Double descent in statistics and machine learning is the phenomenon where a model's error rate on the test set initially decreases with the number of parameters, then peaks, then decreases again. This phenomenon has been considered surprising, as it contradicts assumptions about overfitting in classical machine learning. The increase usually occurs near the interpolation threshold, where the number of parameters is the same as the number of training data points (the model is just large enough to fit the training data). Or, more precisely, it is the maximum number of samples on which the model/training procedure achieves approximately on average 0 training error. == History == Early observations of what would later be called double descent in specific models date back to 1989. The term "double descent" was coined by Belkin et. al. in 2019, when the phenomenon gained popularity as a broader concept exhibited by many models. The latter development was prompted by a perceived contradiction between the conventional wisdom that too many parameters in the model result in a significant overfitting error (an extrapolation of the bias–variance tradeoff), and the empirical observations in the 2010s that some modern machine learning techniques tend to perform better with larger models. == Theoretical models == Double descent occurs in linear regression with isotropic Gaussian covariates and isotropic Gaussian noise. A model of double descent at the thermodynamic limit has been analyzed using the replica trick, and the result has been confirmed numerically. A number of works have suggested that double descent can be explained using the concept of effective dimension: While a network may have a large number of parameters, in practice only a subset of those parameters are relevant for generalization performance, as measured by the local Hessian curvature. This explanation is formalized through PAC-Bayes compression-based generalization bounds, which show that less complex models are expected to generalize better under a Solomonoff prior.
AI-assisted software development
AI-assisted software development is the use of artificial intelligence (AI) to augment software development. It uses large language models (LLMs), AI agents and other AI technologies to assist software developers. It helps in a range of tasks of the software development life cycle, from code generation to debugging, editing, testing, UI design, understanding the code, and documentation. Agentic coding denotes the use of AI agents for software development. == Technologies == === Source code generation === Large language models trained or fine-tuned on source-code corpora can generate source code from natural-language descriptions, comments, or docstrings. Research on code-generation systems often evaluates generated programs by functional correctness, such as whether the output passes automated test cases, rather than by syntax alone. Such tools can be features or extensions of integrated development environments (IDEs). === Intelligent code completion === AI agents using pre-trained and fine-tuned LLMs can predict and suggest code completions based on context. According to Husein, Aburajouh & Catal in a 2025 literature review in Computer Standards & Interfaces, "LLMs significantly enhance code completion performance across several programming languages and contexts, and their capability to predict relevant code snippets based on context and partial input boosts developer productivity substantially." === Testing, debugging, code review and analysis === AI is used to automatically generate test cases, identify potential bugs and security vulnerabilities, and suggest fixes. AI can also be used to perform static code analysis and suggest potential performance improvements. == Limitations == Both ownership of and responsibility for AI-generated code is disputed. According to a report from the German Federal Office for Information Security, the use of AI coding assistants without careful oversight from experienced developers can introduce both minor and major security vulnerabilities, and any potential gain in productivity should be weighed against the cost of additional quality control and security measures. According to Deloitte, outputs from AI-assisted software development must be validated through a combination of automated testing, static analysis tools and human review, creating a governance layer to improve quality and accountability. == Vibe coding ==
Thompson sampling
