CANaerospace

CANaerospace is a higher layer protocol based on Controller Area Network (CAN) which has been developed by Stock Flight Systems in 1998 for aeronautical applications. == Background == CANaerospace supports airborne systems employing the Line-replaceable unit (LRU) concept to share data across CAN and ensures interoperability between CAN LRUs by defining CAN physical layer characteristics, network layers, communication mechanisms, data types and aeronautical axis systems. CANaerospace is an open source project, was initiated to standardize the interface between CAN LRUs on the system level. CANaerospace is continuously being developed further and has also been published by NASA as the Advanced General Aviation Transport Experiments Databus Standard in 2001. It found widespread use in aeronautical research worldwide. A major research aircraft that employs several CANaerospace networks for real-time computer interconnection is the Stratospheric Observatory for Infrared Astronomy (SOFIA), a Boeing 747SP with a 2.5m astronomic telescope. CANaerospace is also frequently used in flight simulation and connects entire aircraft cockpits (i.e. in Eurofighter Typhoon simulators) to the simulation host computers. In Italy CANaerospace is used as UAV data bus technology. Furthermore, CANaerospace serves as communication network in several general aviation avionics systems. The CANaerospace interface definition closes the gap between the ISO/OSI layer 1 and 2 CAN protocol (which is implemented in the CAN controller itself) and the specific requirements of distributed systems in aircraft. It may be used as a primary or ancillary avionics network and was designed to meet the following requirements: Democratic network: CANaerospace does not require any master/slave relationships between LRUs or a "bus controller", thereby avoiding a potential single source of failure. Every node in the network has the same rights for participation in the bus traffic. Self-identifying message format: Each CANaerospace message contains information about the type of the data and the transmitting node. This allows the data to be unambiguously recognized at each receiving node. Continuous Message Numbering: Each CANaerospace message contains a continuously incremented number which allows coherent processing of messages in the receiving stations. Message Status Code: Each CANaerospace message contains information about the integrity of the data is conveying. This allows receiving stations to evaluate the quality of the received data and to react accordingly. Emergency Event Signaling: CANaerospace defines a mechanism that allows each node to transmit information about exception or error situations. This information can be used by other stations to determine the network health. Node Service Interface: As an enhancement to CAN, CANaerospace provides a means for individual stations on the network to communicate with each other using connection-oriented and connectionless services. Predefined CAN Identifier Assignment: CANaerospace offers a predefined identifier assignment list for normal operation data. In addition to the predefined list, user-defined identifier assignment lists may be used. Ease of Implementation: The amount of code to implement CANaerospace is very little by design in order to minimize the effort for testing and certification of flight safety critical systems. Openness to Extensions: All CANaerospace definitions are extendable to provide flexibility for future enhancements and to allow adaptions to the requirements of specific applications. Free Availability: No cost whatsoever apply for the use of CANaerospace. The specification can be downloaded from the Internet == Physical interface == To ensure interoperability and reliable communication, CANaerospace specifies the electrical characteristics, bus transceiver requirements and data rates with the corresponding tolerances based on ISO 11898. The bit timing calculation (baud rate accuracy, sample point definition) and robustness to electromagnetic interference are given special emphasis. Also addressed are CAN connector, wiring considerations and design guidelines to maximize electromagnetic compatibility. == Communication layers == The Bosch CAN specification itself allows messages being transmitted both periodically and aperiodically but does not cover issues like data representation, node addressing or connection-oriented protocols. CAN is entirely based on Anyone-to-Many (ATM) communication which means that CAN messages are always received by all stations in the network. The advantage of the CAN concept is inherent data consistency between all stations, the drawback is that it does not allow node addressing which is the basis for Peer-to-Peer (PTP) communication. Using CAN networks in aeronautical applications, however, demands a standard targeted to the specific requirements of airborne systems which implies that communication between individual stations in the network must be possible to enable the required degree of system monitoring. Consequently, CANaerospace defines additional ISO/OSI layer 3, 4 and 6 functions to support node addressing and unified ATM/PTP communication mechanisms. PTP communication allows to set up client/server interactions between individual stations in the network either temporarily or permanently. More than one of these interactions may be in effect at any given time and each node may be client for one operation and server for another at the same time. This CANaerospace mechanism is called "Node Service Concept" and allows i.e. to distribute system functions over several stations in the network or to control dynamic system reconfiguration in case of failure. The Node Service concept supports both connection-oriented and connectionless interactions like with TCP/IP and UDP/IP for Ethernet. Enabling both ATM and PTP communication for CAN requires the introduction of independent network layers to isolate the different types of communication. This is realized for CANaerospace by forming CAN identifier groups as shown in Figure 1. The resulting structure creates Logical Communication Channels (LCCs) and assigns a specific communication type (ATM, PTP) to each of the LCCs. User-defined LCCs provide the necessary freedom for designers and allow the implementation of CANaerospace according to the needs of specific applications. Figure 1: Logical Communication Channels for CANaerospace As a side effect, the CAN identifier groups in Figure 1 affect the priority of the message transmission in case of bus arbitration. The communication channels are therefore arranged according to their relative importance: Emergency Event Data Channel (EED): This communication channel is used for messages which require immediate action (i.e. system degradation or reconfiguration) and have to be transmitted with very high priority. Emergency Event Data uses ATM communication exclusively. High/Low Priority Node Service Data Channel (NSH/NSL): These communication channels are used for client/server interactions using PTP communication. The corresponding services may be of the connection-oriented as well as the connectionless type. NSH/NSL may also be used to support test and maintenance functions. Normal Operation Data Channel (NOD): This communication channel is used for the transmission of the data which is generated during normal system operation and described in the CANaerospace identifier assignment list. These messages may be transmitted periodically or aperiodically as well as synchronously or asynchronously. All messages which cannot be assigned to other communication channels shall use this channel. High/Low Priority User-Defined Data Channel (UDH/UDL): This channel is dedicated to communication which cannot, due to their specific characteristics, be assigned other channels without violating the CANaerospace specification. As long as the defined identifier range is used, the message content and the communication type (ATM, PTP) for these channels may be specified by the system designer. To ensure interoperability it is highly recommended that the use of these channels is minimized. Debug Service Data Channel (DSD): This channel is dedicated to messages which are used temporarily for development and test purposes only and are not transmitted during normal operation. As long as the defined identifier range is used, the message content and the communication type (ATM, PTP) for these channels may be specified by the system designer. == Data representation == The majority of the real-time control systems used in aeronautics employ "big endian" processor architectures. This data representation was therefore specified for CANaerospace as well. With big endian data representation, the most significant bit of any datum is arranged leftmost and transmitted first on CANaerospace as shown in Figure 2. Figure 2: "Big Endian" Data Representation for CANaerospace CANaerospace uses a self-identifying message

