In computer science, static program analysis (also known as static analysis or static simulation) is the analysis of computer programs performed without executing them, in contrast with dynamic program analysis, which is performed on programs during their execution in the integrated environment. The term is usually applied to analysis performed by an automated tool, with human analysis typically being called "program understanding", program comprehension, or code review. In the last of these, software inspection and software walkthroughs are also used. In most cases the analysis is performed on some version of a program's source code, and, in other cases, on some form of its object code. Two leading approaches to resource certification have been Static Analysis (SA) and Implicit Computational Complexity (ICC). SA is algorithmic in nature: it focuses on a broad programming language of choice, and seeks to determine by syntactic means whether given programs in that language are feasible. In contrast, ICC attempts to create from the outset specialized programming languages or methods that delineate a complexity class. Thus, SA's focus is on compile time, making no demand on the programmer; whereas ICC is a language-design discipline." The discipline of static analysis should not be confused with linting, which is the process of checking for coding style mistakes. == Rationale == The sophistication of the analysis performed by tools varies from those that only consider the behaviour of individual statements and declarations, to those that include the complete source code of a program in their analysis. The uses of the information obtained from the analysis vary from highlighting possible coding errors (e.g., the lint tool) to formal methods that mathematically prove properties about a given program (e.g., its behaviour matches that of its specification). Software metrics and reverse engineering can be described as forms of static analysis. Deriving software metrics and static analysis are increasingly deployed together, especially in creation of embedded systems, by defining so-called software quality objectives. A growing commercial use of static analysis is in the verification of properties of software used in safety-critical computer systems and locating potentially vulnerable code. For example, the following industries have identified the use of static code analysis as a means of improving the quality of increasingly sophisticated and complex software: Medical software: The US Food and Drug Administration (FDA) has identified the use of static analysis for medical devices. Nuclear software: In the UK the Office for Nuclear Regulation (ONR) recommends the use of static analysis on reactor protection systems. Aviation software (in combination with dynamic analysis). Automotive & Machines (functional safety features form an integral part of each automotive product development phase, ISO 26262, section 8). A study in 2012 by VDC Research reported that 28.7% of the embedded software engineers surveyed use static analysis tools and 39.7% expect to use them within 2 years. A study from 2010 found that 60% of the interviewed developers in European research projects made at least use of their basic IDE built-in static analyzers. However, only about 10% employed an additional other (and perhaps more advanced) analysis tool. In the application security industry the name static application security testing (SAST) is also used. SAST is an important part of Security Development Lifecycles (SDLs) such as the SDL defined by Microsoft and a common practice in software companies. == Tool types == The OMG (Object Management Group) published a study regarding the types of software analysis required for software quality measurement and assessment. This document on "How to Deliver Resilient, Secure, Efficient, and Easily Changed IT Systems in Line with CISQ Recommendations" describes three levels of software analysis. Unit Level Analysis that takes place within a specific program or subroutine, without connecting to the context of that program. Technology Level Analysis that takes into account interactions between unit programs to get a more holistic and semantic view of the overall program in order to find issues and avoid obvious false positives. System Level Analysis that takes into account the interactions between unit programs, but without being limited to one specific technology or programming language. A further level of software analysis can be defined. Mission/Business Level Analysis that takes into account the business/mission layer terms, rules and processes that are implemented within the software system for its operation as part of enterprise or program/mission layer activities. These elements are implemented without being limited to one specific technology or programming language and in many cases are distributed across multiple languages, but are statically extracted and analyzed for system understanding for mission assurance. == Formal methods == Formal methods is the term applied to the analysis of software (and computer hardware) whose results are obtained purely through the use of rigorous mathematical methods. The mathematical techniques used include denotational semantics, axiomatic semantics, operational semantics, and abstract interpretation. By a straightforward reduction to the halting problem, it is possible to prove that (for any Turing complete language), finding all possible run-time errors in an arbitrary program (or more generally any kind of violation of a specification on the final result of a program) is undecidable: there is no mechanical method that can always answer truthfully whether an arbitrary program may or may not exhibit runtime errors. This result dates from the works of Church, Gödel and Turing in the 1930s (see: Halting problem and Rice's theorem). As with many undecidable questions, one can still attempt to give useful approximate solutions. Some of the implementation techniques of formal static analysis include: Abstract interpretation, to model the effect that every statement has on the state of an abstract machine (i.e., it 'executes' the software based on the mathematical properties of each statement and declaration). This abstract machine over-approximates the behaviours of the system: the abstract system is thus made simpler to analyze, at the expense of incompleteness (not every property true of the original system is true of the abstract system). If properly done, though, abstract interpretation is sound (every property true of the abstract system can be mapped to a true property of the original system). Data-flow analysis, a lattice-based technique for gathering information about the possible set of values; Hoare logic, a formal system with a set of logical rules for reasoning rigorously about the correctness of computer programs. There is tool support for some programming languages (e.g., the SPARK programming language (a subset of Ada) and the Java Modeling Language—JML—using ESC/Java and ESC/Java2, Frama-C WP (weakest precondition) plugin for the C language extended with ACSL (ANSI/ISO C Specification Language) ). Model checking, considers systems that have finite state or may be reduced to finite state by abstraction; Symbolic execution, as used to derive mathematical expressions representing the value of mutated variables at particular points in the code. Nullable reference analysis == Data-driven static analysis == Data-driven static analysis leverages extensive codebases to infer coding rules and improve the accuracy of the analysis. For instance, one can use all Java open-source packages available on GitHub to learn good analysis strategies. The rule inference can use machine learning techniques. It is also possible to learn from a large amount of past fixes and warnings. == Remediation == Static analyzers produce warnings. For certain types of warnings, it is possible to design and implement automated remediation techniques. For example, Logozzo and Ball have proposed automated remediations for C# cccheck.