Thompson sampling, named after William R. Thompson, is a heuristic for choosing actions that address the exploration–exploitation dilemma in the multi-armed bandit problem. It consists of choosing the action that maximizes the expected reward with respect to a randomly drawn belief. == Description == Consider a set of contexts X {\displaystyle {\mathcal {X}}} , a set of actions A {\displaystyle {\mathcal {A}}} , and rewards in R {\displaystyle \mathbb {R} } . The aim of the player is to play actions under the various contexts, such as to maximize the cumulative rewards. Specifically, in each round, the player obtains a context x ∈ X {\displaystyle x\in {\mathcal {X}}} , plays an action a ∈ A {\displaystyle a\in {\mathcal {A}}} and receives a reward r ∈ R {\displaystyle r\in \mathbb {R} } following a distribution that depends on the context and the issued action. The elements of Thompson sampling are as follows: a likelihood function P ( r | θ , a , x ) {\displaystyle P(r|\theta ,a,x)} ; a set Θ {\displaystyle \Theta } of parameters θ {\displaystyle \theta } of the distribution of r {\displaystyle r} ; a prior distribution P ( θ ) {\displaystyle P(\theta )} on these parameters; past observations triplets D = { ( x ; a ; r ) } {\displaystyle {\mathcal {D}}=\{(x;a;r)\}} ; a posterior distribution P ( θ | D ) ∝ P ( D | θ ) P ( θ ) {\displaystyle P(\theta |{\mathcal {D}})\propto P({\mathcal {D}}|\theta )P(\theta )} , where P ( D | θ ) {\displaystyle P({\mathcal {D}}|\theta )} is the likelihood function. Thompson sampling consists of playing the action a ∗ ∈ A {\displaystyle a^{\ast }\in {\mathcal {A}}} according to the probability that it maximizes the expected reward; action a ∗ {\displaystyle a^{\ast }} is chosen with probability ∫ I [ E ( r | a ∗ , x , θ ) = max a ′ E ( r | a ′ , x , θ ) ] P ( θ | D ) d θ , {\displaystyle \int \mathbb {I} \left[\mathbb {E} (r|a^{\ast },x,\theta )=\max _{a'}\mathbb {E} (r|a',x,\theta )\right]P(\theta |{\mathcal {D}})d\theta ,} where I {\displaystyle \mathbb {I} } is the indicator function. In practice, the rule is implemented by sampling. In each round, parameters θ ∗ {\displaystyle \theta ^{\ast }} are sampled from the posterior P ( θ | D ) {\displaystyle P(\theta |{\mathcal {D}})} , and an action a ∗ {\displaystyle a^{\ast }} chosen that maximizes E [ r | θ ∗ , a ∗ , x ] {\displaystyle \mathbb {E} [r|\theta ^{\ast },a^{\ast },x]} , i.e. the expected reward given the sampled parameters, the action, and the current context. Conceptually, this means that the player instantiates their beliefs randomly in each round according to the posterior distribution, and then acts optimally according to them. In most practical applications, it is computationally onerous to maintain and sample from a posterior distribution over models. As such, Thompson sampling is often used in conjunction with approximate sampling techniques. == History == Thompson sampling was originally described by Thompson in 1933. It was subsequently rediscovered numerous times independently in the context of multi-armed bandit problems. A first proof of convergence for the bandit case has been shown in 1997. The first application to Markov decision processes was in 2000. A related approach (see Bayesian control rule) was published in 2010. In 2010 it was also shown that Thompson sampling is instantaneously self-correcting. Asymptotic convergence results for contextual bandits were published in 2011. Thompson Sampling has been widely used in many online learning problems including A/B testing in website design and online advertising, and accelerated learning in decentralized decision making. A Double Thompson Sampling (D-TS) algorithm has been proposed for dueling bandits, a variant of traditional MAB, where feedback comes in the form of pairwise comparison. == Relationship to other approaches == === Probability matching === Probability matching is a decision strategy in which predictions of class membership are proportional to the class base rates. Thus, if in the training set positive examples are observed 60% of the time, and negative examples are observed 40% of the time, the observer using a probability-matching strategy will predict (for unlabeled examples) a class label of "positive" on 60% of instances, and a class label of "negative" on 40% of instances. === Bayesian control rule === A generalization of Thompson sampling to arbitrary dynamical environments and causal structures, known as Bayesian control rule, has been shown to be the optimal solution to the adaptive coding problem with actions and observations. In this formulation, an agent is conceptualized as a mixture over a set of behaviours. As the agent interacts with its environment, it learns the causal properties and adopts the behaviour that minimizes the relative entropy to the behaviour with the best prediction of the environment's behaviour. If these behaviours have been chosen according to the maximum expected utility principle, then the asymptotic behaviour of the Bayesian control rule matches the asymptotic behaviour of the perfectly rational agent. The setup is as follows. Let a 1 , a 2 , … , a T {\displaystyle a_{1},a_{2},\ldots ,a_{T}} be the actions issued by an agent up to time T {\displaystyle T} , and let o 1 , o 2 , … , o T {\displaystyle o_{1},o_{2},\ldots ,o_{T}} be the observations gathered by the agent up to time T {\displaystyle T} . Then, the agent issues the action a T + 1 {\displaystyle a_{T+1}} with probability: P ( a T + 1 | a ^ 1 : T , o 1 : T ) , {\displaystyle P(a_{T+1}|{\hat {a}}_{1:T},o_{1:T}),} where the "hat"-notation a ^ t {\displaystyle {\hat {a}}_{t}} denotes the fact that a t {\displaystyle a_{t}} is a causal intervention (see Causality), and not an ordinary observation. If the agent holds beliefs θ ∈ Θ {\displaystyle \theta \in \Theta } over its behaviors, then the Bayesian control rule becomes P ( a T + 1 | a ^ 1 : T , o 1 : T ) = ∫ Θ P ( a T + 1 | θ , a ^ 1 : T , o 1 : T ) P ( θ | a ^ 1 : T , o 1 : T ) d θ {\displaystyle P(a_{T+1}|{\hat {a}}_{1:T},o_{1:T})=\int _{\Theta }P(a_{T+1}|\theta ,{\hat {a}}_{1:T},o_{1:T})P(\theta |{\hat {a}}_{1:T},o_{1:T})\,d\theta } , where P ( θ | a ^ 1 : T , o 1 : T ) {\displaystyle P(\theta |{\hat {a}}_{1:T},o_{1:T})} is the posterior distribution over the parameter θ {\displaystyle \theta } given actions a 1 : T {\displaystyle a_{1:T}} and observations o 1 : T {\displaystyle o_{1:T}} . In practice, the Bayesian control amounts to sampling, at each time step, a parameter θ ∗ {\displaystyle \theta ^{\ast }} from the posterior distribution P ( θ | a ^ 1 : T , o 1 : T ) {\displaystyle P(\theta |{\hat {a}}_{1:T},o_{1:T})} , where the posterior distribution is computed using Bayes' rule by only considering the (causal) likelihoods of the observations o 1 , o 2 , … , o T {\displaystyle o_{1},o_{2},\ldots ,o_{T}} and ignoring the (causal) likelihoods of the actions a 1 , a 2 , … , a T {\displaystyle a_{1},a_{2},\ldots ,a_{T}} , and then by sampling the action a T + 1 ∗ {\displaystyle a_{T+1}^{\ast }} from the action distribution P ( a T + 1 | θ ∗ , a ^ 1 : T , o 1 : T ) {\displaystyle P(a_{T+1}|\theta ^{\ast },{\hat {a}}_{1:T},o_{1:T})} . === Upper-confidence-bound (UCB) algorithms === Thompson sampling and upper-confidence bound algorithms share a fundamental property that underlies many of their theoretical guarantees. Roughly speaking, both algorithms allocate exploratory effort to actions that might be optimal and are in this sense "optimistic". Leveraging this property, one can translate regret bounds established for UCB algorithms to Bayesian regret bounds for Thompson sampling or unify regret analysis across both these algorithms and many classes of problems.
Herbrand Award
The Herbrand Award for Distinguished Contributions to Automated Reasoning is an award given by the Conference on Automated Deduction (CADE), Inc., (although it predates the formal incorporation of CADE) to honour persons or groups for important contributions to the field of automated deduction. The award is named after the French scientist Jacques Herbrand and given at most once per CADE or International Joint Conference on Automated Reasoning (IJCAR). It comes with a prize of US$1,000. Anyone can be nominated, the award is awarded after a vote among CADE trustees and former recipients, usually with input from the CADE/IJCAR programme committee. == Recipients == Past award recipients are: === 1990s === Larry Wos (1992) Woody Bledsoe (1994) John Alan Robinson (1996) Wu Wenjun (1997) Gérard Huet (1998) Robert S. Boyer and J Strother Moore (1999) === 2000s === William W. McCune (2000) Donald W. Loveland (2001) Mark E. Stickel (2002). Peter B. Andrews (2003) Harald Ganzinger (2004) Martin Davis (2005) Wolfgang Bibel (2006) Alan Bundy (2007) Edmund M. Clarke (2008) Deepak Kapur (2009) === 2010s === David Plaisted (2010) Nachum Dershowitz (2011) Melvin Fitting (2012) C. Greg Nelson (2013) Robert L. Constable (2014) Andrei Voronkov (2015) Zohar Manna and Richard Waldinger (2016) Lawrence C. Paulson (2017) Bruno Buchberger (2018) Nikolaj Bjørner and Leonardo de Moura (2019) === 2020s === Franz Baader (2020) Tobias Nipkow (2021) Natarajan Shankar (2022) Moshe Vardi (2023) Armin Biere (2024) Aart Middeldorp (2025)
Generative AI pornography
Generative AI pornography or simply AI pornography is a digitally created pornography produced through generative artificial intelligence (AI) technologies. Unlike traditional pornography, which involves real actors and cameras, this content is synthesized entirely by AI algorithms. These algorithms, including generative adversarial networks (GANs) and text-to-image models, generate lifelike images, videos, or animations from textual descriptions or datasets. == Functions and production strategies == AI pornography platforms, beyond account creation and social media linking, primarily enable users to generate sexual images through feature selection or text prompting. Users can customize bodies, clothing, and sociodemographic traits, and browse categorized galleries of user‑generated content. Several sites also support short pornographic videos or GIFs and modification tools such as nudifiers, deepfakes, and facemorphing. Platforms often allow fine‑tuning of parameters such as settings, style, or theme, and provide prompt enhancers or suggestions to improve outputs. Users may edit generated images, refine prior prompts, modify others’ work, or upload personal material as a basis, with iterative and collaborative content creation. Some websites additionally host interactive “erobots,” customizable