Language-theoretic security, or LangSec, is an approach to software security that focuses on input handling, complexity, and program design as strategies to improve the verifiability of computer programs. It was introduced in 2005 by Robert J. Hansen and Meredith L. Patterson at BlackHat and in 2011 by Len Sassaman and Patterson. It aims to create a formal description of which software is likely to have security vulnerabilities of particular classes, and why. It considers programs to have an inherent parser component, whether or not explicit, composed of that part of the program which operates on external input before that input is fully parsed. A central hypothesis of language-theoretic security is that vulnerabilities in software increase according to the computational power of the notional input-accepting automaton equivalent to this parser, using the definitions of automata theory. The lower bound on this computational power is the input language complexity of the program. The extent to which reducing this complexity is possible is a function of the specification of the communication protocol or file format the program takes as input. == Parsing as a security mechanism == The behaviour of a program is defined with reference to its expected input. Unexpected input being used by a program is a factor in numerous security bugs, including the so-called Android master key vulnerability (CVE-2013-4787), because accepting unexpected input renders the program's specification ambiguous. In that instance, the unexpected ambiguity came in the form of a ZIP file with duplicate filenames. If a program fully parses its input and only acts on input that unambiguously meets the specification, it follows that the program will avoid these types of vulnerabilities. This is an intentional inversion of the Postel principle. Accepting only unambiguous and valid input is a more formal requirement than input validation or sanitization, and narrows the number of possible but unanticipated program states that can be induced in an application via user input. Conversely, failure to do this is associated with security vulnerabilities. Input sanitization in particular is held to be an inadequate approach to avoiding malicious input because it inherently ignores context-sensitive properties of the input; it can therefore result in paradoxical effects, such as sanitization code activating otherwise inert cross-site scripting payloads in browsers. === Parser differentials === If the language of accepted program input is sufficiently simple, it is possible to verify that two implementations parse the same input language consistently. This is advantageous because it shows no parser differential exists between the two implementations. The requisite level of simplicity is theoretically that for which there is a solution to the equivalence problem. If the two parsers involved in CVE-2013-4787 were equivalent - that is, if they rendered the same output state given the same input state - the vulnerability could not have existed. One strategy for doing this is to publish machine-readable specifications of a format or protocol, and then use a parser generator to generate the parser code. An example of a parser generator built for this purpose is DaeDaLus. The combination of Lex with any of GNU Bison, ANTLR, or Yacc also accomplishes this. However, many parser generators allow the mixing of general purpose code with the parsing definitions, which weakens the guarantees provided by parsing. === Analysis of injection attacks === Injection attacks are generally the result of differences between the serializer (or "unparser") and the corresponding parser at a layer boundary in a system; therefore, they are a special case of parser differentials. In a SQL injection attack, for example, an attacker is able to cause the application with which they are interacting to serialize a SQL query that has different semantics than intended. In the simplest case where the payload ends a string and adds new code, the payload has crossed the code-data boundary in SQL. In language-theoretic security, this is treated as a bug in the serializer of the SQL query, which should instead be written in a way that constrains its possible outputs to those within the scope of the intended query. === Parser combinators === If a parser generator is not used, it is still possible to avoid implementation bugs by using parser combinator such as Nom to implement the parser code. This has the drawback of relying on a programmer correctly translating the specification into the language of the parser generator library, though this task is still less error-prone than hand-coding a parser. == Input format complexity == Complexity in computer programs is associated with security vulnerabilities. Within the domain of language-theoretic security, complexity is described with reference to the computational power of the abstract machine necessary to implement the program, or more particularly, to implement the parser for its input language. This complexity