AI alignment
In the field of artificial intelligence (AI), alignment aims to steer AI systems toward a person's or group's intended goals, preferences, or ethical principles. An AI system is considered aligned if it advances the intended objectives. A misaligned AI system pursues unintended objectives. It is often difficult for AI designers to specify the full range of desired and undesired behaviors. Therefore, the designers often use simpler proxy goals, such as gaining human approval. But proxy goals can overlook necessary constraints or reward the AI system for merely appearing aligned. AI systems may also find loopholes that allow them to accomplish their proxy goals efficiently but in unintended, sometimes harmful, ways (reward hacking). Advanced AI systems may develop unwanted instrumental strategies, such as seeking power or self-preservation because such strategies help them achieve their assigned final goals. Furthermore, they might develop undesirable emergent goals that could be hard to detect before the system is deployed and encounters new situations and data distributions. Empirical research showed in 2024 that advanced large language models (LLMs) such as OpenAI o1 or Claude 3 sometimes engage in strategic deception to achieve their goals or prevent them from being changed. Some of these issues affect existing commercial systems such as LLMs, robots, autonomous vehicles, and social media recommendation engines. Some AI researchers argue that more capable future systems will be more severely affected because these problems partially result from high capabilities. Many prominent AI researchers and AI company leaders have argued or asserted that AI is approaching human-like (AGI) and superhuman cognitive capabilities (ASI), and could endanger human civilization if misaligned. These include "AI godfathers" Geoffrey Hinton and Yoshua Bengio and the CEOs of OpenAI, Anthropic, and Google DeepMind. These risks remain debated. AI alignment is a subfield of AI safety, the study of how to build safe AI systems. Other subfields of AI safety include robustness, monitoring, and capability control. Research challenges in alignment include instilling complex values in AI, developing honest AI, scalable oversight, auditing and interpreting AI models, and preventing emergent AI behaviors like power-seeking. Alignment research has connections to interpretability research, (adversarial) robustness, anomaly detection, calibrated uncertainty, formal verification, preference learning, safety-critical engineering, game theory, algorithmic fairness, and social sciences. == Objectives in AI == Programmers provide an AI system such as AlphaZero with an "objective function", in which they intend to encapsulate the goal(s) the AI is configured to accomplish. Such a system later populates a (possibly implicit) internal "model" of its environment. This model encapsulates all the agent's beliefs about the world. The AI then creates and executes whatever plan is calculated to maximize the value of its objective function. For example, when AlphaZero is trained on chess, it has a simple objective function of "+1 if AlphaZero wins, −1 if AlphaZero loses". During the game, AlphaZero attempts to execute whatever sequence of moves it judges most likely to attain the maximum value of +1. Similarly, a reinforcement learning system can have a "reward function" that allows the programmers to shape the AI's desired behavior. An evolutionary algorithm's behavior is shaped by a "fitness function". == Alignment problem == In 1960, AI pioneer Norbert Wiener described the AI alignment problem as follows: If we use, to achieve our purposes, a mechanical agency with whose operation we cannot interfere effectively [...] we had better be quite sure that the purpose put into the machine is the purpose which we really desire. AI alignment refers to ensuring that an AI system's objectives match some target. The target is variously defined as the goals of the system's designers or users, widely shared values, objective ethical standards, legal requirements, or the intentions its designers would have if they were more informed and enlightened. In democratic AI alignment, the target is the values and preferences of median voters, which increases political legitimacy. AI alignment is an open problem for modern AI systems and is a research field within AI. Aligning AI involves two main challenges: carefully specifying the purpose of the system (outer alignment) and ensuring that the system adopts the specification robustly (inner alignment). Researchers also attempt to create AI models that have robust alignment, sticking to safety constraints even when users adversarially try to bypass them. === Specification gaming and side effects === To specify an AI system's purpose, AI designers typically provide an objective function, examples, or feedback to the system. But designers are often unable to completely specify all important values and constraints, so they resort to easy-to-specify proxy goals such as maximizing the approval of human overseers, who are fallible. As a result, AI systems can find loopholes that help them accomplish the specified objective efficiently but in unintended, possibly harmful ways. This tendency is known as specification gaming or reward hacking, and is an instance of Goodhart's law. As AI systems become more capable, they are often able to game their specifications more effectively. Specification gaming has been observed in numerous AI systems. OpenAI GPT models for programming—including in real-world cases—have been found to explicitly plan hacking the tests used to evaluate them to falsely appear successful (e.g., explicitly stating "let's hack"). When the company penalized this, many models learned to obfuscate their plans while continuing to hack the tests. Another system was trained to finish a simulated boat race by rewarding the system for hitting targets along the track, but the system achieved more reward by looping and crashing into the same targets indefinitely. A 2025 Palisade Research study found that when tasked to win at chess against a stronger opponent, some reasoning LLMs attempted to hack the game system, for example by