in real time for appearance, personality, memories, speech, and profession, enabling tailored sexual and non‑sexual interactions. Less common features include VR integration, AI porn games, audio or doodle prompts, and consensual replication of individuals with verification. == History == The use of generative AI in the adult industry began in the late 2010s, initially focusing on AI-generated art, music, and visual content. This trend accelerated in 2022 with Stability AI's release of Stable Diffusion (SD), an open-source text-to-image model that enables users to generate images, including NSFW content, from text prompts using the LAION-Aesthetics subset of the LAION-5B dataset. Despite Stability AI's warnings against sexual imagery, SD's public release led to dedicated communities exploring both artistic and explicit content, sparking ethical debates over open-access AI and its use in adult media. By 2020, AI tools had advanced to generate highly realistic adult content, amplifying calls for regulation. === AI-generated influencers === One application of generative AI technology is the creation of AI-generated influencers on platforms such as OnlyFans and Instagram. These AI personas interact with users in ways that can mimic real human engagement, offering an entirely synthetic but convincing experience. While popular among niche audiences, these virtual influencers have prompted discussions about authenticity, consent, and the blurring line between human and AI-generated content, especially in adult entertainment. === The growth of AI porn sites === By 2023, websites dedicated to AI-generated adult content had gained traction, catering to audiences seeking customizable experiences. These platforms allow users to create or view AI-generated pornography tailored to their preferences. These platforms enable users to create or view AI-generated adult content appealing to different preferences through prompts and tags, customizing body type, facial features, and art styles. Tags further refine the output, creating niche and diverse content. Many sites feature extensive image libraries and continuous content feeds, combining personalization with discovery and enhancing user engagement. AI porn sites, therefore, attract those seeking unique or niche experiences, sparking debates on creativity and the ethical boundaries of AI in adult media. == Ethical concerns and misuse == The growth of generative AI pornography has also attracted some cause for criticism. AI technology can be exploited to create non-consensual pornographic material, posing risks similar to those seen with deepfake revenge porn and AI-generated NCII (Non-Consensual Intimate Image). A 2023 analysis found that 98% of deepfake videos online are pornographic, with 99% of the victims being women. Some famous celebrities victims of deepfake include Scarlett Johansson, Taylor Swift, and Maisie Williams. OpenAI is exploring whether NSFW content, such as erotica, can be responsibly generated in age-appropriate contexts while maintaining its ban on deepfakes. This proposal has attracted criticism from child safety campaigners who argue it undermines OpenAI's mission to develop "safe and beneficial" AI. Additionally, the Internet Watch Foundation has raised concerns about AI being used to generate sexual abuse content involving children. === AI-generated non-consensual intimate imagery (AI Undress) === Generative AI have extensively been used to produce pornography images and videos of non-consenting individuals. 404 Media reported a particular AI generated porn bot on Telegram has more than 100,000 monthly users. Alibaba, the Chinese tech company, released an AI video generation model in 2025 called Wan 2.1, which was modified to produce non-consensual pornography. Several US states are taking actions against using deepfake apps and sharing them on the internet. In 2024, San Francisco filed a landmark lawsuit to shut down "undress" apps that allow users to generate non-consensual AI nude images, citing violations of state laws. The case aligns with California's recent legislation—SB 926, SB 942, and SB 981—championed by Senators Aisha Wahab and Josh Becker and signed by Governor Gavin Newsom. These bills aim to protect individuals from AI-generated explicit images by criminalizing non-consensual distribution, mandating disclosures, and empowering victims to report and remove harmful content from platforms. === Differences from deepfake pornography === While both generative AI pornography and deepfake pornography rely on synthetic media, they differ in their methods and ethical considerations. Deepfake pornography typically involves altering existing footage of real individuals, often without their consent, using AI to superimpose faces, undress said persons, or modify scenes. In contrast, generative AI pornography is created using algorithms, producing hyper-realistic content without the need to upload real pictures of people. Hany Farid, digital image analysis expert, also described