Foreground detection is one of the major tasks in the field of computer vision and image processing whose aim is to detect changes in image sequences. Background subtraction is any technique which allows an image's foreground to be extracted for further processing (object recognition etc.). Many applications do not need to know everything about the evolution of movement in a video sequence, but only require the information of changes in the scene, because an image's regions of interest are objects (humans, cars, text etc.) in its foreground. After the stage of image preprocessing (which may include image denoising, post processing like morphology etc.) object localisation is required which may make use of this technique. Foreground detection separates foreground from background based on these changes taking place in the foreground. It is a set of techniques that typically analyze video sequences recorded in real time with a stationary camera. == Description == All detection techniques are based on modelling the background of the image, i.e., setting the background and detecting which changes occur. Defining the background can be difficult when it contains shapes, shadows, and moving objects. In defining the background, it is assumed that stationary objects may vary in color and intensity over time. Scenarios in which these techniques apply tend to be very diverse. There can be highly variable sequences, such as images with different lighting, interiors, exteriors, quality, and noise. In addition to real-time processing, systems need to adapt to these changes. A foreground detection system should be able to: Develop a background model (estimate). Be robust to lighting changes, repetitive movements (leaves, waves, shadows), and long-term changes. == Background subtraction == Background subtraction is a widely used approach for detecting moving objects in videos from static cameras. The rationale in the approach is that of detecting the moving objects from the difference between the current frame and a reference frame, often called "background image", or "background model". Background subtraction is mostly done if the image in question is a part of a video stream. Background subtraction provides important cues for numerous applications in computer vision, for example surveillance tracking or human pose estimation. Background subtraction is generally based on a static background hypothesis which is often not applicable in real environments. With indoor scenes, reflections or animated images on screens lead to background changes. Similarly, due to wind, rain or illumination changes brought by weather, static backgrounds methods have difficulties with outdoor scenes. == Temporal average filter == The temporal average filter is a method that was proposed at the Velastin. This system estimates the background model from the median of all pixels of a number of previous images. The system uses a buffer with the pixel values of the last frames to update the median for each image. To model the background, the system examines all images in a given time period called training time. At this time, we only display images and will find the median, pixel by pixel, of all the plots in the background this time. After the training period for each new frame, each pixel value is compared with the input value of funds previously calculated. If the input pixel is within a threshold, the pixel is considered to match the background model and its value is included in the pixbuf. Otherwise, if the value is outside this threshold pixel is classified as foreground, and not included in the buffer. This method cannot be considered very efficient because they do not present a rigorous statistical basis and requires a buffer that has a high computational cost. == Conventional approaches == A robust background subtraction algorithm should be able to handle lighting changes, repetitive motions from clutter and long-term scene changes. The following analyses make use of the function of V(x,y,t) as a video sequence where t is the time dimension, x and y are the pixel location variables. e.g. V(1,2,3) is the pixel intensity at (1,2) pixel location of the image at t = 3 in the video sequence. === Using frame differencing === A motion detection algorithm begins with the segmentation part where foreground or moving objects are segmented from the background. The simplest way to implement this is to take an image as background and take the frames obtained at the time t, denoted by I(t) to compare with the background image denoted by B. Here using simple arithmetic calculations, we can segment out the objects simply by using image subtraction technique of computer vision meaning for each pixels in I(t), take the pixel value denoted by P[I(t)] and subtract it with the corresponding pixels at the same position on the background image denoted as P[B]. In mathematical equation, it is written as: P [ F ( t ) ] = P [ I ( t ) ] − P [ B ] {\displaystyle P[F(t)]=P[I(t)]-P[B]} The background is assumed to be the frame at time t. This difference image would only show some intensity for the pixel locations which have changed in the two frames. Though we have seemingly removed the background, this approach will only work for cases where all foreground pixels are moving, and all background pixels are static. A threshold "Threshold" is put on this difference image to improve the subtraction (see Image thresholding): | P [ F ( t ) ] − P [ F ( t + 1 ) ] | > T h r e s h o l d {\displaystyle |P[F(t)]-P[F(t+1)]|>\mathrm {Threshold} } This means that the difference image's pixels' intensities are 'thresholded' or filtered on the basis of value of Threshold. The accuracy of this approach is dependent on speed of movement in the scene. Faster movements may require higher thresholds. === Mean filter === For calculating the image containing only the background, a series of preceding images are averaged. For calculating the background image at the instant t: B ( x , y , t ) = 1 N ∑ i = 1 N V ( x , y , t − i ) {\displaystyle B(x,y,t)={1 \over N}\sum _{i=1}^{N}V(x,y,t-i)} where N is the number of preceding images taken for averaging. This averaging refers to averaging corresponding pixels in the given images. N would depend on the video speed (number of images per second in the video) and the amount of movement in the video. After calculating the background B(x,y,t) we can then subtract it from the image V(x,y,t) at time t = t and threshold it. Thus the foreground is: | V ( x , y , t ) − B ( x , y , t ) | > T h {\displaystyle |V(x,y,t)-B(x,y,t)|>\mathrm {Th} } where Th is a threshold value. Similarly, we can also use median instead of mean in the above calculation of B(x,y,t). Usage of global and time-independent thresholds (same Th value for all pixels in the image) may limit the accuracy of the above two approaches. === Running Gaussian average === For this method, Wren et al. propose fitting a Gaussian probabilistic density function (pdf) on the most recent n {\displaystyle n} frames. In order to avoid fitting the pdf from scratch at each new frame time t {\displaystyle t} , a running (or on-line cumulative) average is computed. The pdf of every pixel is characterized by mean μ t {\displaystyle \mu _{t}} and variance σ t 2 {\displaystyle \sigma _{t}^{2}} . The following is a possible initial condition (assuming that initially every pixel is background): μ 0 = I 0 {\displaystyle \mu _{0}=I_{0}} σ 0 2 = ⟨ some default value ⟩ {\displaystyle \sigma _{0}^{2}=\langle {\text{some default value}}\rangle } where I t {\displaystyle I_{t}} is the value of the pixel's intensity at time t {\displaystyle t} . In order to initialize variance, we can, for example, use the variance in x and y from a small window around each pixel. Note that background may change over time (e.g. due to illumination changes or non-static background objects). To accommodate for that change, at every frame t {\displaystyle t} , every pixel's mean and variance must be updated, as follows: μ t = ρ I t + ( 1 − ρ ) μ t − 1 {\displaystyle \mu _{t}=\rho I_{t}+(1-\rho )\mu _{t-1}} σ t 2 = d 2 ρ + ( 1 − ρ ) σ t − 1 2 {\displaystyle \sigma _{t}^{2}=d^{2}\rho +(1-\rho )\sigma _{t-1}^{2}} d = | ( I t − μ t ) | {\displaystyle d=|(I_{t}-\mu _{t})|} Where ρ {\displaystyle \rho } determines the size of the temporal window that is used to fit the pdf (usually ρ = 0.01 {\displaystyle \rho =0.01} ) and d {\displaystyle d} is the Euclidean distance between the mean and the value of the pixel. We can now classify a pixel as background if its current intensity lies within some confidence interval of its distribution's mean: | ( I t − μ t ) | σ t > k ⟶ foreground {\displaystyle {\frac {|(I_{t}-\mu _{t})|}{\sigma _{t}}}>k\longrightarrow {\text{foreground}}} | ( I t − μ t ) | σ t ≤ k ⟶ background {\displaystyle {\frac {|(I_{t}-\mu _{t})|}{\sigma _{t}}}\leq k\longrightarrow {\text{background}}} where the parameter k {\displaystyle k} is a free threshold (usuall
CrocBITE
CrocBITE (currently CrocAttack) was an online database of wild crocodilian attacks reported on humans in the world. The non-profit online research tool helped to scientifically analyze crocodilian behavior via complex models. Users were encouraged to feed information in a crowdsourcing manner. This website excludes captive crocodilian attacks, as well as non-fatal bites on professional handlers, rangers, staff, or researchers, and crocodilian attacks on pets and livestock, because its primary goal is to analyze natural human-crocodilian conflict in the wild for conservation and management purposes, and that these incidents do are not considered indicative of natural species behavior or typical human-wildlife conflict, as well as not providing enough useful data and helping researchers understand wild population behavior or typical human-wildlife conflict dynamics and helps create safety strategies for people living or working near wild crocodilians, rather than tracking workplace accidents in zoos or farms. While fatal incidents involving handlers are sometimes included on the website, typical captive incidents (such as handlers being bitten by them in zoos) are excluded because they are considered manageable professional risks rather than general public safety threats. == About == The online database was established in 2013 (2013) by Dr Adam Britton, a researcher at Charles Darwin University, his student Brandon Sideleau and Erin Britton. It was a compilation of government records, individual reports, registered contributors and historical data. Dr Simon Pooley, Junior Research fellow, Imperial College London joined hands to further the studies. The collaboration culminated when Dr Pooley met Dr Britton at the IUCN Crocodile Specialist Group, in Louisiana in 2014. The program received funds from Economic and Social Research Council, United Kingdom to the tune of