Hard sigmoid

In artificial intelligence, especially computer vision and artificial neural networks, a hard sigmoid is non-smooth function used in place of a sigmoid function. These retain the basic shape of a sigmoid, rising from 0 to 1, but using simpler functions, especially piecewise linear functions or piecewise constant functions. These are preferred where speed of computation is more important than precision. == Examples == The most extreme examples are the sign function or Heaviside step function, which go from −1 to 1 or 0 to 1 (which to use depends on normalization) at 0. Other examples include the Theano library, which provides two approximations: ultra_fast_sigmoid, which is a multi-part piecewise approximation and hard_sigmoid, which is a 3-part piecewise linear approximation (output 0, line with slope 0.2, output 1).

Parity benchmark

Parity problems are widely used as benchmark problems in genetic programming but inherited from the artificial neural network community. Parity is calculated by summing all the binary inputs and reporting if the sum is odd or even. This is considered difficult because: a very simple artificial neural network cannot solve it, and all inputs need to be considered and a change to any one of them changes the answer.

Variational autoencoder

In machine learning, a variational autoencoder (VAE) is an artificial neural network architecture introduced by Diederik P. Kingma and Max Welling in 2013. It is part of the families of probabilistic graphical models and variational Bayesian methods. In addition to being seen as an autoencoder neural network architecture, variational autoencoders can also be studied within the mathematical formulation of variational Bayesian methods, connecting a neural encoder network to its decoder through a probabilistic latent space (for example, as a multivariate Gaussian distribution) that corresponds to the parameters of a variational distribution. Thus, the encoder maps each point (such as an image) from a large complex dataset into a distribution within the latent space, rather than to a single point in that space. The decoder has the opposite function, which is to map from the latent space to the input space, again according to a distribution (although in practice, noise is rarely added during the decoding stage). By mapping a point to a distribution instead of a single point, the network can avoid overfitting the training data. Both networks are typically trained together with the usage of the reparameterization trick, although the variance of the noise model can be learned separately. Although this type of model was initially designed for unsupervised learning, its effectiveness has been proven for semi-supervised learning and supervised learning. == Overview of architecture and operation == A variational autoencoder is a generative model with a prior and noise distribution respectively. Usually such models are trained using the expectation-maximization meta-algorithm (e.g. probabilistic PCA, (spike & slab) sparse coding). Such a scheme optimizes a lower bound of the data likelihood, which is usually computationally intractable, and in doing so requires the discovery of q-distributions, or variational posteriors. These q-distributions are normally parameterized for each individual data point in a separate optimization process. However, variational autoencoders use a neural network as an amortized approach to jointly optimize across data points. In that way, the same parameters are reused for multiple data points, which can result in massive memory savings. The first neural network takes as input the data points themselves, and outputs parameters for the variational distribution. As it maps from a known input space to the low-dimensional latent space, it is called the encoder. The decoder is the second neural network of this model. It is a function that maps from the latent space to the input space, e.g. as the means of the noise distribution. It is possible to use another neural network that maps to the variance, however this can be omitted for simplicity. In such a case, the variance can be optimized with gradient descent. To optimize this model, one needs to know two terms: the "reconstruction error", and the Kullback–Leibler divergence (KL-D). Both terms are derived from the free energy expression of the probabilistic model, and therefore differ depending on the noise distribution and the assumed prior of the data, here referred to as p-distribution. For example, a standard VAE task such as IMAGENET is typically assumed to have a gaussianly distributed noise; however, tasks such as binarized MNIST require a Bernoulli noise. The KL-D from the free energy expression maximizes the probability mass of the q-distribution that overlaps with the p-distribution, which unfortunately can result in mode-seeking behaviour. The "reconstruction" term is the remainder of the free energy expression, and requires a sampling approximation to compute its expectation value. More recent approaches replace Kullback–Leibler divergence (KL-D) with various statistical distances, see "Statistical distance VAE