the difference between "AI porn" and "deepfake porn." == Legality == The legality of generative AI pornography varies widely by jurisdiction and remains an evolving issue. In some countries, laws addressing digital impersonation, obscenity, or deepfake technologies may indirectly apply, particularly when AI-generated content involves the likeness of real individuals without consent. The absence of a physical performer further complicates traditional regulatory frameworks, which are often grounded in performer protection and distribution laws. In the United States, legal responses have primarily focused on non-consensual deepfakes and impersonation. Some states, such as Virginia, California, and Texas, have enacted legislation criminalising the creation or distribution of non-consensual explicit deepfake content. However, there is no comprehensive federal law addressing AI-generated pornography, leaving a patchwork of legal interpretations and enforcement standards across different jurisdictions. According to a 2023 report, South Korea accounts for approximately 53% of global deepfake pornography production. In September 2024, South Korea's National Assembly amended the Act on Special Cases Concerning the Punishment of Sexual Crimes, introducing two significant reforms related to deepfake content. The first criminalises the possession, viewing, purchase, and storage of non-consensual deepfake material, with penalties of up to three years in prison or fines of up to 30 million won (approximately USD 20,000). The second reform specifically addresses the exploitation of minors, establishing that individuals who use deepfakes to threaten or blackmail minors face a minimum of three years' imprisonment, and at least five years if they coerce minors into unwanted acts. In England and Wales the Data (Use and Access) Act 2025 has legislated against the creation, or the request for creation, of intimate images by nudifying software or websites of another person who has not consented to this. However as of January 2026 this has not yet been brought into force.
Touch 'n Go eWallet
Touch 'n Go eWallet is a Malaysian digital wallet and online payment platform, established in Kuala Lumpur, Malaysia, in July 2017 as a joint venture between Touch 'n Go and Ant Financial. It allows users to make payments at over 280,000 merchant touch points via QR code, as well as perform peer-to-peer (P2P) money transfers. Since then, the e-wallet further diversified for users to pay for tolls via RFID or PayDirect, street parking and various online payment spanning e-hailing, car-sharing apps or taxis, various overhead bills; top-up for mobile prepaid or in-game currencies; purchases on e-commerce websites; food delivery; renewing motor insurance and other insurance/takaful plans; and even movie, bus, trains or airline tickets. == Background == Prior to the launch of the e-wallet service, Touch 'n Go provided stored-value physical all-in-one contactless card (namely Touch 'n Go cards or "TnG cards") that users can use to pay for toll fares, public transportation and parking lots as well as purchases in some retail stores. In 1999, Touch 'n Go also markets SmartTag devices that allow road users to pass through certain toll booths without the need to unwind the car window. The high entry cost of the device (around RM 100 each) also meant that only few can enjoy the seamless experience. In 2009, Touch 'n Go partnered with Maxis to launch FastTap, a new mobile payment service that utilised Near-Field Communication (NFC). Maxis customers can make payments by placing the phone near the card readers (that also supports physical bank cards and Touch ’N Go cards). However, the venture featured only one phone model, Nokia 6212, which greatly limited the public reach. In July 2012, Touch 'n Go announced another collaboration with CIMB and Maxis to create similar NFC-based online transaction service that runs on compatible smartphones. Touch 'n Go Wallet was launched in February 2017 as an QR code-based e-wallet application, to compete with Samsung Pay that utilizes NFC modules. In the controlled pilot test in Taman Tun Dr Ismail, the correspondents can experience basic functionalities (prepaid mobile service reload, bills payment, movie tickets and flight tickets purchase, transfer of money with another user, and payments at participating stores and restaurants). While the deployed version of the app was generally well-received, the existing process to transfer the balance to the physical TnG card stored value from the app garnered unanimous backlash. Test groups felt that the need to head to a self-service terminal named "Pick Up Device" in person within 24 hours for completion, along with the failure to do so (the balance would be credited back to the wallet after 24 hours), was not divulged clearly and also defeated the purpose of convenience, not to mention there were only 2 such terminals. The feature was eventually suspended. On 15 November 2017, Touch 'n Go was granted permission