Neuro-Symbolic Programs as Compression, Coordination, and Alignment

computer systems and Synthetic Intelligence, we had established establishments designed to motive systematically about human conduct — the courtroom. The authorized system is one among humanity’s oldest reasoning engines, the place info and proof are taken as enter, related legal guidelines are used as reasoning guidelines and verdicts are the system’s output. The legal guidelines, nonetheless, have been persistently evolving from the very starting of human civilization. The earliest Codified Legislation – the Code of Hammurabi (circa 1750 BCE) – represents one of many first large-scale makes an attempt to formalize ethical and social reasoning into specific symbolic guidelines. Its magnificence lies in readability and uniformity — but it is usually inflexible, incapable of adaptation to context. Centuries later, Frequent Legislation traditions like these formed by the Case of Donoghue v Stevenson (1932), launched the alternative philosophy: reasoning primarily based on precedential expertise and instances. At present’s authorized methods, as we all know, are normally a mix of each, whereas the proportions differ throughout completely different nations.

In distinction to the cohesive mixture in authorized methods, the same paradigm pair in AI — Symbolism and Connectionism — appear to be considerably tougher to unite. The latter has dominated the surge of AI growth in recent times, the place every part is implicitly realized with monumental quantities of knowledge and computing assets and encoded throughout parameters in neural networks. And this course, certainly, has been confirmed very efficient when it comes to benchmark efficiency. So, do we actually want a symbolic element in our AI methods?

Symbolic Programs v.s. Neural Networks: A Perspective of Data Compression

To reply the query above, we have to take a more in-depth take a look at each methods. From a computational standpoint, each symbolic methods and neural networks could be seen as machines of compression — they scale back the huge complexity of the world into compact representations that allow reasoning, prediction, and management. But they achieve this by means of essentially completely different mechanisms, guided by reverse philosophies of what it means to “perceive”.

In essence, each paradigms could be imagined as filters utilized to uncooked actuality. Given enter (X), every learns or defines a change (H(cdot)) that yields a compressed illustration (Y = H(X)), preserving data that it considers significant and discarding the remainder. However the form of this filtering is completely different. Usually talking, symbolic methods behave like high-pass filters — they extract the sharp, rule-defining contours of the world whereas ignoring its clean gradients. Neural networks, in contrast, resemble low-pass filters, smoothing native fluctuations to seize world construction. The distinction isn’t in what they see, however in what they select to neglect.

Symbolic methods compress by discretization. They carve the continual cloth of expertise into distinct classes, relations, and guidelines: a authorized code, a grammar or an ontology. Every image acts as a crisp boundary, a deal with for manipulation inside a pre-defined schema. The method resembles projecting a loud sign onto a set of human-designed foundation vectors — an area spanned by ideas resembling Entity and Relation. A data graph, for example, may learn the sentence “UIUC is a unprecedented college and I like it”, and retain solely (UIUC, is_a, Establishment), discarding every part that falls exterior its schema. The result’s readability and composability, but additionally rigidity: which means exterior the ontological body merely evaporates.

Neural networks, in distinction, compress by smoothing. They forgo discrete classes in favor of clean manifolds the place close by inputs yield comparable activations (normally bounded by some Lipschitz fixed in trendy LLMs). Somewhat than mapping information to predefined coordinates, they be taught a latent geometry that encodes correlations implicitly. The world, on this view, isn’t a algorithm however a area of gradients. This makes neural representations remarkably adaptive: they will interpolate, analogize, and generalize throughout unseen examples. However the identical smoothness that grants flexibility additionally breeds opacity. Data is entangled, semantics turn into distributed, and interpretability is misplaced within the very act of generalization.

In conclusion, we will summarize the distinction between the 2 methods from the knowledge compression perspective in a single sentence: “Neural Networks are blurry photographs of the world, whereas symbolic methods are high-resolution photos with lacking patches.” This truly signifies the explanation why neuro-symbolic methods are an artwork of compromise: they will harness data from each paradigms through the use of them collaboratively at completely different scales, with neural networks offering a world, low-resolution spine and symbolic elements supplying high-resolution native particulars.