A$30,000 and unspecified resourced plus amount from Big Gecko Crocodilian Research, Crocodillian.com and Charles Darwin University. The research yielded pertinent observations that provide inside into crocodile attacks. It was observed that most attacks on humans occur from bites of Saltwater crocodile as against the popular understanding of Nile crocodiles taking the top spot. This is not, however, believed to be the actual case, as most attacks by the Nile crocodile are believed to go unreported or only reported on a local level. The broad category of Nile crocodile attacks were segmented into West African crocodile and Crocodylus niloticus (the Nile Crocodile) species to get a clear understanding of their respective attack zones. The objective was that the information would be used by communities and conservation managers to help inform and educate people about how to keep safe. The information was vital for Australia and Africa where such attacks are more likely than in other parts of the world. This was the only database of its kind with such comprehensive collection of information made available online. The database is no longer online, and its founder Adam Britton is in custody having pleaded guilty to charges of bestiality on September 25, 2023. It has been rebranded and renamed CrocAttack, and serves as a updated database focusing on human-crocodilian conflict and records over 8,500 incidents from the past decades.
Syman
SYMAN is an artificial intelligence technology that uses data from social media profiles to identify trends in the job market. SYMAN is designed to organize actionable data for products and services including recruiting, human capital management, CRM, and marketing. SYMAN was developed with a $21 million series B financing round secured by Identified, which was led by VantagePoint Capital Partners and Capricorn Investment Group.
Meta-Labeling
Meta-labeling, also known as corrective AI, is a machine learning (ML) technique utilized in quantitative finance to enhance the performance of investment and trading strategies, developed in 2017 by Marcos López de Prado at Guggenheim Partners and Cornell University. The core idea is to separate the decision of trade direction (side) from the decision of trade sizing, addressing the inefficiencies of simultaneously learning both side and size predictions. The side decision involves forecasting market movements (long, short, neutral), while the size decision focuses on risk management and profitability. It serves as a secondary decision-making layer that evaluates the signals generated by a primary predictive model. By assessing the confidence and likely profitability of those signals, meta-labeling allows investors and algorithms to dynamically size positions and suppress false positives. == Motivation == Meta-labeling is designed to improve precision without sacrificing recall. As noted by López de Prado, attempting to model both the direction and the magnitude of a trade using a single algorithm can result in poor generalization. By separating these tasks, meta-labeling enables greater flexibility and robustness: Enhances control over capital allocation. Reduces overfitting by limiting model complexity. Allows the use of interpretability tools and tailored thresholds to manage risk. Enables dynamic trade suppression in unfavorable regimes. == Applications == Meta-labeling has been applied in a variety of financial ML contexts, including: Algorithmic trading: Filtering and sizing trades to reduce false positives. Portfolio optimization: Scaling exposure across multiple signals with differing confidence levels. Risk management: Dynamically disabling strategies in adverse market conditions. Model validation: Interpreting when and why a model may be underperforming due to regime shifts. == General architecture == Meta-labeling decouples two core components of systematic trading strategies: directional prediction and position sizing. The process involves training a primary model to generate trade signals (e.g., buy, sell, or hold) and then training a secondary model to determine whether each signal is likely to lead to a profitable trade. The second model outputs a probability that is interpreted as the confidence in the forecast, which can be used to adjust the position size or to filter out unreliable trades. Meta-labeling is typically implemented as a three-stage process: Primary model (M1): Predicts the direction or label of a financial outcome using features such as market prices, returns, or volatility indicators. A typical output is directional, e.g., Y ∈ {−1,0,1}, representing short, neutral, or long positions. Secondary model (M2): A binary classifier trained to predict whether the primary model's prediction will be profitable. The target variable is a binary meta-label F ∈ { 0 , 1 } {\displaystyle F\in \{0,1\}} . Inputs can include features used in the primary model, performance diagnostics, or market regime data. Position sizing algorithm (M3): Translates the output probability of the secondary model into a position size. Higher confidence scores result in larger allocations, while lower confidence leads to reduced or zero exposure. === Stage 1: Forecasting side === Primary model architecture Figure 1 Figure 1 presents the architecture of a primary model. It focuses on forecasting the side of the trade. Following the example, this model (M1) takes in input data – such as