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Thursday, September 5, 2024

AI Threat, Cyber Threat, and Planning for Check and Analysis


Fashionable synthetic intelligence (AI) techniques pose new sorts of dangers, and many of those are each consequential and never effectively understood. Regardless of this, many AI-based techniques are being accelerated into deployment. That is creating nice urgency to develop efficient take a look at and analysis (T&E) practices for AI-based techniques.

This weblog put up explores potential methods for framing T&E practices on the idea of a holistic method to AI threat. In growing such an method, it’s instructive to construct on classes discovered within the many years of wrestle to develop analogous practices for modeling and assessing cyber threat. Cyber threat assessments are imperfect and proceed to evolve, however they supply vital profit nonetheless. They’re strongly advocated by the Cybersecurity and Infrastructure Safety Company (CISA), and the prices and advantages of assorted approaches are a lot mentioned within the enterprise media. About 70% of inside audits for big corporations embrace cyber threat assessments, as do mandated stress assessments for banks.

Threat modeling and assessments for AI are much less effectively understood from each technical and authorized views, however there’s pressing demand from each enterprise adopters and vendor suppliers nonetheless. The industry-led Coalition for Safe AI launched in July 2024 to assist advance {industry} norms round enhancing the safety of recent AI implementations. The NIST AI Threat Administration Framework (RMF) is resulting in proposed practices. Methodologies primarily based on the framework are nonetheless a piece in progress, with unsure prices and advantages, and so AI threat assessments are much less typically utilized than cyber threat assessments.

Threat modeling and evaluation are essential not solely in guiding T&E, but in addition in informing engineering practices, as we’re seeing with cybersecurity engineering and within the rising apply of AI engineering. AI engineering, importantly, encompasses not simply particular person AI parts in techniques but in addition the general design of resilient AI-based techniques, together with the workflows and human interactions that allow operational duties.

AI threat modeling, even in its present nascent stage, can have helpful affect in each T&E and AI engineering practices, starting from total design decisions to particular threat mitigation steps. AI-related weaknesses and vulnerabilities have distinctive traits (see examples within the prior weblog posts), however in addition they overlap with cyber dangers. AI system parts are software program elements, in spite of everything, so that they typically have vulnerabilities unrelated to their AI performance. Nevertheless, their distinctive and sometimes opaque options, each inside the fashions and within the surrounding software program constructions, could make them particularly enticing to cyber adversaries.

That is the third installment in a four-part collection of weblog posts centered on AI for crucial techniques the place trustworthiness—primarily based on checkable proof—is crucial for operational acceptance. The 4 components are comparatively unbiased of one another and tackle this problem in levels:

  • Half 1: What are applicable ideas of safety and security for contemporary neural-network-based AI, together with machine studying (ML) and generative AI, comparable to giant language fashions (LLMs)? What are the AI-specific challenges in growing secure and safe techniques? What are the bounds to trustworthiness with fashionable AI, and why are these limits basic?
  • Half 2: What are examples of the sorts of dangers particular to fashionable AI, together with dangers related to confidentiality, integrity, and governance (the CIG framework), with and with out adversaries? What are the assault surfaces, and what sorts of mitigations are presently being developed and employed for these weaknesses and vulnerabilities?
  • Half 3 (this half): How can we conceptualize T&E practices applicable to fashionable AI? How, extra usually, can frameworks for threat administration (RMFs) be conceptualized for contemporary AI analogous to these for cyber threat? How can a apply of AI engineering tackle challenges within the close to time period, and the way does it work together with software program engineering and cybersecurity concerns?
  • Half 4: What are the advantages of trying past the purely neural-network fashions of recent AI in direction of hybrid approaches? What are present examples that illustrate the potential advantages, and the way, trying forward, can these approaches advance us past the elemental limits of recent AI? What are prospects within the close to and longer phrases for hybrid AI approaches which are verifiably reliable and that may help extremely crucial purposes?

Assessments for Purposeful and High quality Attributes

Purposeful and high quality assessments assist us achieve confidence that techniques will carry out duties accurately and reliably. Correctness and reliability are usually not absolute ideas, nevertheless. They should be framed within the context of meant functions for a element or system, together with operational limits that should be revered. Expressions of intent essentially embody each performance—what the system is meant to perform—and system qualities—how the system is meant to function, together with safety and reliability attributes. These expressions of intent, or techniques specs, could also be scoped for each the system and its function in operations, together with expectations relating to stressors comparable to adversary threats.

