4E cognition refers to a group of theories in (the philosophy of) cognitive science that challenge traditional views of the mind as something that happens only inside the brain. The four Es stand for: embodied, meaning that a brain is found in and, more importantly, vitally interconnected with a larger physical/biological body; embedded, which refers to the limitations placed on the body by the external environment and laws of nature; extended, which argues that the mind is supplemented and even enhanced by the exterior world (e.g., writing, a calculator, etc.); and enactive, which is the argument that without dynamic processes, actions that require reactions, the mind would be ineffectual. It could be argued that the four Es are compounding extensions of cognition or the mind, being part of a body that is, in turn, part of an environment which limits it but also allows for certain extensions, all of which require dynamic actions and reactions. == History == Ideas of embodied cognition, or rather the idea that our physical bodies play a crucial role in our decision making, can be traced back as far as Plato's dialogues and Aristotelian thought. It was, however, in the twentieth century that this debate began to resemble the current discussion, fueled by disagreements between cognitivists and behaviourists. Tensions within cognitivism, as well as the increasing popularity of neurobiology, led, on the one side, to a predominant focus on internal, cognitive processes while neglecting environmental factors, which in turn caused a push-back fuelling our modern understanding of embodied cognition. The term 4E cognition is hard to trace back to its first use, however, some sources attribute it to Shaun Gallagher and the conference on 4E cognition he organised in 2007, while others indicate the term to be first used in 2006 at an 'Embodied mind workshop' at Cardiff University that Gallagher attended. Embodiment or embodied cognition arguably presents the bridge between cognitivism and 4E cognition as the embodiment of cognitive function provides the necessary conditions for embeddedness, enactedness, and extendedness to connect to cognition. 4E cognition was and is heavily influenced by phenomenology. The ideas are still rather fragmented in nature due to their four main components, which can not be neatly divided, causing conceptual questions of internal boundary concepts. As a young field, it is held back both by its fragmented nature and a relative lack of critical evaluations. It is important to acknowledge that 4E cognition, though young, is a broad field containing and combining several different theoretical perspectives that conflict with one another to varying degrees. The somewhat convoluted and competing nature of the theories that can be grouped as 4E cognition, as well as the field's relative youth, make it difficult to put together an exhaustive history beyond the history of its four main theoretical pillars: embodiment, embeddedness, extendedness, and enactedness. == Importance and core tenets of 4E == If there are separate theories of cognition (e.g., embodied, extended, etc.), why group them under this umbrella, causing important epistemological and especially ontological dilemmas? Notably, other theories of 'non-traditional' cognition are not included under the 4E umbrella. The four E's in 4E cognition importantly all reject, or at a minimum draw into question, some of the core tenets of traditional cognitivism. Importantly, 4E cognition is seen as deindividualizing cognition to some extent, allowing for a broader examination of the interplay of personal, social, political, and ethical aspects that shape human cognition. This can be compared to advancements in the field of epigenetics, which have allowed for a broader examination of environmental (both natural and social) factors and their influence on what had previously only been subject to genetic theorizing. In a similar vein, 4E cognition might also help ground cognition in evolutionary theory by extending cognition to a biological account subject to development over time by means of evolution. Overall, the importance of the extension that is 4E cognition aims to reexamine ideas of a self-centered view of cognition, advocating for a more holistic approach. Ideally, this would allow us to reconsider ideas of justice and individual rights and responsibilities that take into account a more nuanced understanding of the relations between people and their context, balancing self-agency with factors beyond it. === Conceptual differences from cognitive psychology === According to the traditional teachings of cognitive psychology, cognition is a type of information processing based on representational mental structures. This idea, as the name suggests, was heavily influenced by computer science. In this light, the brain is a kind of central processing unit that organises and directs all else. The classical cognitivist view draws a strong boundary between 'the internal' and 'the external', where cognition is solely a subject of 'the internal' realm. The four E's, however, break down this boundary. Cognition can not reside solely within the confines of our heads if it is also embodied, embedded, enacted, and extended. In a way, 4E cognition is interested in the extracranial processes affecting cognition. == From embodied cognition to 4E cognition == === The strong and the weak view === ==== Embodied cognition ==== Broadly speaking, there is a strong and a weak perspective of embodied cognition in 4E cognition. The weak understanding refers to mental processes being causally dependent on extracranial processes. This essentially means that there is a cause and effect or action-reaction relationship between the mind and the body and its environment, etc. The strong