variants" below. == Formulation == From the point of view of probabilistic modeling, one wants to maximize the likelihood of the data x {\displaystyle x} by their chosen parameterized probability distribution p θ ( x ) = p ( x | θ ) {\displaystyle p_{\theta }(x)=p(x|\theta )} . This distribution is usually chosen to be a Gaussian N ( x | μ , σ ) {\displaystyle N(x|\mu ,\sigma )} which is parameterized by μ {\displaystyle \mu } and σ {\displaystyle \sigma } respectively, and as a member of the exponential family it is easy to work with as a noise distribution. Simple distributions are easy enough to maximize, however distributions where a prior is assumed over the latents z {\displaystyle z} results in intractable integrals. Let us find p θ ( x ) {\displaystyle p_{\theta }(x)} via marginalizing over z {\displaystyle z} . p θ ( x ) = ∫ z p θ ( x , z ) d z , {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x,z})\,dz,} where p θ ( x , z ) {\displaystyle p_{\theta }({x,z})} represents the joint distribution under p θ {\displaystyle p_{\theta }} of the observable data x {\displaystyle x} and its latent representation or encoding z {\displaystyle z} . According to the chain rule, the equation can be rewritten as p θ ( x ) = ∫ z p θ ( x | z ) p θ ( z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x|z})p_{\theta }(z)\,dz} In the vanilla variational autoencoder, z {\displaystyle z} is usually taken to be a finite-dimensional vector of real numbers, and p θ ( x | z ) {\displaystyle p_{\theta }({x|z})} to be a Gaussian distribution. Then p θ ( x ) {\displaystyle p_{\theta }(x)} is a mixture of Gaussian distributions. It is now possible to define the set of the relationships between the input data and its latent representation as Prior p θ ( z ) {\displaystyle p_{\theta }(z)} Likelihood p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} Posterior p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} Unfortunately, the computation of p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} is expensive and in most cases intractable. To speed up the calculus to make it feasible, it is necessary to introduce a further function to approximate the posterior distribution as q ϕ ( z | x ) ≈ p θ ( z | x ) {\displaystyle q_{\phi }({z|x})\approx p_{\theta }({z|x})} with ϕ {\displaystyle \phi } defined as the set of real values that parametrize q {\displaystyle q} . This is sometimes called amortized inference, since by "investing" in finding a good q ϕ {\displaystyle q_{\phi }} , one can later infer z {\displaystyle z} from x {\displaystyle x} quickly without doing any integrals. In this way, the problem is to find a good probabilistic autoencoder, in which the conditional likelihood distribution p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} is computed by the probabilistic decoder, and the approximated posterior distribution q ϕ ( z | x ) {\displaystyle q_{\phi }(z|x)} is computed by the probabilistic encoder. Parametrize the encoder as E ϕ {\displaystyle E_{\phi }} , and the decoder as D θ {\displaystyle D_{\theta }} . == Evidence lower bound (ELBO) == Like many deep learning approaches that use gradient-based optimization, VAEs require a differentiable loss function to update the network weights through backpropagation. For variational autoencoders, the idea is to jointly optimize the generative model parameters θ {\displaystyle \theta } to reduce the reconstruction error between the input and the output, and ϕ {\displaystyle \phi } to make q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} as close as possible to p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . As reconstruction loss, mean squared error and cross entropy are often used. The Kullback–Leibler divergence D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) {\displaystyle D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))} can be used as a loss function to squeeze q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} under p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . This divergence loss expands to D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) = E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( z | x ) ] = E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( x ) p θ ( x , z ) ] = ln ⁡ p θ ( x ) + E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( x , z ) ] . {\displaystyle {\begin{aligned}D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }(z|x)}{p_{\theta }(z|x)}}\right]\\&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})p_{\theta }(x)}{p_{\theta }(x,z)}}\right]\\&=\ln p_{\theta }(x)+\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})}{p_{\theta }(x,z)}}\right].\end{aligned}}} Now, define the evidence lower bound (ELBO): L θ , ϕ ( x ) := E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ p θ ( x , z ) q ϕ ( z | x ) ] = ln ⁡ p θ ( x ) − D K L ( q ϕ ( ⋅ | x ) ∥ p θ ( ⋅ | x ) ) {\displaystyle L_{\theta ,\phi }(x):=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\ln p_{\theta }(x)-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }({\cdot |x}))} Maximizing the ELBO θ ∗ , ϕ ∗ = argmax θ , ϕ L θ , ϕ ( x ) {\dis

