In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection. Take that same function, but do a two-dimensional Fourier transform first, and then slice the function through its origin, parallel to the projection line. In operator terms, if F1 and F2 are the 1- and 2-dimensional Fourier transform operators mentioned above, P1 is the projection operator (which projects a 2-D function onto a 1-D line), S1 is a slice operator (which extracts a 1-D central slice from a function), then F 1 P 1 = S 1 F 2 . {\displaystyle F_{1}P_{1}=S_{1}F_{2}.} This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medical CT scans where a "projection" is an x-ray image of an internal organ. The Fourier transforms of these images are seen to be slices through the Fourier transform of the 3-dimensional density of the internal organ, and these slices can be interpolated to build up a complete Fourier transform of that density. The inverse Fourier transform is then used to recover the 3-dimensional density of the object. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem. == The projection-slice theorem in N dimensions == In N dimensions, the projection-slice theorem states that the Fourier transform of the projection of an N-dimensional function f(r) onto an m-dimensional linear submanifold is equal to an m-dimensional slice of the N-dimensional Fourier transform of that function consisting of an m-dimensional linear submanifold through the origin in the Fourier space which is parallel to the projection submanifold. In operator terms: F m P m = S m F N . {\displaystyle F_{m}P_{m}=S_{m}F_{N}.\,} == The generalized Fourier-slice theorem == In addition to generalizing to N dimensions, the projection-slice theorem can be further generalized with an arbitrary change of basis. For convenience of notation, we consider the change of basis to be represented as B, an N-by-N invertible matrix operating on N-dimensional column vectors. Then the generalized Fourier-slice theorem can be stated as F m P m B = S m B − T | B − T | F N {\displaystyle F_{m}P_{m}B=S_{m}{\frac {B^{-T}}{|B^{-T}|}}F_{N}} where B − T = ( B − 1 ) T {\displaystyle B^{-T}=(B^{-1})^{T}} is the transpose of the inverse of the change of basis transform. == Proof in two dimensions == The projection-slice theorem is easily proven for the case of two dimensions. Without loss of generality, we can take the projection line to be the x-axis. There is no loss of generality because if we use a shifted and rotated line, the law still applies. Using a shifted line (in y) gives the same projection and therefore the same 1D Fourier transform results. The rotated function is the Fourier pair of the rotated Fourier transform, for which the theorem again holds. If f(x, y) is a two-dimensional function, then the projection of f(x, y) onto the x axis is p(x) where p ( x ) = ∫ − ∞ ∞ f ( x , y ) d y . {\displaystyle p(x)=\int _{-\infty }^{\infty }f(x,y)\,dy.} The Fourier transform of f ( x , y ) {\displaystyle f(x,y)} is F ( k x , k y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − 2 π i ( x k x + y k y ) d x d y . {\displaystyle F(k_{x},k_{y})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\,e^{-2\pi i(xk_{x}+yk_{y})}\,dxdy.} The slice is then s ( k x ) {\displaystyle s(k_{x})} s ( k x ) = F ( k x , 0 ) = ∫ − ∞ ∞ ∫ − ∞ ∞ f ( x , y ) e − 2 π i x k x d x d y {\displaystyle s(k_{x})=F(k_{x},0)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,y)\,e^{-2\pi ixk_{x}}\,dxdy} = ∫ − ∞ ∞ [ ∫ − ∞ ∞ f ( x , y ) d y ] e − 2 π i x k x d x {\displaystyle =\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }f(x,y)\,dy\right]\,e^{-2\pi ixk_{x}}dx} = ∫ − ∞ ∞ p ( x ) e − 2 π i x k x d x {\displaystyle =\int _{-\infty }^{\infty }p(x)\,e^{-2\pi ixk_{x}}dx} which is just the Fourier transform of p(x). The proof for higher dimensions is easily generalized from the above example. == The FHA cycle == If the two-dimensional function f(r) is circularly symmetric, it may be represented as f(r), where r = |r|. In this case the projection onto any projection line will be the Abel transform of f(r). The two-dimensional Fourier transform of f(r) will be a circularly symmetric function given by the zeroth-order Hankel transform of f(r), which will therefore also represent any slice through the origin. The projection-slice theorem then states that the Fourier transform of the projection equals the slice or F 1 A 1 = H , {\displaystyle F_{1}A_{1}=H,} where A1 represents the Abel-transform operator, projecting a two-dimensional circularly symmetric function onto a one-dimensional line, F1 represents the 1-D Fourier-transform operator, and H represents the zeroth-order Hankel-transform operator. == Extension to fan beam or cone-beam CT == The projection-slice theorem is suitable for CT image reconstruction with parallel beam projections. It does not directly apply to fanbeam or conebeam CT. The theorem was extended to fan-beam and conebeam CT image reconstruction by Shuang-ren Zhao in 1995.