perspective views extracranial processes as a (partial) constitutive aspect of cognition. An example here could be using a calculator to solve math problems. The calculator is not part of your brain or mind, but it supports your cognitive processes. === Extracranial processes: bodily or extrabodily === In addition to the weak and the strong reading of 4E cognition, there is also the distinction between bodily and extrabodily extracranial processes. Bodily extracranial processes refer to processes within the body, e.g., sensory perception. Extrabodily extracranial processes refer to processes outside of the body, like the aforementioned calculator example. === Four claims of embodied cognition === ==== Embedded and extended cognition ==== When combining the weak/strong reading of embodied cognition and bodily/extrabodily extracranial process, four claims about embodied cognition emerge: strongly embodied and bodily processes strongly embodied and extrabodily processes weakly embodied and bodily processes weakly embodied and extrabodily processes The first and third claims signify a strong and a weak reading of embodied cognition in the more classical sense. The second claim fits almost perfectly with embedded cognition. Claim two is most compatible with extended cognition. ==== Enacted cognition ==== Finally, enacted cognition refers to cognition being connected to active interaction between a conscious agent and their environment. Here, too, there can be a weak and a strong reading. == Criticisms == Given the divided nature of the field, much criticism surrounding the lack of unity within the field has emerged. In particular, the claims of embodied cognition centering around the body appear to conflict with the tenets of extended cognition, which also appear to conflict with the body/environment distinction that is central to enactivism. Some theoreticians argue that the umbrella of 4E theories is still lacking a common language that might bridge the gaps between the theories that constitute it. There is also the concern that the grouping of such variable theories results in an important loss of nuance and complexity, which is a part of human cognition. Another concern raised is the "dogma of harmony". The criticism contained there regards the notion that within 4E theorizing, there is generally an optimistic and harmonic expectation of the extension between humans and their technologies, ignoring the possibility of those extensions detracting from cognition in some way rather than adding to it. Recent attempts to incorporate embodied cognitive neuroscience have been argued to hold the potential to resolve internal issues within 4E cognition. Overall, a concern often voiced regarding 4E cognition is that its proponents are at best only vaguely interested in cognition. More broadly, this concern reflects the arguably too distracted nature of this emerging field.
EfficientNet
EfficientNet is a family of convolutional neural networks (CNNs) for computer vision published by researchers at Google AI in 2019. Its key innovation is compound scaling, which uniformly scales all dimensions of depth, width, and resolution using a single parameter. EfficientNet models have been adopted in various computer vision tasks, including image classification, object detection, and segmentation. == Compound scaling == EfficientNet introduces compound scaling, which, instead of scaling one dimension of the network at a time, such as depth (number of layers), width (number of channels), or resolution (input image size), uses a compound coefficient ϕ {\displaystyle \phi } to scale all three dimensions simultaneously. Specifically, given a baseline network, the depth, width, and resolution are scaled according to the following equations: depth multiplier: d = α ϕ width multiplier: w = β ϕ resolution multiplier: r = γ ϕ {\displaystyle {\begin{aligned}{\text{depth multiplier: }}d&=\alpha ^{\phi }\\{\text{width multiplier: }}w&=\beta ^{\phi }\\{\text{resolution multiplier: }}r&=\gamma ^{\phi }\end{aligned}}} subject to α ⋅ β 2 ⋅ γ 2 ≈ 2 {\displaystyle \alpha \cdot \beta ^{2}\cdot \gamma ^{2}\approx 2} and α ≥ 1 , β ≥ 1 , γ ≥ 1 {\displaystyle \alpha \geq 1,\beta \geq 1,\gamma \geq 1} . The α ⋅ β 2 ⋅ γ 2 ≈ 2 {\displaystyle \alpha \cdot \beta ^{2}\cdot \gamma ^{2}\approx 2} condition is such that increasing ϕ {\displaystyle \phi } by a factor of ϕ 0 {\displaystyle \phi _{0}} would increase the total FLOPs of running the network on an image approximately 2 ϕ 0 {\displaystyle 2^{\phi _{0}}} times. The hyperparameters α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } are determined by a small grid search. The original paper suggested 1.2, 1.1, and 1.15, respectively. Architecturally, they optimized the choice of modules by neural architecture search (NAS), and found that the inverted bottleneck convolution (which they called MBConv) used in MobileNet worked well. The EfficientNet family is a stack of MBConv layers, with shapes determined by the compound scaling. The original publication consisted of 8 models, from EfficientNet-B0 to EfficientNet-B7, with increasing model size and accuracy. EfficientNet-B0 is the baseline network, and subsequent models are obtained by scaling the baseline network by increasing ϕ {\displaystyle \phi } . == Variants == EfficientNet has been adapted for fast inference on edge TPUs and centralized TPU or GPU clusters by NAS. EfficientNet V2 was published in June 2021. The architecture was improved by further NAS search with more types of convolutional layers. It also introduced a training method, which progressively increases image size during training, and uses regularization techniques like dropout, RandAugment, and Mixup. The authors claim this approach mitigates accuracy drops often associated with progressive resizing.