Bondy's theorem

In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972. == Statement == The theorem is as follows: Let X be a set with n elements and let A1, A2, ..., An be distinct subsets of X. Then there exists a subset S of X with n − 1 elements such that the sets Ai ∩ S are all distinct. In other words, if we have a 0-1 matrix with n rows and n columns such that each row is distinct, we can remove one column such that the rows of the resulting n × (n − 1) matrix are distinct. == Example == Consider the 4 × 4 matrix [ 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 ] {\displaystyle {\begin{bmatrix}1&1&0&1\\0&1&0&1\\0&0&1&1\\0&1&1&0\end{bmatrix}}} where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix [ 1 0 1 1 0 1 0 1 1 1 1 0 ] {\displaystyle {\begin{bmatrix}1&0&1\\1&0&1\\0&1&1\\1&1&0\end{bmatrix}}} no longer has this property: the first row is identical to the second row. Nevertheless, by Bondy's theorem we know that we can always find a column that can be deleted without introducing any identical rows. In this case, we can delete the third column: all rows of the 3 × 4 matrix [ 1 1 1 0 1 1 0 0 1 0 1 0 ] {\displaystyle {\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\\0&1&0\end{bmatrix}}} are distinct. Another possibility would have been deleting the fourth column. == Learning theory application == From the perspective of computational learning theory, Bondy's theorem can be rephrased as follows: Let C be a concept class over a finite domain X. Then there exists a subset S of X with the size at most |C| − 1 such that S is a witness set for every concept in C. This implies that every finite concept class C has its teaching dimension bounded by |C| − 1.

FedRAMP

The Federal Risk and Authorization Management Program (FedRAMP) is a United States federal government-wide compliance program that provides a standardized approach to security assessment, authorization, and continuous monitoring for cloud products and services. The US government describes FedRAMP as FISMA for the cloud. == Overview == The FedRAMP PMO mission is to promote the adoption of secure cloud services across the federal government by providing a standardized approach to security and risk assessment. Per the OMB memorandum, any cloud services that hold federal data must be FedRAMP authorized. FedRAMP prescribes the security requirements and processes that cloud service providers must follow in order for the government to use their service. There are two ways to authorize a cloud service through FedRAMP: a Joint Authorization Board (JAB) provisional authorization (P-ATO), and through individual agencies. FedRAMP provides accreditation for cloud services for the various cloud offering models which are Infrastructure as a Service (IaaS), Platform as a Service (PaaS), and Software as a Service, (SaaS). == History == In 2011, the Office of Management and Budget (OMB) released a memorandum establishing FedRAMP "to provide a cost-effective, risk-based approach for the adoption and use of cloud services to Executive departments and agencies." The General Services Administration (GSA) established the FedRAMP Program Management Office (PMO) in June 2012. Before the introduction of FedRAMP, individual federal agencies managed their own assessment methodologies following guidance set by the Federal Information Security Management Act of 2002. == Governance and applicable laws == FedRAMP is governed by different Executive Branch entities that collaborate to develop, manage, and operate the program. These entities include: The Office of Management and Budget (OMB): The governing body that issued the FedRAMP policy memo, which defines the key requirements and capabilities of the program The Joint Authorization Board (JAB): The primary governance and decision-making body for FedRAMP comprises the chief information officers (CIOs) from the Department of Homeland Security (DHS), General Services Administration (GSA), and Department of Defense (DOD) The National Institute of Standards and Technology (NIST): Advises FedRAMP on FISMA compliance requirements and assists in developing the standards for the accreditation of independent 3PAOs The Department of Homeland Security (DHS): Manages the FedRAMP continuous monitoring strategy including data feed criteria, reporting structure, threat notification coordination, and incident response The Federal Chief Information Officers (CIO) Council: Disseminates FedRAMP information to Federal CIOs and other representatives through cross-agency communications and events The FedRAMP PMO: Established within GSA and responsible for the development of the FedRAMP program, including the management of day-to-day operations There are several laws, mandates, and policies that are foundational to FedRAMP. FISMA–the Federal Information Security Modernization Act–requires that agencies authorize the information systems that they use. The US government describes FedRAMP as FISMA for the cloud. The FedRAMP Policy Memo requires federal agencies to use FedRAMP when assessing, authorizing, and continuously monitoring cloud services in order to aid agencies in the authorization process as well as save government resources and eliminate duplicative efforts. FedRAMP's security baselines are derived from NIST SP 800-53 (as revised) with a set of control enhancements that pertain to the unique security requirements of cloud computing. == Third-party assessment organizations == Third-party assessment organizations (3PAOs) play a critical role in the FedRAMP security assessment process, as they are the independent assessment organizations that verify cloud providers' security implementations and provide the overall risk posture of a cloud environment for a security authorization decision. Accredited by the American Association for Laboratory Accreditation (A2LA), these assessment organizations must demonstrate independence and the technical competence required to test security implementations and collect representative evidence. == FedRAMP Marketplace == The FedRAMP Marketplace provides a searchable, sortable database of Cloud Service Offerings (CSOs) that have achieved a FedRAMP designation. 3PAOs, accredited auditors that can perform the FedRAMP assessment, are listed within the Marketplace. The FedRAMP Marketplace is maintained by the FedRAMP Program Management Office (PMO). == Security and authorization concerns == A 2026 ProPublica investigation found that FedRAMP entered into a partnership with Microsoft despite considerable concerns about the security of its cloud technology.