Fashionable AI-based techniques pose vital technical challenges in all these points, starting from expressing specs to acceptance analysis and operational monitoring. What does it imply, for instance, to specify intent for a educated ML neural community, past inventorying the coaching and testing information?

We should take into account, in different phrases, the conduct of a system or an related workflow underneath each anticipated and surprising inputs, the place these inputs could also be significantly problematic for the system. It’s difficult, nevertheless, even to border the query of how one can specify behaviors for anticipated inputs that aren’t precisely matched within the coaching set. A human observer could have an intuitive notion of similarity of latest inputs with coaching inputs, however there isn’t a assurance that this aligns with the precise that includes—the salient parameter values—inside to a educated neural community.

We should, moreover, take into account assessments from a cybersecurity perspective. An knowledgeable and motivated attacker could intentionally manipulate operational inputs, coaching information, and different points of the system growth course of to create circumstances that impair appropriate operation of a system or its use inside a workflow. In each instances, the absence of conventional specs muddies the notion of “appropriate” conduct, additional complicating the event of efficient and reasonably priced practices for AI T&E. This specification problem suggests one other commonality with cyber threat: facet channels, that are potential assault surfaces which are unintended to implementation and that might not be a part of a specification.

Three Dimensions of Cyber Threat

This alignment within the rising necessities for AI-focused T&E with strategies for cybersecurity analysis is obvious when evaluating NIST’s AI threat administration playbook with the extra mature NIST Cybersecurity Framework, which encompasses an enormous range of strategies. On the threat of oversimplification, we are able to usefully body these strategies within the context of three dimensions of cyber threat.

  • Menace considerations the potential entry and actions of adversaries in opposition to the system and its broader operational ecosystem.
  • Consequence pertains to the magnitude of influence on a company or mission ought to an assault on a system achieve success.
  • Vulnerability pertains to intrinsic design weaknesses and flaws within the implementation of a system.

Each risk and consequence carefully rely on the operational context of use of that system, although they are often largely extrinsic to the system itself. Vulnerability is attribute of the system, together with its structure and implementation. The modeling of assault floor—apertures right into a system which are uncovered to adversary actions—encompasses risk and vulnerability, as a result of entry to vulnerabilities is a consequence of operational setting. It’s a significantly helpful factor of cyber threat evaluation.

Cyber threat modeling is not like conventional probabilistic actuarial threat modeling. That is primarily as a result of usually nonstochastic nature of every of the three dimensions, particularly when threats and missions are consequential. Menace, for instance, is pushed by the operational significance of the system and its workflow, in addition to potential adversary intents and the state of their information. Consequence, equally, is decided by decisions relating to the position of a system in operational workflows. Changes to workflows—and human roles—is a mitigation technique for the consequence dimension of threat. Dangers might be elevated when there are hidden correlations. For cyber threat, these may embrace widespread parts with widespread vulnerabilities buried in provide chains. For AI threat, these may embrace widespread sources inside giant our bodies of coaching information. These correlations are a part of the explanation why some assaults on LLMs are moveable throughout fashions and suppliers.

CISA, MITRE, OWASP, and others provide handy inventories of cyber weaknesses and vulnerabilities. OWASP, CISA, and the Software program Engineering Institute additionally present inventories of secure practices. Lots of the generally used analysis standards derive, in a bottom-up method, from these inventories. For weaknesses and vulnerabilities at a coding stage, software program growth environments, automated instruments, and continuous-integration/continuous-delivery (CI/CD) workflows typically embrace evaluation capabilities that may detect insecure coding as builders sort it or compile it into executable elements. Due to this quick suggestions, these instruments can improve productiveness. There are numerous examples of standalone instruments, comparable to from Veracode, Sonatype, and Synopsys.

