):  
Maria Luisa Lozano-Camargo ◽  
Christian Hugo Rodríguez-Gómez ◽  
Laura Galicia-Luis ◽  
Fernando Talavera-Romero
Keyword(s):  
Air Pollution ◽  
Air Quality ◽  
Cardiovascular Diseases ◽  
Electrochemical Sensor ◽  
Fixed Points ◽  
Anthropogenic Activities ◽  
The State ◽  
Physical And Chemical

As humanity evolves, air pollution has increased due to the various anthropogenic activities that man carries out, alienating emissions of gases and polluting particles to the environment, seriously affecting the health of living beings and the planet, since these cause Irreversible physical and chemical alterations in the environment, becoming a major problem worldwide. In Mexico there are meteorological stations that measure air quality and they are only found at fixed points, however, they do not cover all areas of the State of Mexico, the deterioration of air quality brings with it an increase in respiratory and cardiovascular diseases ; That is why this project's main objective is to build a portable electrochemical sensor using an Arduino board, using customizable software capable of quantifying and analyzing three polluting gases CO, CO2 and O3, especially in the municipality of Chimalhuacán located in the area East of the State of México.

scholarly journals A Formalisation of Weak Normalisation (with Respect to Permutations) of Sequent Calculus Proofs

2000 ◽  
Vol 3 ◽  
pp. 1-26
Author(s):  
A.A. Adams
Keyword(s):  
Fixed Points ◽  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
The State ◽  
Original Presentation

AbstractDyckhoff and Pinto present a weakly normalising system of reductions on derivations are characterised as the fixed points of the composition of the Prawitz translations into natural deduction and back. This paper presents a formalisation of the system, including a proof of the Weak normalisation property for the formalisation. More details can be found in earlier work by the author. The formalisation has been kept as closes as possible to the original presentation to allow an evaluation of the state of proof assistance for such methods, and to ensure similarity of methods, and not merely similarly of results. The formalisation is restricted to the implicational fragment of intuitionistic logic.

scholarly journals Fixed Points on Abstract Structures without the Equality Test

2002 ◽  
Vol 9 (26) ◽  
Author(s):  
Margarita Korovina
Keyword(s):  
Fixed Points ◽  
Recursive Definitions ◽  
Equality Test ◽  
Effective Operators

In this paper we present a study of definability properties of fixed points of effective operators on abstract structures without the equality test. In particular we prove that Gandy theorem holds for abstract structures. This provides a useful tool for dealing with recursive definitions using Sigma-formulas.<br /> <br />One of the applications of Gandy theorem in the case of the reals without the equality test is that it allows us to define universal Sigma-predicates. It leads to a topological characterisation of Sigma-relations on |R.

Some restrictions on simple fixed points of the integers

1989 ◽  
Vol 54 (4) ◽  
pp. 1324-1345 ◽  
Author(s):  
G. L. McColm
Keyword(s):  
Fixed Points ◽  
Recursive Functions

AbstractA function is recursive (in given operations) if its values are computed explicitly and uniformly in terms of other “previously computed” values of itself and (perhaps) other “simultaneously computed” recursive functions. Here, “explicitly” includes definition by cases.We investigate those recursive functions on the structure N = 〈ω, 0, succ, pred〉 that are computed in terms of themselves only, without other simultaneously computed recursive functions.

A recursion principle for linear orderings

1992 ◽  
Vol 57 (1) ◽  
pp. 82-96 ◽  
Author(s):  
Juha Oikkonen
Keyword(s):  
Fixed Points ◽  
Initial Segment ◽  
Theoretical Methods ◽  
Linear Orderings ◽  
Direct Limits ◽  
Game Theoretic ◽  
Recursive Definitions ◽  
Natural Way

AbstractThe idea of this paper is to approach linear orderings as generalized ordinals and to study how they are made from their initial segments. First we look at how the equality of two linear orderings can be expressed in terms of equality of their initial segments. Then we shall use similar methods to define functions by recursion with respect to the initial segment relation. Our method is based on the use of a game where smaller and smaller initial segments of linear orderings are considered. The length of the game is assumed to exceed that of the descending sequences of elements of the linear orderings considered. By use of such game-theoretical methods we can for example extend the recursive definitions of the operations of sum, product and exponentiation of ordinals in a unique and natural way for arbitrary linear orderings. Extensions coming from direct limits do not satisfy our game-theoretic requirements in general. We also show how our recursive definitions allow very simple constructions for fixed points of functions, giving rise to certain interesting linear orderings.