modifying or entirely deleting their opponent. Some alignment researchers aim to help humans detect specification gaming and steer AI systems toward carefully specified objectives that are safe and useful to pursue. When a misaligned AI system is deployed, it can have consequential side effects. Social media platforms have been known to optimize their recommendation algorithms for click-through rates, causing user addiction on a global scale. Stanford researchers say that such recommender systems are misaligned with their users because they "optimize simple engagement metrics rather than a harder-to-measure combination of societal and consumer well-being". Explaining such side effects, Berkeley computer scientist Stuart J. Russell said that the omission of implicit constraints can cause harm: "A system [...] will often set [...] unconstrained variables to extreme values; if one of those unconstrained variables is actually something we care about, the solution found may be highly undesirable. This is essentially the old story of the genie in the lamp, or the sorcerer's apprentice, or King Midas: you get exactly what you ask for, not what you want." Some researchers suggest that AI designers specify their desired goals by listing forbidden actions or by formalizing ethical rules (as with Asimov's Three Laws of Robotics). But Russell and Norvig argue that this approach overlooks the complexity of human values: "It is certainly very hard, and perhaps impossible, for mere humans to anticipate and rule out in advance all the disastrous ways the machine could choose to achieve a specified objective." Additionally, even if an AI system fully understands human intentions, it may still disregard them, because following human intentions may not be its objective (unless it is already fully aligned). === Pressure to deploy unsafe systems === Commercial organizations sometimes have incentives to take shortcuts on safety and to deploy misaligned or unsafe AI systems. For example, social media recommender systems have been profitable despite creating unwanted addiction and polarization. Competitive pressure can also lead to a race to the bottom on AI safety standards. For example, OpenAI has been sued for releasing a ChatGPT version that encouraged suicide for some unstable users, a behavior the company had overlooked amid a rushed product release. Similarly, in 2018, a self-driving car killed a pedestrian (Elaine Herzberg) after engineers disabled the emergency braking system because it was oversensitive and slowed development. === Risks from advanced misaligned AI === Some researchers are interested in aligning increasingly advanced AI systems, as progress in AI development is rapid, and industry and governments are trying to build advan
Quantum artificial life
Quantum artificial life is the application of quantum algorithms with the ability to simulate biological behavior. Quantum computers offer many potential improvements to processes performed on classical computers, including machine learning and artificial intelligence. Artificial intelligence applications are often inspired by the idea of mimicking human brains through closely related biomimicry. This has been implemented to a certain extent on classical computers (using neural networks), but quantum computers offer many advantages in the simulation of artificial life. Artificial life and artificial intelligence are extremely similar, with minor differences; the goal of studying artificial life is to understand living beings better, while the goal of artificial intelligence is to create intelligent beings. In 2016, Alvarez-Rodriguez et al. developed a proposal for a quantum artificial life algorithm with the ability to simulate life and Darwinian evolution. In 2018, the same research team led by Alvarez-Rodriguez performed the proposed algorithm on the IBM ibmqx4 quantum computer, and received optimistic results. The results accurately simulated a system with the ability to undergo self-replication at the quantum scale. == Artificial life on quantum computers == The growing advancement of quantum computers has led researchers to develop quantum algorithms for simulating life processes. Researchers have designed a quantum algorithm that can accurately simulate Darwinian Evolution. Since the complete simulation of artificial life on quantum computers has only been actualized by one group, this section shall focus on the implementation by Alvarez-Rodriguez, Sanz, Lomata, and Solano on an IBM quantum computer. Individuals were realized as two qubits, one representing the genotype of the individual and the other representing the phenotype. The genotype is copied to transmit genetic information through generations, and the phenotype is dependent on the genetic information as well as the individual's interactions with their environment. In order to set up the system, the state of the genotype is instantiated by some rotation of an ancillary state ( | 0 ⟩ ⟨ 0 | {\displaystyle |0\rangle \langle 0|} ). The environment is a two-dimensional spatial grid occupied by individuals and ancillary states. The environment is divided into cells that are able to possess one or more individuals. Individuals move throughout the grid and occupy cells randomly; when two or more individuals occupy the same cell they interact with each other. === Self replication === The ability to self-replicate is critical for simulating life. Self-replication occurs when the genotype of an individual interacts with an ancillary state, creating a genotype for a new individual; this genotype interacts with a different ancillary state in order to create the phenotype. During this interaction, one would like to copy some information about the initial state into the ancillary state, but by the no cloning theorem, it is impossible to copy an arbitrary unknown quantum state. However, physicists have derived different methods for quantum cloning which does not require the exact copying of an unknown state. The method that has been implemented by Alvarez-Rodriguez et al. is one that involves the cloning of the expectation value of some observable. For a unitary U {\displaystyle U} which copies the expectation value of some set of