Microsoft Whiteboard
Microsoft Whiteboard is a free multi-platform application, as well as an online service and a feature in Microsoft Teams, which simulates a virtual whiteboard and enables real-time collaboration between users. == Overview and features == Microsoft Whiteboard allows users to draw on a virtual whiteboard using input methods such as a stylus pen or a mouse and keyboard, and write down notes, draw connections between shareable ideas, and interact in real time. Microsoft Whiteboard is available to download on the following platforms and devices: Microsoft Windows (on Windows 10 or above) Android Apple iOS Surface Hub devices It is also available on the web and as a feature in Microsoft Teams. Microsoft Whiteboard allows users with Microsoft accounts to view, edit, and share whiteboards using the provided tools and options. The feature set includes tools for drawing, shapes, and media. Drawing in Microsoft Whiteboard is called inking. It works both on mobile devices and computers. The inking toolbar has customizable pencils, a ruler, a highlighter, an eraser, and an object selector. Whiteboard can recognize shapes drawn by hand and straighten them. Holding the Shift key on a computer while inking draws straight lines. Microsoft Whiteboard has keyboard shortcuts for some functions. Additional features include inserting sticky notes, text boxes, stickers, as well as images. Grid lines and colors are adjustable. Different templates can be inserted into the whiteboard. Users can also share their reactions. A feature limited to boards created in Microsoft Teams, is the ability to make them read-only; other participants from the meeting cannot edit them. == Reviews == PC Magazine gave Microsoft Whiteboard a score of 3.5 out of 5, praising the app's free availability and plentiful templates. It compared it to other, paid whiteboarding solutions, and concluded that Microsoft offers the best free one. Some of the cons, described by PCMag, include the inability to view boards without a Microsoft account and the inability to create custom templates.
Federation of International Robot-soccer Association
The Federation of International Robot-soccer Association (FIRA) is an international organisation organising competitive soccer competitions between autonomous robots. The matches are usually five-a-side. == History == In 1996 and 1997, this competition was known as MiroSot and was held in Daejeon, Korea. The 1996 competition offered a challenging arena to the younger generation and researchers working with autonomous mobile robotic systems. From 1998 through 2008, it was called the FIRA Cup, and in 2009, it became the FIRA RoboWorld Cup & Congress. The 15th RoboWorld Cup was held at Amrita Vishwa Vidyapeetham, Bangalore, India in September 2010. In 2013, it took place in Kuala Lumpur, Malaysia. The championship started on August 24, 2013, and ended on August 29. At that time, it involved five categories: Micro-Robot Soccer Tournament, Amire, Naro, Simulated Robot, Android, Robo and Humanoid Robot. It attracted teams from Singapore, Indonesia, Taiwan, India, China, South Korea, the United Kingdom, Mexico, Canada, Russia and Malaysia. 80 teams from 11 countries participated. In 2018, the competition had 277 teams participating from 12 countries. === Past Events === == FIRA RoboWorld Cup & Congress == This competition has 4 leagues: FIRA AIR, FIRA Sports, FIRA Challenges, and FIRA Youth. Each league has its own competitions, and each competition can have several events. === FIRA AIR === The FIRA AIR league has two associated competitions, Autonomous Race and Emergency Service. === FIRA Sports === The FIRA Sports league has four associated competitions, HuroCup, RoboSot, SimuroSot, and AndroSot. This the robot soccer league. HuroCup consists of single events for bipedal humanoid robots. The events are: archery, sprint, marathon, united soccer, obstacle run, long jump, spartan race, marathon, weightlifting, and basketball. There is an all-round competition for the single robot that performs the best overall. === FIRA Challenges === The FIRA Challenges league has three associated competitions, Autonomous Cars, Autonomous Cars Simulation, Innovation and Business. === FIRA Youth === The FIRA Youth league has six associated challenges, Sport Robots, HuroCup Junior, CityRacer, DRV_Explorer, Cliff Hanger, and Mission Impossible.