Quotient automaton
In computer science, in particular in formal language theory, a quotient automaton can be obtained from a given nondeterministic finite automaton by joining some of its states. The quotient recognizes a superset of the given automaton; in some cases, handled by the Myhill–Nerode theorem, both languages are equal. == Formal definition == A (nondeterministic) finite automaton is a quintuple A = ⟨Σ, S, s0, δ, Sf⟩, where: Σ is the input alphabet (a finite, non-empty set of symbols), S is a finite, non-empty set of states, s0 is the initial state, an element of S, δ is the state-transition relation: δ ⊆ S × Σ × S, and Sf is the set of final states, a (possibly empty) subset of S. A string a1...an ∈ Σ is recognized by A if there exist states s1, ..., sn ∈ S such that ⟨si-1,ai,si⟩ ∈ δ for i=1,...,n, and sn ∈ Sf. The set of all strings recognized by A is called the language recognized by A; it is denoted as L(A). For an equivalence relation ≈ on the set S of A’s states, the quotient automaton A/≈ = ⟨Σ, S/≈, [s0], δ/≈, Sf/≈⟩ is defined by the input alphabet Σ being the same as that of A, the state set S/≈ being the set of all equivalence classes of states from S, the start state [s0] being the equivalence class of A’s start state, the state-transition relation δ/≈ being defined by δ/≈([s],a,[t]) if δ(s,a,t) for some s ∈ [s] and t ∈ [t], and the set of final states Sf/≈ being the set of all equivalence classes of final states from Sf. The process of computing A/≈ is also called factoring A by ≈. == Example == For example, the automaton A shown in the first row of the table is formally defined by ΣA = {0,1}, SA = {a,b,c,d}, sA0 = a, δA = { ⟨a,1,b⟩, ⟨b,0,c⟩, ⟨c,0,d⟩ }, and SAf = { b,c,d }. It recognizes the finite set of strings { 1, 10, 100 }; this set can also be denoted by the regular expression "1+10+100". The relation (≈) = { ⟨a,a⟩, ⟨a,b⟩, ⟨b,a⟩, ⟨b,b⟩, ⟨c,c⟩, ⟨c,d⟩, ⟨d,c⟩, ⟨d,d⟩ }, more briefly denoted as a≈b,c≈d, is an equivalence relation on the set {a,b,c,d} of automaton A’s states. Building the quotient of A by that relation results in automaton C in the third table row; it is formally defined by ΣC = {0,1}, SC = {a,c}, sC0 = a, δC = { ⟨a,1,a⟩, ⟨a,0,c⟩, ⟨c,0,c⟩ }, and SCf = { a,c }. It recognizes the finite set of all strings composed of arbitrarily many 1s, followed by arbitrarily many 0s, i.e. { ε, 1, 10, 100, 1000, ..., 11, 110, 1100, 11000, ..., 111, ... }; this set can also be denoted by the regular expression "10". Informally, C can be thought of resulting from A by glueing state a onto state b, and glueing state c onto state d. The table shows some more quotient relations, such as B = A/a≈b, and D = C/a≈c. == Properties == For every automaton A and every equivalence relation ≈ on its states set, L(A/≈) is a superset of (or equal to) L(A). Given a finite automaton A over some alphabet Σ, an equivalence relation ≈ can be defined on Σ by x ≈ y if ∀ z ∈ Σ: xz ∈ L(A) ↔ yz ∈ L(A). By the Myhill–Nerode theorem, A/≈ is a deterministic automaton that recognizes the same language as A. As a consequence, the quotient of A by every refinement of ≈ also recognizes the same language as A.