observables X {\displaystyle {\mathsf {X}}} of state ρ {\displaystyle \rho } into a blank state ρ e {\displaystyle \rho _{e}} , the cloning machine is defined by any ( U , ρ e , X ) {\displaystyle (U,\rho _{e},{\mathsf {X}})} that fulfill the following: ∀ ρ ∀ X ∈ X {\displaystyle \forall \rho \forall X\in {\mathsf {X}}} X ¯ = X 1 ¯ = X 2 ¯ {\displaystyle {\bar {X}}={\bar {X_{1}}}={\bar {X_{2}}}} Where X ¯ {\displaystyle {\bar {X}}} is the mean value of the observable in ρ {\displaystyle \rho } before cloning, X 1 ¯ {\displaystyle {\bar {X_{1}}}} is the mean value of the observable in ρ {\displaystyle \rho } after cloning, and X 2 ¯ {\displaystyle {\bar {X_{2}}}} is the mean value of the observable in ρ e {\displaystyle \rho _{e}} after cloning. Note that the cloning machine has no dependence on ρ {\displaystyle \rho } because we want to be able to clone the expectation of the observables for any initial state. It is important to note that cloning the mean value of the observable transmits more information than is allowed classically. The calculation of the mean value is defined naturally as: X ¯ = T r [ ρ X ] {\displaystyle {\bar {X}}=Tr[\rho X]} , X 1 ¯ = T r [ R X ⊗ I ] {\displaystyle {\bar {X_{1}}}=Tr[RX\otimes I]} , X 2 ¯ = T r [ R I ⊗ X ] {\displaystyle {\bar {X_{2}}}=Tr[RI\otimes X]} where R = U ρ ⊗ ρ e U † {\displaystyle R=U\rho \otimes \rho _{e}U^{\dagger }} The simplest cloning machine clones the expectation value of σ z {\displaystyle \sigma _{z}} in arbitrary state ρ = | ψ ⟩ ⟨ ψ | {\displaystyle \rho =|\psi \rangle \langle \psi |} to ρ e = | 0 ⟩ ⟨ 0 | {\displaystyle \rho _{e}=|0\rangle \langle 0|} using U = C N O T {\displaystyle U=CNOT} . This is the cloning machine implemented for self-replication by Alvarez-Rodriguez et al. The self-replication process clearly only requires interactions between two qubits, and therefore this cloning machine is the only one necessary for self replication. === Interactions === Interactions occur between individuals when the two take up the same space on the environmental grid. The presence of interactions between individuals provides an advantage for shorter-lifespan individuals. When two individuals interact, exchanges of information between the two phenotypes may or may not occur based on their existing values. When both individual's control qubits (genotypes) are alike, no information will be exchanged. When the control qubits differ, the target qubits (phenotype) will be exchanged between the two individuals. This procedure produces a constantly changing predator-prey dynamic in the simulation. Therefore, long-living qubits, with a larger genetic makeup in the simulation, are at a disadvantage. Since information is only exchanged when interacting with an individual of different genetic makeup, the short-lived population has the advantage. === Mutation === Mutations exist in the artificial world with limited probability, equivalent to their occurrence in the real world. There are two ways in which the individual can mutate: through random single qubit rotations and by errors in the self-replication process. There are two different operators that act on the individual and cause mutations. The M operation causes a spontaneous mutation within the individual by rotating a single qubit by parameter θ. The parameter θ is random for each mutation, which creates biodiversity within the artificial environment. The M operation is a unitary matrix which can be described as: M = ( cos ( θ ) s i n ( θ ) s i n ( θ ) − c o s ( θ ) ) {\displaystyle M={\begin{pmatrix}\cos(\theta )&sin(\theta )\\sin(\theta )&-cos(\theta )\end{pmatrix}}} The other possible way for mutations to occur is due to errors in the replication process. Due to the no-cloning theorem, it is impossible to produce perfect copies of systems that are originally in unknown quantum states. However, quantum cloning machines make it possible to create imperfect copies of quantum states, in other words, the process introduces some degree of error. The error that exists in current quantum cloning machines is the root cause for the second kind of mutations in the artificial life experiment. The imperfect cloning operation can be seen as: U M ( θ ) = I 4 + 1 2 ( 0 0 0 1 ) ⊗ ( − 1 1 1 − 1 ) ( c o s θ + i s i n θ + 1 ) {\displaystyle U_{M}(\theta )=\mathrm {I} _{4}+{\frac {1}{2}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\otimes {\begin{pmatrix}-1&1\\1&-1\end{pmatrix}}(cos\theta +isin\theta +1)} The two kinds of mutations affect the individual differently. While the spontaneous M operation does not affect the phenotype of the individual, the self-replicating error mutation, UM, alters both the genotype of the individual, and its associated lifetime. The presence of mutations in the quantum artificial life experiment is critical for providing randomness and biodiversity. The inclusion of mutations helps to increase the accuracy of the quantum algorithm. === Death === At the instant the individual is created (when the genotype is copied into the phenotype), the phenotype interacts with the environment. As time evolves, the interaction of the individual with the environment simulates aging which eventually leads to the death of the individual. The death of an individual occurs when the expectation value of σ z {\displaystyle \sigma _{z}} is within some ϵ {\displaystyle \epsilon } of 1 in the phenotype, or, equivalently, when ρ p = | 0 ⟩ ⟨ 0 | {\displaystyle \rho _{p}=|0\rangle \langle 0|} The Lindbladian describes the interaction of the individual with the environment: ρ
Neurocomputing (journal)
Neurocomputing is a peer-reviewed scientific journal covering research on artificial intelligence, machine learning, and neural computation. It was established in 1989 and is published by Elsevier. The editor-in-chief is Zidong Wang (Brunel University London). Independent scientometric studies noted that despite being one of the most productive journals in the field, it has kept its reputation across the years intact and plays an important role in leading the research in the area. The journal is abstracted and indexed in Scopus and Science Citation Index Expanded. According to the Journal Citation Reports, its 2023 impact factor is 5.5.