Genetic representation

In computer programming, genetic representation is a way of presenting solutions/individuals in evolutionary computation methods. The term encompasses both the concrete data structures and data types used to realize the genetic material of the candidate solutions in the form of a genome, and the relationships between search space and problem space. In the simplest case, the search space corresponds to the problem space (direct representation). The choice of problem representation is tied to the choice of genetic operators, both of which have a decisive effect on the efficiency of the optimization. Genetic representation can encode appearance, behavior, physical qualities of individuals. Difference in genetic representations is one of the major criteria drawing a line between known classes of evolutionary computation. Terminology is often analogous with natural genetics. The block of computer memory that represents one candidate solution is called an individual. The data in that block is called a chromosome. Each chromosome consists of genes. The possible values of a particular gene are called alleles. A programmer may represent all the individuals of a population using binary encoding, permutational encoding, encoding by tree, or any one of several other representations. == Representations in some popular evolutionary algorithms == Genetic algorithms (GAs) are typically linear representations; these are often, but not always, binary. Holland's original description of GA used arrays of bits. Arrays of other types and structures can be used in essentially the same way. The main property that makes these genetic representations convenient is that their parts are easily aligned due to their fixed size. This facilitates simple crossover operation. Depending on the application, variable-length representations have also been successfully used and tested in evolutionary algorithms (EA) in general and genetic algorithms in particular, although the implementation of crossover is more complex in this case. Evolution strategy uses linear real-valued representations, e.g., an array of real values. It uses mostly gaussian mutation and blending/averaging crossover. Genetic programming (GP) pioneered tree-like representations and developed genetic operators suitable for such representations. Tree-like representations are used in GP to represent and evolve functional programs with desired properties. Human-based genetic algorithm (HBGA) offers a way to avoid solving hard representation problems by outsourcing all genetic operators to outside agents, in this case, humans. The algorithm has no need for knowledge of a particular fixed genetic representation as long as there are enough external agents capable of handling those representations, allowing for free-form and evolving genetic representations. === Common genetic representations === binary array integer or real-valued array binary tree natural language parse tree directed graph == Distinction between search space and problem space == Analogous to biology, EAs distinguish between problem space (corresponds to phenotype) and search space (corresponds to genotype). The problem space contains concrete solutions to the problem being addressed, while the search space contains the encoded solutions. The mapping from search space to problem space is called genotype-phenotype mapping. The genetic operators are applied to elements of the search space, and for evaluation, elements of the search space are mapped to elements of the problem space via genotype-phenotype mapping. == Relationships between search space and problem space == The importance of an appropriate choice of search space for the success of an EA application was recognized early on. The following requirements can be placed on a suitable search space and thus on a suitable genotype-phenotype mapping: === Completeness === All possible admissible solutions must be contained in the search space. === Redundancy === When more possible genotypes exist than phenotypes, the genetic representation of the EA is called redundant. In nature, this is termed a degenerate genetic code. In the case of a redundant representation, neutral mutations are possible. These are mutations that change the genotype but do not affect the phenotype. Thus, depending on the use of the genetic operators, there may be phenotypically unchanged offspring, which can lead to unnecessary fitness determinations, among other things. Since the evaluation