P4-metric
The P4 metric (also known as FS or Symmetric F ) enables performance evaluation of a binary classifier. The P4 metric is calculated from precision, recall, specificity, and NPV (negative predictive value). The definition of the P4 metric is similar to that of the F1 metric, however the P4 metric definition addresses criticisms leveled against the definition of the F1 metric. The definition of the P4 metric may, therefore, be understood as an extension of the F1 metric. Like the other known metrics, the P4 metric is a function of: TP (true positives), TN (true negatives), FP (false positives), FN (false negatives). == Justification == The key concept of the P4 metric is to leverage the four key conditional probabilities: P ( + ∣ C + ) {\displaystyle P(+\mid C{+})} — the probability that the sample is positive, provided the classifier result was positive. P ( C + ∣ + ) {\displaystyle P(C{+}\mid +)} — the probability that the classifier result will be positive, provided the sample is positive. P ( C − ∣ − ) {\displaystyle P(C{-}\mid -)} — the probability that the classifier result will be negative, provided the sample is negative. P ( − ∣ C − ) {\displaystyle P(-\mid C{-})} — the probability the sample is negative, provided the classifier result was negative. The main assumption behind this metric is that all the probabilities mentioned above are close to 1 for a properly designed binary classifier. Indeed, P 4 = 1 {\displaystyle \mathrm {P} _{4}=1} if, and only if, all of the probabilities above are equal to 1. Another important feature is that P 4 {\displaystyle \mathrm {P} _{4}} tends to zero any of the above probabilities tend to zero. == Definition == P4 is defined as a harmonic mean of four key conditional probabilities: P 4 = 4 1 P ( + ∣ C + ) + 1 P ( C + ∣ + ) + 1 P ( C − ∣ − ) + 1 P ( − ∣ C − ) = 4 1 p r e c i s i o n + 1 r e c a l l + 1 s p e c i f i c i t y + 1 N P V . {\displaystyle \mathrm {P} _{4}={\frac {4}{{\frac {1}{P(+\mid C{+})}}+{\frac {1}{P(C{+}\mid +)}}+{\frac {1}{P(C{-}\mid -)}}+{\frac {1}{P(-\mid C{-})}}}}={\frac {4}{{\frac {1}{\mathit {precision}}}+{\frac {1}{\mathit {recall}}}+{\frac {1}{\mathit {specificity}}}+{\frac {1}{\mathit {NPV}}}}}.} In terms of TP,TN,FP,FN it can be calculated as follows: P 4 = 4 ⋅ T P ⋅ T N 4 ⋅ T P ⋅ T N + ( T P + T N ) ⋅ ( F P + F N ) . {\displaystyle \mathrm {P} _{4}={\frac {4\cdot \mathrm {TP} \cdot \mathrm {TN} }{4\cdot \mathrm {TP} \cdot \mathrm {TN} +(\mathrm {TP} +\mathrm {TN} )\cdot (\mathrm {FP} +\mathrm {FN} )}}.} == Evaluation of the binary classifier performance == Evaluating the performance of binary classifiers is a multidisciplinary concept. It spans from the evaluation of medical tests, psychiatric tests to machine learning classifiers from a variety of fields. Thus, many of the metrics in use exist under several names, some defined independently. == Properties of P4 metric == Symmetry — contrasting to the F1 metric, P4 is symmetrical. It means - it does not change its value when dataset labeling is changed - positives named negatives and negatives named positives. Range: P 4 ∈ [ 0 , 1 ] {\displaystyle \mathrm {P} _{4}\in [0,1]} . Achieving P 4 ≈ 1 {\displaystyle \mathrm {P} _{4}\approx 1} requires all the key four conditional probabilities being close to 1. For P 4 ≈ 0 {\displaystyle \mathrm {P} _{4}\approx 0} it is sufficient that one of the key four conditional probabilities is close to 0. == Examples, comparing with the other metrics == Dependency table for selected metrics ("true" means depends, "false" - does not depend): Metrics that do not depend on a given probability are prone to misrepresentation when the probability approaches 0. === Example 1: Rare disease detection test === Let us consider a medical test used to detect a rare disease. Suppose a population size of 100000 and 0.05% of the population is infected. Further suppose the following test performance: 95% of all positive individuals are classified correctly (TPR=0.95) and 95% of all negative individuals are classified correctly (TNR=0.95). In such a case, due to high population imbalance and in spite of having high test accuracy (0.95), the probability that an individual who has been classified as positive is in fact positive is very low: P ( + ∣ C + ) = 0.0095. {\displaystyle P(+\mid C{+})=0.0095.