describes whether it is possible to show that there is no unintended or undesired functionality in the program which might be exploitable by an attacker. To be bounded in complexity, the program's input must be well-defined both in terms of form and of semantics. === Weird machines === A weird machine is a model of computation in a program that exists in parallel with, but is distinct from, the intended abstract model of computation in that program. Some classes of weird machine arise from the multi-layered nature of computer programs, or the context in which the programs run; others result from the unanticipated functionality a program has due to its complexity or to software bugs. The more complex the computation model of a program, the more likely it is to implement a weird machine. Depending on context, the weird machine may or may not be concretely useful for an attacker. Since the space of weird machines in the context of some program is the universe of all possible states that are not within the program's intended states, many exploited states including remote code execution and injection attacks belong to the domain of weird machines. A reduction in weird machines is therefore a likely correlate with reduced program vulnerability. === SafeDocs project === SafeDocs is a DARPA project undertaken in 2018 to take existing file formats, create safer subsets of them, and develop programming tools to work for the safer formats. The initial test case for this was PDF. The purpose of creating safer subsets in this case is to lower the minimum bound on parser complexity so that it becomes possible to create tools that will generate correct, normative parsers for them. == Relation to programming languages == The analytic framework of language-theoretic security assumes programs to be virtual machines that execute their input. A document that is read by an application is in this sense a form of machine code, in a generalization of the data as code idea, following the automata theory description of parsers. === Type-safe programming languages === Parsing input and serializing output are operations that consume one data type and emit another. A programming language can therefore check that data is correctly parsed and contains the expected structure by checking data types, and correct serializing (or unparsing) can be implemented as operations on the data types that are relevant to the program's output. This approach can be used to show that the recognizer and unparser patterns have been implemented. It is also possible to implement type checking across a distributed system to enforce parsing and unparsing of the expected structures and to verify that the assumptions made in designing the compositional properties of a distributed system have been followed. === Memory-safe programming languages === In the general case, spatial memory correctness is undecidable. If any proof of spatial memory correctness is to be made, it is therefore necessary to bound the complexity of the code. Interpreted languages such as Java and Python effectively accomplish this via runtime bounds checking, and frameworks for runtime bounds checking also exist for C. The effect of these strategies for spatial memory correctness are to create a halt state in place of a spatial memory correctness violation; therefore, it can be shown that the program will not violate spatial memory correctness, but in exchange, it cannot be shown in the general case that programs will not have runtime bounds checking exceptions. Some programming languages, such as Rust, accomplish this using borrow checking. The borrow checker acts to assure spatial memory correctness by compile-time reference counting. Code for which spatial memory correctness cannot be shown to not be violated therefore does not compile, inherently limiting the complexity of the spatial memory correctness of the program to what is decidable. Thi
Kolmogorov–Arnold Networks
Kolmogorov–Arnold Networks (KANs) are a type of artificial neural network architecture inspired by the Kolmogorov–Arnold representation theorem, also known as the superposition theorem. Unlike traditional multilayer perceptrons (MLPs), which rely on fixed activation functions and linear weights, KANs replace each weight with a learnable univariate function, often represented using splines. == History == KANs (Kolmogorov–Arnold Networks) were proposed by Liu et al. (2024) as a generalization of the Kolmogorov–Arnold representation theorem (KART), aiming to outperform MLPs in small-scale AI and scientific tasks. Before KANs, numerous studies explored KART's connections to neural networks or used it as a basis for designing new network architectures. In the 1980s and 1990s, early research applied KART to neural network design. Kůrková et al. (1992), Hecht-Nielsen (1987), and Nees (1994) established theoretical foundations for multilayer networks based on KART. Igelnik et al. (2003) introduced the Kolmogorov Spline Network using cubic splines to model complex functions. Sprecher (1996, 1997) introduced