open-high-low-close data and determines the side of the position to take: a negative number is a short position, and positive number is a long position, the range is set between −1 and 1 (the closer it is to −1 or 1, the stronger the models conviction is). When training the model, the labels are −1 and 1, based on the direction of forward returns for some predefined investment horizon. The researcher may decide to apply a recall check (τ: "Tau") by setting a minimum threshold that the initial output needs to be to qualify of a short or long position (if the threshold is not met, no side forecast is predicted, leading to closing of any open positions), this leads to the primary model output which is one of three possible side forecasts: −1, 0, or 1. The primary model also generates evaluation data which can be used by the secondary model, to improve performance of size forecasts. Some examples of evaluation data include rolling accuracy, F1, recall, precision, and AUC scores. === Stage 2: Filtering out false positives === General meta-labeling architecture Figure 2 Next comes the phase of filtering out false positives, by applying a secondary machine learning model (M2), which is a binary classifier trained to determine if the trade will be profitable or not. The model takes as input four general groupings of data: General input data which is predictive of a false positive. For example the last 30 days rolling volatility of the underlying asset. Evaluation data. Market state and regime data, one may find that macro economic data or clustering the market into regimes may help as specific trading strategies are known to perform better in particular regimes. Example: momentum based strategies perform best in periods with low volatility and strong directional moves. Primary models initial input which is a value between −1 and 1. This highlights the strength of the primary models conviction. The output of the model is a value between −1 and 1 (if using a Tanh function) which will indicate the strength of the conviction that a short or long position is profitable, or it could simply be between 0 and 1 (using a sigmoid function) if one only wanted to know if it made money or not. This output allows filtering out trades that are likely to lead to losses. One could stop at this point or use the outputs of the secondary model as inputs to a position sizing algorithm (M3) which could further enhance strategy performance metrics by translating the output probability of the secondary model into a position size. Higher confidence scores result in larger allocations, while lower confidence leads to reduced or zero exposure. === Stage 3: Optimizing position sizes === ==== Position sizing methods (M3) ==== Various algorithms have been proposed for transforming predicted probabilities into trade sizes: All-or-nothing: Allocate 100% of capital if the probability exceeds a predefined threshold (e.g., 0.5); otherwise, do not trade. Model confidence: Use the probability score directly as the fraction of capital allocated. Linear scaling: Rescale the model's probabilities using min-max normalization based on the training data. Normal CDF (NCDF): Use a normal cumulative distribution function applied to a z-statistic derived from the predicted probability. Empirical CDF (ECDF): Rank probabilities based on their percentile in the training data to ensure relative allocation. Sigmoid Optimal Position Sizing (SOPS): Applies a smooth non-linear sigmoid transformation optimized to maximize risk-adjusted returns (Sharpe ratio). ==== Model calibration ==== Each machine learning algorithm used in meta-labeling tends to produce outputs with different characteristic distributions; for example, some are approximately normally distributed, whereas others exhibit a pronounced U-shape, concentrating probabilities near the extremes. Due to these varying distributions, simply summing the outputs of different models can inadvertently lead to uneven weighting of signals, biasing trade decisions. To address this, model calibration techniques are essential to adjust the predicted probabilities towards frequentist probabilities, ensuring that model outputs reflect true likelihoods more accurately. Two common calibration techniques are: Platt scaling (Sigmoid scaling): Suitable for correcting S-shaped calibration plots typically produced by models such as support vector machines (SVMs). Isotonic regression: Fits a non-decreasing step function to probabilities and is effective particularly with larger datasets, though it can sometimes lead to overfitting. Transforming predictions to frequentist probabilities is crucial as it provides probabilistic outputs that are directly interpretable as the actual likelihood of an event occurring. Such calibration significantly enhances the effectiveness of fixed position sizing methods, reducing maximum drawdowns and increasing risk-adjusted returns. However, calibration has less impact on position sizing methods that directly estimate parameters from the training data, such as ECDF and SOPS, suggesting that calibration is a critical step mainly for fixed methods that rely heavily on raw model outputs. =
Psychology of reasoning