Importantly, cyber threat is only one factor within the total analysis of a system’s health to be used, whether or not or not it’s AI-based. For a lot of built-in hardware-software techniques, acceptance analysis will even embrace, for instance, conventional probabilistic reliability analyses that mannequin (1) sorts of bodily faults (intermittent, transient, everlasting), (2) how these faults can set off inside errors in a system, (3) how the errors could propagate into numerous sorts of system-level failures, and (4) what sorts of hazards or harms (to security, safety, efficient operation) may end in operational workflows. This latter method to reliability has a protracted historical past, going again to John von Neumann’s work within the Nineteen Fifties on the synthesis of dependable mechanisms from unreliable elements. Curiously, von Neumann cites analysis in probabilistic logics that derive from fashions developed by McCulloch and Pitts, whose neural-net fashions from the Nineteen Forties are precursors of the neural-network designs central to fashionable AI.

Making use of These Concepts to Framing AI Threat

Framing AI threat might be thought-about as an analog to framing cyber threat, regardless of main technical variations in all three points—risk, consequence, and vulnerability. When adversaries are within the image, AI penalties can embrace misdirection, unfairness and bias, reasoning failures, and so forth. AI threats can embrace tampering with coaching information, patch assaults on inputs, immediate and fine-tuning assaults, and so forth. Vulnerabilities and weaknesses, comparable to these inventoried within the CIG classes (see Half 2), usually derive from the intrinsic limitations of the structure and coaching of neural networks as statistically derived fashions. Even within the absence of adversaries, there are a number of penalties that may come up as a result of specific weaknesses intrinsic to neural-network fashions.

From the angle of conventional threat modeling, there’s additionally the issue, as famous above, of surprising correlations throughout fashions and platforms. For instance, there might be comparable penalties attributable to diversely sourced LLMs sharing basis fashions or simply having substantial overlap in coaching information. These surprising correlations can thwart makes an attempt to use strategies comparable to range by design as a method to enhance total system reliability.

We should additionally take into account the precise attribute of system resilience. Resilience is the capability of a system that has sustained an assault or a failure to nonetheless proceed to function safely, although maybe in a degraded method. This attribute is typically known as swish degradation or the flexibility to function by means of assaults and failures. On the whole, this can be very difficult, and sometimes infeasible, so as to add resilience to an present system. It’s because resilience is an emergent property consequential of system-level architectural choices. The architectural objective is to cut back the potential for inside errors—triggered by inside faults, compromises, or inherent ML weaknesses—to trigger system failures with pricey penalties. Conventional fault-tolerant engineering is an instance of design for resilience. Resilience is a consideration for each cyber threat and AI threat. Within the case of AI engineering, resilience might be enhanced by means of system-level and workflow-level design choices that, for instance, restrict publicity of weak inside assault surfaces, comparable to ML inputs, to potential adversaries. Such designs can embrace imposing energetic checking on inputs and outputs to neural-network fashions constituent to a system.

As famous in Half 2 of this weblog collection, an extra problem to AI resilience is the issue (or maybe lack of ability) to unlearn coaching information. Whether it is found {that a} subset of coaching information has been used to insert a vulnerability or again door into the AI system, it turns into a problem to take away that educated conduct from the AI system. In apply, this continues to stay troublesome and will necessitate retraining with out the malicious information. A associated difficulty is the other phenomenon of undesirable unlearning—known as catastrophic forgetting—which refers to new coaching information unintentionally impairing the standard of predictions primarily based on earlier coaching information.

Business Considerations and Responses Relating to AI Threat

There’s a broad recognition amongst mission stakeholders and corporations of the dimensionality and problem of framing and evaluating AI threat, regardless of speedy development in AI-related enterprise actions. Researchers at Stanford College produced a 500-page complete enterprise and technical evaluation of AI-related actions that states that funding for generative AI alone reached $25.2 billion in 2023. That is juxtaposed in opposition to a seemingly infinite stock of new sorts of dangers related to ML and generative AI. Illustrative of this can be a joint examine by the MIT Sloan Administration Overview and the Boston Consulting Group that signifies that corporations are having to broaden organizational threat administration capabilities to deal with AI-related dangers, and that this example is more likely to persist as a result of tempo of technological advance. A separate survey indicated that solely 9 p.c of corporations stated they had been ready to deal with the dangers. There are proposals to advance obligatory assessments to guarantee guardrails are in place. That is stimulating the service sector to reply, with unbiased estimates of a marketplace for AI mannequin threat administration value $10.5 billion by 2029.