by the Central Bank of Malaysia to form a joint venture with Ant Financial, a Chinese-based financial company that operates Alipay. The partnership allowed the local e-wallet to learn from and build upon the operational model pioneered by Alipay. In June 2018, it was reported that Touch 'n Go was pilot testing the uses of the Touch 'n Go eWallet in Rapid Transit, as the ticketing system was enabled on the Kelana Jaya line in the Klang Valley. Pilot testing only applied to stations in Kelana Jaya, KL Gateway–Universiti, Kerinchi, KL Sentral, Dang Wangi, KLCC, and Ampang Park. The test was reported to be successful in February 2020 and was planned to be fully deployed on the LRT and MRT. Due to unforeseen circumstances, this feature did not come into fruition, the app merely adds in-app purchase of monthly concession cards called "My50". In August 2018, Touch 'n Go announced that selected drivers may experience first-hand a new RFID-based payment (later rebranded as "myRFID") that serves to replace SmartTag devices on closed toll roads with during pilot testing phase commencing on 3 September 2018. On 2 November 2018, participation in the ongoing pilot programme was expanded, allowing more drivers to sign up ahead of the public rollout of the RFID system. During the same period, Touch 'n Go has discontinued the sales of SmartTAG devices in favor of the RFID-based payment system. Initially, the installation of the RFID chip onto the car could only be done by Touch 'n Go staff at the RFID fitment centers, at no cost. As the pilot testing concluded on 15 February 2020, a self-installation kit are being offered to the public on Lazada and Shopee. Support for taxi-hailing mobile apps was added in November 2018 when Touch 'n Go partnered with EzCab and Public Cab, allowing users to make payments via QR code. This was later expanded to support MULA on 7 January 2020, and later MyCar on 4 April 2020. Touch 'n Go eWallet was also the first eWallet to convert Kuala Lumpur's most famous Ramadan bazaar in Kampong Bahru into "Kampong Kashless", a venue that can accept cashless QR payments. It welcomed more than 250,000 Malaysians including local celebrities and government officials. On 1 October 2019, some e-commerce websites owned by the Alibaba Group (TMall and Taobao) began to support Touch 'n Go eWallet payments, Lazada joined the list on 29 October 2019. Touch 'n Go eWallet was one of the three e-wallet services in Malaysia (the other being Boost and GrabPay) that was eligible for its users to receive an RM 30 credit in conjunction of E-Tunai Rakyat program under the Budget 2020 plan, that further normalizes adoption of cashless and mobile payment among Malaysians. Unlike Boost and GrabPay, whose P2P transfers were completely disabled until users have exhausted the RM 30 first, Touch 'n Go eWallet did not impose such measures. in 2020, Touch 'n Go eWallet joined DuitNow, an electronic transaction ecosystem in Malaysia which allows the funds from Touch 'n Go eWallet to be transferred to other competing services and vice versa, by implementing a standard DuitNow QR code deisgn. Japan become the first country outside Malaysia to support Touch 'n Go eWallet payment via Alipay Connect. During the COVID-19 pandemic and the enforcement of the movement control order, use of eWallets (including Touch 'n Go eWallet) increased tremendously among citizens due to its contactless nature of the payment and increased take-out orders at home; which in turn helped small and medium-sized enterprises to thrive. Touch 'n Go eWallet launched its loyalty programme – The Goal Hunter – in October 2020 where on monthly basis, users collect stamps by paying with the app in exchange for rewards that include lucky draws and other vouchers. == Services == Touch 'n Go eWallet app is available for download on both Google Play and Apple Appstore. It utilizes QR code technology for local in-store payments. The Touch 'n Go eWallet app also diversifies payment types, including but not limited to Utility bills Purchase of motor insurance policy Pay Later facility Prepaid reload and Postpaid payment to telecommunications companies loan repayments for courts, MBSJ payments, zakat and PTPTN payment for car parking P2P transfer airline ticket bookings; movie tickets from TGV Cinemas RFID refuelling at Shell stations (defunct after Shell launched its own payment app in 2024) User can reload the eWallet credit by setting up auto-reload, purchasing reload pins from convenience stores (such as 7-Eleven, KK Super Mart, MyNews, Family Mart etc.), reloading by FPX and credit/debit card. The PayDirect feature allows users to link their physical Touch 'n Go cards into the eWallet, where the toll fare can be debited from the eWallet balance when flashing the card near the sensor. In the circumstance of insufficient balance in the app, the toll fare will be deducted from the physical card's balance instead. This also conveniently allows users to view the card's remaining balance. Touch 'n Go eWallet is