Digital organism
A digital organism is a self-replicating computer program that mutates and evolves. Digital organisms are used as a tool to study the dynamics of Darwinian evolution, and to test or verify specific hypotheses or mathematical models of evolution. The study of digital organisms is closely related to the area of artificial life. == History == Digital organisms can be traced back to the game Darwin, developed in 1961 at Bell Labs, in which computer programs had to compete with each other by trying to stop others from executing . A similar implementation that followed this was the game Core War. In Core War, it turned out that one of the winning strategies was to replicate as fast as possible, which deprived the opponent of all computational resources. Programs in the Core War game were also able to mutate themselves and each other by overwriting instructions in the simulated "memory" in which the game took place. This allowed competing programs to embed damaging instructions in each other that caused errors (terminating the process that read it), "enslaved processes" (making an enemy program work for you), or even change strategies mid-game and heal themselves. Steen Rasmussen at Los Alamos National Laboratory took the idea from Core War one step further in his core world system by introducing a genetic algorithm that automatically wrote programs. However, Rasmussen did not observe the evolution of complex and stable programs. It turned out that the programming language in which core world programs were written was very brittle, and more often than not mutations would completely destroy the functionality of a program. The first to solve the issue of program brittleness was Thomas S. Ray with his Tierra system, which was similar to core world. Ray made some key changes to the programming language such that mutations were much less likely to destroy a program. With these modifications, he observed for the first time computer programs that did indeed evolve in a meaningful and complex way. Later, Chris Adami, Titus Brown, and Charles Ofria started developing their Avida system, which was inspired by Tierra but again had some crucial differences. In Tierra, all programs lived in the same address space and could potentially execute or otherwise interfere with each other's code. In Avida, on the other hand, each program lives in its own address space. Because of this modification, experiments with Avida became much cleaner and easier to interpret than those with Tierra. With Avida, digital organism research has begun to be accepted as a valid contribution to evolutionary biology by a growing number of evolutionary biologists. Evolutionary biologist Richard Lenski of Michigan State University has used Avida extensively in his work. Lenski, Adami, and their colleagues have published in journals such as Nature and the Proceedings of the National Academy of Sciences (USA). In 1996, Andy Pargellis created a Tierra-like system called Amoeba that evolved self-replication from a randomly seeded initial condition. More recently REvoSim - a software package based around binary digital organisms - has allowed evolutionary simulations of large populations that can be run for geological timescales.