The Problem of Scalability

Although it is vitally tempting so as to add symbolic elements into neural networks to harness advantages from each, scalability is an enormous drawback getting in the way in which of our makes an attempt, particularly within the period of Basis Fashions. Conventional neuro-symbolic methods depend on a set of expert-defined ontology / schema / symbols, which is assumed to have the ability to cowl all potential enter instances. That is acceptable for domain-specific methods (for instance, a pizza order chatbot); nonetheless, you can’t apply comparable approaches to open-domain methods, the place you will want specialists to assemble trillions of symbols and their relations.

A pure response is to go absolutely data-driven: as an alternative of asking people to handcraft an ontology, we let the mannequin induce its personal “symbols” from inside activations. Sparse autoencoders (SAEs) are a outstanding incarnation of this concept. By factorizing hidden states into a big set of sparse options, they seem to present us a dictionary of neural ideas: every function fires on a selected sample, is (usually) human-interpretable, and behaves like a discrete unit that may be turned on or off. At first look, this appears like an ideal escape from the professional bottleneck: we not design the image set; we be taught it.

Right here (D) is named the dictionary matrix the place every column shops a semantically significant idea; the primary time period is the reconstruction loss of the hidden state (h), whereas the second is a sparsity penalty encouraging minimal activated neurons within the code.

Nonetheless, an SAE-only method runs into two basic points. The primary is computational: utilizing SAEs as a stay symbolic layer would require multiplying each hidden state by an infinite dictionary matrix, paying a dense computation price even when the ensuing code is sparse. This makes them not possible for deployment at Basis Mannequin scales. The second is conceptual: SAE options are symbol-like representations, however they don’t seem to be a symbolic system — they lack an specific formal language, compositional operators, and executable guidelines. They inform us what ideas exist within the mannequin’s latent house, however not the right way to motive with them.

This doesn’t imply we should always abandon SAEs altogether — they supply substances, not a completed meal. Somewhat than asking SAEs to be the symbolic system, we will deal with them as a bridge between the mannequin’s inside idea house and the numerous symbolic artefacts we have already got: data graphs, ontologies, rule bases, taxonomies, the place reasoning can occur by definition. And a high-quality SAE skilled on a big mannequin’s hidden states then turns into a shared “idea coordinate system”: completely different symbolic methods can then be aligned inside this coordinate system by associating their symbols with the SAE options which are persistently activated when these symbols are invoked in context.

Doing this has a number of benefits over merely inserting symbolic methods aspect by aspect and querying them independently. First, it permits image merging and aliasing throughout methods: if two symbols from completely different formalisms repeatedly mild up virtually the identical set of SAE options, we’ve got robust proof that they correspond to the identical underlying neural idea, and could be linked and even unified. Second, it helps cross-system relation discovery: symbols which are far aside in our hand-designed schemas however persistently shut in SAE house level to bridges we did not encode — new relations, abstractions, or mappings between domains. Third, SAE activations give us a model-centric notion of salience: symbols that by no means discover a clear counterpart within the neural idea house are candidates for pruning or refactoring, whereas robust SAE options with no matching image in any system spotlight blind spots shared by all of our present abstractions.

Crucially, this use of SAEs stays scalable. The costly SAE is skilled offline, and the symbolic methods themselves don’t must develop to “Basis Mannequin measurement” — they will stay as small or as massive as their respective duties require. At inference time, the neural community continues to do the heavy lifting in its steady latent house; the symbolic artefacts solely form, constrain, or audit behaviour on the factors the place specific construction and accountability are Most worthy. SAEs assist by tying all these heterogeneous symbolic views again to a single realized conceptual map of the mannequin, making it potential to match, merge, and enhance them with out ever setting up a monolithic, expert-designed symbolic twin.

When Can an SAE Function a Symbolic Bridge?