The psychology of reasoning (also known as the cognitive science of reasoning) is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory. Psychological experiments on how humans and other animals reason have been carried out for over 100 years. An enduring question is whether or not people have the capacity to be rational. Current research in this area addresses various questions about reasoning, rationality, judgments, intelligence, relationships between emotion and reasoning, and development. == Everyday reasoning == One of the most obvious areas in which people employ reasoning is with sentences in everyday language. Most experimentation on deduction has been carried out on hypothetical thought, in particular, examining how people reason about conditionals, e.g., If A then B. Participants in experiments make the modus ponens inference, given the indicative conditional If A then B, and given the premise A, they conclude B. However, given the indicative conditional and the minor premise for the modus tollens inference, not-B, about half of the participants in experiments conclude not-A and the remainder concludes that nothing follows. The ease with which people make conditional inferences is affected by context, as demonstrated in the well-known selection task developed by Peter Wason. Participants are better able to test a conditional in an ecologically relevant context, e.g., if the envelope is sealed then it must have a 50 cent stamp on it compared to one that contains symbolic content, e.g., if the letter is a vowel then the number is even. Background knowledge can also lead to the suppression of even the simple modus ponens inference Participants given the conditional if Lisa has an essay to write then she studies late in the library and the premise Lisa has an essay to write make the modus ponens inference 'she studies late in the library', but the inference is suppressed when they are also given a second conditional if the library stays open then she studies late in the library. Interpretations of the suppression effect are controversial Other investigations of propositional inference examine how people think about disjunctive alternatives, e.g., A or else B, and how they reason about negation, e.g., It is not the case that A and B. Many experiments have been carried out to examine how people make relational inferences, including comparisons, e.g., A is better than B. Such investigations also concern spatial inferences, e.g. A is in front of B and temporal inferences, e.g. A occurs before B. Other common tasks include categorical syllogisms, used to examine how people reason about quantifiers such as All or Some, e.g., Some of the A are not B. For example if all A are B and some B are C, what (if anything) follows? == Theories of reasoning == There are several alternative theories of the cognitive processes that human reasoning is based on. One view is that people rely on a mental logic consisting of formal (abstract or syntactic) inference rules similar to those developed by logicians in the propositional calculus. Another view is that people rely on domain-specific or content-sensitive rules of inference. A third view is that people rely on mental models, that is, mental representations that correspond to imagined possibilities. A fourth view is that people compute probabilities. One controversial theoretical issue is the identification of an appropriate competence model, or a standard against which to compare human reasoning. Initially classical logic was chosen as a competence model. Subsequently, some researchers opted for non-monotonic logic and Bayesian probability. Research on mental models and reasoning has led to the suggestion that people are rational in principle but err in practice. Connectionist approaches towards reasoning have also been proposed. Despite the ongoing debate about the cognitive processes involved in human reasoning, recent research has shown that multiple approaches can be useful in modeling human thinking. For instance, studies have found that people's reasoning is often influenced by their prior beliefs, which can be modeled using Bayesian probability theory. Additionally, research on mental models has shown that people tend to reason about problems by constructing multiple mental representations of the situation, which can help them to identify relevant features and make inferences based on their understanding of the problem. Moreover, connectionist approaches to reasoning have also gained attention, which focus on the neural network models that can learn from data and generalize to new situations. == Development of reasoning == It is an active question in psychology how, why, and when the ability to reason develops from infancy to adulthood. Jean Piaget's theory of cognitive development posited general mechanisms and stages in the development of reasoning from infancy to adulthood. According to the neo-Piagetian theories of cognitive development, changes in reasoning with development come from increasing working memory capacity, increasing speed of processing, and enhanced executive functions and control. Increasing self-awareness is also an important factor. In their book The Enigma of Reason, the cognitive scientists Hugo Mercier and Dan Sperber put forward an "argumentative" theory of reasoning, claiming that humans evolved to reason primarily to justify our beliefs and actions and to convince others in a social environment. Key evidence for their theory includes the errors in reasoning that solitary individuals are prone to when their arguments are not criticized, such as logical fallacies, and how groups become much better at performing cognitive