} We can observe how this low probability is reflected in some of the metrics: P 4 = 0.0370 {\displaystyle \mathrm {P} _{4}=0.0370} , F 1 = 0.0188 {\displaystyle \mathrm {F} _{1}=0.0188} , J = 0.9100 {\displaystyle \mathrm {J} =\mathbf {0.9100} } (Informedness / Youden index), M K = 0.0095 {\displaystyle \mathrm {MK} =0.0095} (Markedness). === Example 2: Image recognition — cats vs dogs === Consider the problem of training a neural network based image classifier with only two types of images: those containing dogs (labeled as 0) and those containing cats (labeled as 1). Thus, the goal is to distinguish between the cats and dogs. Suppose that the classifier overpredicts in favour of cats ("positive" samples): 99.99% of cats are classified correctly and only 1% of dogs are classified correctly. Further, suppose that the image dataset consists of 100000 images, 90% of which are pictures of cats and 10% are pictures of dogs. In this situation, the probability that the picture containing dog will be classified correctly is pretty low: P ( C − | − ) = 0.01. {\displaystyle P(C-|-)=0.01.} Not all metrics are notice this low probability: P 4 = 0.0388 {\displaystyle \mathrm {P} _{4}=0.0388} , F 1 = 0.9478 {\displaystyle \mathrm {F} _{1}=\mathbf {0.9478} } , J = 0.0099 {\displaystyle \mathrm {J} =0.0099} (Informedness / Youden index), M K = 0.8183 {\displaystyle \mathrm {MK} =\mathbf {0.8183} } (Markedness).
Permutation automaton
In automata theory, a permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states. Formally, a deterministic finite automaton A may be defined by the tuple (Q, Σ, δ, q0, F), where Q is the set of states of the automaton, Σ is the set of input symbols, δ is the transition function that takes a state q and an input symbol x to a new state δ(q,x), q0 is the initial state of the automaton, and F is the set of accepting states (also: final states) of the automaton. A is a permutation automaton if and only if, for every two distinct states qi and qj in Q and every input symbol x in Σ, δ(qi,x) ≠ δ(qj,x). A formal language is p-regular (also: a pure-group language) if it is accepted by a permutation automaton. For example, the set of strings of even length forms a p-regular language: it may be accepted by a permutation automaton with two states in which every transition replaces one state by the other. == Applications == The pure-group languages were the first interesting family of regular languages for which the star height problem was proved to be computable. Another mathematical problem on regular languages is the separating words problem, which asks for the size of a smallest deterministic finite automaton that distinguishes between two given words of length at most n – by accepting one word and rejecting the other. The known upper bound in the general case is O ( n 2 / 5 ( log n ) 3 / 5 ) {\displaystyle O(n^{2/5}(\log n)^{3/5})} . The problem was later studied for the restriction to permutation automata. In this case, the known upper bound changes to O ( n 1 / 2 ) {\displaystyle O(n^{1/2})} .
Signal-to-noise ratio (imaging)
Signal-to-noise ratio (SNR) is used in imaging to characterize image quality. The sensitivity of a (digital or film) imaging system is typically described in the terms of the signal level that yields a threshold level of SNR. Industry standards define sensitivity in terms of the ISO film speed equivalent, using SNR thresholds (at average scene luminance) of 40:1 for "excellent" image quality and 10:1 for "acceptable" image quality. SNR is sometimes quantified in decibels (dB) of signal power relative to noise power, though in the imaging field the concept of "power" is sometimes taken to be the power of a voltage signal proportional to optical power; so a 20 dB SNR may mean either 10:1 or 100:1 optical power, depending on which definition is in use. == Definition of SNR == Traditionally, SNR is defined to be the ratio of the average signal value μ s i g {\displaystyle \mu _{\mathrm {sig} }} to the standard deviation of the signal σ s i g {\displaystyle \sigma _{\mathrm {sig} }} : S N R = μ s i g σ s i g {\displaystyle \mathrm {SNR} ={\frac {\mu _{\mathrm {sig} }}{\sigma _{\mathrm {sig} }}}} when the signal is an optical intensity, or as the square of this value if the signal and noise are viewed as amplitudes (field quantities).