The image above quietly assumes that our SAE is “adequate” to function a significant coordinate system. What does that truly require? We don’t want perfection, nor do we want the SAE to outperform human symbolic methods on each axis. As an alternative, we want a number of extra modest however essential properties:

– Semantic Continuity: Inputs that categorical the identical underlying idea ought to induce comparable help patterns within the sparse code: the identical subset of SAE options ought to are usually non-zero, quite than flickering on and off underneath small paraphrases or context shifts. In different phrases, semantic equivalence ought to be mirrored in a secure sample of lively ideas.

– Partial Interpretability: We wouldn’t have to know each function, however a nontrivial fraction of them ought to admit strong human descriptions, in order that merging and debugging are potential on the idea degree.

– Behavioral Relevance: The options that the SAE discovers should truly matter for the mannequin’s outputs: intervening on them, or conditioning on their presence, ought to change or predict the mannequin’s choices in systematic methods.

– Capability and Grounding: An SAE can solely refactor no matter construction already exists within the base mannequin; it can not conjure wealthy ideas out of a weak spine. For the “idea coordinate system” image to make sense, the bottom mannequin itself needs to be massive and well-trained sufficient that its hidden states already encode a various, non-trivial set of abstractions. In the meantime, the SAE will need to have enough dimensionality and overcompleteness: if the code house is just too small, many distinct ideas will likely be compelled to share the identical options, resulting in entangled and unstable representations.

Now we talk about the primary three properties intimately.

Semantic Continuity

On the degree of pure perform approximation, a deep neural community with ReLU- or GELU-type activations implements a Lipschitz-continuous map: small perturbations within the enter can not trigger arbitrarily unbounded jumps within the output logits. However this sort of continuity could be very completely different from what we want in a sparse autoencoder. For the bottom mannequin, a number of neurons flipping on or off can simply be absorbed by downstream layers and redundancy; so long as the ultimate logits change easily, we’re happy.

In an SAE, in contrast, we’re not simply a clean output — we’re treating the help sample of the sparse code reconstructed over the residual stream as a proto-symbolic object. A “idea” is recognized with a selected code subset being lively. That makes the geometry way more brittle: if a small change within the underlying illustration pushes a pre-activation throughout the ReLU threshold within the SAE layer, a neuron within the code will all of the sudden flip from off to on (or vice versa), and from the symbolic perspective the idea has appeared or disappeared. There is no such thing as a downstream community to common this out; the code itself is the illustration we care about.

Sparsity penalty in setting up the SAE even exacerbates this. The standard SAE goal combines a reconstruction loss with an (ell_1) penalty on the activations, which explicitly encourages most neuron values to be as near zero as potential. In consequence, even many helpful neurons find yourself sitting close to the activation boundary: simply above zero when they’re wanted, just under zero when they don’t seem to be — this is named “activation shrinkage” in SAEs. That is unhealthy for semantic continuity on the help sample degree: tiny perturbations within the enter can change non-zero neurons, even when the underlying which means has barely modified. Subsequently, Lipschitz continuity of the bottom mannequin doesn’t routinely give us a secure non-zero subset of code within the SAE house, and support-level stability needs to be handled as a separate design goal and evaluated explicitly.

Partial Interpretability

SAE defines an overcomplete dictionary to retailer potential options realized from information. Subsequently, we solely want a subset of those dictionary entries to be interpretable options. Even for that subset, meanings of the options are solely required to be roughly correct. Once we align current symbols to the SAE house, it’s the activation patterns within the SAE layer that we depend on: we probe the mannequin in contexts the place a logo is “in play”, report the ensuing sparse codes, and use the aggregated code as an embedding for that image. Symbols from completely different methods whose embeddings are shut could be linked or merged, even when we by no means assign human-readable semantics to each particular person function.

Interpretable options then play a extra centered function: they supply human-facing anchors inside this activation geometry. If a selected function has a fairly correct description, all symbols that load closely on it inherit a shared semantic trace (e.g. “these are all duty-of-care-like issues”), making it simpler to examine, debug, and set up the merged symbolic house. In different phrases, we don’t want an ideal, absolutely named dictionary. We want (i) sufficient capability in order that vital ideas can get their very own instructions, and (ii) a sizeable, behaviorally related subset of options whose approximate meanings are secure sufficient to function anchors. The remainder of the overcomplete code can stay as nameless background; it nonetheless contributes to distances and clusters within the SAE house, even when we by no means identify it.