numerical methods for building network layers, while Nakamura et al. (1993) created activation functions with guaranteed approximation accuracy. These works linked KART's theoretical potential with practical neural network implementation. KART has also been used in other computational and theoretical fields. Coppejans (2004) developed nonparametric regression estimators using B-splines, Bryant (2008) applied it to high-dimensional image tasks, Liu (2015) investigated theoretical applications in optimal transport and image encryption, and more recently, Polar and Poluektov (2021) used Urysohn operators for efficient KART construction, while Fakhoury et al. (2022) introduced ExSpliNet, integrating KART with probabilistic trees and multivariate B-splines for improved function approximation. == Architecture == KANs are based on the Kolmogorov–Arnold representation theorem, which was linked to the 13th Hilbert problem. Given x = ( x 1 , x 2 , … , x n ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{n})} consisting of n variables, a multivariate continuous function f ( x ) {\displaystyle f(x)} can be represented as: f ( x ) = f ( x 1 , … , x n ) = ∑ q = 1 2 n + 1 Φ q ( ∑ p = 1 n φ q , p ( x p ) ) {\displaystyle f(x)=f(x_{1},\dots ,x_{n})=\sum _{q=1}^{2n+1}\Phi _{q}\left(\sum _{p=1}^{n}\varphi _{q,p}(x_{p})\right)} (1) This formulation contains two nested summations: an outer and an inner sum. The outer sum ∑ q = 1 2 n + 1 {\displaystyle \sum _{q=1}^{2n+1}} aggregates 2 n + 1 {\displaystyle 2n+1} terms, each involving a function Φ q : R → R {\displaystyle \Phi _{q}:\mathbb {R} \to \mathbb {R} } . The inner sum ∑ p = 1 n {\displaystyle \sum _{p=1}^{n}} computes n terms for each q, where each term φ q , p : [ 0 , 1 ] → R {\displaystyle \varphi _{q,p}:[0,1]\to \mathbb {R} } is a continuous function of the single variable x p {\displaystyle x_{p}} . The inner continuous functions φ q , p {\displaystyle \varphi _{q,p}} are universal, independent of f {\displaystyle f} , while the outer functions Φ q {\displaystyle \Phi _{q}} depend on the specific function f {\displaystyle f} being represented. The representation (1) holds for all multivariate functions f {\displaystyle f} as proved in . If f {\displaystyle f} is continuous, then the outer functions Φ q {\displaystyle \Phi _{q}} are continuous; if f {\displaystyle f} is discontinuous, then the corresponding Φ q {\displaystyle \Phi _{q}} are generally discontinuous, while the inner functions φ q , p {\displaystyle \varphi _{q,p}} remain the same universal functions. Liu et al. proposed the name KAN. A general KAN network consisting of L layers takes x to generate the output as: K A N ( x ) = ( Φ L − 1 ∘ Φ L − 2 ∘ ⋯ ∘ Φ 1 ∘ Φ 0 ) x {\displaystyle \mathrm {KAN} (x)=(\Phi ^{L-1}\circ \Phi ^{L-2}\circ \cdots \circ \Phi ^{1}\circ \Phi ^{0})x} (3) Here, Φ l {\displaystyle \Phi ^{l}} is the function matrix of the l-th KAN layer or a set of pre-activations. Let i denote the neuron of the l-th layer and j the neuron of the (l+1)-th layer. The activation function φ j , i l {\displaystyle \varphi _{j,i}^{l}} connects (l, i) to (l+1, j): φ j , i l , l = 0 , … , L − 1 , i = 1 , … , n l , j = 1 , … , n l + 1 {\displaystyle \varphi _{j,i}^{l},\quad l=0,\dots ,L-1,\;i=1,\dots ,n_{l},\;j=1,\dots ,n_{l+1}} (4) where nl is the number of nodes of the l-th layer. Thus, the function matrix Φ l {\displaystyle \Phi ^{l}} can be represented as an n l + 1 × n l {\displaystyle n_{l+1}\times n_{l}} matrix of activations: x l + 1 = ( φ 1 , 1 l ( ⋅ ) φ 1 , 2 l ( ⋅ ) ⋯ φ 1 , n l l ( ⋅ ) φ 2 , 1 l ( ⋅ ) φ 2 , 2 l ( ⋅ ) ⋯ φ 2 , n l l ( ⋅ ) ⋮ ⋮ ⋱ ⋮ φ n l + 1 , 1 l ( ⋅ ) φ n l + 1 , 2 l ( ⋅ ) ⋯ φ n l + 1 , n l l ( ⋅ ) ) x l {\displaystyle x^{l+1}={\begin{pmatrix}\varphi _{1,1}^{l}(\cdot )&\varphi _{1,2}^{l}(\cdot )&\cdots &\varphi _{1,n_{l}}^{l}(\cdot )\\\varphi _{2,1}^{l}(\cdot )&\varphi _{2,2}^{l}(\cdot )&\cdots &\varphi _{2,n_{l}}^{l}(\cdot )\\\vdots &\vdots &\ddots &\vdots \\\varphi _{n_{l+1},1}^{l}(\cdot )&\varphi _{n_{l+1},2}^{l}(\cdot )&\cdots &\varphi _{n_{l+1},n_{l}}^{l}(\cdot )\end{pmatrix}}x^{l}} == Implementations == To make the KAN layers optimizable, the inner function is formed by the combination of spline and basic functions as the formula: φ ( x ) = w b b ( x ) + w s spline ( x ) {\displaystyle \varphi (x)=w_{b}\,b(x)+w_{s}\,{\text{spline}}(x)} where b ( x ) {\displaystyle b(x)} is the basic function, usually defined as s i l u ( x ) = x / ( 1 + e x ) {\displaystyle silu(x)=x/(1+e^{x})} and w b {\displaystyle w_{b}} is the base weight matrix. Also, w s {\displaystyle w_{s}} is the spline weight matrix and spline ( x ) {\displaystyle {\text{spline}}(x)} is the spline function. The spline function can be a sum of B-splines. spline ( x ) = ∑ i c i B i ( x ) {\displaystyle {\text{spline}}(x)=\sum _{i}c_{i}B_{i}(x)} Many studies suggested to use other polynomial and curve functions instead of B-spline to create new KAN variants. == Functions used == The choice of functional basis strongly influences the performance of KANs. Common function families