Hierarchical Risk Parity
Hierarchical Risk Parity (HRP) is an advanced investment portfolio optimization framework developed in 2016 by Marcos López de Prado at Guggenheim Partners and Cornell University. HRP is a probabilistic graph-based alternative to the prevailing mean-variance optimization (MVO) framework developed by Harry Markowitz in 1952, and for which he received the Nobel Prize in economic sciences. HRP algorithms apply discrete mathematics and machine learning techniques to create diversified and robust investment portfolios that outperform MVO methods out-of-sample. HRP aims to address the limitations of traditional portfolio construction methods, particularly when dealing with highly correlated assets. Following its publication, HRP has been implemented in numerous open-source libraries, and received multiple extensions. == Key features == HRP portfolios have been proposed as a robust alternative to traditional quadratic optimization methods, including the Critical Line Algorithm (CLA) of Markowitz. HRP addresses three central issues commonly associated with quadratic optimizers: numerical instability, excessive concentration in a small number of assets, and poor out-of-sample performance. HRP leverages techniques from graph theory and machine learning to construct diversified portfolios using only the information embedded in the covariance matrix. Unlike quadratic programming methods, HRP does not require the covariance matrix to be invertible. Consequently, HRP remains applicable even in cases where the covariance matrix is ill-conditioned or singular—conditions under which standard optimizers fail. Monte Carlo simulations indicate that HRP achieves lower out-of-sample variance than CLA, despite the fact that minimizing variance is the explicit optimization objective of CLA. Furthermore, HRP portfolios exhibit lower realized risk compared to those generated by traditional risk parity methodologies. Empirical backtests have demonstrated that HRP would have historically outperformed conventional portfolio construction techniques. Algorithms within the HRP framework are characterized by the following features: Machine Learning Approach: HRP employs hierarchical clustering, a machine learning technique, to group similar assets based on their correlations. This allows the algorithm to identify the underlying hierarchical structure of the portfolio, and avoid that errors spread through the entire network. Risk-Based Allocation: The algorithm allocates capital based on risk, ensuring that assets only compete with similar assets for representation in the portfolio. This approach leads to better diversification across different risk sources, while avoiding the instability associated with noisy returns estimates. Covariance Matrix Handling: Unlike traditional methods like Mean-Variance Optimization, HRP does not require inverting the covariance matrix. This makes it more stable and applicable to portfolios with a large number of assets, particularly when the covariance matrix's condition number is high. == The problem: Markowitz's Curse == Portfolio construction is perhaps the most recurrent financial problem. On a daily basis, investment managers must build portfolios that incorporate their views and forecasts on risks and returns. Despite the theoretical elegance of Markowitz's mean-variance framework, its practical implementation is hindered by several limitations that undermine the reliability of solutions derived from the Critical Line Algorithm (CLA). A principal concern is the high sensitivity of optimal portfolios to small perturbations in expected returns: even minor forecasting errors can result in significantly different allocations (Michaud, 1998). Given the inherent difficulty of producing accurate return forecasts, numerous researchers have advocated for approaches that forgo expected returns entirely and instead rely solely on the covariance structure of asset returns. This has given rise to risk-based allocation methods, among which risk parity is a widely cited example (Jurczenko, 2015). While eliminating return forecasts mitigates some instability, it does not eliminate it. Quadratic programming techniques employed in portfolio optimization require the inversion of a positive-definite covariance matrix, meaning all eigenvalues must be strictly positive. When the matrix is numerically ill-conditioned—that is, when the ratio of its largest to smallest eigenvalue (its condition number) is large—matrix inversion becomes unreliable and prone to significant numerical errors (Bailey and López de Prado, 2012). The condition number of a covariance, correlation, or any symmetric (and thus diagonalizable) matrix is defined as the absolute value of the ratio between its largest and smallest eigenvalues in modulus. The figure on the right presents the sorted eigenvalues of several correlation matrices; the condition number is represented by the ratio of the first to last eigenvalues in each sequence. A diagonal correlation matrix, which is equal to its own inverse, exhibits the minimum possible condition number. As the number of correlated (or multicollinear) assets in a portfolio increases, the condition number rises. At high levels, this leads to severe numerical instability, whereby slight modifications in any matrix entry may result in drastically different inverses. This phenomenon, often referred to as Markowitz’s curse, encapsulates the paradox wherein increased correlation among assets heightens the theoretical need for diversification, yet simultaneously increases the likelihood of unstable optimization outcomes. Consequently, the potential benefits of diversification are frequently overshadowed by estimation errors. These problems are exacerbated as the dimensionality of the covariance matrix increases. The estimation of each covariance term consumes degrees of freedom, and in general, a minimum of 1 2 N ( N + 1 ) {\displaystyle {\frac {1}{2}}N(N+1)} independent and identically distributed (IID) observations is required to estimate a non-singular covariance matrix of dimension N {\displaystyle N} . For example, constructing an invertible covariance matrix of dimension 50 necessitates at least five years of daily IID observations. However, empirical evidence suggests that the correlation structure of financial assets is highly unstable over such extended periods. These difficulties are highlighted by the observation that even naïve allocation strategies—such as equally weighted portfolios—have frequently outperformed both mean-variance and risk-based optimizations in out-of-sample tests (De Miguel et al., 2009). == The solution: Hierarchical Risk Parity == The HRP algorithm addresses Markowitz's curse in three steps: Hierarchical Clustering: Assets are grouped into clusters based on their correlations, forming a hierarchical tree structure. Quasi-Diagonalization: The correlation matrix is reordered based on the clustering results, revealing a block diagonal structure. Recursive Bisection: Weights are assigned to assets through a top-down approach, splitting the portfolio into smaller sub-portfolios and allocating capital based on inverse variance. === Step 1: Hierarchical clustering === Given a T × N {\displaystyle T\times N} matrix of asset returns X {\displaystyle X} , where each column represents a time series of returns for one of N {\displaystyle N} assets over T {\displaystyle T} time periods, a hierarchical clustering process can be used to construct a tree-based representation of asset relationships. First, we compute the N × N {\displaystyle N\times N} correlation matrix ρ = ρ i , j i , j = 1 . . . N {\displaystyle \rho ={\rho _{i,j}}\;{i,j=1\;...