Behavioral Relevance by way of Counterfactuals

A function is just attention-grabbing, as a part of a bridge, if it truly influences the mannequin’s conduct — not simply if it correlates with a sample within the information. In causal phrases, we care about whether or not the function lies on a causal path within the community’s computation from enter to output: if we perturb the function whereas holding every part else fastened, does the mannequin’s behaviour change in the way in which that its believed which means would predict?

Formally, altering a function is much like an intervention of the shape (textual content{do}(z = c)) within the causal sense, the place we overwrite that inside variable and rerun the computation. However in contrast to classical causal inference modeling, we don’t really want Pearl’s do-calculus to determine (P(y mid textual content{do}(z))). The neural community is a absolutely observable and intervenable system, so we will merely execute the intervention on the inner nodes and observe the brand new output. On this sense, neural networks give us the posh of performing idealized interventions which are not possible in most real-world social or financial methods.

Intervening on SAE options is conceptually comparable however applied otherwise. We sometimes have no idea the which means of an arbitrary worth within the function house, so the onerous intervention talked about above will not be significant. As an alternative, we amplify or suppress the magnitude of an current function, which behaves extra like a smooth intervention: the structural graph is left untouched, however the function’s efficient affect is modified. As a result of SAE reconstructs hidden activations as a linear mixture of a small variety of semantically significant options, we will change the coefficients of these options to implement significant, localized interventions with out affecting different options.

Symbolic-System Primarily based Compression as an Alignment Course of

Now let’s take a barely completely different view. Whereas neural networks compress the world into some extremely summary, steady manifolds, symbolic methods compress it right into a human-defined house with semantically significant axes alongside which the system’s behaviors could be judged. From this angle, compressing data into the symbolic house is an alignment course of, the place a messy, high-dimensional world is projected onto an area whose coordinates mirror human ideas, pursuits, and values.

Once we introduce symbols like “obligation of care”, “risk of violence”, or “protected attribute” right into a symbolic system, we’re not simply inventing labels. This compression course of does three issues without delay:

– It selects which facets of the world the system is obliged to care about (and which it’s purported to ignore).

– It creates a shared vocabulary in order that completely different stakeholders can reliably level to “the identical factor” in disputes and audits.

– It turns these symbols into dedication factors: as soon as written down, they are often cited, challenged, and reinterpreted, however not quietly erased.

In contrast, a purely neural compression lives completely contained in the mannequin. Its latent axes are unnamed, its geometry is personal, and its content material can drift as coaching information or fine-tuning targets change. Such a illustration is superb for generalization, however poor as a locus of obligation. It’s onerous to say, in that house alone, what the system owes to anybody, or which distinctions it’s purported to deal with as invariant. In different phrases, neural compression serves prediction, whereas symbolic compression serves alignment with a human normative body.

When you see symbolic methods as alignment maps quite than mere rule lists, the connection to accountability turns into direct. To say “the mannequin should not discriminate on protected attributes”, or “the mannequin should apply a duty-of-care customary”, is to insist that sure symbolic distinctions be mirrored, in a secure means, inside its inside idea house — and that we have the ability to find, probe, and, if essential, appropriate these reflections. And this accountability is normally desired, even at the price of compromising a part of the mannequin functionality.

From Hidden Legislation to Shared Symbols

In Zuo Zhuan, the Jin statesman Shu-Xiang as soon as wrote to Zi-Chan of Zheng: “When punishment is unknown, deterrence turns into unfathomable.” For hundreds of years, the ruling class maintained order by means of secrecy, believing that worry thrived the place understanding ended. That’s why it grew to become a milestone in historical Chinese language historical past when Zi-Chan shattered that custom, forged the felony code onto bronze tripods and displayed it publicly in 536 BCE. Now AI methods are dealing with the same drawback. Who would be the subsequent Zi-Chan?

References

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