Diagnosis (artificial intelligence)
As a subfield in artificial intelligence, diagnosis is concerned with the development of algorithms and techniques that are able to determine whether the behaviour of a system is correct. If the system is not functioning correctly, the algorithm should be able to determine, as accurately as possible, which part of the system is failing, and which kind of fault it is facing. The computation is based on observations, which provide information on the current behaviour. The expression diagnosis also refers to the answer of the question of whether the system is malfunctioning or not, and to the process of computing the answer. This word comes from the medical context where a diagnosis is the process of identifying a disease by its symptoms. == Example == An example of diagnosis is the process of a garage mechanic with an automobile. The mechanic will first try to detect any abnormal behavior based on the observations on the car and his knowledge of this type of vehicle. If he finds out that the behavior is abnormal, the mechanic will try to refine his diagnosis by using new observations and possibly testing the system, until he discovers the faulty component; the mechanic plays an important role in the vehicle diagnosis. == Expert diagnosis == The expert diagnosis (or diagnosis by expert system) is based on experience with the system. Using this experience, a mapping is built that efficiently associates the observations to the corresponding diagnoses. The experience can be provided: By a human operator. In this case, the human knowledge must be translated into a computer language. By examples of the system behaviour. In this case, the examples must be classified as correct or faulty (and, in the latter case, by the type of fault). Machine learning methods are then used to generalize from the examples. The main drawbacks of these methods are: The difficulty acquiring the expertise. The expertise is typically only available after a long period of use of the system (or similar systems). Thus, these methods are unsuitable for safety- or mission-critical systems (such as a nuclear power plant, or a robot operating in space). Moreover, the acquired expert knowledge can never be guaranteed to be complete. In case a previously unseen behaviour occurs, leading to an unexpected observation, it is impossible to give a diagnosis. The complexity of the learning. The off-line process of building an expert system can require a large amount of time and computer memory. The size of the final expert system. As the expert system aims to map any observation to a diagnosis, it will in some cases require a huge amount of storage space. The lack of robustness. If even a small modification is made on the system, the process of constructing the expert system must be repeated. A slightly different approach is to build an expert system from a model of the system rather than directly from an expertise. An example is the computation of a diagnoser for the diagnosis of discrete event systems. This approach can be seen as model-based, but it benefits from some advantages and suffers some drawbacks of the expert system approach. == Model-based diagnosis == Model-based diagnosis is an example of abductive reasoning using a model of the system. In general, it works as follows: We have a model that describes the behaviour of the system (or artefact). The model is an abstraction of the behaviour of the system and can be incomplete. In particular, the faulty behaviour is generally little-known, and the faulty model may thus not be represented. Given observations of the system, the diagnosis system simulates the system using the model, and compares the observations actually made to the observations predicted by the simulation. The modelling can be simplified by the following rules (where A b {\displaystyle Ab\,} is the Abnormal predicate): ¬ A b ( S ) ⇒ I n t 1 ∧ O b s 1 {\displaystyle \neg Ab(S)\Rightarrow Int1\wedge Obs1} A b ( S ) ⇒ I n t 2 ∧ O b s 2 {\displaystyle Ab(S)\Rightarrow Int2\wedge Obs2} (fault model) The semantics of these formulae is the following: if the behaviour of the system is not abnormal (i.e. if it is normal), then the internal (unobservable) behaviour will be I n t 1 {\displaystyle Int1\,} and the observable behaviour O b s 1 {\displaystyle Obs1\,} . Otherwise, the internal behaviour will be I n t 2 {\displaystyle Int2\,} and the observable behaviour O b s 2 {\displaystyle Obs2\,} . Given the observations O b s {\displaystyle Obs\,} , the problem is to determine whether the system behaviour is normal or not ( ¬ A b ( S ) {\displaystyle \neg Ab(S)\,} or A b ( S ) {\displaystyle Ab(S)\,} ). This is an example of abductive reasoning. == Diagnosability == A system is said to be diagnosable if whatever the behavior of the system, we will be able to determine without ambiguity a unique diagnosis. The problem of diagnosability is very important when designing a system because on one hand one may want to reduce the number of sensors to reduce the cost, and on the other hand one may want to increase the number of sensors to increase the probability of detecting a faulty behavior. Several algorithms for dealing with these problems exist. One class of algorithms answers the question whether a system is diagnosable; another class looks for sets of sensors that make the system diagnosable, and optionally comply to criteria such as cost optimization. The diagnosability of a system is generally computed from the model of the system. In applications using model-based diagnosis, such a model is already present and doesn't need to be built from scratch.