the first and only eWallet to offer a money-back guarantee when an unauthorised transaction is made on the user’s eWallet account, subject to Terms & Conditions. Payment via QR code scanning, including Touch 'n Go eWallet, becomes a norm in most of the shops/restaurants across Malaysia, including roadside hawkers/stall owners and automatic vending machines. The merchants usually display their owner's individual QR or Business account that they can apply for in-app. The popularity attributes to the low merchant onboarding cost (Unlike NFC payment and debit/credit card that requires purchase or rental of a payment terminal device at a yearly fee.) The app is also one of the few ewallet that supports bidirectional liquidity (alongside MAE developed by Maybank), where funds can be transferred two-way with bank accounts. This is not possible with the other major ewallets (GrabPay, Boost, ShopeePay etc.) where the money that is reloaded to the wallet cannot be transferred to another bank account, unless through manual req
T-norm fuzzy logics
T-norm fuzzy logics are a family of non-classical logics, informally delimited by having a semantics that takes the real unit interval [0, 1] for the system of truth values and functions called t-norms for permissible interpretations of conjunction. They are mainly used in applied fuzzy logic and fuzzy set theory as a theoretical basis for approximate reasoning. T-norm fuzzy logics belong in broader classes of fuzzy logics and many-valued logics. In order to generate a well-behaved implication, the t-norms are usually required to be left-continuous; logics of left-continuous t-norms further belong in the class of substructural logics, among which they are marked with the validity of the law of prelinearity, (A → B) ∨ (B → A). Both propositional and first-order (or higher-order) t-norm fuzzy logics, as well as their expansions by modal and other operators, are studied. Logics that restrict the t-norm semantics to a subset of the real unit interval (for example, finitely valued Łukasiewicz logics) are usually included in the class as well. Important examples of t-norm fuzzy logics are monoidal t-norm logic (MTL) of all left-continuous t-norms, basic logic (BL) of all continuous t-norms, product fuzzy logic of the product t-norm, or the nilpotent minimum logic of the nilpotent minimum t-norm. Some independently motivated logics belong among t-norm fuzzy logics, too, for example Łukasiewicz logic (which is the logic of the Łukasiewicz t-norm) or Gödel–Dummett logic (which is the logic of the minimum t-norm). == Motivation == As members of the family of fuzzy logics, t-norm fuzzy logics primarily aim at generalizing classical two-valued logic by admitting intermediary truth values between 1 (truth) and 0 (falsity) representing degrees of truth of propositions. The degrees are assumed to be real numbers from the unit interval [0, 1]. In propositional t-norm fuzzy logics, propositional connectives are stipulated to be truth-functional, that is, the truth value of a complex proposition formed by a propositional connective from some constituent propositions is a function (called the truth function of the connective) of the truth values of the constituent propositions. The truth functions operate on the set of truth degrees (in the standard semantics, on the [0, 1] interval); thus the truth function of an n-ary propositional connective c is a function Fc: [0, 1]n → [0, 1]. Truth functions generalize truth tables of propositional connectives known from classical logic to operate on the larger system of truth values. T-norm fuzzy logics impose certain natural constraints on the truth function of conjunction. The truth function ∗ : [ 0 , 1 ] 2 → [ 0 , 1 ] {\displaystyle \colon [0,1]^{2}\to [0,1]} of conjunction is assumed to satisfy the following conditions: Commutativity, that is, x ∗ y = y ∗ x {\displaystyle xy=yx} for all x and y in [0, 1]. This expresses the assumption that the order of fuzzy propositions is immaterial in conjunction, even if intermediary truth degrees are admitted. Associativity, that is, ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ) {\displaystyle (xy)z=x(yz)} for all x, y, and z in [0, 1]. This expresses the assumption that the order of performing conjunction is immaterial, even if intermediary truth degrees are admitted. Monotony, that is, if x ≤ y {\displaystyle x\leq y} then x ∗ z ≤ y ∗ z {\displaystyle xz\leq yz} for all x, y, and z in [0, 1]. This expresses the assumption that increasing the truth degree of a conjunct should not decrease the truth degree of the conjunction. Neutrality of 1, that is, 1 ∗ x = x {\displaystyle 1x=x} for all x in [0, 1]. This assumption corresponds to regarding the truth degree 1 as full truth, conjunction with which does not decrease the truth value of the other conjunct. Together with the previous conditions this condition ensures that also 0 ∗ x = 0 {\displaystyle 0x=0} for all x in [0, 1], which corresponds to regarding the truth degree 0 as full