include: B-splines: Provide locality, smoothness, and interpretability; they are the most widely used in current implementations. RBFs (include Gaussian RBFs): Capture localized features in data and are effective in approximating functions with non-linear or clustered structures. Chebyshev polynomials: Offer efficient approximation with minimized error in the maximum norm, making them useful for stable function representation. Rational function: Useful for approximating functions with singularities or sharp variations, as they can model asymptotic behavior better than polynomials. Fourier series: Capture periodic patterns effectively and are particularly useful in domains such as physics-informed machine learning. Wavelet functions (DoG, Mexican hat, Morlet, and Shannon): Used for feature extraction as they can capture both high-frequency and low-frequency data components. Piecewise linear functions: Provide efficient approximation for multivariate functions in KANs. == Usage == In some modern neural architectures like convolutional neural networks (CNNs), recurrent neural networks (RNNs), and Transformers, KANs are typically used as drop-in substitutes for MLP layers. Despite KANs' general-purpose design, researchers have created and used them for a number of tasks: Scientific machine learning (SciML): Function fitting, partial differential equations (PDEs) and physical/mathematical laws. Continual learning: KANs better preserve previously learned information during incremental updates, avoiding catastrophic forgetting due to the locality of spline adjustments. Graph neural networks: Extensions such as Kolmogorov–Arnold Graph Neural Networks (KA-GNNs) integrate KAN modules into message-passing architectures, showing improvements in molecular property prediction tasks. Sensor data processing: Kolmogorov–Arnold Networks (KANs) have recently been applied to sensor data processing due to their ability to model complex nonlinear relationships with relatively few parameters and improved interpretability compared to conventional multilayer perceptrons. Applications include industrial soft sensors, biomedical signal analysis, remote sensing, and environmental monitoring systems. == Drawbacks == KANs can be computationally intensive and require a large number of parameters due to their use of polynomial functions to capture data.
Lymphater's Formula
"Lymphater's Formula" (Polish: "Formula Lymphatera") is a 1961 science fiction short story by Polish writer Stanisław Lem. It is a story of a "mad scientist", mathematician Ammon Lymphater, who invents an artificial intelligence, and then he realizes that it is capable of rendering the humankind obsolete. It was first published in the 1961 collection Księga robotów (Book of Robots) with the pre-annotation "from the memoirs of Ijon Tichy". The story was never republished with this pre-annotation, and nothing in the novel gives any indication at Ijon Tichy. Piotr Krywak tried to figure out possible explanations for this, apart from a typographical error. == Plot == Ammon Lymphater became interested in the emerging science of cybernetics and information theory, and started studying the works of an animal brain, the ant's brain in particular. He took note that the inherited knowledge is an evolutionary advantage somehow not exploited in full by the evolution. Eventually he came to a conclusion that only by pure biological restrictions that adaptive abilities of insects were stopped in their tracks by the evolution. He went on further wondering whether the ants have an ability to apriori knowledge, i.e., knowledge neither inherited nor learned. He decided to consult a famous myrmecologist, who told him about a rare ant species Acanthis Rubra Willinsoniana with an exceptionally high adaptability. Eventually Lymphater devised and constructed "It" capable of instant precognition of everything within "Its" rapidly expanding range of perception. From "It" Lymphater learns that the humanity is not the "crown of evolution", but rather evolution's tool to create "It", because the evolution could not create "It" directly (confirming Lymphater's reasoning about ants). Realizing that the Superentity "It" renders the human civilization redundant and obsolete, Lymphater destroys "It". "It" already knew Lymphater's intentions, but was not worried, knowing that sooner or later someone else will create "It" again and again. "It" was only the first variant of Lymphater's formula and the second variant is possible. Lyphater wonders whether the second one would be capable to create the third stage of the evolution which would amount to an artificial God. == Publication history == It was translated in Russian (as "Формула Лимфатера") in 1963, in Hungarian (as "Lymphater utolsó képlete") in 1966, and in Bulgarian (as "Формулата на Лимфатер" by Георги Димитров Георгиев) in 1969. In 1973 an audiobook was released in German (as "Die lymphatersche Formel"), narrated by Martin Held. It was also republished (and translated) in some other collections of Lem's short stories.