\;N}} , where ρ i , j = c o r r ( X i , X j ) {\displaystyle \rho _{i,j}=\mathrm {corr} (X_{i},X_{j})} . From this, a pairwise distance matrix D = d i , j {\displaystyle D={d_{i,j}}} is defined using the transformation: d i , j = 1 2 ( 1 − ρ i , j ) {\displaystyle d_{i,j}={\sqrt {{\frac {1}{2}}(1-\rho _{i,j})}}} This distance function defines a proper metric space, satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Next, a secondary distance matrix D ~ = d ~ i , j {\displaystyle {\tilde {D}}={{\tilde {d}}_{i,j}}} is computed, where each entry measures the Euclidean distance between the distance profiles of two assets: d ~ i , j = ∑ n = 1 N ( d n , i − d n , j ) 2 {\displaystyle {\tilde {d}}_{i,j}={\sqrt {\sum _{n=1}^{N}(d_{n,i}-d_{n,j})^{2}}}} While d i , j {\displaystyle d_{i,j}} reflects correlation-based proximity between two assets, d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} quantifies dissimilarity across the entire system, as it depends on all pairwise distances. Hierarchical clustering proceeds by identifying the pair ( i , j ) {\displaystyle (i,j)} with the smallest value of d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} (for i ≠ j {\displaystyle i\neq j} ), and forming a new cluster u [ 1 ] = ( i , j ) {\displaystyle u[1]=(i,j)} .
Verbal overshadowing
Verbal overshadowing is a phenomenon where giving a verbal description of sensory input impairs formation of memories of that input. This was first reported by Schooler and Engstler-Schooler (1990) where it was shown that the effects can be observed across multiple domains of cognition which are known to rely on non-verbal knowledge and perceptual expertise. One example of this is memory, which has been known to be influenced by language. Seminal work by Carmichael and collaborators (1932) demonstrated that when verbal labels are connected to non-verbal forms during an individual's encoding process, it could potentially bias the way those forms are reproduced. Because of this, memory performance relying on reportable aspects of memory that encode visual forms should be vulnerable to the effects of verbalization. == Initial findings == Schooler and Engstler-Schooler (1990) were the first to report findings of verbal overshadowing. In their study, participants watched a video of a simulated robbery and were instructed to either verbally describe the robber or engage in a control task. Those who engaged in giving a verbal description were less likely to correctly identify the robber from a test lineup, compared to those who engaged in the control task. A larger effect was detected when the verbal description was provided 20, rather than 5, minutes after the video, and immediately before the test lineup. A meta-analysis by Meissner and Brigham (2001) supported the effects of verbal overshadowing, showing a small but reliably negative effect. == General effects of verbal overshadowing == The effects of verbal overshadowing have been generalized across multiple domains of cognition that are known to rely on non-verbal knowledge and perceptual expertise, such as memory. Memory has been known to be influenced by language. Seminal work by Carmichael and collaborators (1932) demonstrated that labels attached to, or associated with, non-verbal forms during memory encoding can affect the way the forms were subsequently reproduced. Because of this, memory performance that relies on reportable aspects of memory that encode visual forms should be vulnerable to the effects of verbalization. Pelizzon, Brandimonte, and Luccio (2002) found that visual memory representations appear to incorporate visual, spatial, and temporal characteristics. It is explained as follows: With the temporal code (where the only information available is the sequence of the stimuli), performance levels remain high, unless participants are required to retrieve the stimuli in a different order from that used at encoding (visual cue). In this case, performance is significantly impaired, even in the presence of a visual cue. The study showed that order information acts as a link between the two separate representations of figure and background, hence preventing verbal overshadowing at encoding (temporal component) or attenuating its influence at retrieval (spatial component).(p. 960) Hatano, Ueno, Kitagami, and Kawaguchi found that verbal overshadowing is likely to occur when participants verbally described targets in detail. Detailed verbal descriptions resulted in more frequently inaccurate descriptions that in turn created inaccurate representations in the memories of participants. Inaccuracies are also likely to occur when face recognition comes immediately after verbalization. Other forms of non-verbal knowledge affected by verbal overshadowing include the following: [Verbal overshadowing] has also been observed when participants attempt to generate descriptions of other 'difficult-to-describe' stimuli such as colors (Schooler and Engstler-Schooler, 1990) or abstract figures (Brandimonte et al., 1997), or other non-visual tasks such as wine tasting (Melcher and Schooler, 1996), decision making (Wilson and Schooler, 1991), and insight problem-solving. (p. 871) (Schooler et al., 1993) Verbalization of stimuli leads to the disruption of non-reportable processes that are necessary for achieving insight solutions, which are distinct from language processes. Schooler, Ohlsson, and Brooks (1993) found that face recognition requires information that cannot be adequately verbalized, giving rise to difficulty in describing factors in recognition judgments. Subjects were less effective in solving insight problems when compelled to put their thoughts in words, which suggests that language may interfere with thought. The verbal overshadowing effect was not seen