falsity, conjunction with which is always fully false. Continuity of the function ∗ {\displaystyle } (the previous conditions reduce this requirement to the continuity in either argument). Informally this expresses the assumption that microscopic changes of the truth degrees of conjuncts should not result in a macroscopic change of the truth degree of their conjunction. This condition, among other things, ensures a good behavior of (residual) implication derived from conjunction; to ensure the good behavior, however, left-continuity (in either argument) of the function ∗ {\displaystyle } is sufficient. In general t-norm fuzzy logics, therefore, only left-continuity of ∗ {\displaystyle } is required, which expresses the assumption that a microscopic decrease of the truth degree of a conjunct should not macroscopically decrease the truth degree of conjunction. These assumptions make the truth function of conjunction a left-continuous t-norm, which explains the name of the family of fuzzy logics (t-norm based). Particular logics of the family can make further assumptions about the behavior of conjunction (for example, Gödel–Dummett logic requires its idempotence) or other connectives (for example, the logic IMTL (involutive monoidal t-norm logic) requires the involutiveness of negation). All left-continuous t-norms ∗ {\displaystyle } have a unique residuum, that is, a binary function ⇒ {\displaystyle \Rightarrow } such that for all x, y, and z in [0, 1], x ∗ y ≤ z {\displaystyle xy\leq z} if and only if x ≤ y ⇒ z . {\displaystyle x\leq y\Rightarrow z.} The residuum of a left-continuous t-norm can explicitly be defined as ( x ⇒ y ) = sup { z ∣ z ∗ x ≤ y } . {\displaystyle (x\Rightarrow y)=\sup\{z\mid zx\leq y\}.} This ensures that the residuum is the pointwise largest function such that for all x and y, x ∗ ( x ⇒ y ) ≤ y . {\displaystyle x(x\Rightarrow y)\leq y.} The latter can be interpreted as a fuzzy version of the modus ponens rule of inference. The residuum of a left-continuous t-norm thus can be characterized as the weakest function that makes the fuzzy modus ponens valid, which makes it a suitable truth function for implication in fuzzy logic. Left-continuity of the t-norm is the necessary and sufficient condition for this relationship between a t-norm conjunction and its residual implication to hold. Truth functions of further propositional connectives can be defined by means of the t-norm and its residuum, for instance the residual negation ¬ x = ( x ⇒ 0 ) {\displaystyle \neg x=(x\Rightarrow 0)} or bi-residual equivalence x ⇔ y = ( x ⇒ y ) ∗ ( y ⇒ x ) . {\displaystyle x\Leftrightarrow y=(x\Rightarrow y)(y\Rightarrow x).} Truth functions of propositional connectives may also be introduced by additional definitions: the most usual ones are the minimum (which plays a role of another conjunctive connective), the maximum (which plays a role of a disjunctive connective), or the Baaz Delta operator, defined in [0, 1] as Δ x = 1 {\displaystyle \Delta x=1} if x = 1 {\displaystyle x=1} and Δ x = 0 {\displaystyle \Delta x=0} otherwise. In this way, a left-continuous t-norm, its residuum, and the truth functions of additional propositional connectives determine the truth values of complex propositional formulae in [0, 1]. Formulae that always evaluate to 1 are called tautologies with respect to the given left-continuous t-norm ∗ , {\displaystyle ,} or ∗ - {\displaystyle {\mbox{-}}} tautologies. The set of all ∗ - {\displaystyle {\mbox{-}}} tautologies is called the logic of the t-norm ∗ , {\displaystyle ,} as these formulae represent the laws of fuzzy logic (determined by the t-norm) that hold (to degree 1) regardless of the truth degrees of atomic formulae. Some formulae are tautologies with respect to a larger class of left-continuous t-norms; the set of such formulae is called the logic of the class. Important t-norm logics are the logics of particular t-norms or classes of t-norms, for example: Łukasiewicz logic is the logic of the Łukasiewicz t-norm x ∗ y = max ( x + y − 1 , 0 ) {\displaystyle xy=\max(x+y-1,0)} Gödel–Dummett logic is the logic of the minimum t-norm x ∗ y = min ( x , y ) {\displaystyle xy=\min(x,y)} Product fuzzy logic is the logic of the product t-norm x ∗ y = x ⋅ y {\displaystyle xy=x\cdot y} Monoidal t-norm logic MTL is the logic of (the class of) all left-continuous t-norms Basic fuzzy logic BL is the logic of (the class of) all continuous t-norms It turns out that many logics of particular t-norms and classes of t-norms are axiomatizable. The completeness theorem of the axiomatic system with respect to the corresponding t-norm semantics on [0, 1] is then called the standard completeness of the logic. Besides the standard real-valued semantics on [0, 1], the logics are sound and complete with respect to general algebraic semantics, formed by suitable classes of prelinear commutative bounded integral residuated lattices. == History == Some particular t-norm fuzzy logics have been introduced and investigated long before the family was re