List of artificial intelligence artists
Many notable artificial intelligence artists have created a wide variety of artificial intelligence art from the 1960s to today. These include: == 20th century == Harold Cohen, active from 1960s to 2010s. Cohen's work is primarily with AARON, a series of computer programs that autonomously create original images. Eric Millikin, active from 1980s to present. Millikin's work includes AI-generated virtual reality, video art, poetry, music, and performance art, on topics such as animal rights, climate change, anti-racism, witchcraft, and the occult. Karl Sims, active from 1980s to present. Sims is best known for using particle systems and artificial life in computer animation. == 21st century == Refik Anadol, active from 2010s to present. Anadol's work includes video installations based on generative algorithms with artificial intelligence. Sougwen Chung, active from 2010s to present. Chung's work includes performances with a robotic arm that uses AI to attempt to draw in a manner similar to Chung. Stephanie Dinkins, active from 2010s to present. Dinkins' work includes recordings of conversations with an artificially intelligent robot that resembles a black woman, discussing topics such as race and the nature of being. Jake Elwes, active from 2010s to present. Their practice is the exploration of artificial intelligence, queer theory and technical biases. Libby Heaney, active from 2010s to present. Heaney's practice includes work with chatbots. Mario Klingemann, active from 2010s to present. Klingemann's works examine creativity, culture, and perception through machine learning and artificial intelligence. Mauro Martino, active from 2010s to present. Martino's work includes design, data visualization and infographics. Trevor Paglen, active from 2000s to present. Paglen's practice includes work in photography and geography, on topics like mass surveillance and data collection. Anna Ridler, active from 2010s to present. Ridler works with collections of information, including self-generated data sets, often working with floral photography.
Shy Girl
Shy Girl is a horror novel initially self-published in February 2025 by Mia Ballard. Publishing rights for the book were acquired by Hachette Book Group, which released the book in the United Kingdom in November 2025 and planned to publish it in the United States in 2026. Its US release was cancelled and its UK release was discontinued after it faced accusations of being created with generative AI. Ballard denied having personally used AI in the book's writing, claiming that a freelance editor had introduced AI-generated changes. She also stated that she would take legal action against the editor. == Premise == The novel follows Gia, a depressed woman with obsessive–compulsive disorder, who encounters a mysterious man named Nathan while looking for a sugar daddy to ease her financial troubles. Nathan offers to erase all of Gia's debts in exchange for her agreeing to live as his pet. Living like an animal convinces her that she is becoming an animal, making her behave like one. == Publication and cancellation == Shy Girl was first self-published online by Mia Ballard in February 2025. Marketing material described the book as a "buzzy BookTok sensation" and "bloody and unforgiving". The self-published edition of the book was highly successful and had over 4,900 ratings on Goodreads and an average score of 3.52 stars. In an interview, Ballard described her writing style as lyrical, feverish, and introspective, and stated she was more interested in "what it feels like to live inside a body" than in plot-driven storylines. Publishing rights were acquired by Hachette Book Group and it was published by its Wildfire imprint in the United Kingdom in November 2025. By March 2026, the book had sold 1,800 copies in the United Kingdom. A US release was planned for 2026 by the imprint Orbit Books. After the British publication, critics and readers began to make claims that the book appeared to have been written by generative AI. A January 2026 post on Reddit claimed that the book had many of the hallmarks of having been written with a large language model, and stated that it was "repulsive" that the book was accepted by Hachette. A two-and-a-half-hour video essay covering the book, titled "i'm pretty sure this book is ai slop", received 1.2 million views on YouTube by March 2026. In response, Hachette Book Group announced in March 2026 that it would cancel the book's US publication and discontinue its UK publication. It told The Wall Street Journal that it had made "a lengthy investigation" before deciding to cancel the book. Ballard told The New York Times that she had not used AI when writing the book, but that AI-generated elements were added by a freelance editor without her knowledge. She also stated that she could not elaborate on her claim because she was pursuing legal action against the editor. Writer Andrea Bartz opined that the situation "raises many concerns about trust, authenticity and publishing's readiness for a new, A.I.-assisted world", but that "readers made it abundantly clear they want books by humans, not machines".