Mealy machine
In the theory of computation, a Mealy machine is a finite-state machine whose output values are determined both by its current state and the current inputs. This is in contrast to a Moore machine, whose output values are determined solely by its current state. A Mealy machine is a deterministic finite-state transducer: for each state and input, at most one transition is possible. == History == The Mealy machine is named after George H. Mealy, who presented the concept in a 1955 paper, "A Method for Synthesizing Sequential Circuits". == Formal definition == A Mealy machine is a 6-tuple ( S , S 0 , Σ , Λ , T , G ) {\displaystyle (S,S_{0},\Sigma ,\Lambda ,T,G)} consisting of the following: a finite set of states S {\displaystyle S} a start state (also called initial state) S 0 {\displaystyle S_{0}} which is an element of S {\displaystyle S} a finite set called the input alphabet Σ {\displaystyle \Sigma } a finite set called the output alphabet Λ {\displaystyle \Lambda } a transition function T : S × Σ → S {\displaystyle T:S\times \Sigma \rightarrow S} mapping pairs of a state and an input symbol to the corresponding next state. an output function G : S × Σ → Λ {\displaystyle G:S\times \Sigma \rightarrow \Lambda } mapping pairs of a state and an input symbol to the corresponding output symbol. In some formulations, the transition and output functions are coalesced into a single function T : S × Σ → S × Λ {\displaystyle T:S\times \Sigma \rightarrow S\times \Lambda } . "Evolution across time" is realized in this abstraction by having the state machine consult the time-changing input symbol at discrete "timer ticks" t 0 , t 1 , t 2 , . . . {\displaystyle t_{0},t_{1},t_{2},...} and react according to its internal configuration at those idealized instants, or else having the state machine wait for a next input symbol (as on a FIFO) and react whenever it arrives. == Comparison of Mealy machines and Moore machines == Mealy machines tend to have fewer states: Different outputs on arcs (n2) rather than states (n). When implemented as electronic circuits (rather than as mathematical abstractions or code): Moore machines are safer to use than Mealy machines: Outputs change at the clock edge (always one cycle later). In Mealy machines, input change can cause output change as soon as logic is done — a big problem when two machines are interconnected – asynchronous feedback may occur if one isn't careful. Mealy machines react faster to inputs: React in the same cycle—they don't need to wait for the clock. In Moore machines, more logic may be necessary to decode state into outputs—more gate delays after clock edge. == Diagram == The state diagram for a Mealy machine associates an output value with each transition edge, in contrast to the state diagram for a Moore machine, which associates an output value with each state. When the input and output alphabet are both Σ, one can also associate to a Mealy automata a Helix directed graph (S × Σ, (x, i) → (T(x, i), G(x, i))). This graph has as vertices the couples of state and letters, each node is of out-degree one, and the successor of (x, i) is the next state of the automata and the letter that the automata output when it is instate x and it reads letter i. This graph is a union of disjoint cycles if the automaton is bireversible. == Examples == === Simple === A simple Mealy machine has one input and one output. Each transition edge is labeled with the value of the input (shown in red) and the value of the output (shown in blue). The machine starts in state Si. (In this example, the output is the exclusive-or of the two most-recent input values; thus, the machine implements an edge detector, outputting a 1 every time the input flips and a 0 otherwise.) === Complex === More complex Mealy machines can have multiple inputs as well as multiple outputs. == Applications == Mealy machines provide a rudimentary mathematical model for cipher machines. Considering the input and output alphabet the Latin alphabet, for example, then a Mealy machine can be designed that given a string of letters (a sequence of inputs) can process it into a ciphered string (a sequence of outputs). However, although a Mealy model could be used to describe the Enigma, the state diagram would be too complex to provide feasible means of designing complex ciphering machines. Moore/Mealy machines are DFAs that have also output at any tick of the clock. Modern CPUs, computers, cell phones, digital clocks and basic electronic devices/machines have some kind of finite state machine to control it. Simple software systems, particularly ones that can be represented using regular expressions, can be modeled as finite state machines. There are many such simple systems, such as vending machines or basic electronics. By finding the intersection of two finite state machines, one can design in a very simple manner concurrent systems that exchange messages for instance. For example, a traffic light is a system that consists of multiple subsystems, such as the different traffic lights, that work concurrently.