Enhancing Threat Administration inside AI Engineering Follow

Because the neighborhood advances threat administration practices for AI, it can be crucial have in mind each the various points of threat, as illustrated within the earlier put up of this collection, and in addition the feasibility of the completely different approaches to mitigation. It’s not a simple course of: Evaluations must be carried out at a number of ranges of abstraction and construction in addition to a number of levels within the lifecycles of mission planning, structure design, techniques engineering, deployment, and evolution. The various ranges of abstraction could make this course of troublesome. On the highest stage, there are workflows, human-interaction designs, and system architectural designs. Selections made relating to every of those points have affect over the chance parts: attractiveness to risk actors, nature and extent of penalties of potential failures, and potential for vulnerabilities attributable to design choices. Then there’s the architecting and coaching for particular person neural-network fashions, the fine-tuning and prompting for generative fashions, and the potential publicity of assault surfaces of those fashions. Beneath this are, for instance, the precise mathematical algorithms and particular person traces of code. Lastly, when assault surfaces are uncovered, there might be dangers related to decisions within the supporting computing firmware and {hardware}.

Though NIST has taken preliminary steps towards codifying frameworks and playbooks, there stay many challenges to growing widespread parts of AI engineering apply—design, implementation, T&E, evolution—that might evolve into helpful norms—and broad adoption pushed by validated and usable metrics for return on effort. Arguably, there’s a good alternative now, whereas AI engineering practices are nonetheless nascent, to shortly develop an built-in, full-lifecycle method that {couples} system design and implementation with a shift-left T&E apply supported by proof manufacturing. This contrasts with the apply of safe coding, which was late-breaking within the broader software program growth neighborhood. Safe coding has led to efficient analyses and instruments and, certainly, many options of recent memory-safe languages. These are nice advantages, however safe coding’s late arrival has the unlucky consequence of an unlimited legacy of unsafe and sometimes weak code which may be too burdensome to replace.

Importantly, the persistent problem of instantly assessing the safety of a physique of code hinders not simply the adoption of finest practices but in addition the creation of incentives for his or her use. Builders and evaluators make choices primarily based on their sensible expertise, for instance, recognizing that guided fuzzing correlates with improved safety. In lots of of those instances probably the most possible approaches to evaluation relate to not the precise diploma of safety of a code base. As an alternative they concentrate on the extent of compliance with a strategy of making use of numerous design and growth strategies. Precise outcomes stay troublesome to evaluate in present apply. As a consequence, adherence to codified practices such because the safe growth lifecycle (SDL) and compliance with the Federal Data Safety Modernization Act (FISMA) has grow to be important to cyber threat administration.

Adoption will also be pushed by incentives which are unrelated however aligned. For instance, there are intelligent designs for languages and instruments that improve safety however whose adoption is pushed by builders’ curiosity in enhancing productiveness, with out intensive coaching or preliminary setup. One instance from internet growth is the open supply TypeScript language as a secure various to JavaScript. TypeScript is sort of equivalent in syntax and execution efficiency, however it additionally helps static checking, which might be carried out virtually instantly as builders sort in code, quite than surfacing a lot later when code is executing, maybe in operations. Builders could thus undertake TypeScript on the idea of productiveness, with safety advantages alongside for the experience.

Potential optimistic alignment of incentives might be essential for AI engineering, given the issue of growing metrics for a lot of points of AI threat. It’s difficult to develop direct measures for common instances, so we should additionally develop helpful surrogates and finest practices derived from expertise. Surrogates can embrace diploma of adherence to engineering finest practices, cautious coaching methods, assessments and analyses, decisions of instruments, and so forth. Importantly, these engineering strategies embrace growth and analysis of structure and design patterns that allow creation of extra reliable techniques from much less reliable parts.

The cyber threat realm provides a hybrid method of surrogacy and selective direct measurement by way of the Nationwide Data Assurance Partnership (NIAP) Widespread Standards: Designs are evaluated in depth, however direct assays on lower-level code are carried out by sampling, not comprehensively. One other instance is the extra broadly scoped Constructing Safety In Maturity Mannequin (BSIMM) mission, which features a strategy of ongoing enhancement to its norms of apply. In fact, any use of surrogates should be accompanied by aggressive analysis each to repeatedly assess validity and to develop direct measures.