International Conference on Language Resources and Evaluation
The International Conference on Language Resources and Evaluation is an international conference organised by the ELRA Language Resources Association every other year (on even years) with the support of institutions and organisations involved in Natural language processing. The series of LREC conferences was launched in Granada in 1998. == History of conferences == The survey of the LREC conferences over the period 1998-2013 was presented during the 2014 conference in Reykjavik as a closing session. It appears that the number of papers and signatures is increasing over time. The average number of authors per paper is higher as well. The percentage of new authors is between 68% and 78%. The distribution between male (65%) and female (35%) authors is stable over time. The most frequent technical term is "annotation", then comes "part-of-speech". == The LRE Map == The LRE Map was introduced at LREC 2010 and is now a regular feature of the LREC submission process for both the conference papers and the workshop papers. At the submission stage, the authors are asked to provide some basic information about all the resources (in a broad sense, i.e. including tools, standards and evaluation packages), either used or created, described in their papers. All these descriptors are then gathered in a global matrix called the LRE Map. This feature has been extended to several other conferences.
Computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program. Attempts have also been made in the area of artificial intelligence research to create smaller, explicit, new proofs of mathematical theorems from the bottom up using automated reasoning techniques such as heuristic search. Such automated theorem provers have proved a number of new results and found new proofs for known theorems. Additionally, interactive proof assistants allow mathematicians to develop human-readable proofs which are nonetheless formally verified for correctness. Since these proofs are generally human-surveyable (albeit with difficulty, as with the proof of the Robbins conjecture) they do not share the controversial implications of computer-aided proofs-by-exhaustion. == Methods == One method for using computers in mathematical proofs is by means of so-called validated numerics or rigorous numerics. This means computing numerically yet with mathematical rigour. One uses set-valued arithmetic and inclusion principle in order to ensure that the set-valued output of a numerical program encloses the solution of the original mathematical problem. This is done by controlling, enclosing and propagating round-off and truncation errors using for example interval arithmetic. More precisely, one reduces the computation to a sequence of elementary operations, say ( + , − , × , / ) {\displaystyle (+,-,\times ,/)} . In a computer, the result of each elementary operation is rounded off by the computer precision. However, one can construct an interval provided by upper and lower bounds on the result of an elementary operation. Then one proceeds by replacing numbers with intervals and performing elementary operations between such intervals of representable numbers. == Philosophical objections == Computer-assisted proofs are the subject of some controversy in the mathematical world, with Thomas Tymoczko first to articulate objections. Those who adhere to Tymoczko's arguments believe that lengthy computer-assisted proofs are not, in some sense, 'real' mathematical proofs because they involve so many logical steps that they are not practically verifiable by human beings, and that mathematicians are effectively being asked to replace logical deduction from assumed axioms with trust in an empirical computational process, which is potentially affected by errors in the computer program, as well as defects in the runtime environment and hardware. Other mathematicians believe that lengthy computer-assisted proofs should be regarded as calculations, rather than proofs: the proof algorithm itself should be proved valid, so that its use can then be regarded as a mere "verification". Arguments that computer-assisted proofs are subject to errors in their source programs, compilers, and hardware can be resolved by providing a formal proof of correctness for the computer program (an approach which was successfully applied to the four color theorem in 2005) as well as replicating the result using different programming languages, different compilers, and different computer hardware. Another possible way of verifying computer-aided proofs is to generate their reasoning steps in a machine readable form, and then use a proof checker program to demonstrate their correctness. Since validating a given proof is much easier than finding a proof, the checker program is simpler than the original assistant program, and it is correspondingly easier to gain confidence into its correctness. However, this approach of using a computer program to prove the output of another program correct does not appeal to computer proof skeptics, who see it as adding another layer of complexity without addressing the perceived need for human understanding. Another argument against computer-aided proofs is that they lack mathematical elegance—that they provide no insights or new and useful concepts. In fact, this is an argument that could be advanced against any lengthy proof by exhaustion. An additional philosophical issue raised by computer-aided proofs is whether they make mathematics into a quasi-empirical science, where the scientific method becomes more important than the application of pure reason in the area of abstract mathematical concepts. This directly relates to the argument within mathematics as to whether mathematics is based on ideas, or "merely" an exercise in formal symbol manipulation. It also raises the question whether, if according to the Platonist view, all possible mathematical objects in some sense "already exist", whether computer-aided mathematics is an observational science like astronomy, rather than an experimental one like physics or chemistry. This controversy within mathematics is occurring at the same time as questions are being asked in the physics community about whether twenty-first century theoretical physics is becoming too mathematical, and leaving behind its experimental roots. The emerging field of experimental mathematics is confronting this debate head-on by focusing on numerical experiments as its main tool for mathematical exploration. == Theorems proved with the help of computer programs == Inclusion in this list does not imply that a formal computer-checked proof exists, but rather, that a computer program has been involved in some way. See the main articles for details.