reasoning tasks when they communicate with one another and can evaluate each other's arguments. Sperber and Mercier offer one attempt to resolve the apparent paradox that the confirmation bias is so strong despite the function of reasoning naively appearing to be to come to veridical conclusions about the world. The study of the development of reasoning abilities is an ongoing area of research in psychology, and multiple factors have been proposed to explain how, why, and when reasoning develops from infancy to adulthood. Recent research has suggested that early experiences and social interactions play a critical role in the development of reasoning abilities. For example, studies have shown that infants as young as six months old can engage in basic logical reasoning, such as reasoning about the relationship between objects and their properties. Furthermore, research has highlighted the importance of parental interaction and cognitive stimulation in the development of children's reasoning abilities. Additionally, studies have suggested that cultural factors, such as educational practices and the emphasis on critical thinking, can also influence the development of reasoning skills across different populations. == Different sorts of reasoning == Philip Johnson-Laird trying to taxonomize thought, distinguished between goal-directed thinking and thinking without goal, noting that association was involved in unrelated reading. He argues that goal directed reasoning can be classified based on the problem space involved in a solution, citing Allen Newell and Herbert A. Simon. Inductive reasoning makes broad generalizations from specific cases or observations. In this process of reasoning, general assertions are made based on past specific pieces of evidence. This kind of reasoning allows the conclusion to be false even if the original statement is true. For example, if one observes a college athlete, one makes predictions and assumptions about other college athletes based on that one observation. Scientists use inductive reasoning to create theories and hypotheses. Philip Johnson-Laird distinguished inductive from deductive reasoning, in that the former creates semantic information while the later does not . In opposition, deductive reasoning is a basic form of valid reasoning. In this reasoning process a person starts with a known claim or a general belief and from there asks what follows from these foundations or how will these premises influence other beliefs. In other words, deduction starts with a hypothesis and examines the possibilities to reach a conclusion. Deduction helps people understand why their predictions are wrong and indicates that their prior knowledge or beliefs are off track. An example of deduction can be seen in the scientific method when testing hypotheses and theories. Although the conclusion usually corresponds and therefore proves the hypothesis, there are some cases where the conclusion is logical, but the generalization is not. For example, the argument, "All young girls wear skirts; Julie is a young
ASR-complete
ASR-complete is, by analogy to "NP-completeness" in complexity theory, a term to indicate that the difficulty of a computational problem is equivalent to solving the central automatic speech recognition problem, i.e. recognize and understanding spoken language. Unlike "NP-completeness", this term is typically used informally. Such problems are hypothesised to include: Spoken natural language understanding Understanding speech from far-field microphones, i.e. handling the reverbation and background noise These problems are easy for humans to do (in fact, they are described directly in terms of imitating humans). Some systems can solve very simple restricted versions of these problems, but none can solve them in their full generality.
Algorithmic probability
In algorithmic information theory, algorithmic probability, also known as Solomonoff probability, is a mathematical method of assigning a prior probability to a given observation. It was invented by Ray Solomonoff in the 1960s. It is used in inductive inference theory and analyses of algorithms. In his general theory of inductive inference, Solomonoff uses the method together with Bayes' rule to obtain probabilities of prediction for an algorithm's future outputs. In the mathematical formalism used, the observations have the form of finite binary strings viewed as outputs of Turing machines, and the universal prior is a probability distribution over the set of finite binary strings calculated from a probability distribution over programs (that is, inputs to a universal Turing machine). The prior is universal in the Turing-computability sense, i.e. no string has zero probability. It is not computable, but it can be approximated. Formally, the probability P {\displaystyle P} is not a probability and it is not computable. It is only "lower semi-computable" and a "semi-measure". By "semi-measure", it means that 0 ≤ ∑ x P ( x ) < 1 {\displaystyle 0\leq \sum _{x}P(x)<1} . That is, the "probability" does not actually sum up to one, unlike actual probabilities. This is because some inputs to the Turing machine causes it to never halt, which means the probability mass allocated to those inputs is lost. By "lower semi-computable", it means there is a Turing machine that, given an input string x {\displaystyle x} , can print out a sequence y 1 < y 2 < ⋯ {\displaystyle y_{1}