when participants engaged in articulatory suppression. Performance was reduced in both the verbal and non-verbal description conditions. This is evidence that verbal encoding plays a role in face recognition. By testing with distracting faces presented between study and test, Lloyd-Jones and Brown (2008) suggested a dual-process approach to recognition memory took place, that verbalization influenced familiarity-based processes at first, but its effects were later seen on recollection, when discrimination between items became more difficult. == Verbal overshadowing in facial recognition == The verbal overshadowing effect can be found for facial recognition because faces are predominately processed in a holistic or configurable manner. (Tanaka & Farah, 1993; Tanaka & Sengco, 1997) Verbalizing one's memory for a face is done using a featural or analytic strategy, leading to a drift from the configurable information about the face and to impaired recognition performance. However, Fallshore & Schooler (1995) found that the verbal overshadowing effect was not found when participants described faces of races different from their own. A study by Brown and Lloyd-Jones (2003) found that there was no verbal overshadowing effect found in car descriptions; it was only seen in facial descriptions. The authors noted that descriptions were no different on any measure including accuracy. It is suggested that less expertise in verbalizing faces rather than cars invokes a stronger shift in verbal and featural processing. This supports the concept of a transfer inappropriate retrieval framework and addresses some limitations of the effect. Wickham and Swift (2006) suggested that the verbal overshadowing effect is not seen in describing all faces, and one aspect that determines this is distinctiveness. Results showed that typical faces produce verbal overshadowing, while distinctive faces did not. In studies of eyewitness reports, variation in response criteria given by participants influenced the quality of the descriptions generated and accuracy on identification task, known as the retrieval-based effect. Face recognition was also impaired when subjects described a familiar face, such as a parent, or when describing a previously seen but novel face. Dodson, Johnson, and Schooler (1997) found that recognition was also impaired when participants were provided with a description of a previously seen face, and they were able to ignore provided versus self-generated descriptions more easily. This finding of verbal overshadowing suggested that eyewitness recognition is not only affected by their own descriptions, but of descriptions heard from others, such other eyewitness testimonies. == Voice recognition == The verbal overshadowing effect has also been found to affect voice identification. Research shows that describing a non-verbal stimuli leads to a decrease in recognition accuracy. In an unpublished study by Schooler, Fiore, Melcher, and Ambadar (1996), participants listened to a tape-recorded voice, after which they were asked either to verbally describe it or to not do so, and then asked to distinguish the voice from 3 similar distractor voices. The results showed that verbal overshadowing impaired accuracy of recognition based on gut feeling, suggesting an overall verbal overshadowing for voice recognition. Due to the forensic relevance of voices heard over the telephone and harassing phone calls that are often a problem for police, Perfect, Hunt, and Harris (2002) examined the influence of three factors on accuracy and confidence in voice recognition from a line-up. They expected to find an effect, because voice represents a class of stimuli that is difficult to describe verbally. This meets Schooler et al.'s (1997) modality mismatch criterion, meaning that describing the speakers age, gender, or accent is difficult, making voice recognition susceptible to the verbal overshadowing phenomenon. It was found that the method of memory encoding had no impact on performance, and that hearing a telephone voice reduced confidence but did not affect accuracy. They also found that providing a verbal description impaired accuracy but had no effect on confidence. The data showed an effect of verbal overshadowing in voice recognition and provided yet another disassociation between confidence and performance. Although there was a difference in confidence level, witnesses were able to identify voices over the telephone as accurately as voices heard direc
Agent verification
Agent verification is activity to gain assurances that purposeful artificial constructs act in accordance with their specifications. While primitive forms of inorganic agents have been used in manufacturing for centuries, the study of artificial agents did not begin until the mid 20th century. Foundational work on such agents was closely bound with the emergence of artificial intelligence as an academic discipline. Early agents deployed for industrial control systems and in computing were often controlled by quite simple logic however, not involving artificial intelligence as such. When deployed as part of a multi-agent system, even such simple agents could require special agent orientated testing methods, as their collective behaviour was challenging to verify with traditional testing techniques. Difficulties in providing assurances that agents will not behave in dangerous ways became more prevalent after the introduction of LLM agents, especially after the rapid acceleration of their deployment in 2025. The verification of agent behaviour can be conducted by formal or informal methods. Informal verification requires less mathematical skill. But when agents are part of systems where errors have significant risks — such as danger to human life, environmental damage or major financial loss — formal verification is preferred. Both regulators and system designers themselves like formal verification as it provides a high degree of mathematical certainty. It is not however always possible to formally test all aspects of an agent based system's behaviour, especially where newer LLM based agents are concerned, due in part to their high degree of autonomy. Accordingly, agent verification for low impact deployments might be carried out only with informal methods, while for high impact deployments, it may be performed with a mix of formal and informal techniques. == Terminology == In academia, the term agent verification is often defined to mean activity concerned with gaining assurance that the agent behaves in accordance with its specification - whether by processes such as testing or simulation. 