Linguatec
The Linguatec Sprachtechnologien GmbH is a language technology provider, specialized in the field of machine translation, speech synthesis and speech recognition. Linguatec was founded in Munich in 1996 and its headquarters are in Pasing. Linguatec has won the European Information Society Technologies Prize three times. On their website, they are now using the online service Voice Reader Web, so that the information can be read out in every language by means of a text-to-speech function. == Core areas == Machine translation The different versions of Personal Translator (seven language pairs) can be used "for home use" or for professional business use in the company network. In addition to this, specialist dictionaries are offered to broaden standard vocabulary. Speech synthesis The Voice Reader text-to-speech program reads in twelve languages: German, British English, American English, French, Quebec French, Spanish, Mexican Spanish, Italian, Dutch, Portuguese, Czech, Chinese. Speech recognition Voice Pro is based on ViaVoice technology from IBM. There are special software programs for doctors and lawyers. == Patents == 2005 pending patent application for a newly developed hybrid technology that uses the intelligence of neural networks for machine translation. == Awards == 2004 European IT Prize for Beyond Babel 2004 test winner Stiftung Warentest – best voice recognition 1998 European IT Prize – applied voice recognition 1996 European IT Prize – automated translation == Studies == 2005 University of Regensburg: Voice Reader user test 2002 Fraunhofer Institute for Industrial Engineering and Organization IAO: user study on the efficiency of machine translation
Possibility theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier, in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise. == Formalization of possibility == For simplicity, assume that the universe of discourse Ω is a finite set. A possibility measure is a function Π {\displaystyle \Pi } from 2 Ω {\displaystyle 2^{\Omega }} to [0, 1] such that: Axiom 1: Π ( ∅ ) = 0 {\displaystyle \Pi (\varnothing )=0} Axiom 2: Π ( Ω ) = 1 {\displaystyle \Pi (\Omega )=1} Axiom 3: Π ( U ∪ V ) = max ( Π ( U ) , Π ( V ) ) {\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)} for any disjoint subsets U {\displaystyle U} and V {\displaystyle V} . It follows that, like probability on finite probability spaces, the possibility measure is determined by its behavior on singletons: Π ( U ) = max ω ∈ U Π ( { ω } ) . {\displaystyle \Pi (U)=\max _{\omega \in U}\Pi (\{\omega \}).} Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω. Axiom 2 could be interpreted as the assumption that the evidence from which Π {\displaystyle \Pi } was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1. Axiom 3 corresponds to the additivity axiom in probabilities. However, there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that: Π ( U ∪ V ) = max ( Π ( U ) , Π ( V ) ) {\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)} for any subsets U {\displaystyle U} and V {\displaystyle V} . Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is compositional with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally: Π ( U ∩ V ) ≤ min ( Π ( U ) , Π ( V ) ) ≤ max ( Π ( U ) , Π ( V ) ) . {\displaystyle \Pi (U\cap V)\leq \min \left(\Pi (U),\Pi (V)\right)\leq \max \left(\Pi (U),\Pi (V)\right).} When Ω is not finite, Axiom 3 can be replaced by: For all index sets I {\displaystyle I} , if the subsets U i , i ∈ I {\displaystyle U_{i,\,i\in I}} are pairwise disjoint, Π ( ⋃ i ∈ I U i ) = sup i ∈ I Π ( U i ) . {\displaystyle \Pi \left(\bigcup _{i\in I}U_{i}\right)=\sup _{i\in I}\Pi (U_{i}).