Analysis Practices: Trying Forward

Classes for AI Purple Teaming from Cyber Purple Teaming

The October 2023 Govt Order 14110 on the Secure, Safe, and Reliable Improvement and Use of Synthetic Intelligence highlights using purple teaming for AI threat analysis. Within the army context, a typical method is to make use of purple groups in a capstone coaching engagement to simulate extremely succesful adversaries. Within the context of cyber dangers or AI dangers, nevertheless, purple groups will typically interact all through a system lifecycle, from preliminary mission scoping, idea exploration, and architectural design by means of to engineering, operations, and evolution.

A key query is how one can obtain this sort of integration when experience is a scarce useful resource. One of many classes of cyber purple teaming is that it’s higher to combine safety experience into growth groups—even on a part-time or rotating foundation—than to mandate consideration to safety points. Research counsel that this may be efficient when there are cross-team safety consultants instantly collaborating with growth groups.

For AI purple groups, this means that bigger organizations may keep a cross-team physique of consultants who perceive the stock of potential weaknesses and vulnerabilities and the state of play relating to measures, mitigations, instruments, and related practices. These consultants can be briefly built-in into agile groups so they might affect operational decisions and engineering choices. Their targets are each to maximise advantages from use of AI and in addition to attenuate dangers by means of making decisions that help assured T&E outcomes.

There could also be classes for the Division of Protection, which faces specific challenges in integrating AI threat administration practices into the techniques engineering tradition, as famous by the Congressional Analysis Service.

AI purple groups and cyber purple groups each tackle the dangers and challenges posed by adversaries. AI purple groups should additionally tackle dangers related to AI-specific weaknesses, together with all three CIG classes of weaknesses and vulnerabilities: confidentiality, integrity, and governance. Purple group success will rely on full consciousness of all dimensions of threat in addition to entry to applicable instruments and capabilities to help efficient and reasonably priced assessments.

On the present stage of growth, there’s not but a standardized apply for AI purple groups. Instruments, coaching, and actions haven’t been totally outlined or operationalized. Certainly, it may be argued that the authors of Govt Order 14110 had been smart to not await technical readability earlier than issuing the EO! Defining AI purple group ideas of operation is an huge, long-term problem that mixes technical, coaching, operational, coverage, market, and lots of different points, and it’s more likely to evolve quickly because the know-how evolves. The NIST RMF is a crucial first step in framing this dimensionality.

Potential Practices for AI Threat

A broad range of technical practices is required for the AI purple group toolkit. Analogously with safety and high quality evaluations, AI stakeholders can count on to depend on a mixture of course of compliance and product examination. They will also be introduced with various sorts of proof starting from full transparency with detailed technical analyses to self-attestation by suppliers, with decisions difficult by enterprise concerns regarding mental property and legal responsibility. This extends to provide chain administration for built-in techniques, the place there could also be various ranges of transparency. Legal responsibility is a altering panorama for cybersecurity and, we are able to count on, additionally for AI.

Course of compliance for AI threat can relate, for instance, to adherence to AI engineering practices. These practices can vary from design-level evaluations of how AI fashions are encapsulated inside a techniques structure to compliance with finest practices for information dealing with and coaching. They’ll additionally embrace use of mechanisms for monitoring behaviors of each techniques and human operators throughout operations. We be aware that process-focused regimes in cyber threat, such because the extremely mature physique of labor from NIST, can contain a whole lot of standards which may be utilized within the growth and analysis of a system. Programs designers and evaluators should choose and prioritize among the many many standards to develop aligned mission assurance methods.

We will count on that with a maturing of strategies for AI functionality growth and AI engineering, proactive practices will emerge that, when adopted, are inclined to end in AI-based operational capabilities that decrease key threat attributes. Direct evaluation and testing might be advanced and dear, so there might be actual advantages to utilizing validated process-compliance surrogates. However this may be difficult within the context of AI dangers. For instance, as famous in Half 1 of this collection, notions of take a look at protection and enter similarity standards acquainted to software program builders don’t switch effectively to neural-network fashions.

Product examination can pose vital technical difficulties, particularly with rising scale, complexity, and interconnection. It may possibly additionally pose business-related difficulties, attributable to problems with mental property and legal responsibility. In cybersecurity, sure points of merchandise at the moment are turning into extra readily accessible as areas for direct analysis, together with use of exterior sourcing in provide chains and the administration of inside entry gateways in techniques. That is partially a consequence of a cyber-policy focus that advances small increments of transparency, what we may name translucency, comparable to has been directed for software program payments of supplies (SBOM) and nil belief (ZT) architectures. There are, in fact, tradeoffs regarding transparency of merchandise to evaluators, and this can be a consideration in using open supply software program for mission techniques.