in real-world applications usually accounts for the lion's share of the computation time, it can slow down the optimization process. In addition, this can cause the population to have higher genotypic diversity than phenotypic diversity, which can also hinder evolutionary progress. In biology, the Neutral Theory of Molecular Evolution states that this effect plays a dominant role in natural evolution. This has motivated researchers in the EA community to examine whether neutral mutations can improve EA functioning by giving populations that have converged to a local optimum a way to escape that local optimum through genetic drift. This is discussed controversially and there are no conclusive results on neutrality in EAs. On the other hand, there are other proven measures to handle premature convergence. === Locality === The locality of a genetic representation corresponds to the degree to which distances in the search space are preserved in the problem space after genotype-phenotype mapping. That is, a representation has a high locality exactly when neighbors in the search space are also neighbors in the problem space. In order for successful schemata not to be destroyed by genotype-phenotype mapping after a minor mutation, the locality of a representation must be high. === Scaling === In genotype-phenotype mapping, the elements of the genotype can be scaled (weighted) differently. The simplest case is uniform scaling: all elements of the genotype are equally weighted in the phenotype. A common scaling is exponential. If integers are binary coded, the individual digits of the resulting binary number have exponentially different weights in representing the phenotype. Example: The number 90 is written in binary (i.e., in base two) as 1011010. If now one of the front digits is changed in the binary notation, this has a significantly greater effect on the coded number than any changes at the rear digits (the selection pressure has an exponentially greater effect on the front digits). For this reason, exponential scaling has the effect of randomly fixing the "posterior" locations in the genotype before the population gets close enough to the optimum to adjust for these subtleties. == Hybridization and repair in genotype-phenotype mapping == When mapping the genotype to the phenotype being evaluated, domain-specific knowledge can be used to improve the phenotype and/or ensure that constraints are met. This is a commonly used method to improve EA performance in terms of runtime and solution quality. It is illustrated below by two of the three examples. == Examples == === Example of a direct representation === An obvious and commonly used encoding for the traveling salesman problem and related tasks is to number the cities to be visited consecutively and store them as integers in the chromosome. The genetic operators must be suitably adapted so that they only change the order of the cities (genes) and do not cause deletions or duplications. Thus, the gene order corresponds to the city order and there is a simple one-to-one mapping. === Example of a complex genotype-phenotype mapping === In a scheduling task with heterogeneous and partially alternative resources to be assigned to a set of subtasks, the genome must contain all necessary information for the individual scheduling operations or it must be possible to derive them from it. In addition to the order of the subtasks to be executed, this includes information about the resource selection. A phenotype then consists of a list of subtasks with their start times and assigned resources. In order to be able to create this, as many allocation matrices must be created as resources can be allocated to one subtask at most. In the simplest case this is one resource, e.g., one machine, which can perform the subtask. An allocation matrix is a two-dimensional matrix, with one dimension being the available time units and the other being the resources to be allocated. Empty matrix cells indicate availability, while an entry indicates the number of the assigned subtask. The creation of allocation matrices ensures firstly that there are no inadmissible multiple allocations. Secondly, the start times of the subtasks can be read from it as well as the assigned resources. A common constraint when scheduling resources to subtasks is that a resource can only be allocated once per time unit and that the reservation must be for a contiguous period of time. To achieve this in a timely manner, which is a c