} == Necessity == Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity of the event. For any set U {\displaystyle U} , the necessity measure is defined by N ( U ) = 1 − Π ( U ¯ ) {\displaystyle N(U)=1-\Pi ({\overline {U}})} . In the above formula, U ¯ {\displaystyle {\overline {U}}} denotes the complement of U {\displaystyle U} , that is the elements of Ω {\displaystyle \Omega } that do not belong to U {\displaystyle U} . It is straightforward to show that: N ( U ) ≤ Π ( U ) {\displaystyle N(U)\leq \Pi (U)} for any U {\displaystyle U} and that: N ( U ∩ V ) = min ( N ( U ) , N ( V ) ) {\displaystyle N(U\cap V)=\min(N(U),N(V))} . Note that contrary to probability theory, possibility is not self-dual. That is, for any event U {\displaystyle U} , we only have the inequality: Π ( U ) + Π ( U ¯ ) ≥ 1 {\displaystyle \Pi (U)+\Pi ({\overline {U}})\geq 1} However, the following duality rule holds: For any event U {\displaystyle U} , either Π ( U ) = 1 {\displaystyle \Pi (U)=1} , or N ( U ) = 0 {\displaystyle N(U)=0} Accordingly, beliefs about an event can be represented by a number and a bit. == Interpretation == There are four cases that can be interpreted as follows: N ( U ) = 1 {\displaystyle N(U)=1} means that U {\displaystyle U} is necessary. U {\displaystyle U} is certainly true. It implies that Π ( U ) = 1 {\displaystyle \Pi (U)=1} . Π ( U ) = 0 {\displaystyle \Pi (U)=0} means that U {\displaystyle U} is impossible. U {\displaystyle U} is certainly false. It implies that N ( U ) = 0 {\displaystyle N(U)=0} . Π ( U ) = 1 {\displaystyle \Pi (U)=1} means that U {\displaystyle U} is possible. I would not be surprised at all if U {\displaystyle U} occurs. It leaves N ( U ) {\displaystyle N(U)} unconstrained. N ( U ) = 0 {\displaystyle N(U)=0} means that U {\displaystyle U} is unnecessary. I would not be surprised at all if U {\displaystyle U} does not occur. It leaves Π ( U ) {\displaystyle \Pi (U)} unconstrained. The intersection of the last two cases is N ( U ) = 0 {\displaystyle N(U)=0} and Π ( U ) = 1 {\displaystyle \Pi (U)=1} meaning that I believe nothing at all about U {\displaystyle U} . Because it allows for indeterminacy like this, possibility theory relates to the graduation of a many-valued logic, such as intuitionistic logic, rather than the classical two-valued logic. Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. The relationship with fuzzy theory can be explained with the following classic example. Fuzzy logic: When a bottle is half full, it can be said that the level of truth of the proposition "The bottle is full" is 0.5. The word "full" is seen as a fuzzy predicate describing the amount of liquid in the bottle. Possibility theory: There is one bottle, either completely full or totally empty. The proposition "the possibility level that the bottle is full is 0.5" describes a degree of belief. One way to interpret 0.5 in that proposition is to define its meaning as: I am ready to bet that it's empty as long as the odds are even (1:1) or better, and I would not bet at any rate that it's full. == Possibility theory as an imprecise probability theory == There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator. A possibility measure can be seen as a consonant plausibility measure in the Dempster–Shafer theory of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence. Possibility can be seen as an upper probability: any possibility distribution defines a unique credal set of admissible probability distributions by K = { P ∣ ∀ S P ( S ) ≤ Π ( S ) } . {\displaystyle K=\{\,P\mid \forall S\ P(S)\leq \Pi (S)\,\}.} This allows one to study possibility theory using the tools of imprecise probabilities. == Necessity logic == We call generalized possibility every function satisfying Axiom 1 and Axiom 3. We call generalized necessity the dual of a generalized possibility. The generalized necessities are related to a very simple and interesting fuzzy logic called necessity logic. In the deduction apparatus of necessity logic the logical axioms are the usual classical tautologies. Also, there is only a fuzzy inference rule extending the usual modus ponens. Such a rule says that if α and α → β are proved at degree λ and μ, respectively, then we can assert β at degree min{λ,μ}. It is easy to see that the theories of such a logic are the generalized necessities and that the completely consistent theories coincide with the necessities (see for example Gerla 2001).