scholarly journals INFLUENCE OF REFRACTORY PERIODS IN THE HOPFIELD MODEL

1996 ◽  
Vol 07 (01) ◽  
pp. 43-56 ◽  
Author(s):  
CRISÓGONO R. DA SILVA ◽  
FRANCISCO A. TAMARIT ◽  
EVALDO M. F. CURADO
Keyword(s):  
Fixed Points ◽  
Associative Memory ◽  
Dynamical Behavior ◽  
The State ◽  
Hopfield Model ◽  
Refractory Periods ◽  
Previous Time

We study both analytically and numerically the effects of including refractory periods in the Hopfield model for associative memory. These periods are introduced in the dynamics of the network as thresholds that depend on the state of the neuron at the previous time. Both the retrieval properties and the dynamical behavior are analyzed, and we found that, depending on the value of the thresholds and on the the ratio α between the number of stored memories (p) and the total number of neurons (N), the system presents not only fixed points but also chaotic or cyclic orbits. The recognizing capability of the network in the cyclic region is also studied.

Dynamic behavior of a neural network model of locomotor control in the lamprey

1996 ◽  
Vol 75 (3) ◽  
pp. 1074-1086 ◽  
Author(s):  
R. Jung ◽  
T. Kiemel ◽  
A. H. Cohen
Keyword(s):  
Neural Network ◽  
Fixed Points ◽  
Dynamic Behavior ◽  
Network Model ◽  
Neural Network Model ◽  
Pattern Generator ◽  
The State ◽  
Model Network ◽  
Reticular Neurons ◽  
Parameter Values

1. Experimental studies have shown that a central pattern generator in the spinal cord of the lamprey can produce the basic rhythm for locomotion. This pattern generator interacts with the reticular neurons forming a spinoreticulospinal loop. To better understand and investigate the mechanisms for locomotor pattern generation in the lamprey, we examine the dynamic behavior of a simplified neural network model representing a unit spinal pattern generator (uPG) and its interaction with the reticular system. We use the techniques of bifurcation analysis and specifically examine the effects on the dynamic behavior of the system of 1) changing tonic drives to the different neurons of the uPG; 2) altering inhibitory and excitatory interconnection strengths among the uPG neurons; and 3) feedforward-feedback interactions between the uPG and the reticular neurons. 2. The model analyzed is a qualitative left-right symmetric network based on proposed functional architecture with one class of phasic reticular neurons and three classes of uPG neurons: excitatory (E), lateral (L), and crossed (C) interneurons. In the model each class is represented by one left and one right neuron. Each neuron has basic passive properties akin to biophysical neurons and receives tonic synaptic drive and weighted synaptic input from other connecting neurons. The neuron's output as a function of voltage is given by a nonlinear function with a strict threshold and saturation. 3. With an appropriate set of parameter values, the voltage of each neuron can oscillate periodically with phase relationships among the different neurons that are qualitatively similar to those observed experimentally. The uPG alone can also oscillate, as observed experimentally in isolated lamprey spinal cords. Varying the parameters can, however, profoundly change the state of the system via different kinds of bifurcations. Change in a single parameter can move the system from nonoscillatory to oscillatory states via different kinds of bifurcations. For some parameter values the system can also exhibit multistable behavior (e.g., an oscillatory state and a nonoscillatory state). The analysis also shows us how the amplitudes of the oscillations vary and the periods of limit cycles change as different bifurcation points are approached. 4. Altering tonic drive to just one class of uPG neurons (without altering the interconnections) can change the state of the system by altering the stability of fixed points, converting fixed points to oscillations, single oscillations to two stable oscillations, etc. Two-parameter bifurcation diagrams show the critical regions in which a balance between the tonic drives is necessary to maintain stable oscillations. A minimum tonic drive is necessary to obtain stable oscillatory output. With appropriate changes in the tonic drives to the L and C neurons, stable oscillatory output can be obtained even after eliminating the E neurons. Indeed, the presence of active E neurons in the biological system does not prove they play a functional role in the system, because tonic drive from other sources can substitute for them. On the other hand, very high excitation of any one class of neurons can terminate oscillations. Appropriate balance of tonic drives to different neuron classes can help sustain stable oscillations for larger tonic drives. Published experimental results concerning changes in amplitude and swimming frequency with increased tonic drives are mimicked by the model's responses to increased tonic drive. 5. Interconnectivity among the neurons plays a crucial role. The analysis indicates that the C and L classes of neurons are essential components of the model network. Sufficient inhibition from the L to C neurons as well as mutual inhibition between the left and right halves is necessary to obtain stable oscillatory output. When the E neurons are present in the model network, they must receive appropriate tonic drive and provide appropriate excitation

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