Mockingly, for contemporary AI techniques, even full transparency of a mannequin with billions of parameters could not yield a lot helpful data to evaluators. This pertains to the conflation of code and information in fashionable AI fashions famous on the outset of this collection. There may be vital analysis, nevertheless, in extracting associational maps from LLMs by taking a look at patterns of neuron activations. Conversely, black field AI fashions could reveal much more about their design and coaching than their creators could intend. The perceived confidentiality of coaching information might be damaged by means of mannequin inversion assaults for ML and memorized outputs for LLMs.

To be clear, direct analysis of neural-network fashions will stay a major technical problem. This provides further impetus to AI engineering and the appliance of applicable rules to the event and analysis of AI-based techniques and the workflows that use them.

Incentives

The proliferation of process- and product-focused standards, as simply famous, generally is a problem for leaders in search of to maximise profit whereas working affordably and effectively. The balancing of decisions might be extremely specific to the operational circumstances of a deliberate AI-based system in addition to to the technical decisions made relating to the inner design and growth of that system. That is one cause why incentive-based approaches can typically be fascinating over detailed process-compliance mandates. Certainly, incentive-based approaches can provide extra levels of freedom to engineering leaders, enabling threat discount by means of variations to operational workflows in addition to to engineered techniques.

Incentives might be each optimistic and detrimental, the place optimistic incentives could possibly be provided, for instance, in growth contracts, when assertions regarding AI dangers are backed with proof or accountability. Proof may relate to a variety of early AI-engineering decisions starting from techniques structure and operational workflows to mannequin design and inside guardrails.

An incentive-based method additionally has the benefit of enabling assured techniques engineering—primarily based on rising AI engineering rules—to evolve particularly contexts of techniques and missions at the same time as we proceed to work to advance the event of extra common strategies. The March 2023 Nationwide Cybersecurity Technique highlights the significance of accountability relating to information and software program, suggesting one essential attainable framing for incentives. The problem, in fact, is how one can develop dependable frameworks of standards and metrics that may inform incentives for the engineering of AI-based techniques.

Here’s a abstract of classes for present analysis apply for AI dangers:

  1. Prioritize mission-relevant dangers. Primarily based on the precise mission profile, establish and prioritize potential weaknesses and vulnerabilities. Do that as early as attainable within the course of, ideally earlier than techniques engineering is initiated. That is analogous to the Division of Protection technique of mission assurance.
  2. Determine risk-related targets. For these dangers deemed related, establish targets for the system together with related system-level measures.
  3. Assemble the toolkit of technical measures and mitigations. For those self same dangers, establish technical measures, potential mitigations, and related practices and instruments. Monitor the event of rising technical capabilities.
  4. Modify top-level operational and engineering decisions. For the upper precedence dangers, establish changes to first-order operational and engineering decisions that might result in seemingly threat reductions. This may embrace adapting operational workflow designs to restrict potential penalties, for instance by elevating human roles or lowering assault floor on the stage of workflows. It may additionally embrace adapting system architectures to cut back inside assault surfaces and to constrain the influence of weaknesses in embedded ML capabilities.
  5. Determine strategies to evaluate weaknesses and vulnerabilities. The place direct measures are missing, surrogates should be employed. These strategies may vary from use of NIST-playbook-style checklists to adoption of practices comparable to DevSecOps for AI. It may additionally embrace semi-direct evaluations on the stage of specs and designs analogous to Widespread Standards.
  6. Search for aligned attributes. Search optimistic alignments of threat mitigations with probably unrelated attributes that supply higher measures. For instance, productiveness and different measurable incentives can drive adoption of practices favorable to discount of sure classes of dangers. Within the context of AI dangers, this might embrace use of design patterns for resilience in technical architectures as a approach to localize any hostile results of ML weaknesses.

The subsequent put up on this collection examines the potential advantages of trying past the purely neural-network fashions in direction of approaches that hyperlink neural-network fashions with symbolic strategies. Put merely, the objective of those hybridizations is to attain a form of hybrid vigor that mixes the heuristic and linguistic virtuosity of recent neural networks with the verifiable trustworthiness attribute of many symbolic approaches.

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