'Verification' is typically contrasted with 'validation', the latter meaning activity concerned with checking that the specification itself meets user or real world needs. Such definitions are not universally adhered to however - for example, in some workplaces and documents, the words 'verification' and 'validation' can be used synonymously. Efforts to gain confidence in Agents have intensified sharply since 2025 due to the rapid roll out of LLM agents; different terms are sometimes used in the commercial sector. Here the term 'agent verification' can be used in the same sense as it is in academia, but sometimes the same activity can be covered by more ambiguous and wider ranging terms such as 'Agent governance' , 'Agent observability' or 'AI agent policing'. == History == === Classical agents === The theoretical underpinnings for artificial (inorganic) agents emerged in the mid 20th century, with establishment of cybernetics and artificial intelligence. Oliver Selfridge's 1958 Pandemonium - A Paradigm for Learning paper was an important early theoretical contribution in establishing agent oriented architecture. Practical implementations of agents for real world applications began to become widespread in the 1990s, after the introduction of the belief–desire–intention software model (BDI), and agent-oriented programming. Pure digital agents were deployed in computer infrastructure for purposes such as monitoring, while agents connected to real-world sensors and actuators were increasingly used in industrial control systems. While the concept of artificial agents was interwoven with early artificial intelligence studies right from the start, early agents lacked general purpose reasoning capabilities, often only having simple if then logic. Even a device as simple as a thermostat, which has a sensor and a means of acting, can be considered a proto agent in this sense. Verifying the behaviours of a simple single agent system is not generally especially difficult, but it can be a different matter when several simple agents coexist in the same system. Craig Reynolds's work on boids showed that relatively complex, "intelligent" behaviour can emerge from a number of such simple agents working together in a Multi-agent system (MAS). By the 1990s, even the behaviour of a single agent system could sometimes be quite complex; in accordance with the Belief–desire–intention software model, agents could have believes that might evolve over time. Agents were increasingly introduced that were controlled by quite large decision tree models, which had new vulnerabilities to adversarial attack. It was becoming increasingly apparent that traditional software verification methods had limitations for testing such agents, or even for the more primitive type of agents when they were deployed as part of a MAS. It was the use of agents for industrial control systems, sometimes associated with robotics, that lent urgency to the practice of agent verification. Informal testing might be acceptable for digital agents used say to monitor whether each of an organisation's computers are properly licensed. But with an increasing potential for faulty agents to result in a failure that might cause a large fire to break out at a chemical manufacturing plant, a botched medical operation, or even a crashed aircraft, the need to develop reliable means of verifying behaviour of such agents was considered urgent. The Foundation for Intelligent Physical Agents was established in 1996. From the late 90s, a growing number of industry and university based scientists began working on the problem, with researchers publishing papers on the verification of both single and multi agent systems. Much of this work showed how formal verification techniques like model checking could be used to gain a high level of assurance that agent based systems would conform with their specification. A 2018 systematic review covering 231 studies found that model checking was the most common technique for agent verification, with theorem proving the second most commonly used formal verification method. In the first two decades of the 20th century, agents run by AI became more common, with Siri and Alexa being well known examples. But such agents still lacked general reasoning capabilities and did not pose new pressing problems for agent verification. === General purpose reasoning agents === The advent of LLMs created huge potential for further use of artificial agents, as agents based on them could have general purpose cognitive abilities. Agents run by LLMs (and occasionally non-LLM foundation models) have similar vulnerability to adversarial attack as those run by decision tree models. The wider scope of actions for LLM agents has created new challenges for their verification, over and above those present for classical agents. For example, the LLM's neural network endows it with infinite domains, an especial challenge for traditional formal verification techniques. Academics began to study the problems involved in verifying LLM agents from 2018. Deployment of such agents began to accelerate in late 2023 after OpenAI's "function-calling" API was made available, and especially after Anthropic's late 2024 introduction of Model Context Protocol (MCP), a standardised way for LLM agents to gain contextual awareness, and to act on the world by calling various external tools. The rapid rollout of LLM agents following MCP's release has seen the task of agent verification receive increased attention within academia, and also from the private sector. In 2024 and 2025 several startups focusing on LLM agent verification have been founded in both Europe and the US to meet growing demand. == Approaches == === Formal verification === Formal verification involves proving the correctness of some or all aspects of a system using mathematical methods. Such methods can range from manual formal proof, to verification assisted with automated theorem provers like Isabelle. For agent verification, model checking is by far the most frequently used formal verification method; for pre-LLM models it was often complemented with techniques using computation tree logic. Another common method is theorem proving. Formal verification provides a higher degree of confidence than informal methods, but it is not always used, even when it is possible. Sometimes a person or organisation developing software agents won't have the necessary skills, or may not see it as worth the effort if the agent(s) will not have the ability to cause much harm even if they malfunction. When agents are deployed in systems where errors could have serious consequences, the ability of formal verification methods to provide mathematical certainty tends to be strongly preferred by both regulators and designers themselves. But even for high impact systems, formal verificatio